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Second Sound and Ballistic Heat Conduction: Naf Experiments Revisited ⇑ R International Journal of Heat and Mass Transfer 117 (2018) 682–690 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Second sound and ballistic heat conduction: NaF experiments revisited ⇑ R. Kovács , P. Ván Department of Theoretical Physics, Wigner Research Centre for Physics, Institute for Particle and Nuclear Physics, Budapest, Hungary Department of Energy Engineering, BME, Budapest, Hungary Montavid Thermodynamic Research Group, Hungary article info abstract Article history: Second sound phenomenon and ballistic heat conduction, the two wave like propagation modes of heat, Received 25 August 2017 are the two most prominent, experimentally observed non-Fourier effects of heat conduction. In this Received in revised form 4 October 2017 paper we compare three related theories by quantitatively analyzing the crucial NaF experiments of Accepted 8 October 2017 Jackson, Walker and McNelly, where these effects were observed together. We conclude that with the Available online 17 October 2017 available information the best comparison and insight is provided by non-equilibrium thermodynamics with internal variables. However, the available data and information is not the best, and further, new Keywords: experiments are necessary. Non-equilibrium thermodynamics Ó 2017 Elsevier Ltd. All rights reserved. Ballistic propagation NaF experiments Kinetic theory 1. Introduction reflected and scattered only on the boundaries. In the framework of non-equilibrium thermodynamics with internal variables [5], Due to the technological development, manufacturing and the ballistic propagation is represented as a coupling between material designing achieved the level where the classical laws of the thermal and mechanical fields and propagates with the elastic physics and the related engineering methodologies do not hold. wave. In the complex viscosity model of Rogers it is a phenomenon The challenging areas are the low temperatures and nano-scales. of pure mechanical origin [6]. The propagation speed is common in This is most relevant for heat conduction where the deviation from these three theories, it is the speed of sound. The experimental the classical Fourier law are well known since decades but the cir- detection of second sound and ballistic modes is not easy and cumstances leading to non-Fourier phenomena are not yet clear. In requires a theoretical insight, too. The theoretical predictions of particular the discovery of Guyer-Krumhansl-type heat conduction second sound by Tisza and Landau have been based on their in heterogeneous materials [1,2] at room temperature shows, that respective two fluid theories [7,8]. It was measured first by Pesh- the traditional view of the validity of these theories is too narrow, kov [9] in super-fluid He-4. The so-called window condition and requires further investigations. This is also important for the derived by Guyer and Krumhansl [10] indicated the frequency two prominent non-Fourier phenomena, for the second sound range, where the dissipation of the wave propagation is minimal and for the ballistic propagation. and significantly aided the detection of second sound in solids. The second sound denotes the wave like propagation of heat However, such kind of theoretical tool for ballistic type conduction modelled by the Maxwell-Cattaneo-Vernotte (MCV) equation. Here does not exist yet. Moreover, it is not clear what really influences the propagation speed of heat waves is material dependent, but the existence of ballistic signals [11]. less than the speed of sound wave. In case of ballistic conduction Let us shortly review the essential information of experiments the heat propagates exactly at the speed of mechanical sound of Jackson, Walker and McNelly [12–14]. According to McNelly’s waves. The later one can be understood and interpreted in different PhD thesis [14], each NaF sample is identified by a number. Here ways depending on the underlying theory. measurements on two different NaF crystals are analyzed, these In phonon hydrodynamics and in particular in Rational are called as ‘‘#607167J” and ‘‘#7204205W” [14]. Fig. 1 shows Extended Thermodynamics [3,4], ballistic conduction is under- the experimental results related to these samples. stood as a non-interactive propagation of phonons. They are These curves in Fig. 1 correspond to a so-called heat pulse experiment. Their schematic arrangement is presented in Fig. 2 [14]. The rear side temperature history is measured and presented ⇑ Corresponding author at: Department of Theoretical Physics, Wigner Research previously in Fig. 1. Centre for Physics, Institute for Particle and Nuclear Physics, Budapest, Hungary. E-mail address: [email protected] (R. Kovács). https://doi.org/10.1016/j.ijheatmasstransfer.2017.10.041 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved. R. Kovács, P. Ván / International Journal of Heat and Mass Transfer 117 (2018) 682–690 683 Fig. 1. NaF experiment results: the left one (‘‘#607167J”) was published in [12] and the right one (‘‘#7204205W”) in [13]. complex viscosity based hybrid phonon gas model of Ma [15], non-equilibrium thermodynamics with internal variables and Nyíri-multipliers [5,11]. These models provided results regarding the reproduction of NaF experiments. One should mention also the work of Frischmuth and Cimmelli based on thermoelastic approach [16], Cimmelli et al. [17–19] and Bargmann and Steinmann [20,21] in spite of the poor reproduction of the measured results. 2.1. Kinetic theory and phonon hydrodynamics It is a particle based approach with phonons. The interactions between phonons can be classified as [3,4]: Normal (N) processes: the total momentum of phonons is con- served during the interaction, Resistive (R) processes: the opposite of N processes, total momentum is not conserved. Fig. 2. Arrangement of NaF experiment, original figure from [14]. Common in both cases is that the energy is conserved during the interactions. A third type, called as Umklapp process also exists In the next section we shortly review the relevant theories. where neither the energy nor the momentum is conserved [4]. The Then we collect and compare the performance of these theories thermal conductivity k is expected to be in connection with R pro- considering both qualitative and quantitative aspects. cesses and their frequencies 1 , i.e. sR 2. Models of ballistic heat conduction c2 k ¼ cv s ; ð1Þ 3 R Here we mention three different approaches to describe ballis- tic type propagation: where c is the Debye speed of phonons, cv is the isochoric specific heat, and sR is the characteristic time of the R processes. In this kinetic theory based phonon hydrodynamics of Dreyer and approach, Fourier’s law can be applied only when the R processes Struchtrup [3,4], are significantly dominant which results the diffusive kind of heat 684 R. Kovács, P. Ván / International Journal of Heat and Mass Transfer 117 (2018) 682–690 propagation. However, if one decreases the temperature then N pro- One has to apply M ffi 30 equations to obtain a good approxima- cesses become dominant and the wave nature of heat conduction tion of the ballistic propagation speed of phonons but it would reveals itself. In case of ballistic propagation there is no interaction require infinitely many momentum equations to obtain its exact between phonons, these particles just go through the sample without value. Naturally, it is difficult to solve (9) for practical problems, colliding. In order to include these propagation modes into phonon but M ¼ 3 equations give an acceptable approximation for a qual- hydrodynamics, one needs to distinguish between the phase density itative reproduction of the whole heat conduction process, while for the R and N processes. The resistive processes tend to the function one accepts that the predicted value of the ballistic propagation ¼ f R and the normal processes tend to the distribution function f N: speed is not correct. For M 3 one obtains a 3 field theory: y 2 ¼ ; ð Þ @te þ c @xp ¼ 0; f R 2 exp hck À 1 1 1 kBT @ p þ @ e þ @ N ¼ p; ð10Þ y t 3 x x s f ¼ no; ð3Þ R N cp n hck À 3 i i À 4 2 1 1 exp 1 4 1 @ þ @ ¼ þ : kBT 4 aT tN c xp N 15 sR sN where h is the Planck constant, kB is the Boltzmann constant, k denotes the wavenumber, n is the unit vector in the direction of i 2.2. Hybrid phonon gas model ; ¼ =ð 3Þ k pi is the momentum, y 3 8p is a constant, a is also a con- stant, see below. The temperature T is defined as the energy density Ma developed this approach based on the work of Rogers [6] by Debye law for phonons [3]: and Landau [22] to describe the longitudinal and transversal ballis- 4 5 k4 tic signals at the same time [15,23]. In this model, the internal ¼ 4; ¼ p B : ð Þ e aT a 3 4 energy E is splitted into two parts: 5 h c3 ¼ þ 0; ð Þ The phase density f of one phonon evolves according to the E E0 E 11 Boltzmann equation, 0 where E0 corresponds to the equilibrium part and E is the perturba- b tion. It is supposed to consist of the longitudinal and transversal @tf þ cni@if ¼ S; ð5Þ parts as b where S is the collision integral and n is a unit vector in the direc- 0 0 0 i E ¼ E þ 2E : ð12Þ tion of propagation. In the Callaway model the previous equilibrium l t distributions f R and f N are considered and combined in relaxation Let us consider the classical equation of motion of a viscous terms: fluid: b 1 1 S ¼ ðf À f Þ ðf À f Þ: ð6Þ 2 1 R N q@tv þ qðv Á gradÞv ¼gradP þ gr v þ n þ g grad divv; sR sN 3 This assumption implies that two different equilibrium distribu- ð13Þ tions exist in this system.
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