Friction and Radiative Heat Exchange in a System of Two Parallel Plates Moving Sideways : Levin-Polevoy-Rytov Theory Revisited
FFriction and Radiative Heat Exchange in a System of Two Parallel Plates Moving Sideways : Levin-Polevoy-Rytov Theory Revisited
G.V. Dedkov and A.A. Kyasov Nanoscale Physics Group, Kabardino-Balkarian State University, Nalchik, 360004 Russia It is shown that the fundamental results obtained in the works by Levine, Polevoi, Rytov (1980) and Polevoi (1990), based on the fluctuation-electromagnetic theory by Levine and Rytov, adequately describe the rate of radiative heat exchange and the frictional force in a system of two parallel thick plates in relative lateral motion. A numerically calculated friction force for good metals and thin gaps turns out to be by a factor 10 higher than earlier obtained by Polevoi and increases with increasing conductivity of the metals.
1. Introduction
Fluctuation-electromagnetic theory formulated by Levine and Rytov [1] is an extension of Rytov’s theory [2]. In [1], the spectrum of electromagnetic fluctuations of a heated body at an arbitrary distance from the surface of the body is expressed through mixed losses of the two point-dipole sources nearby it, which are calculated from the solution of regular electrodynamic problem. This is the essence of the generalized Kirchhoff’s law, which represents the form of a fluctuation-dissipation theorem. Within the framework of theory [1], expressions for the rate of heat exchange between the two semi-infinite media (thick plates) separated by a vacuum gap of finite width were obtained [3, 4], as well as the dissipative frictional force, arising in the case of relative lateral motion of one of the plates [5]. The first calculation of radiation heat exchange (within the framework of theory [2]) between the two plates in rest was carried out by Polder and van Hove [6] assuming a simpler case of two identical plates and a small temperature difference between them. In contrast to this, in [3, 4] media 1 and 2 were assumed to be homogeneous and isotropic with permittivities and permeabilities , µε and , µε , being the complex functions of the frequency ω . Moreover, the general case of anisotropic media was also examined. Later, similar problems were solving by a number of authors, but the formula for the thermal energy flux was reproduced in many cases either without references to [3, 4] (see, for example, [7, 8]) or in another equivalent form [9, 10]. The situation with work [5] turned out to be much more dramatic: in the final formulas for dissipative frictional force in the linear velocity approximation, reported by Polevoi [5], the dependence / cVF ∝ has appeared ( c is the speed of light in vacuum), whereas later several authors obtained linear in V and independent of c expressions for this force (at a finite temperature of the plates) [11, 12], or dependence ~ VF in the quantum zero-temperature limit [13]. These contradictions "poured fuel to the fire" of a lengthy discussion on the magnitude of the dissipative force, which has been started even earlier [14] and is being not completed so far [10-13, 15-17] (for many other references see in [18, 19]). In this work we show that the basic results for the friction force obtained in [3-5] are completely consistent with all results obtained by other authors later, while the dependence / cVF ∝ and a very low numerical value of the frictional stress (about /10 mN − for metallic plates at room temperature, at a gap width of 10 nm and a relative velocity of sm /1 ), is due to a special form of material properties of interacting bodies. We have recalculated numerically the frictional force for good nonmagnetic metals using the dielectric permittivity ωπσωε /4i)( = ( −σ the conductivity) and obtained much higher values (by 10 times) in the case of thin gaps. Moreover, a striking fact is that the friction force between the metallic plates increases with increasing conductivity.
2. Problem statement and general expression for the tangential force by Polevoi
In a system configuration used by Polevoi [5], the Cartesian laboratory coordinate system fixed in (plate) 1 (Fig.1) is chosen so that the axis xz = is orthogonal to the boundaries of the plates, the axis xx = , without loss of generality, is parallel to the velocity V of plate 2, the axis xy = (not shown in Fig.1) is orthogonal to the x and z axes. The temperatures of the plates are held constant at T and T , respectively. Following [5], the resulting force densities F and F acting on a unit surface area of plates 1 and 2 differ only in sign: VF / VFF −=−= , where F is the modulus of the dissipative tangential force F per unit area of the moving plate 2 in the laboratory reference system associated with resting plate 1. It is expressed in terms of the heat fluxes P and P from plates 1 and 2 (per unit area), the flow P being calculated through the Poynting vector in the reference frame of plate 1, and the flow P from plate 2 is calculated in its rest system: ( ) γ /1 PPVF += , (1) where cVuu /,)1/(1 =−= γ . The heat fluxes P and P are given by ( ) ( ) ukuk ,,~ )~,(),(41 ,,~~8 ωωωωωωωπ ωωωωωωωπ MTTkdd MkddP ⎥⎦⎤⎢⎣⎡ Π−Π+ +⎟⎟⎠⎞⎜⎜⎝⎛ −= ∫∫ ∫∫ ∞∞− ∞∞− h , (2) ( )( ) ukuk ,,~~ )~,(),(41 ,,~~~8 ωωωωωωωπ ωωωωωωωπ MTTkdd MkddP ⎥⎦⎤⎢⎣⎡ Π−Π− −⎟⎟⎠⎞⎜⎜⎝⎛ −−= ∫∫ ∫∫ ∞∞− ∞∞− h , (3) where − h the Planck constant, −= ),( kk k a two-dimensional wave vector coplanar to the plates, )(~ kV −= ωγω , uV == uV , , ( ) hh /,1)/exp(/),( TT TT =−=Π ωωωωω , T is the temperature in energy units and the integration is performed over the entire space of wave vectors. The function ( ) uk ,, ω M has the form ⎥⎦⎤⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+ +⎥⎦⎤⎢⎣⎡ +⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛++⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛= ~~ImIm~~ImIm4 )1(~~ImIm)1(~~ImIm4 µεεµ εµ βεµβµε βµµβεε QqqQqqQq QqqQqqQqM (4) where ( ) )/( µεω ckq −= , ( ) )/( µεω ckq −= , ( ) )/( ckq ω−= . The branches of the square roots are chosen to satisfy > q and parameter β is given by
22 2222 ~ kk kqu ⊥ = γβ . (5) Here )( ⎥⎦⎤⎢⎣⎡ −= ⊥ uk kuuk and ukuukk ku γγ −−+= )()1(~ ( k ~ – wave vector in the rest frame of plate 2), the tilde means that the corresponding quantities depending on ω and k are taken at ω ~ and k ~ . The quantities QQQQQ ,,,, εµµεµε in (4) are given by ( a is the gap width in Fig. 1) ( )( ) ( )( ) )exp(~/~/)exp(~/~/ qaqqqqqaqqqqQ −−−−++= εεεε ε , (6) ( )( ) ( )( ) )exp(~/~/)exp(~/~/ qaqqqqqaqqqqQ −−−−++= µµµµ µ , (7) ( )( ) ( )( ) )exp(~/~/)exp(~/~/ qaqqqqqaqqqqQ −−−−++= µεµε εµ , (8) ( )( ) ( )( ) )exp(~/~/)exp(~/~/ qaqqqqqaqqqqQ −−−−++= εµεµ εµ , (9) ( )( ) )~~(1)(1~4 −− −−−= µεµεµβ µε kQQQ . (10) We retained all the notation used in [5] with a single replacement κ , → , κκ k , , kk .
3. Transformation of the general formula for <<= cVu In the case <<= cVu , we have kk == ~,0 β , and (4) with allowance for (10) reduces to ⎥⎦⎤⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛= −− ~~ImIm~~ImIm4 µε µµεε QqqQqqqM (11) Using (11) and the identity ⎟⎟⎠⎞⎜⎜⎝⎛ −⎟⎟⎠⎞⎜⎜⎝⎛ −−≡⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ *2*222*1*1112211 ~~~~~~ImIm4 εεεεεε qqqqqq , (12) as well the analogous identity with the permutation ~~, µεµε ↔↔ , Eq. (1) for P reduces to Eq. (6) in [3] for the spectral energy flux density of the thermal field (since ~ qq = , ~ εε = ). In this case, PP −= , PP = , and the different sign of these quantities is due to the different direction of the flow of thermal energy relative to the plates. At ≠ V , using (6), the expression for −ε Q can be rewritten in the form )2exp(~)2exp(~1~~ qaqaqqqqQ ee −−∆∆−++= −−−− εεεε ε (13) where
22 22211 111 ~~ ~~~, qq qqqq qq ee −−=∆+−=∆ εεεε . For −µ Q we have the same expression as for −ε Q with allowance for the obvious permutations meme ~~,,~~, ∆→∆∆→∆↔↔ µεµε . The integrals in (2), (3) contains the contributions from inhomogeneous (evanescent) waves ( ck / ω> ) and from traveling waves ck / ω≤ . At ck / ω> we have ),2exp()2exp(, qaqaqq −=−= and (12) takes the form ~~~~ImIm~~~~ qqqqqqqqq ee ++∆∆−=⎟⎟⎠⎞⎜⎜⎝⎛ −⎟⎟⎠⎞⎜⎜⎝⎛ − εεεεεεεε . (14) At ck / ω≤ , correspondingly, 1)2exp(,i =−−= qaqq and ( ) ( ) ~~~4 ~11~~~~ qqqqqqqqq ee ++∆−∆−−=⎟⎟⎠⎞⎜⎜⎝⎛ −⎟⎟⎠⎞⎜⎜⎝⎛ − εεεεεεεε (15) Identities analogous to (14), (15) can be easily written with the permutations meme ~~,,~~, ∆→∆∆→∆↔↔ µεµε . Substituting (11)–(15) into (1)–(3), and taking into account the analytical properties of the function ( ) uk ,, ω M , namely [5] ( )( ) << >> ωωω ωωω uk uk MM (16) we obtain the following expression for the tangential force acting on the moving plate 2 in the laboratory coordinate system (negative values of x F correspond to the dissipative frictional force) [ ] ( ) ( ) [ ] )()2/coth()2/~coth(~1116 )2/coth()2/~coth(~ImIm)2exp(4 meTTDkkdd TTDqakkddF eck eex eeck ex ↔+−∆−∆−− −−∆∆−−= −≤∞ −>∞ ∫∫ ∫∫ ωωωπ ωωωπ ω ω hhh hhh (17) where )2exp(~1 qaD eee −∆∆−= and the terms )( me ↔ are determined by the same integrals by replacing meme ~~,,~~, ∆→∆∆→∆↔↔ µεµε . It should be emphasized once again that in this case Vk x −= ωω ~ . Formula (17) completely includes all the results of other authors [10–13, 17–19] at 1/ <<= cVu , obtained in the nonretarded and retarded limits . In particular, at →→ TT from (16) one obtains the formula for the quantum frictional force between the two smooth plates [13, 17] )(~ImIm)2exp(4 meDkadkdkdkF eeeVkxxy x ↔+∆∆−= −∞∞∞− ∫∫∫ ωπ h . (18) In turn, in the case of resting plates ( 0 = V ), the formula for the resultant energy flux P of the thermal field from plate 1 (for definiteness), taking into account (11)–(16), reduces to [8-10] [ ] ( )( ) [ ] ,)()()(118 )()(ImIm)2exp(2 mennDkdd nnDqakddP eck ee eeck e ↔+−∆−∆−+ +−∆∆−= −≤∞ −>∞ ∫∫ ∫∫ ωωωωπ ωωωωπ ω ω h h (19) where ( ) =−= iTn ii ωω h . Another useful expression for the frictional force stems from (17) with allowance for (13)–(15): ⎥⎦⎤⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛⋅ ⋅⎥⎥⎦⎤⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−= ∫∫ ∞
12 222211222211203
TT QqqqQqqqkkddF x ωω µµεεωπ µε hhh (20) It is worth noting that the inner integral in (20) includes the contribution from both evanescent and traveling modes. It turns out that Eq. (20) is most convenient when calculating the frictional force between normal metals. In the same way, the expression for P (in the case 1/0 <<< cV ) can be written in the form ⎥⎦⎤⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛⋅ ⋅⎥⎥⎦⎤⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛= ∫∫ ∞
12 2222112222112031
TT QqqqQqqqkddP ωω µµεεωωπ µε hhh (21) Obviously, Eq. (21) reduces to (19) at = V . Thus, the fundamental results of theory [3-5] (Eqs. (1)–(10)) fully include all the results of other authors for the rate of heat transfer and frictional force in the configuration of two arbitrary semi-infinite media (thick plates) in relative nonrelativistic motion.
4. Certain consequences and particular cases
In addition to Eqs. (17), (18) and (20) following from (1)–(3), it is expedient to examine some other known limits. In the case of low sliding velocity and a rather high temperature
TTT == , 1/ << wx Vk ω ( h / T w =ω is the Wien temperature), the temperature factor in (17) and (20) is −=−≈− wx nddnVkTT ωωωωωω hh (22) On the other hand, in the first-order expansion by V , the quantities with “tilde” in (17) and (20) will contain the velocity-independent terms, and the velocity-proportional ones. Therefore, according to (20), (22), the first-order-velocity approximation to the frictional force is given by ∫∫ ∞∞ ⎥⎥⎦⎤⎢⎢⎣⎡ ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛= ImImImIm)(2 µε µµεεωωωπ QqqqQqqqdkkddndVF h (23) It should be emphasized that all quantities in −ε Q and −µ Q (see (13)) must be free of “tilde”. In the same way, Eq. (17) reduces to ( )( ) )(11)(8 ImIm)2exp()(2
2/ 22212203 22/ 12203 meDkkdddndV DqakkdddndVF eck eex eeck ex ↔+∆−∆−+ +∆∆−= −≤∞ −>∞ ∫∫ ∫∫ ω ω ωωωπ ωωωπ h h (24) In the limiting case of absolutely black bodies, 1 ==== µµεε , the magnitude of the integrand in square brackets in (23) equals (–2) at ck / ω< and zero otherwise. The resultant integral yields ⎟⎠⎞⎜⎝⎛−= hh TcVF π (25) The same result stems from (24) since 0 =∆=∆=∆=∆ mmee . At KT = and smV /1 = Eq. (25) yields /105~ mNF − ⋅ . In the particular case 1/ << ca w ω (i. e. ma µ << at KT = ), if use is made of the approximation kck ≈− )/( ω , one obtains )1/()1( +−≈∆ iiie εε and ck iim −≈∆ εω ( = i ). Then, for dielectrics and poor conductors, 0 ≈∆ im , and Eq. (24) transforms to ∫∫ ∞ −∞ −⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +−≈ )2exp(11Im11Im)(2 e DkadkkdddnVF εεεεωωωπ h , (26) where )2exp(11111 kaD e −⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +−−= εεεε (27) For conductors with permittivities ωπσε /4i1 += , assuming << πσω w , we have 1 ≈ e D and Eq. (26) yields
421 24 )/(32 )3( aVTF hh σσπς−≈ , (28) where = ς – Riemann’s zeta-function. At − ==== ssmVKT σσ (graphite), and nma = Eq. (28) yields /106~ mNF − ⋅ . We note that the value of F decreases with increasing conductivities. For good metals, however, the above argumentation is not valid and a more accurate calculation is required. In [5], the impedance approximation with the factor iiii cck ςεωµεωµε i2/1i2/1222 i/)i()/( =≈− was used ( i ς – the impedance). This led to the dependence aVTF /~ σ (at a small gap width) for good metals with σσσ == and 1 == µµ . The corresponding numerical assessment results in ~ /103 mN − ⋅ [5] at the same conditions as above and assuming that − ⋅= s σ . However, as we will show in what follows, a more accurate numerical calculation with the use of the exact factor )/( ck ii ωµε − for good nonmagnetic metals leads to a considerably higher frictional force.
5. The case of good nonmagnetic metals
Let us transform Eq. (20) to a form convenient for further numerical computation. We consider the case of identical metals σσσ == , == µµ , ωπσωε /4i)( = . Then =− )/( ck εω )iexp()4( φπσω ⎟⎟⎠⎞⎜⎜⎝⎛ + ck , ⎟⎠⎞⎜⎝⎛−= ckarctg πσωφ (29) To find the contribution from evanescent waves ck / ω> , we introduce the new variables x w ωω= ( h / T w =ω ) and /)( axyk w λ+= ( ca ww / ωλ = ), / aydykdk = . The most important contribution is related to the second term in square brackets of (23). The corresponding inner integral transforms to
140 221212223/ 42222113 µµω µ φλµµ
IaQPxydyyaQqqqdkk wc ≡+=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ ∫∫ ∞ −∞ , (30) ( ) [ ] xxyT w λλ ++= , c a w πσωλ= (31) )cosh()iexp(2)sinh())i2exp(( yyPyPyQ φφ µ ++= (32) ⎟⎟⎠⎞⎜⎜⎝⎛ +−= xy xarctg m λλφ (33) The contribution from propagating waves ck / ω< is obtained with substitutions x w ωω= and ( ) /, cydykdkyxck ww ωω −=−= . Then we obtain )(sin)(4ImIm µµω µ ωφωµµ IcQPyxdyycQqqqdkk wxc w ≡−=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ ∫∫ − (34) ( ) ~)( xyxT λ+−= , w ωπσλ = (35) )cos()iexp(2)sin())i2exp(( yyPyPyQ ww λφλφ µ −−= (36) ⎟⎟⎠⎞⎜⎜⎝⎛ −−= ~5.0 yx xarctg λφ (37) Making use the same substitution for variables k , ω , the integrals corresponding to the first term in square brackets of (23) transform to: i) evanescent-wave branch
140 22121222223/ 42222113 εεω ε φλλεε
IaQPxxydyyaQqqqdkk wc ≡+=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ ∫∫ ∞ −−∞ , (38) )cosh()iexp(~2i)sinh())i2exp(~( yyPxyPxyQ φλφλ ε −− −−= (39) ii) propagating-wave branch )(cos)(~4ImIm εεω ε ωφλωεε IcQPyxxdyycQqqqdkk wxc w ≡−=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ ∫∫ −− (40) )cos()iexp(~2)sin())i2exp(~(i yyPxyPxyQ ww λφλλφλ ε −− −+= (41) Finally, substituting (30), (34), (38), (40) into (23) yields ( ) ( ) ww IIIIx xdxaVF λλπ εεµµ +++−−= ∫ ∞ h (42) As follows from (34), (40) and (42), the contributions from propagating modes in the force F are independent of distance a , and therefore these terms will be insignificant at small gap widths. Unlike this, the contributions from evanescent modes will be negligible at large gap widths. The calculation results are shown in Figs. 2–5 (the values of x F are given with a positive sign). Solid curves 1 and 2 in Fig. 2 correspond to the terms µ I and ε I in (42) (i. e. the contributions from evanescent modes). The dashed curve was calculated according to [5], namely aVcTF P h σπς ⋅−= (43) where = ς . The contribution from propagating modes in (42) (not shown in Fig. 2) are negligible and become more noticeable only at the distances more than m µ . These contributions are shown separately in Fig. 3. As one can see from Fig. 2, the frictional force for the plates of good metals proves to be 10 times higher in comparison with the original calculation by Polevoi [5]. In addition, the dominating contribution stems from the magnetic terms in (41). In the range of the gap widths nma ≤≤ the force goes down slightly slower than a /1 , but with a further increase of the gap width, the slope of the curve is closer to /1 a . The increase of the frictional force at a small gap width is physically due to the high values of the reflection coefficients of the metal plates, as a result of which the electromagnetic waves that acquire the Doppler shift due to their relative motion are repeatedly reflected from plates, resulting in an increase in the frictional force. In this case, the magnitude of the frictional force is 7-8 orders of magnitude higher than in the case of absolutely black plates. Another striking fact is that the frictional force increases with increasing conductivity of metals (approximately as σ ∝ x F ). This is illustrated in Fig. 4 at various temperatures, assuming nma = , smV /1 = . Note that the relative conductivity − =σσ ( − ⋅= s σ ) corresponds to poor conductors like graphite. In the same way, Fig. 5 shows the frictional force as a function of temperature and conductivity. From Fig. 4, 5 it follows the fallacy of the claim that the frictional force is maximal in the case of poor conductors of the type of graphite [10, 13]. It is interesting to compare the calculated values of x F with the measured dissipative force in experiment [20], corresponding to the geometry of the spherical probing tip (of gold) with a curvature radius of m µ , moving above a flat Au-coated mica surface: ~ N − ⋅ at smVnma /1,10 == . Assuming that the tip has the cylindrical form, its end-face has a radius of m µ , and the gap width is 10 nm , the calculated force x F proves to be N − ⋅ , i. e. it is too low to explain the results in [20]. At the same time, the distance dependence ( α− dF ~ ) and the temperature dependence )( TF of the force prove to be close to those observed in [20]. So, according to [20], 3.03.1 ±=α and ≈ FF at nma = , whereas from our calculation it follows 21 ÷=α and = FF . Conclusions
We have proved that the fundamental expressions for the frictional force and the rate of radiative heat exchange between two halfspaces (thick plates) separated by a thin gap, obtained in the works by Levine, Rytov and Polevoi , are in full agreement with the works by other authors in the case of nonrelativistic relative velocities. The case of relativistic velocities needs a special consideration. We also came to the conclusion, that in the case of good metals the frictional force is higher by a factor 10 as compared to the earlier assessment by Polevoi. Though the absolute value of the frictional force is small compared to the dissipative force observed in [20], its temperature and distance dependences agree well with the experiment. Another important result is that the frictional force increases with increasing conductivity of the metals. In our opinion, the measurement of the frictional force will be more realistic when using the tips with a radius of ~100 m µ . Moreover, it would be interesting to examine the behavior of this force at temperatures close to the temperature of superconducting transition, since the growth in the conductivity can compensate its drop with decreasing temperature. References [1] M.L. Levin and S.M. Rytov,
Theory of equilibrium thermal fluctuations in electrodynamics , Moscow, Nauka,1967. [2] S.M. Rytov,
Theory of electric fluctuations and thermal radiation , Moscow, Acad.Sci. USSR, 1953 (in Russian). [3] M.L. Levin, V.G. Polevoi, and S.M. Rytov, Sov. Phys. JETP 52(6) (1980) 1054. [4] V.G. Polevoi,
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