# Thermo-Mechanical and Mechano-Thermal Effects in Liquids Explained by Means of the Dual Model of Liquids

## Abstract

**:**

## 1. Introduction

- The quasi-equilibrium postulate, requiring that gradients (driving forces) are not too large.
- The linearity postulate, stating that all fluxes are linear functions of the relevant gradients (this is a consequence of the previous postulate, which allows a linear dependence of fluxes upon driving forces).
- Curie’s postulate, constraining the tensor rank of coupled fluxes and forces.
- Onsager’s reciprocal relations, requiring symmetric coefficient matrices in the force-flux relations.

## 2. The Thermodiffusion and the Soret Equilibrium in NET

_{T}becomes a vector. Figure 1 represents the typical situation one is dealing with in analyzing the thermodiffusion phenomenon. At the beginning the two chemical species are uniformly distributed; when the heat starts crossing the medium, from the hot to the cold side, one of the two species, for instance, the spheres of Figure 1, moves towards the cold side, while the other is pushed back towards the hot side (for the Newton’ third law): this is the phenomenon of thermodiffusion, i.e., a mass flow produced by a heat flow. As soon as the thermodiffusion begins, as the spheres move along the heat current (and the cubes in the opposite direction), the back diffusion acts on the chemical species to balance the concentration gradient that is being established. Definitively, the displacement of the components of the solution generates at the steady state a concentration gradient for each of them, which in turn generates two mass flows of the chemical species, coupled with the heat flux due to the external temperature gradient. At the steady-state, or the Soret equilibrium, the thermodiffusion is balanced by the ordinary diffusion; one may observe a temperature gradient (external constraint) responsible for a heat flow, and two opposite concentration gradients (crossed effect), for the spheres and the cubes. Generally, the attention is focused on the less concentrated component of the mixture, the solute. Such dynamical equilibrium, the Soret, is described by means of the Soret coefficient ${S}_{T}$, which is positive when the solute drifts towards the cold side, while it is negative when it drifts towards the hot one. The Dufour effect is the opposite of the Soret. When there is an initial gradient of concentration in a fluid mixture, i.e., a gradient of chemical potential, it is known that the system spontaneously evolves towards the equilibrium due to the ordinary Fick diffusion. A careful evaluation of the thermal evolution of the system shows that the spontaneous diffusion of the chemical species of a mixture to establish a uniform distribution generates a temperature gradient: this phenomenon is named the Dufour effect in NET.

_{T}and makes it evident that the equilibrium is attained by balancing the thermodiffusion and the ordinary diffusion. Comparing Equations (1)–(4), one may easily get the following expressions for the two thermal diffusion coefficients:

## 3. The Thermodiffusion and the Soret Equilibrium in the DML

^{th}, dimensionally a force per unit of volume, responsible for momentum transport associated with the wave-packets propagation. Equation (12) is valid when the characteristics of the medium change continuously along the wave-packets propagation direction. The same argument may be used when wave-packets impinge on an “obstacle”, as for instance a liquid particle, either of solute or of solvent: the radiant vector changes and a thermal radiation pressure Π

^{th}is produced on the boundary. Observing that the inelastic character of the collision allows the exchange of momentum, this leads to the appearance of the force f

^{th}, acting on molecular clusters. By rewriting Equation (12) in terms of discrete quantities, one yields the following expression for the net Π

^{th}:

_{T}through Equation (25) is not a simple task, of course, for this reason, dedicated numerical evaluations are in due course and their results will be reported in a separate paper [Peluso, F. et al., in preparation]. However, Equation (25) contains very useful and important information about the meaning of S

_{T}in the DML framework. Recalling that $\langle \Lambda \rangle $ is the liquid particle drift caused by the $\langle {v}_{p}\rangle $ collisions per second with the lattice particle, Equation (25) becomes:

^{wp}, or Π

^{th}, along the gradient. Besides being an energy density, ${q}_{T}^{wp}$ represents in fact also the pressure Π

^{wp}exerted by elastic wave-packets on the liquid particles. Compiling Equation (13) with Equations (20) and (21) we get:

## 4. An “Unexpected” Mechano-Thermal Effect in Pure Liquids Explained by Means of DML

## 5. Discussion

_{T}or D

^{th}are negative means that the local virtual gradients are oriented along the direction opposite to that of the external gradient (see Equations (25) and (37)). This circumstance produces a thermodiffusive drift along the direction opposite to that of the external gradient, resulting in a negative value of ${S}_{T}$. In other words, the thermodiffusive drift of the solvent prevails on that of the solute, consequently we will observe the solute migrating to the hot side and the solvent to the cold one because the force due to the radiation pressure exerted by wave-packets on the solvent particles is higher than that produced on the solute particles.

**k**-gap.

_{ϕ}’s. As it concerns the validation of Equation (25), dedicated numerical analysis will be the core of another work.

## 6. Conclusions and Future Perspectives

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Callen, H.B. Thermodynamics and an Introduction to Thermostatics, 2nd ed.; Wiley: Philadelphia, PA, USA, 1991; ISBN 978-0-471-86256-7. [Google Scholar]
- Onsager, L. Reciprocal relations in irreversible processes I. Phys. Rev.
**1931**, 37, 405–426. [Google Scholar] [CrossRef] - Onsager, L. Reciprocal relations in irreversible processes II. Phys. Rev.
**1931**, 38, 2265–2279. [Google Scholar] [CrossRef] - Ludwig, C. Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl.
**1856**, 20, 539. [Google Scholar] - Soret, M.C. Sur l’état d’équilibre que prend au point de vue de sa concentration une dissolution saline primitivement homohéne dont deux parties sont portées à des températures différentes. Ann. Chim. Phys.
**1881**, 22, 293–297. [Google Scholar] - Kume, E.; Zaccone, A.; Noirez, L. Unexpected thermoelastic effects in liquid glycerol by mechanical deformation. Phys. Fluids
**2021**, 33, 072007. [Google Scholar] [CrossRef] - Kume, E.; Noirez, L. Identification of thermal response of mesoscopic liquids under mechanical excitations: From harmonic to nonharmonic thermal wave. J. Phys. Chem. B
**2021**, 125, 8652–8658. [Google Scholar] [CrossRef] - Noirez, L.; Baroni, P. Identification of a low-frequency elastic behaviour in liquid water. J. Phys. Condens. Matter
**2012**, 24, 372101–372107. [Google Scholar] [CrossRef] - Kume, E.; Baroni, P.; Noirez, L. Strain-induced violation of temperature uniformity in mesoscale liquids. Sci. Rep.
**2020**, 10, 13340. [Google Scholar] [CrossRef] - Kume, E.; Noirez, L. Thermal Shear Waves Induced in Mesoscopic Liquids at Low Frequency Mechanical Deformation. J. Non-Equilib. Thermodyn.
**2022**, 47, 155–163. [Google Scholar] [CrossRef] - Noirez, L.; Baroni, P. Revealing the solid-like nature of glycerol at ambient temperature. J. Mol. Struct.
**2010**, 972, 16–21. [Google Scholar] [CrossRef] - Eslamian, M.; Saghir, Z. A critical review of the thermodiffusion model: Role and significance of the heat of transport and the activation energy of viscous flow. Non-Equilib. Thermodyn.
**2009**, 34, 97–131. [Google Scholar] [CrossRef] - Rahman, M.A.; Saghir, M.Z. Thermodiffusion or Soret effect: Historical review. Int. J. Heat Mass Tranfer
**2014**, 73, 693–705. [Google Scholar] [CrossRef] - Gittus, O.R.; Bresme, F. On the microscopic origin of the Soret coefficient minima in liquid mixtures. arXiv
**2022**, arXiv:2207.12864vl. [Google Scholar] [CrossRef] - Zaccone, A.; Noirez, L. Universal G’ ≈ L
^{−3}law for the low-frequency shear modulus of confined liquids. J. Phys. Chem. Lett.**2021**, 12, 650–657. [Google Scholar] [CrossRef] [PubMed] - Phillips, A.E.; Baggioli, M.; Sirk, T.W.; Trachenko, K.; Zaccone, A. Universal L
^{−3}finite-size effects in the viscoelasticity of amorphous systems. Phys. Rev. Mater.**2021**, 5, 035602. [Google Scholar] [CrossRef] - Peluso, F. Mesoscopic collective dynamics in liquids and the Dual Model ASME. J. Heat Transf.
**2022**, 144, 112502. [Google Scholar] [CrossRef] - Peluso, F. Isochoric specific heat in the Dual Model of Liquids. Liquids
**2021**, 1, 77–95. [Google Scholar] [CrossRef] - Peluso, F. How Does Heat Propagate in Liquids? Liquids
**2023**, 3, 92–117. [Google Scholar] [CrossRef] - Stewart, B. Temperature Equilibrium of an Enclosure in which there is a Body in Visible Motion. In British Association Reports; 41st Meeting, Notes and Abstracts; John Murray, Albemare St.: London, UK, 1871; p. 45. [Google Scholar]
- Maxwell, J.C. A Treatise on Electricity and Magnetism; Clarendon Press: London, UK, 1873; Volume 2. [Google Scholar]
- Lebedev, P. Untersuchen ueber die Druckkraefte des Lichtes. Ann. Phys.
**1901**, 6, 433. [Google Scholar] [CrossRef] - Joyce, W.B. Classical-particle description of photons and phonons. Phys. Rev. D
**1974**, 9, 3234–3256. [Google Scholar] [CrossRef] - Westervelt, P.J. Acoustic Radiation Pressure. J. Acoust. Soc. Am.
**1957**, 29, 26–29. [Google Scholar] [CrossRef] - Mercier, J. De la pression de radiation dans le fluides. J. Phys. Rad.
**1956**, 17, 401–404. [Google Scholar] [CrossRef] - Joyce, W.B. Radiation force and the classical mechanics of photons and phonons. Am. J. Phys.
**1974**, 43, 245–255. [Google Scholar] [CrossRef] - Brillouin, L. Tensors in Mechanics and Elasticity; Academic Press: New York, NY, USA, 1964; pp. 240–243. [Google Scholar]
- Smith, W.E. Generalization of the Boltzmann-Ehrenfest Adiabatic Theorem in Acoustics. J. Acoust. Soc. Am.
**1971**, 50, 386–388. [Google Scholar] [CrossRef] - Gaeta, F.S.; Ascolese, E.; Tomicki, B. Radiation forces associated with heat propagation in nonisothermal systems. Phys. Rev. A
**1991**, 44, 5003–5017. [Google Scholar] [CrossRef] - Gaeta, F.S.; Mita, D.G. Nonisothermal mass transport in porous media. J. Membr. Sci.
**1978**, 3, 191–214. [Google Scholar] [CrossRef] - Braun, D.; Libchaber, A. Thermal force approach to molecular evolution. Phys. Biol.
**2004**, 1, 1. [Google Scholar] - Budin, I.; Bruckner, R.J.; Szostack, J.W. Formation of Protocell-like vesicles in a thermal diffusion column. J. Am. Chem. Soc.
**2009**, 131, 9628. [Google Scholar] [CrossRef] [PubMed] - Vance, F.H.; de Goey, P.; van Oijen, J.A. Development of a flashback correlation for burner-stabilized hydrogen-air premixed flame. Combust. Flame
**2020**, 216, 45. [Google Scholar] [CrossRef] - Demirel, Y.; Gerbaud, V. Nonequilibrium Thermodynamics; Elsevier: Amsterdam, The Netherlands, 2019; pp. 337–379. [Google Scholar]
- Huang, F.; Chakraborty, P.; Lundstrom, C.C.; Holmden, C.; Glessner, J.J.G.; Kieffer, S.W.; Lesher, C.E. Isotope fractionation in silicate melts by thermal diffusion. Nature
**2010**, 464, 396. [Google Scholar] [CrossRef] - Parola, R.; Piazza, A. Thermophoresis in colloidal suspensions. J. Phys. Condens. Matter
**2008**, 20, 153102. [Google Scholar] - Kania, H.; Sipa, J. Microstructure characterization and corrosion resistance of zinc coating obtained on high-stength grade 10.9 bolts using a new thermal diffusion process. Materials
**2019**, 12, 1400. [Google Scholar] [CrossRef] [PubMed] - Clusius, K.; Dickel, G. Neues Verfahren zur Gasentmischung. Naturwiss
**1938**, 26, 546–550. [Google Scholar] [CrossRef] - Agar, J.N.; Turner, J.C.R. Thermal diffusion in solutions of electrolytes. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1960**, 255, 307. [Google Scholar] - de Mezquia, D.A.; Bou-Ali, M.M.; Madariaga, J.A.; Santamaría, C. Mass effect on the Soret coefficient in n-alkane mixtures. J. Chem. Phys.
**2014**, 140, 084503. [Google Scholar] [CrossRef] - Bou-Ali, M.M. Soret coefficient of some binary liquid mixtures. J. Non-Equilib. Thermodyn.
**1999**, 24, 228–233. [Google Scholar] [CrossRef] - Gaeta, F.S.; Perna, G.; Scala, G.; Bellucci, F. Non-isothermal matter transport in sodium chloryde and potassium chloryde aqueous solutions. I: Homogeneous systems (thermal diffusion). J. Phys. Chem.
**1982**, 86, 2967–2974. [Google Scholar] [CrossRef] - Farber, M.; Libby, W.F. Effect of gravitational field on the thermal diffusion separation method. J. Chem. Phys.
**1940**, 8, 965–970. [Google Scholar] [CrossRef] - Costeseque, P.; Gaillard, S.; Gachet, Y.; Jamet, P. Determination of the apparent negative Soret coefficient of water-10% alcohol solutions by experimental methods in packed cells. Philos. Mag.
**2003**, 83, 2039–2044. [Google Scholar] [CrossRef] - Dougherty, E.L.; Drickamer, H.G. Thermal diffusion and molecular motion in liquids. J. Chem. Phys.
**1954**, 22, 443–449. [Google Scholar] [CrossRef] - Emery, A.H.; Drickamer, H.G. Thermal diffusion in polymer solutions. J. Chem. Phys.
**1955**, 23, 2252–2257. [Google Scholar] [CrossRef] - Najafi, A.; Golestanian, R. Forces Induced by Non-Equilibrium Fluctuations: The Soret-Casimir Effect. Europhys. Lett.
**2004**, 68, 776. [Google Scholar] [CrossRef] - Shukla, K.; Firoozabadi, A. A New Model of Thermal Diffusion Coefficients in Binary Hydrocarbon Mixtures. Ind. Eng. Chem. Res.
**1998**, 37, 3331–3342. [Google Scholar] [CrossRef] - Kempers, L.J.T.M. A thermodynamic theory of the Soret effect in a multicomponent liquid. J. Chem. Phys.
**1989**, 90, 6541–6548. [Google Scholar] [CrossRef] - Kempers, L.J.T.M. A comprehensive thermodynamic theory of the Soret effect in a multicomponent gas, liquid or solid. J. Chem. Phys.
**2001**, 115, 6330–6341. [Google Scholar] [CrossRef] - Artola, P.-A.; Rousseau, B.; Galliéro, G. A new model for thermal diffusion: Kinetic approach. J. Am. Chem. Soc.
**2008**, 130, 10963–10969. [Google Scholar] [CrossRef] - Duhr, S.; Braun, D. Why molecules move along a temperature gradient. Proc. Natl. Acad. Sci. USA
**2006**, 103, 19678–19682. [Google Scholar] [CrossRef] [PubMed] - Gaeta, F.S. Radiation pressure theory of thermal diffusion in liquids. Phys. Rev. A
**1969**, 182, 289–296. [Google Scholar] [CrossRef] - Gaeta, F.S.; Peluso, F.; Mita, D.G.; Albanese, C.; Castagnolo, D. Phonon-particle interactions and transport processes in liquids. Phys. Rev. E
**1993**, 47, 1066–1077. [Google Scholar] [CrossRef] - Gaeta, F.S.; Albanese, C.; Mita, D.G.; Peluso, F. Phonons in liquids, Onsager’s reciprocal relations and the heats of transport. Phys. Rev. E
**1994**, 49, 433–444. [Google Scholar] [CrossRef] [PubMed] - Chapman, S. The characteristics of thermal diffusion. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1940**, 177, 38. [Google Scholar] - Baghooee, H.; Shapiro, A. Unified thermodynamic modelling of diffusion and thermodiffusion coefficients. Fluid Phase Equilibria
**2022**, 558, 113445. [Google Scholar] [CrossRef] - Gonzalez-Bagnoli, M.G.; Shapiro, A.; Stenby, E.H. Evaluation of the thermodynamic models for the thermal diffusion factor. Philos. Mag.
**2003**, 83, 2171–2183. [Google Scholar] [CrossRef] - Abbasi, A.; Saghir, A.Z.; Kawaji, M. Theoretical and experimental comparison of the Soret effect for binary mixtures of toluene and n-hexane, and benzene and n-heptane. J. Non-Equilib. Thermodyn.
**2010**, 35, 1–14. [Google Scholar] [CrossRef] - Saghir, A.Z.; Jiang, C.G.; Derawi, S.O.; Stenby, E.H.; Kawaji, M. Theoretical and experimental comparison of the Soret coefficient for water-methanol and water-ethanol binary mixtures. Eur. Phys. J. E
**2004**, 15, 241–247. [Google Scholar] [CrossRef] [PubMed] - Hoang, H.; Galliero, G. Predicting thermodiffusion in simple binary fluid mixtures. Eur. Phys. J. E
**2022**, 45, 42. [Google Scholar] [CrossRef] [PubMed] - Piazza, R. Thermal force: Colloids in temperature gradients. J. Phys. Condens. Matter
**2004**, 16, S4195–S4211. [Google Scholar] [CrossRef] - Guy, A.G. Prediction of thermal diffusion in binary mixtures of nonelectrolyte liquids by the use of Non-Equilibrium thermodynamics. Int. J. Thermophys.
**1986**, 7, 563–572. [Google Scholar] [CrossRef] - Parola, A.; Piazza, R. A microscopic approach to thermophoresis in colloidal suspensions. J. Phys. Condens. Matter
**2005**, 17, S3639–S3643. [Google Scholar] [CrossRef] - Peluso, F.; Ceriello, A.; Albanese, C.; Gaeta, F.S. Behaviour of solid particles suspended in non-isothermal liquid. Entropie
**2002**, 239–240, 45–51. [Google Scholar] - Albanese, C.; Peluso, F.; Castagnolo, D. Thermal Radiation Forces in Microgravity: The TRUE and TRAMP Experiments: Results and Future Perspectives. In Microgravity Research and Aplications in Physical Sciences and Biotechnology; ESA–European Space Agency, Directorate of Manned Spaceflight and Microgravity, Noordwjik (NL): Sorrento, Italy, 2001; Volume 454, pp. 755–769. [Google Scholar]
- Vailati, A.; Cerbino, R.; Mazzoni, S.; Takacs, C.J.; Cannell, D.S.; Giglio, M. Fractal fronts of diffusion in microgravity. Nat. Commun.
**2011**, 2, 290. [Google Scholar] [CrossRef] [PubMed] - Van Vaerenberg, S.; Le Gros, C.; Dupin, J.C. First results of Soret coefficients measurement experiment. Adv. Space Res.
**1995**, 16, 205. [Google Scholar] - Van Vaerenberg, S.; Le Gros, C. Soret coefficients of organic solutions measured in the microgravity SCM experiment and by the flow and Bénard cells. J. Phys. Chem. B
**1998**, 102, 4426. [Google Scholar] [CrossRef] - Prodi, F.; Santachiara, G.; Travaini, S.; Vedernikov, A.; Dubois, F.; Minetti, C.; Legros, J. Measurement of phoretic velocities of aerosols particles in microgravity conditions. Atmos. Res.
**2006**, 82, 183. [Google Scholar] [CrossRef] - Shevtsova, V.; Mialdun, A.; Melnikov, D.; Ryzhkov, I.; Gaponenko, Y.; Saghir, Z.; Lyubimova, T.; Legros, J.C. The IVIDIL experiment onboard the ISS: Thermodiffusion in the presence of controlled vibrations. Comptes Rendus Mécanique
**2011**, 339, 310–317. [Google Scholar] [CrossRef] - Shevtsova, V.; Lyubimova, T.; Saghir, Z.; Melnikov, D.; Gaponenko, Y.; Sechenyh, V.; Legros, J.C.; Mialdun, A. IVIDIL: On-board g-jitters and diffusion controlled phenomena. J. Phys. Conf. Ser.
**2011**, 327, 012031. [Google Scholar] [CrossRef] - Van Vaerenberg, S.; Shapiro, A.; Galliero, G.; Montel, F.; Legros, J.C.; Caltagirone, J.P.; Daridon, J.L.; Saghir, Z. Multicomponent processes in crudes. In Microgravity Applications Programme: Successful Teaming of Science and and Industry; ESA—Directorate of Manned Spaceflight and Microgravity, Noordwjik (NL): Sorrento, Italy, 2005; Volume SP-1290, pp. 202–213. [Google Scholar]
- Van Vaerenberg, S.; Legros, J.C.; Daridon, J.L.; Karapantsios, T.; Kostoglou, M.; Saghir, Z.M. Multicomponent transport studies of crude oils and asphaltene in DSC program. Microgravity Sci. Technol.
**2006**, XVIII, 150–154. [Google Scholar] [CrossRef] - Praizey, J.P. Benefits of microgravity for measuring thermotransport coefficients in liquid metallic alloys. Int. J. Heat Mass Transfer
**1989**, 32, 2385. [Google Scholar] [CrossRef] - Bert, J.; Dupuy-Philon, J. Microgravity measurement of the Soret effect in a molten salts mixture. J. Phys. Condens. Matter
**1997**, 9, 11045. [Google Scholar] [CrossRef] - Praizey, J.P.; Van Vaerenberg, S.; Garandet, J.P. Thermomigration experiment onboard EURECA. Adv. Space Res.
**1995**, 16, 205. [Google Scholar] [CrossRef] - Sköld, K. Small energy transfer scattering of cold neutrons from liquid argon. Phys. Rev. Lett.
**1967**, 19, 1023–1025. [Google Scholar] [CrossRef] - Cunsolo, A. The terahertz spectrum of density fluctuations of water: The viscoelastic regime. Adv. Condens. Matter Phys.
**2015**, 2015, 137435–137459. [Google Scholar] [CrossRef] - Cunsolo, A.; Kodituwakku, C.N.; Bencivenga, F.; Frontzek, M.; Leu, B.M.; Said, A.H. Transverse dynamics of water across the melting point: A parallel neutron and X-ray inelastic scattering study. Phys. Rev. B
**2012**, 85, 174305. [Google Scholar] [CrossRef] - Ruocco, G.; Sette, F.; Bergmann, U.; Krisch, M.; Masciovecchio, C.; Mazzacurati, V.; Signorelli, G.; Verbeni, R. Equivalence of the sound velocity in water and ice at mesoscopic lengths. Nature
**1996**, 379, 521–523. [Google Scholar] [CrossRef] - Ruocco, G.; Sette, F. The history of fast sound in liquid water. Condens. Matter Phys.
**2008**, 11, 29–46. [Google Scholar] [CrossRef] - Cunsolo, A. Onset of a transverse dynamics in the THz spectrum of liquid water. Mol. Phys.
**2013**, 111, 455–463. [Google Scholar] [CrossRef] - Cunsolo, A. The terahertz dynamics of simplest fluids probed by X-ray scattering. Int. Rev. Phys. Chem.
**2017**, 36, 433–539. [Google Scholar] [CrossRef] - Sette, F.; Ruocco, G.; Krisch, M.; Bergmann, U.; Masciovecchio, C.; Mazzacurati, V.; Signorelli, G.; Verbeni, R. Collective dynamics in water by high-energy resolution inelastic X-ray scattering. Phys. Rev. Lett.
**1995**, 75, 850–854. [Google Scholar] [CrossRef] - Sette, F.; Ruocco, G.; Krisch, M.; Masciovecchio, C.; Verbeni, R. Collective dynamics in water by inelastic X-ray scattering. Phys. Scr.
**1996**, T66, 48–56. [Google Scholar] [CrossRef] - Sette, F.; Ruocco, G.; Krisch, M.; Masciovecchio, C.; Verbeni, R.; Bergmann, U. Transition from normal to fast sound in liquid water. Phys. Rev. Lett.
**1996**, 77, 83–86. [Google Scholar] [CrossRef] [PubMed] - Ruocco, G.; Sette, F.; Krisch, M.; Bergmann, U.; Masciovecchio, C.; Verbeni, R. Line broadening in the collective dynamics of liquid and solid water. Phys. Rev. B
**1996**, 54, 14892–14895. [Google Scholar] [CrossRef] - Sampoli, M.; Ruocco, G.; Sette, F. Mixing of longitudinal and transverse dynamics in liquid water. Phys. Rev. Lett.
**1997**, 79, 1678–1681. [Google Scholar] [CrossRef] - Sette, F.; Krisch, M.; Masciovecchio, C.; Ruocco, G.; Monaco, G. Dynamics of glasses and glass-forming liquids studied by inelastic X-ray scattering. Science
**1998**, 280, 1550–1555. [Google Scholar] [CrossRef] - Ruocco, G.; Sette, F. The high-frequency dynamics of liquid water. J. Phys. Condens. Matter
**1999**, 11, R259–R293. [Google Scholar] [CrossRef] - Monaco, G.; Cunsolo, A.; Ruocco, G.; Sette, F. Viscoelastioc behaviour of water in the THz frequency range: An inelastic X-ray study. Phys. Rev. E
**1999**, 60, 5505–5521. [Google Scholar] [CrossRef] [PubMed] - Scopigno, T.; Balucani, U.; Ruocco, G.; Sette, F. Inelastic X-ray scattering and the high-frequency dynamics of disordered systems. Phys. B
**2002**, 318, 341–349. [Google Scholar] [CrossRef] - Cunsolo, A.; Ruocco, G.; Sette, F.; Masciovecchio, C.; Mermet, A.; Monaco, G.; Sampoli, M.; Verbeni, R. Experimental determination of the structural relaxation in liquid water. Phys. Rev. Lett.
**1999**, 82, 775–778. [Google Scholar] [CrossRef] - Cunsolo, A. Inelastic X-ray scattering as a probe of the transition between the hydrodynamic and the single-particle regimes in simple fluids. In X-ray Scattering; Ares, A.E., Ed.; Intech Open: London, UK, 2017; Chapter 1. [Google Scholar] [CrossRef]
- Grimsditch, M.; Bhadra, R.; Torell, L.M. Shear Waves Through the Glass-Liquid Transformation. Phys. Rev. Lett.
**1989**, 62, 2616–2619. [Google Scholar] [CrossRef] - Giordano, V.M.; Monaco, G. Fingerprints of Order and Disorder on High-Frequency Dynamics of Liquids. Proc. Natl. Acad. Sci. USA
**2010**, 107, 21985–21989. [Google Scholar] [CrossRef] - Debye, P. Zur Theorie des Specifische Wärmer. Ann. Phys.
**1912**, 344, 798–839. [Google Scholar] [CrossRef] - Debye, P. Vorträge Über die Kinetische Gastheorie; B.G. Teubner: Leipzig, Germany, 1914; pp. 46–60. [Google Scholar]
- Brillouin, L. Diffusion de la Lumière Wet des Rayon X par un Corps Transaparent Homogéne—Influence de L’agitation Thermique. Ann. Phys.
**1922**, 9, 88–120. [Google Scholar] [CrossRef] - Brillouin, L. La chaleur spécifique des liquides et leur constitution. Serie VII, Tome VII. J. Phys. Rad.
**1936**, 4, 153–157. [Google Scholar] [CrossRef] - Frenkel, J. Kinetic Theory of Liquids; Oxford University Press: Oxford, UK, 1946. [Google Scholar]
- Landau, L.D. Theory of superfluidity of Helium II. Phys. Rev.
**1941**, 60, 356–358. [Google Scholar] [CrossRef] - Glasstone, S.; Laidler, K.J.; Eyring, H. The theory of rate processes. In The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; McGraw-Hill: New York, NY, USA; London, UK, 1941. [Google Scholar]
- Ewell, R.H.; Eyring, H. The viscosity of liquids as function of temperature and pressure. J. Chem. Phys.
**1937**, 5, 726–736. [Google Scholar] [CrossRef] - Kauzmann, W.; Eyring, H. The viscous flow of large molecules. J. Am. Chem. Soc.
**1940**, 62, 3113. [Google Scholar] [CrossRef] - Andreev, A.F. Two-Liquid effects in a normal liquid. JEPT
**1971**, 32, 987–990. [Google Scholar] - Baroni, P.; Bouchet, P.; Noirez, L. Highlighting a cooling regime in liquids under submillimeter flows. J. Phys. Chem. Lett.
**2013**, 4, 2026–2029. [Google Scholar] [CrossRef] - Noirez, L.; Baroni, P. Identification of thermal shear bands in a low molecular weight polymer melt using oscillatory strain field. Colloid Polym. Sci.
**2017**, 296, 713–720. [Google Scholar] [CrossRef] - Landau, L.D. Mechanique des Fluides; Edition MIR: Moscow, Russia, 1971; Volume Tome VI, p. §7. [Google Scholar]
- Bolmatov, D. The phonon theory of liquids and biological fluids: Developments and applications. Phys. Chem. Lett.
**2022**, 13, 7121–7129. [Google Scholar] [CrossRef] [PubMed] - Baggioli, M.; Vasin, M.; Brazhkin, V.; Trachenko, K. Gapped Momentum States. Phys. Rep.
**2020**, 865, 1–44. [Google Scholar] [CrossRef] - Gaeta, F.S.; Scala, G.; Brescia, G.; Di Chiara, A. Characterization of macromolecules in liquid solutions by thermal diffusion. I. Dependence of the Soret coefficient on the nature of the dispersing medium. J. Polym. Sci. Polym. Phys. Ed.
**1975**, 13, 177–202. [Google Scholar] [CrossRef] - Experimental Data for K
_{l}’s and u_{ϕ}’s Have Been Collected from Several Fonts, in Particular: (a) Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology-Group II. Atomic and Molecular Physics, Vol. 5, Molecular Acoustics, W. Schaafs, FA.; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1967. 2. (b) Tsederberg, N.V. Thermal Conductivity of Gases and Liquids; for the Thermal Conductivities of Liquid Mixtures and Solutions, See 220–221; M.I.T. Press: Cambridge, MA, USA, 1965. 3. (c) Hodgman, C.H.; Weast, R.C.; Shankland, R.S.; Selby, S.M. (Eds.) Handbook of Chemistry and Physics; The Chemical Rubber Publishing Co.: Cleveland, OH, USA, 1962. 4. (d) Lide, D.R., (Ed.) Handbook of Chemistry and Physics; 80th ed.; CRC Publisher: New York, NY, USA, 1999. 5. (e) (c). Available online: theengineeringtoolbox.com (accessed on 24 August 2023). - Prigogine, I.; De Brouckere, L.; Amand, M.R. Recherche sur la thermodiffusion en phase liquid. Physica
**1950**, 16, 577–598. [Google Scholar] [CrossRef] - Bierlein, A.; Finch, C.R.; Bowers, H.E. Coefficients de soret dans le système benzène=heptane normal. J. Chim. Phys.
**1957**, 54, 872–878. [Google Scholar] [CrossRef] - Chen, G. Non-Fourier phonon heat conduction at the microscale and nanoscale. Nat. Rev. Phys.
**2021**, 3, 555–569. [Google Scholar] [CrossRef] - Ziman, J.M. Elctrons and Phonons: Theory of Transport Phenomena in Solids; Oxford University Press: Oxford, UK, 2001. [Google Scholar]
- Griffin, A. Brilluoin light scattering fromn crystals in the hydrodynamic regime. Rev. Mod. Phys.
**1968**, 40, 167–205. [Google Scholar] [CrossRef] - Ward, J.C.; Wilks, J. Second sound and the thermo-mechanical effect at very low temperatures. London Edinburgh Dublin Philos. Mag. J. Sci.
**1952**, 43, 48–50. [Google Scholar] [CrossRef] - Eslamian, M.; Saghir, Z. Dynamic thermodiffusion theory for ternary liquid mixtures. J. Non-Equilibr. Thermodyn.
**2010**, 35, 51–73. [Google Scholar] [CrossRef] - Andrade, J.N. The Viscosity of Liquids. Nature
**1930**, 125, 582–584. [Google Scholar] [CrossRef] - Bondi, A. Notes on the rate process theory of flow. J. Chem. Phys.
**1946**, 14, 591. [Google Scholar] [CrossRef] - Macedo, P.B.; Litovitz, T.A. On the relative roles of free volume and activation energy in the viscosity of liquids. J. Chem. Phys.
**1965**, 42, 245. [Google Scholar] [CrossRef] - Kayanattil, M.; Huang, Z.; Gitaric, D.; Epp, S.W. Rubber-like elasticity in laser-driven free surface flow of a Newtonian fluid. Proc. Natl. Acad. Sci. USA
**2023**, 120, e2301956120. [Google Scholar] [CrossRef] [PubMed] - Prigogine, I. Non Equilibrium Statistical Mechanics; Wiley Interscience: New York, NY, USA, 1962. [Google Scholar]

**Figure 1.**Schematic representation of a mixture made by two different chemical species, 1 and 2, ideally represented by spheres and cubes. The solids represent the icebergs that in the DML constitute the liquid matrix. A temperature gradient $\nabla {T}_{ext}$ is applied to the medium generating a heat flux ${J}_{q}$. At the beginning of the experiment (

**a**) the two species are uniformly distributed. As far as the heat crosses the medium, the two species separate, until a steady state is reached, characterized by the separation of the two chemical species (

**b**). The steady state is the Soret equilibrium, characterized by the dynamical equilibrium reached by the two diffusive mechanisms, one driven by the temperature gradient, the other by the concentration gradients of the chemical species. The arrows represent the direction of the related vectors indicated at their tips.

**Figure 2.**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. [81] of about 3200 m/s for the case of water. Average sizes of icebergs $\langle {\Lambda}_{0}\rangle $ have been estimated of the order of magnitude of some nanometers. When perturbations cross the boundary between two icebergs, ${f}^{th}$ develops and energy and momentum are transmitted from one to the nearest-neighbor 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.

**Figure 3.**Schematic representation of inelastic collisions between wave-packets and liquid particles. The dots represent the molecules, arranged in a metastable liquid particle, the small springs indicate the forces involving the internal DoF and responsible for the temporary stability of the cluster. In the event represented in (

**a**), an energetic wave-packet transfers energy and momentum to a liquid particle; it is commuted upon time reversal into the one represented in (

**b**), where a liquid particle transfers energy and momentum to a wave-packet. The particle changes velocity and the frequency of wave-packet is shifted by the amount $\left({\nu}_{2}-{\nu}_{1}\right)$. Due to its time symmetry, this mechanism has been assumed the equivalent of Onsager’s reciprocity law at microscopic level [17,19]. This elementary interaction has an activation threshold for potential energy and, depending on how much energy is absorbed by internal DoF, the rest will become kinetic energy of the liquid particle. When a system is subjected to an external temperature gradient, the first effect that occurs is the establishment of the internal temperature gradient, according to the Cattaneo–Fourier equation. Once the internal DoF have reached the statistical equilibrium in terms of distribution of their degree of excitation depending on the local temperature, the phenomenon of matter diffusion develops due to the temperature gradient and at the expense of the kinetic energy reservoir. In other words, inelastic effects dominate the dynamics during the transient phase, leaving the control to the kinetic reservoir at the steady-state, where only the elastic effects of the elementary interactions matter.

**Figure 4.**Close-up of the wave-packet–liquid particle interaction shown in Figure 3a. Only the first part of the interaction is represented, i.e., that during which the wave packet transfers momentum and energy to the liquid particle.

**Figure 5.**Schematic (drawing not to scale) of the experimental device adopted in [11]. The liquid, glycerol in this case, lies in between the two plates, one of which may rotate, the other is fixed.

**Figure 6.**(

**a**) Values of Soret coefficient deduced from [116] and their best fit lines. (

**b**) Values of the ratios $K/u$ and their best fit lines for the same couple of liquids as in (

**a**). The two plots cross at about the same temperature as where ${S}_{T}$ crosses the “zero” in (

**a**).

**Figure 7.**(

**a**) Closed system constituted by a mixture of two substances at thermal equilibrium with the environment; only thermal energy can be exchanged with the environment. (

**b**) How the mass ${S}_{0}$, ${S}_{1}$, ${S}_{2}$ and energy $E$ barycenters of the same system as in (

**a**) appear at the Soret equilibrium, by supposing that the species “1” has and higher density than species “2”.

**Table 1.**Experimental values for Soret coefficient (last column, units ${10}^{-3}{\mathrm{K}}^{-1}$) for mixtures of polyvinylpyrrolidone K90 of 360,000 amu in various solvents as listed in first column, measured and reported in [113]. Thermodiffusion of this macromolecule was experimentally studied in several solvents; in particular, an inversion of the sign of the Soret coefficient was detected in buthanol and propanol. The columns 2 and 3 report the values of the ratios $K/u$ for the several substances (units ${10}^{-3}\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$). Their differences are reported in fourth column. It is interesting to note the sign inversion for the same two solvents as experimentally detected. Data for ${K}^{\prime}s$ and ${u}^{\prime}s$ are deduced from [114].

Mixture | ${\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{\mathit{l}}$ | ${\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{\mathit{p}}$ | ${\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{\mathit{l}}-{\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{\mathit{p}}$ | ${\mathit{S}}_{\mathit{T}}$ |
---|---|---|---|---|

K-90 in water | 0.4031 | 0.137 | 0.266 | 19.82 |

K-90 in methanol | 0.1830 | 0.137 | 0.046 | 0.38 |

K-90 in ethanol | 0.1440 | 0.137 | 0.007 | 2.31 |

K-90 in buthanol | 0.1205 | 0.137 | −0.016 | −5.78 |

K-90 in propanol | 0.1197 | 0.137 | −0.017 | −6.01 |

**Table 2.**Values of the ratios $K/u$ elaborated from experimental data of thermodiffusion in liquid mixtures obtained with the Klusius-Deckel device [115]. The columns 2 and 3 report the values of the ratios $K/u$ for the substances (units ${10}^{-3}\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$). Their differences are reported in fourth column. It is interesting to note the sign inversion for the same two solvents as experimentally detected. Data for ${K}^{\prime}s$ and ${u}^{\prime}s$ are deduced from [114]. Data from the two Tables, in particular for the ratios $K/u$, must however be taken cum grano salis and considered with caution, a thorough evaluation of ${S}_{T}$ in the DML should be elaborated from Equation (25).

Mixture | ${\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{1}$ | ${\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{2}$ | ${\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{1}-{\left(\raisebox{1ex}{$\mathit{K}$}\!\left/ \!\raisebox{-1ex}{$\mathit{u}$}\right.\right)}_{2}$ | Component Drifting to Cold Plate |
---|---|---|---|---|

Hexane-Carbontetrachloride | 0.1130 | 0.1120 | 0.0010 | 2 |

Hexane-Cyclohexane | 0.1130 | 0.0978 | 0.0152 | 2 |

Toluene-Cyclohexane | 0.1100 | 0.0978 | 0.0122 | 2 |

Toluene-Benzene | 0.1100 | 0.1028 | 0.0072 | 2 |

Benzene-Chlorobenzene | 0.1028 | 0.1021 | 0.0007 | 2 |

Benzene-Nitrobenzene | 0.1028 | 0.0990 | 0.0038 | 2 |

Bromobenzene-Carbontetrachloride | 0.0973 | 0.1120 | −0.0157 | 1 |

m-Xylene-o-Xylene | 0.0978 | 0.0974 | 0.0004 | 2 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Peluso, F.
Thermo-Mechanical and Mechano-Thermal Effects in Liquids Explained by Means of the Dual Model of Liquids. *Thermo* **2023**, *3*, 625-656.
https://doi.org/10.3390/thermo3040037

**AMA Style**

Peluso F.
Thermo-Mechanical and Mechano-Thermal Effects in Liquids Explained by Means of the Dual Model of Liquids. *Thermo*. 2023; 3(4):625-656.
https://doi.org/10.3390/thermo3040037

**Chicago/Turabian Style**

Peluso, Fabio.
2023. "Thermo-Mechanical and Mechano-Thermal Effects in Liquids Explained by Means of the Dual Model of Liquids" *Thermo* 3, no. 4: 625-656.
https://doi.org/10.3390/thermo3040037