VOLUME 79, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 24 NOVEMBER 1997 Experimental Detection of Forces Produced by the Flow of Heat C. Albanese,1 P. Dell'Aversana,1 and F. S. Gaeta1,2,* 1MARS Center, Via Comunale Tavernola, 80144 Napoli, Italy 2IIGB of CNR, Via Marconi 10, 80125 Napoli, Italy (Received 2 May 1997) We detected thermal radiation forces produced by heat flow through solid slabs suspended in nonisothermal liquids. Lateral effects are avoided, allowing a unidimensional approach. The results confirm expectations. Numerically simulated gravity-dependent perturbations are smaller than the measured forces. More precise measurements will be attempted on a European Space Agency suborbital flight in 1998. [S0031-9007(97)04544-4] PACS numbers: 44.30.+v The existence in liquids of thermal excitations consist- Only when a symmetry-breaking factor such as a tempera- ing of high frequency wave packets was postulated by ture gradient is applied does the energy and momentum Debye in 1912 [1]. Naturally occurring hyperfrequency exchange among excitations and particles become unbal- waves were experimentally detected fifty years later, by anced. In these conditions, the drift of wave packets along their interactions with light [2­6], and the low frequency the temperature gradient causes, in the average, an excess part of the spectrum of thermal waves in liquids could be transfer of energy and momentum to the particles. explored. Propagation velocities, up to about 10 GHz, are Neglecting the tensorial nature of thermal conductivity, only slightly higher than those of ordinary sound. we shall consider a unidimensional case. The momentum More recently, collective, longitudinal excitations in a flux Jp coupled to a flux of thermal energy Jq will be much higher frequency range, where wavelengths become Jp Jq ug, ug being the group velocity of propagation comparable with intermolecular distances, were observed of the wave packets. Inelastic scattering processes transfer in liquid D2O at room temperature by coherent inelas- momentum to the medium at an a priori unknown rate, tic neutron scattering [7]. Their propagation velocity is producing a volume force, i.e., a pressure gradient in the 3310 6 250 m s21. By a different technique, based on in- medium: elastic x-ray scattering, Sette et al. [8,9] were able to study µ µ P th K dT thermal waves in liquid H2O in an energy-momentum re- r HJp 2 H r , (1) gion wider than the one accessible to neutron scattering z z z ug dz in D r being the unit vector in the direction of the temperature 2O. Naturally occurring ultrahigh frequency elastic waves exhibited a solidlike measured velocity of 3200 6 gradient, K is the thermal conductivity, and H is a nu- 100 m s21. From these studies a fundamental analogy merical quantity ranging between zero and unity, which is emerges in the dynamical responses of liquid H the reflection coefficient of mechanical waves [11­14]. If 2O and D the rate of momentum transfer to the medium is exceed- 2O with those of the respective ices. A fundamental difference between the properties of ingly small H 0 , there will be no observable pressure wavelike thermal excitations in solids and liquids should, gradient. however, be kept in mind, due to the ubiquitous presence of Let us consider a system in which the heat flows across disorder in dense fluids over mesoscopic distances. This a surface of discontinuity between two adjacent media- has the consequence of distorting the harmonic potential say, two immiscible liquids or a solid and a liquid. The within which the molecules oscillate. In consequence of heat flux is continuous through the limiting surface, but the skewness of the potential wells, the oscillations of the there is a discontinuity in the momentum flux, due to particles are anharmonic. Furthermore, there will be a the abrupt change of the propagation velocity of thermal multiplicity of nearby modes of slightly different en- waves-an extreme case of anharmonicity. From Eq. (1), ergy. Interactions among such anharmonic modes are well a thermal radiation pressure is produced at the interface, known in solids [10]. given by µ µ We propose that liquids may be viewed as dual systems Jq Jq constituted by a "gas" of thermal excitations and a popu- Pth 1,2 D1,2 HJp H1,2 2 r ug 2 ug 1 lation of particles, interacting with a mutual exchange of µ µ µ ć energy and momentum. dT 1 1 H1,2 K 2 r , (2) I. Thermal radiation forces: a quantitative evalua- dz ug 1 ug 2 tion.-In the course of the interactions, when the sys- where subscripts 1 and 2 refer to the heat-emitting and tem is isotropic and homogeneous, energy and momentum heat-receiving surface. It should be observed that the sign may be gained or lost with equal probability by the gas of the thermal force per unit of surface depends on whether of excitations and the local groups of oscillating particles. 1 ug 2 _ 1 ug 1. 0031-9007 97 79(21) 4151(4)$10.00 © 1997 The American Physical Society 4151 VOLUME 79, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 24 NOVEMBER 1997 An expression analogous to Eq. (2) was derived by an around a copper cylinder, dipping in the liquid, is fixed extension of Rayleigh's concept of acoustic radiation pres- above the slab. Bath and heater temperatures are con- sure to nonisothermal systems [11]. Subsequently, it was trolled independently; the heater temperature can rise up deduced from an extension of a Boltzmann theorem and to 90 ±C. The slab is connected with the balance by means from considerations of rational mechanics [12]. These ap- of a couple of very thin tungsten wires diam 50 mm . proaches have led to a physically coherent interpretation Care has been used to reduce undesired effects; in our ap- of thermomechanical and mechanothermal effects in liq- paratus, the horizontal position of the slab and its paral- uids and to new insight on the issue of microscopic time lelism with the heater surface are optically ensured within reversibility [13,14]. 620 min of arc, sufficient to achieve the desired repro- II. Experimental.-To validate the theory, we mea- ducibility in the tests. The distance between the slab and sured the forces produced by heat flowing through a solid the heater has been fixed with an accuracy of 0.1 mm. disk immersed in a nonisothermal liquid. According to the The liquids used in the experiments are water, ethylene theory, forces will be produced at the liquid-solid upper in- glycol, glycerol, and Fluorinert®, while slabs have been terface, as well as at the lower interface, over all of the disk made with Teflon® and Plexiglas®. The geometry of the surface crossed by the heat flow. (Here Pth 1,2 should be un- apparatus ensures that border effects are avoided at least derstood as Pth l,s, the media being liquid and solid.) The for a few minutes. Temperature readings are performed above approximation holds until the advancing tempera- by thermocouples distributed in key points inside the ture front reaches the lateral surface of the disk. experimental cell with particular attention to the space near Experimentation with various couples of solids and liq- the solid disk. uids may, thus, yield a check of the existence of thermal As the heater is switched on, the temperature rises radiation forces (TRFs) and their modus operandi. Infor- following an imposed ramp until it reaches, after a short mation may also be obtained on the propagation veloci- overshoot, the imposed temperature. Simultaneously, the ties of wave packets through Eq. (2), which, however, force measurements are started; a timer is used as a requires accurate, perturbation-free measurements, impos- common time base for the heater temperature profile and sible on ground. the signals coming from the thermocouples inserted in the A dedicated experimental apparatus, sketched in Fig. 1, liquid. In Fig. 2, the heater temperature profile and the has been designed to measure TRFs. The measuring de- force signals have been plotted for solid disks, 4 cm in vice has been realized by integrating an analytical balance having a force sensitivity of about 1021 dyn, with a ther- mostated container filled with a liquid in which the slab is suspended. A heater, formed by a Ni-Cr wire wrapped FIG. 2. Force and temperature profiles in two experiments with disks 4 cm in diameter of Plexiglas® (0.2 cm thick) and Teflon® (0.3 cm thick) in water and in ethylene glycol. The temperature profile and the initial temperature 120 ±C are FIG. 1. Schematic representation of the apparatus used for the always the same. Some properties of the materials are listed; measurement of the TRFs on ground. sound velocity refers to ultrasonic frequencies. 4152 VOLUME 79, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 24 NOVEMBER 1997 diameter, immersed in various liquids. (Since the force is laterally, near the border of the heater; they generate a force always negative, i.e., opposite to Jq, we refer to its absolute pulling the disk up, i.e., in the same direction of the TRF. value.) From a sequence of images, the mean convective velocity After an initial steep rise of both the temperature and of 0.07 cm sec in the radial direction was deduced. force profiles, the forces continue to increase for a while, Another disturbance due to the change of buoyancy after the heater temperature has reached the steady plateau comes from the different thermal expansions of liquid and level. The force profile, indeed, follows the building up of solid. From the instantaneous temperature distribution, the temperature gradient across the disk, through an area this force contribution, Dfbuoy, may be approximately cal- which widens while the heat diffuses to an increasingly culated at any instant. Finally, capillary forces, which pull large region around the heat source. A plateau value of the tungsten wires down, decrease as the upper surface of the force is reached, followed, under normal gravity, by a the liquid warms up; a differential force, pointing up, thus slow decrement, due to the decrease of the intensity of local appears at the beginning of a measurement in the first part temperature gradient with time, to the decreasing buoyancy of the heating phase and disappears during the plateau. in the progressively warmer liquid and, finally, to the onset The hydrostatic pressure variation connected with thermal of lateral effects, when the advancing temperature front expansion of the heated liquid gives equal contributions reaches the border of the disk. Experiments with various on the upper and lower surfaces of the slab, so that it has solid-liquid couples evidenced a delay of the force signal no net effect on the measured force. These effects act increasing with the distance from the heater, and dependent more or less independently; each contributes a force nor- on the heat diffusivity of the materials. The decrease of mal to the surface of the disk, some pushing it up, others the forces measured after switching off the heating also pulling down as shown in Fig. 4. The resultant, thus, is fits well the approach discussed so far. One example of given by the algebraic sum of the respective components. a registration inclusive of the phase after the switch off of A numerical simulation has been done to evaluate the the heat source is given in Fig. 3. system evolution from the fluid-dynamic point of view. The reproducibility of results is fair, typical variations The geometry used in the simulation is axisymmetric, in distinct runs executed under the same experimental and reproduces quite well the experimental cell geometry. conditions being congruous with the fluctuations in the Using the simulation results, the entity of the convective course of a single experiment, amounting to less than 5% effects near the heater and the buoyancy variations, due to of the measured forces. the heating of the liquid surrounding the solid slab, have On ground, the various perturbing factors are difficult to been calculated. We found that the contribution of TRF to assess, owing to the overlap of all of the gravity-induced the measured force is preeminent, and that the measured disturbances. An evaluation of the most important per- total forces represent the intensity of the respective TRF turbing hydrodynamical effects is now presented. Ap- with a 10% overall margin of imprecision. plying heat from above, convection is established in the III. Discussion and conclusions.-An extensive dis- liquid gap between the heater and the solid slab, and be- cussion of the theoretical implications of our findings tween the hot zone beneath the heater and the surrounding would be out of place here, since this paper is a report cold zone near the border of the slab. Convective motions have been observed by introducing a dye, methylene blue, FIG. 4. Hydrodynamic force, buoyancy force, and capillary effect on the suspender wires, calculated with the aid of FIG. 3. Temperature and force profiles in an experiment with numerical simulations, together with the total measured force a Plexiglas® disk 4 cm in diameter and 0.2 cm thick in ethylene on a Plexiglas® disk 4 cm in diameter and 0.2 cm thick, in glycol. The post-heater switch-off events are shown. water. 4153 VOLUME 79, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 24 NOVEMBER 1997 on preliminary experimental measurements. However, we a precise value for the propagation velocity of hyper- would like to make a concise comment of a general nature. frequency elastic waves in liquids. On the other hand, The existence of well defined-though strongly with our method, there are no limitations concerning the damped-sound waves, of frequencies up to very high val- nature of the liquids employed. Thus, a very extensive in- ues, i.e., to wavelengths comparable with interatomic dis- vestigation of the dynamical behavior of the liquid state at tances, was evidenced by de Schepper et al. also in liquid mesoscopic lengths appears feasible, if gravity-dependent argon [15]. Compiling these results with those from neu- perturbations may be eliminated. The method per se tron scattering and x-ray scattering in associated hydrogen- doesn't meet any of the serious limitations encountered by bonded liquids such as D2O and H2O mentioned above coherent inelastic neutron [7,15] or inelastic x-ray scatter- [7­9], it seems possible to conclude that thermal ex- ing [8,9]. citations of frequencies extending up to the 1000 GHz Further experimentation will be carried out in condi- range exist in all types of liquids. Their propagation tions of weightlessness on a European Space Agency sub- speed at very high frequencies ("fast sound") might be orbital Maser flight scheduled for the spring of 1998. substantially higher than group velocity in the ultrasonic range. Solidlike velocities are not surprising, in view of the similarity of the values of intermolecular forces at *Person to whom all correspondence should be ad- short range in the two states of aggregation, which suggest dressed. analogy of dynamical responses of liquids and solids to [1] P. Debye, Ann. Phys. (Paris) 39, 798 (1912). periodic solicitations having mesoscopic wavelengths. [2] I. L. Fabelinskii, Sov. Phys. Usp. 77, 649 (1962) [Sov. From the experiments performed until now, it is already Phys. JETP 5, 667 (1963)]. possible to assess the existence and order of magnitude [3] G. B. Benedek, J. B. Lastovka, K. Fritsch, and T. Greytak, of the TRF, notwithstanding the presence of various per- J. Opt. Soc. Am. 54, 1284 (1964). turbations of gravitational origin. The analysis of our re- [4] R. Y. Chiao and B. P. Stoicheff, J. Opt. Soc. Am. 54, 1286 sults only allows calculation of approximate values for the (1964). TRF of each solid-liquid couple of materials. These val- [5] W. J. 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