Casimir Friction Between Dense Polarizable Media

The present paper - a continuation of our recent series of papers on Casimir friction for a pair of particles at low relative particle velocity - extends the analysis so as to include dense media. The situation becomes in this case more complex due to induced dipolar correlations, both within planes, and between planes. We show that the structure of the problem can be simplified by regarding the two half-planes as a generalized version of a pair of particles. It turns out that macroscopic parameters such as permittivity suffice to describe the friction also in the finite density case. The expression for the friction force per unit surface area becomes mathematically well-defined and finite at finite temperature. We give numerical estimates, and compare them with those obtained earlier by Pendry (1997) and by Volokitin and Persson (2007). We also show in an appendix how the statistical methods that we are using, correspond to the field theoretical methods more commonly in use.


14
The typical situation envisaged in connection with Casimir friction is the one where two parallel semi-15 infinite dielectric nonmagnetic plates at micron or semi-micron separation are moving longitudinally 16 with respect to each other, one plate being at rest, the other having a nonrelativistic velocity v. Usually For reasons of readability we begin in the next section by summarizing essential points of the theory 48 for dilute media [18,19]. Then, after giving an account of Fourier methods in Sect. 3 we embark, as 49 mentioned, on the general case of finite particle density in Sect. 4.
where the summation convention for repeated indices i and j is implied. The s 1i and s 2j are components of the fluctuation dipole moments of the two particles (i, j = 1, 2, 3). With electrostatic dipole-dipole interaction we can write (i.e. ψ ij = −(3x i x j /r 5 − δ ij /r 3 )). Here r = r(t) with components x i = x i (t) is the separation between 52 the particles. The time dependence in Eq. (1) is due to the varation of r with time t, and the interaction 53 will vary as where v l are the components of the relative velocity v. The components of the force B between the oscillators are The friction force is due to the second term of the right hand side of Eq. (3), and for dilute media the 55 first term can be neglected. However, for the more general situation to be considered below, this will no 56 longer be the case since correlations will be induced.

57
For the time dependent part of Eq. (3) we may write −AF (t) → −A l F l (t) where A l = B l and F l (t) = v l t. According to Kubo [20,21,25] the perturbing term leads to a response in the thermal average of B l given by where the response function is (t > 0) Here ρ is the density matrix and B l (t) is the Heisenberg operator B l (t) = e itH/ B l e −itH/ where B l like A q are time independent operators. With Eqs. (3) and (4) the expression (6) can be rewritten as where (the i in the denominator is the imaginary unit). 58 Here as in Ref. [18] it is convenient to use imaginary time λ and consider the correlation function g ijnm (λ) = Tr[ρs 1n (t)s 2m (t)s 1i s 2j ], with λ = it/ .

60
The φ is related to the g via andφ where the Fourier transforms areφ with K the imaginary frequency, Here β = 1/k B T , where T is the temperature and k B Boltzmann's constant.

61
With Eqs. (12) and (13) we have by which theg(K) can be written as a convolutioñ K 0 = 2πn/β (n is integer) being the Matsubara frequencies. Theg a (K) can be identified with the frequency dependent polarizability α aK of oscillator a (=1,2), which for a simple harmonic oscillator is where α a is the zero-frequency polarizability.
Further following Ref.

64
The treatment above can be extended to a more general polarizability where it can be shown that the function f (K 2 ) satisfies the relation [26] With this one finds This is obtained by replacing α a with α Ia (m 2 a ) d(m 2 a ) (a = 1, 2) in expression (26) which is then inserted in Eq. (24) and then integrated with the δ-function included.

66
Finally by integrating G lq over space one obtains for dilute media the friction force F h between a particle and a half-plane, and the friction force F (per unit area) between two half-planes that move parallel to each other Here v is the relative velocity in the x direction, and one finds [18] Here ρ 1 and ρ 2 are the particle densities in the half-planes, z 0 is the separation between the particle and 67 one half-plane, and d is the separation between the half-planes.

Use of Fourier Methods
For higher densities of polarizable particles the above results will be modified due to induced dipolar 70 correlations within planes, and between planes. This affects the evaluations of G h and G which become 71 more complex and demanding. Further, the expression (31) for H 0 will be modified where we will find 72 that the imaginary parts of the polarizabilities will be replaced by functions based upon the frequency 73 dependent permittivity only. Thus the friction will depend solely on this macroscopic property, and be 74 independent of the explicit relation between the permittivity and the polarizability. On physical grounds 75 we find this reasonable.

76
To facilitate the analysis we find it convenient to evaluate the integral (33) by use of a Fourier transform in the xand y-directions. Then the quantities in (2) -(4) should be transformed. The three-dimensional Fourier transform of the Coulomb potential ψ = 1/r is where k i (i = 1, 2, 3 or x, y, z) is the Fourier variable. This can be transformed back with respect to z to obtainψ where q = k ⊥ , k 2 ⊥ = k 2 x + k 2 y , and ik z = ±q for z > 0 or z < 0. The variable k z may seem 77 unnecessary here, but it is kept for convenience in order to obtain simple and compact expressions for  Thus with Eqs.
With this we findĜ Symmetry with respect to x and y means that k 2 x can be replaced by 1 2 (k 2 x + k 2 y ) = 1 2 k 2 ⊥ = 1 2 q 2 in the integral (39), so we get (z > 0) By further insertion into Eqs. (33) and (34) the results of those integrals are recovered. For higher densities, separate oscillators both within each plane and between planes will be correlated.

84
This will add to the complexity of the problem. However, the structure of the problem can be simplified 85 by regarding the two half-planes as a generalized version of a pair of particles. Some details of this 86 approach are given a closer treatment in Appendix A.

87
The expression (11) is a thermal average of four oscillating dipole moments. They have Gaussian distributions since they represent coupled harmonic oscillators. This means that averages can be divided into averages of pairs of dipole moments. Thus we have Now the first term on the right hand side is equal-time average of the operators A and B ∼ s 1i s 2j and should be subtracted from (9) to obtain the proper response function. Thus we need here the first average represents correlations within the same half-plane while the second is the same for 88 different planes.

89
To better see the structure or formal contributions to these correlations one may consider the free energy of a pair of one-dimensional oscillators.The harmonic fluctuations result in a distribution function for the dipole moment of each molecule, assuming a static polarizability α, If the configuration becomes perturbed by an interaction energy φs 1 s 2 (not necessarily equal to the Coulomb potential called ψ above), the partition function is Thus the change in the free energy F due to the mutual interaction φs 1 s 2 becomes 1 1 In Ref.
where α 1 and α 2 are the polarizabilities of the two oscillators. The correlation functions s 2 a (a = 1, 2) and where the prime means differentiation with respect to φ. Further, When extended to half-planes one may interpret this expression in the following way: In the first term 90 on the right hand side α 1 and α 2 are the correlations within each half-plane for φ = 0. When the 91 planes interact, the denominator represents induced correlations due to the presence of the second plane.

92
Likewise, the second term in Eq. (47) represents correlations between the planes.  For our problem we also need correlations in imaginary time, λ = it/ as in (42) and (43). This generalizes Eq. (47) into (s a = s a (0)) Its Fourier transform in imaginary time is, similarly to Eq. (21), In view of the graph interpretation the structure of Eq. (50) will be similar to the one of Eq. (47), sõ With static interactions the φ will not vary with K.

110
With three-dimensional polarizations the formalism will be modified, but will still be manageable.

111
Some of the points will be verified more explicitly in Appendix A.

112
As established in Ref.
where d is the separation between the planes.

113
In Eq. (51), α aK alone represents correlations within each plane. For dilute media it corresponds to the polarizability whose relation to the permittivity is consistent with Eq. (52). As verified in Appendix A it follows that for a general permittivity the α aK in Eq. (51) will generalize to and by that φ → 2π(ρ 1 ρ 2 ) 1/2 e −qd .
Consistent with Eq. (52), the Eq. (51) is modified to The functionh(K) given by Eq. (50) will in the general case replace theg(K) given by Eq. (21). where For one particle outside a half-plane one has ρ 1 → 0 and thus A 1K → 0 (when ρ 2 is the density in the half-plane). With this the numerators in the expressions are replaced by 1 andh 12 (K) vanishes, so in this case the H 0 (u) will no longer vary with x. The same situation occurs when medium 1 is dilute. Due to this the expressions (32) for the friction force will be the same except that in the expression (31) for H 0 the role of α 2K is replaced by A 2K as given by Eq. (54). Thus while α 1K is kept.

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For larger α 1K the u-dependence will be present. This will modify the integral (41) as H 0 (u) has to be included. However, the zand z 0 -integration over the half-planes that led to the results (33) and (34) can be evaluated first. With G ⊥ given by integral (41) we then find (u = qd) With this the H 0 can be included along with the integrand of (59), and the friction force F between two half-planes can again be written in the form (32) as where G again is given by Eq. (34) as G = 3πρ 1 ρ 2 /(8d 4 ), but where now with H 0 (u) given by Eq. (56). with frequencies that match the energy difference (ω 1 − ω 2 ), while low constant velocity represents the limit of zero frequency. Moreover, the expansion (3) requires the displacement to be small and 130 comparable to molecular diameters, a situation that may be of minor interest.

131
Due to these problems we find it more appropriate and realistic to consider the whole mutual 132 interaction φ as belonging to the perturbing force. An obvious justification for this is that the mutual  Next, it is noteworthy that the expression (58) can be given a direct physical interpretation. This is most obvious for metals where the dielectric constant is given by the plasma relation (ε = ε a ) the ω p being the plasma frequency. Then with Eq. (58) which corresponds to the polarizability of a harmonic oscillator with eigenfrequency Thus each half-plane represents a set of harmonic oscillators for each wavevector k ⊥ in the xy-plane. In 139 the idealized situation considered here they all have the same eigenfrequency.

140
As mentioned above result (64) can be given a simple physical interpretation as it represents the 141 frequency of surface plasma waves [23]. Thus each half-plane can be regarded as an assembly of 142 independent harmonic oscillators. There is an oscillator for each wavevector k ⊥ , and they all oscillate

148
Casimir friction is a complicated topic; it is approached from different points of view by various 149 researchers, and the obtained results are not always so easy to compare with each other.

150
The basic method used by us is to employ statistical mechanical methods based upon the Kubo 151 formula. The simplest systems to compare, are clearly those of low particle density. The approach 152 most closely related to ours for such a case, appears to be that of Barton  Now let us consider metal plates for which the permittivity ε is given by the plasma relation (62) as an idealized case. As a physical quantity, the ε with dispersion will have to include also an imaginary part for ω real. The simplest generalized version of the expression (62) is the one of the Drude model where ζ = iω, and where ν represents damping of plasma oscillations due to finite conductivity of the 163 medium. [Note that a different convention implying ζ = −iω is frequently used, the sign being dictated 164 by the Fourier transform used. We here assumef (ω) = f (t)e −iωt dt, such that singularities will only 165 be present for ℑ(ω) > 0, with f (t) = 0 for t < 0 due to causality.]

166
In terms of the variable K = i ω = ζ, Eq. (65) can be written as where q 2 = ( ω p ) 2 2 and σ = ν have been introduced. The ε as function of either ω or K is the same in the common region where ε has 167 no singularities, i.e. for ℑ(ω) < 0 or ℜ(K) > 0 [21]. Further, ε(K) is symmetric in K, and |K| is to be 168 interpreted as |K| = lim(K 2 + γ 2 ) 1/2 , γ → 0. The singularities representing the singularities of ε(K) 169 will be along the imaginary K axis or for K 2 negative.
which gives the frequency spectrum For small σ we can simplify this, as the expression will be sharply peaked around m = q. Thus one might assume to get the main contribution from around this value. However, unless βm is small the sinh(·) term in Eq. (31) will very large, by which values around m = q can be fully neglected; and for small m Eq. (69) can be replaced by (70) where the substitution x = βm has been made.

172
Finally with Eqs. (32) and (34) we find for the friction force per unit area (ρ 1 = ρ 2 = ρ, and k B is Let us consider a numerical example. Assume room temperature, T = 300 K, corresponding to k B T = 25.86 meV. Choose gold as medium, for which ω p = 9.0 eV and ν = 35 meV. Then choose v = 100 m/s for the relative velocity and a small separation d = 10 nm between the plates. With = 1.054 · 10 −34 Js we then find for the friction force (72) This is a very small force. However, by changing parameters this will change rapidly by which F can become very large instead. But first let us compare this force with the result obtained by Pendry [2] for T = 0 where the friction linear in velocity was found to be zero, assuming constant conductivity. Instead a non-zero force, proportional to v 3 , was found. The influence of relative velocity is to create an oscillating force between the particles; these oscillations will create excitations from the T = 0 ground states of low frequency oscillations and thus contribute to friction. According to the derivations of Ref.
[2] the friction for low and not too high velocities for a dielectric function ε = 1 + iσ/ωε 0 was found to be F P = 5 ε 2 0 v 3 2 8 π 2 σ 2 d 6 (74) (here and henceforth σ is the conductivity in SI units, not ν as above). When compared with our dielectric function Eq. (65) for small ω one sees that the σ/ε 0 of this equation is our ω 2 p /ν. This can then be inserted in Eq. (74), and one finds the ratio between the friction forces (72) and (74) to be This simple expression can be given a direct physical interpretation. The k B T is the energy quanta that can be excited due to thermal energies or fluctuations while v/d are the energy quanta generated due to frequencies v/d generated by the finite velocity. Due to the physical interpretation above and arguments used in Ref.
[2], there is reason to expect its result for finite velocity and zero temperature to be consistent with ours, obtained for small velocity and nonzero temperature. For the numerical values that gave the force (73) one finds the ratio F F P = 1.95 · 10 9 .
[2] the result F P ≈ 175 3·10 3 Pa was found by substitution into its high velocity formula (the formula least sensitive to velocity).

176
Anyway, the latter force (77) is unrealistically large. One reason is that a separation d = 10 −10 m would 177 more or less imply direct contact between the particles of the two half-planes. Further the force decreases 178 very rapidly with increasing separation.

179
Another factor of influence here is that the effective separation between the planes will increase when

186
Among other previous approaches with which it seems natural to compare our results, we shall focus on those of Volokitin and Persson. As mentioned earlier, they have written a series of papers on this topic [5][6][7][8][9]. In contrast to the paper of Pendry [2], they considered finite temperatures. In their review article [7] a variety of situations were considered. One such situation is for parallel relative motion of metal plates. The friction coefficient according to their Eq. (97) is then where here the conductivity σ is given in Gaussian units such that 4πσ = ω 2 p /ν. This follows from their Eq. (38) which is proportional to our expression (69). The friction force F V P = γ evan p v is thus directly comparable to our result (72), and we find the ratio Thus we can conclude that the two expressions for the friction force are consistent except for a small 187 difference in numerical prefactor.

188
A closer look at integral (92) for γ evan p in Ref.
[2] the |γ| seems to be a misprint for the integration variable q.) Then one has integrals of the form ∞ 0 nu n q 3 dq ∝ n −3 d −4 . This gives the sum where ζ(z) is the ζ-function. This sum may be precisely the ratio (79). Here it can be noted that the 189 term above is also present in our expressions (47) and (55) with u = α 1 α 2 φ 2 and u = A 1K A 2K e −2qd 190 respectively, but was later disregarded by further explicit evaluations as discussed in Sec. 4.2.

191
In view of this, observing the similarity of the expressions, the results of Ref.
[7] seem consistent with 192 our results as they agree numerically except from some uncertainty in the prefactor.

194
Here we will justify expressions (55) for the correlation functions and the arguments that led to them.

195
As discussed in Ref.
[27], the correlation functions can be identified as solutions of Maxwell's equations.

196
In [27] the solution was derived for two half-planes with equal permittivities. Here we will consider two 197 half-planes with permittivities ε 1 and ε 2 , separated by a distance d. For the electrostatic case, which we 198 will assume, the Coulomb interaction for a point charge as given in the form (36) will be the basis.

210
Thus we will not try to perform it here. Instead we note that the result of all this should merely produce where K = i ω is the imaginary frequency.

217
It turns out that the quantum statistical mechanics for particles, and the quantum theory for fields, 218 are closely related although the correspondence is not always so easy to see from a mere inspection.

219
Therefore, we found it useful to point out how a parallel to Eq. (B.1) reads in the conventional QFT for 220 the electromagnetic field.

221
Consider, for definiteness, Schwinger's source theory in the form presented, for instance, in Ref.
[31]. For simplicity, as common in field theory, we work with natural units so that = c = k B = 1. The electric field components E i (x) are related to the polarization components P k (x ′ ) via a tensor Γ ik , called the generalized susceptibility, where x = (r, t). Stationarity of the system means that Γ ik depends on time only through the difference Introduce the Fourier transform Γ ik (r, r ′ , ω) via The function Γ ik (r, r ′ , ω) is known to be one-valued in the upper half frequency plane; it has no 224 singularity on the real axis (omitting metals), and it does not take real values at any finite point in 225 the upper half plane except on the imaginary axis.

226
From Kubo's formula we can now write It means that the generalized susceptibility can be identified with the retarded Green function: Consider now the the correlation E i (x)E k (x ′ ) . Its Fourier transform E i (r, ω)E k (r ′ , ω ′ ) (in field theory commonly called the two-point function) can be expressed in terms of the spectral correlation E i (r)E k (r ′ ) ω as E i (r, ω)E k (r ′ , ω ′ ) = 2π E i (r)E k (r ′ ) ω δ(ω + ω ′ ). there is a close relationship between the response (or Green function) and the correlation.

232
To make the connection to the statistical mechanical method more explicit, one may consider expressions (B.3) -B.6) for an oscillator which has a frequency spectrum. Its response function (6)