Isochoric specic heat in the Dual Model of Liquids
and comparison with the Phonon theory of Liquid
Thermodynamics
Fabio Peluso ( fpeluso65@gmail.com )
Leonardo Finmeccanica https://orcid.org/0000-0003-0890-9772
Research Article
Keywords: Liquid model, Phonons in liquids, Mesoscopic model of liquids, Phonon – particle interaction,
Specic heat
Posted Date: March 31st, 2021
DOI: https://doi.org/10.21203/rs.3.rs-357792/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
Read Full License
1
Isochoric specific heat in the Dual Model of Liquids and
comparison with the Phonon theory of Liquid Thermodynamics.
Fabio Peluso
Leonardo SpA – Electronics Division – Defense Systems LoB
Via Monterusciello 75, 80078 Pozzuoli (NA) – Italy
Onorary Member of Marscenter Scientific Commettee
mailto: fpeluso65@gmail.com.
Abstract.
We continue in this paper to illustrate the implications of the Dual Model of Liquids (DML)
by deriving the expression for the isochoric specific heat as function of the collective degrees of
freedom available at a given temperature and comparing it with the analogous expression obtained
in the Phonon Theory of Liquid Thermodynamics.
The Dual Model of Liquids has been recently proposed as a model describing the dynamics of
liquids at mesoscopic level. Bringing together the early pictures of Brillouin and Frenkel and the
recent experimental outcomes obtained by means of high energy scattering, liquids are considered
in the DML as constituted by a population of wave packets, responsible for the propagation of
elastic and thermal perturbations, and of dynamic aggregates of molecules, in continuous re-
arrangement, diving in an ocean of amorphous, disordered liquid. The collective degrees of freedom
contribute to the exchange of energy and momentum between the material particles and the lattice
particles, which the liquids are supposed to be composed of in the DML.
First, we show that the expression obtained for the specific heat in the DML is in line with the
experimental results. Second, its comparison with that of the Phonon Theory of Liquid
Thermodynamics allows to get interesting insights about the limiting values of the collective
degrees of freedom and on that of the isobaric thermal expansion coefficient, two quantities that
appear related each other in this framework.
Keywords: Liquid model, Phonons in liquids, Mesoscopic model of liquids; Phonon – particle interaction; Specific
heat.
1. Introduction
In a previous paper [1] the mesoscopic model of liquids dubbed Dual Model of Liquids
(DML) has been introduced. It assumes a liquid be a system made up of molecules arranged in
solid-like dynamic structures in continuous rearrangement, swimming in an ocean of amorphous
2
liquid. These solid-like structures, that we like to call icebergs, interact with lattice particles, the
phonons, present in liquids and responsible for the propagation of elastic and thermal energy. As
consequence of the interactions, the dynamic icebergs of molecules and the phonons exchange
among them energy and momentum. This interaction is described in Figure 1, where two
elementary events are represented. In events of type (a), an energetic wave-packet interact with a
solid-like structure, the liquid particle, (i.e. a mesoscopic aggregate of molecules as described
above) transferring to it energy and momentum. Events of type (b) are just the opposite, a wave-
packet interact with a liquid particle and emerges from the interaction with increased energy. The
two events are commuted one into the other by time reversal. These interactions are responsible for
momentum, energy and mass propagation in a liquid, and in this frame thermal energy is seen as a
form of elastic energy. The interaction lasts
p
, at the end of which the liquid particle relaxes the
energy stored into internal degrees of freedom (DoF); then it travels by
R
during
R
. Figure 2 is a
close-up of the first part of the wave-packet
particle interaction shown in Figure 1a, that lasting
p
.
wp
is the extension of the wave-packet and
p
d that of the liquid particle [1].
Starting from this considerations and applying a simple kinetic model, we have calculated,
among others, the expressions for several macroscopic quantities in terms of elementary parameters.
Examples are the thermal conductivity, the thermal diffusivity, the specific heat, etc.. The
evaluation of the order of magnitude of the particle and phonon mean free paths and in particular of
the relaxation times involved in their interactions, have also been provided [1].
The internal energy associated to the pool of wave-packets,
wp
T
q , is represented in the DML as
a fraction of the total one,
T
q , namely:
1.
wpwp
T
VT
wp
T
dCmmqq
Ν
0
where
is the medium density,
V
C the isochoric specific heat per unit mass,
wp
Ν
the number of
wave-packets per unit of volume
1
, and
wp
their average energy (here and in the rest of the paper,
the two brackets
indicate the average over a statistical ensemble of the quantity inside them).
The dynamics described above occurs at high frequencies and involves only the DoF of the lattice;
the parameter
m
that we have introduced accounts for the ratio between the number of collective
DoF surviving at temperature T, and the total number of available DoF. We will return on the
meaning and values of
m
in the course of the paper, for now it is relevant to note that 10
m .
1
wp
Ν
is the average density of the wave-packets statistical distribution. As such
wp
Ν
is represented by a Bose-Einstein
distribution function
3
Figure 1a shows that the energy acquired by the liquid particle is given back to the pool of
elastic excitation a
Rp
step forward and a
Rp
time lapse later; this
last assumes the right meaning of a relaxation time. One of the immediate consequences of this
model is that heat propagation in liquids is naturally described by a Cattaneo equation, as shown in
another recent paper [2], the delay time introduced into the propagation equation being physically
interpreted as the time taken by icebergs to displace from one site to the next, as in Figure 1.
Few years ago the “Phonon theory of Liquid Thermodynamics” (PLT) has been presented [3-
12]. One of the main achievements of the PLT is that, starting from a solid-like Hamiltonian of the
liquid, modified to account for the presence in liquids of anharmonic DoF, provides an expression
for the isochoric specific heat
V
C covering the solid, glassy, liquid, gas and quantum liquids states
of matter. Two expressions are obtained for the specific heat, one in harmonic approximation, the
other including the anharmonic contributions to the pool of energy. The theoretical expressions
arrived at have been confirmed experimentally in 21 different liquids. Moreover, PLT has also
provided a theoretical interpretation of the elastic, visco-elastic and viscous behaviour response of
matter to external disturbances as due to the value of the product
2
, where
is the frequency of
the excitation of the DoF, and
the relaxation time as defined above. It depends on whether
is
larger, comparable or smaller than
1 , respectively.
Here in this paper we shortly recall the calculations that allow to get in the frame of reference
of the DML the expression for the isochoric specific heat as function of the number of collective
DoF available at a given temperature, and compare it with the analogous expression arrived at in the
Phonon Theory of Liquid Thermodynamics (PLT) [3-12]. This comparison allows to get interesting
insights about the limiting values of the collective DoF, and on that of the isobaric thermal
expansion coefficient,
.
The paper is organized as follows. In Section 2 the DML is shortly summarized and compared
with the PLT. In Section 3 the expressions for the isochoric specific heat derived along with the two
approaches, DML and PLT, are compared among them, and their similarities, insights and the
implications are discussed. Finally in the last Section some further consequences and comparisons
even between the two theories are discussed.
2
This product is normally reported as
by many authors instead of
. It is clear that there is no ambiguity among
the two definitions.
4
2. Dual Model of Liquids and Phonon theory of Liquid Thermodynamics: the
two sides of the same coin.
It is known that historically the liquid state has been initially dealt with as a sort of extension
of the gaseous one, this mainly because of its property of flowing like gases, of the lacking of the
capability of transmitting shear stresses, and of adapting to the shapes of their containers. However,
this approach has never allowed to go very far in the formulation of theories that were able to
provide a fairly general picture of the physical quantities characterizing the liquid state. Let us take
for instance the specific heat, for which strong theoretical basis exist for gaseous and solid states,
but not for the liquid one.
The alternative approach, to consider the liquid state in the same way as that of a solid, has
illustrious predecessors, such as Debye [13-14], Brillouin [15-16], and Frenkel [17]. However, they
too have never gone so far as to provide a model that went beyond an intuitive description, this
because of the lack of experimental evidences that could guide and/or validate it. Nevertheless,
recent experimental discoveries have given credit again to the ideas of Frenkel and Brillouin. At
low frequencies, i.e. large wavelengths, it is not possible to "see" the mesoscopic structure of
liquids. The entire liquid oscillates, compressing and expanding under the effect of pressure waves,
which travel at the speed of sound. When one investigates the behaviour of a liquid at very high
frequencies, and therefore at small wavelengths, it is discovered that it has a mesoscopic structure
similar to that of a solid, the one identified here in a picturesque and imaginative way with the
icebergs, elsewhere simply called pseudo-crystalline structures (see for instance [13-29]).
The first experimental clear evidence of the presence in liquids of solid-like local structures
with a pseudo-periodicity came in 1996 [25], when Ruocco and Sette measured by IXS
experiments, therefore at high frequency, in liquid water at ambient conditions, the propagation
speed of elastic waves, and found it equal to 3200 m/s, i.e. more than double that known at
traditional frequencies, 1500 m/s, and very close to that of solid water, 4000 m/s. Starting from the
results of experiments performed on various liquids with INS and IXS techniques, [13-29], the
evidence that pseudo-crystalline structures “exist and persist” in liquids has gradually consolidated,
their size and number depending on the liquid temperature (and pressure). These structures can
obviously be discovered only when high frequency measurements are carried out, typical of INS
and IXS techniques, the wavelengths of the radiation involved being small enough to reveal their
presence. In the DML these pseudo-crystalline structures (the icebergs) organized by means of
phonons, interact with the rest of the (amorphous) liquid through anharmonic interactions which
arise at their border. Ultimately, a quasi-elastic propagation within icebergs is transmitted to the
amorphous matrix in the form of anharmonic wave-packets, and vice versa. Unlike in crystalline
![Figure 1. Schematic representation of inelastic collisions between wave-packets and liquid particles. The event represented in (a), in which an energetic wave-packet transfers energy and momentum to a liquid particle, is commuted upon time reversal into the one represented in (b), where a liquid particle transfers energy and momentum to a wavepacket. The particle changes velocity and the frequency of wave-packet is shifted by the amount 12 . Due to its time symmetry, we assume this mechanism is the equivalent of Onsager’ reciprocity law at microscopic level [31-33].](/figures/figure-1-schematic-representation-of-inelastic-collisions-1zlfnkge.webp)
![Figure 3. Icebergs of solid lattice fluctuating and interacting within the liquid global system at equilibrium. As far as elastic (thermal) perturbations propagate within an iceberg, they behave as in solids. Propagation velocity has then the values typical as those of the solid lattice, as found by Ruocco et al. [25] of about 3200 m/s for the case of water. Average sizes of icebergs 0 have been found of some nanometers. When perturbations cross the boundary between two icebergs, thf develops and energy and momentum are transmitted from one to the nearest-neighbour iceberg. This pictorial model of liquids at mesoscopic scale, on which the DML is based, reflects also what may be deduced from experiments performed with IXS techniques, able to observe liquids at such scale-lengths. In a solution solute particles may be considered as icebergs having elastic impedance different from that of the solvent. Energy and momentum exchanged between the two types of icebergs produce a net effect resulting in the diffusion of the solute along the concentration gradient. If a temperature gradient is imposed externally, the net effect will depend on the prevailing flux of wave packets, which will give rise to thermal diffusion of one species with respect to the other.](/figures/figure-3-icebergs-of-solid-lattice-fluctuating-and-2ho8v40e.webp)