Quantum Friction and van der Waals Friction in Configuration Particle–Plate and Plate–Plate: Nonlocal Effects

Abstract

In the nonrelativistic approximation of fluctuation electrodynamics, using the specular reflection model and the nonlocal dielectric permittivity of a metal, simple analytical expressions are obtained for the friction forces in the particle-plate and plate-plate configurations at relative motion of bodies with constant velocity. It has been shown that at distances of about 1–10 nm, for an Au nanoparticle (or a gold plate) moving near another similar plate at rest, the dissipative forces are 2–4 orders of magnitude higher than when using the Drude local dielectric function.

APPENDIX A

In [20], transforming the general expression for the friction force obtained in the Rytov–Levin–Polevoy theory [19] with the local dielectric permittivity ε(ω), we obtained a formula for the friction force of two parallel thick plates in the case of their relative motion with nonrelativistic velocity V/c $\ll$ 1 (values related to plates 1 and 2 are provided with indices 1, 2)

$${{F}_{x}} = - \frac{\hbar }{{4{{\pi }^{3}}}}\int\limits_0^\infty {d\omega } \int {{{d}^{2}}k{{k}_{x}}\left[ {\operatorname{Im} \left( {\frac{{{{q}_{1}}}}{{{{\varepsilon }_{1}}}}} \right)\operatorname{Im} \left( {\frac{{{{{\tilde {q}}}_{2}}}}{{{{{\tilde {\varepsilon }}}_{2}}}}} \right)\frac{{{{{\left| q \right|}}^{2}}}}{{{{{\left| {{{Q}_{e}}} \right|}}^{2}}}}} \right.} $$

$$\left. { + \;\operatorname{Im} \left( {\frac{{{{q}_{1}}}}{{{{\mu }_{1}}}}} \right)\operatorname{Im} \left( {\frac{{{{{\tilde {q}}}_{2}}}}{{{{{\tilde {\mu }}}_{2}}}}} \right)\frac{{{{{\left| q \right|}}^{2}}}}{{{{{\left| {{{Q}_{\mu }}} \right|}}^{2}}}}} \right]$$

(A.1)

$$ \times \;[\coth (\hbar {{\omega }_{ - }}{\text{/}}2{{T}_{2}}) - \coth (\hbar \omega {\text{/}}2{{T}_{1}})].$$

Here q = $\sqrt {{{k}^{2}} - {{{(\omega {\text{/}}c)}}^{2}}}$, q₁ = $\sqrt {{{k}^{2}} - {{\varepsilon }_{1}}{{\mu }_{1}}{{{(\omega {\text{/}}c)}}^{2}}}$, q₂ = $\sqrt {{{k}^{2}} - {{\varepsilon }_{2}}{{\mu }_{2}}{{{(\omega {\text{/}}c)}}^{2}}}$, “tilde” means that arguments of the dielectric permittivities and magnetic permeabilities ε_1, 2, μ_1, 2 are taken at ω_– = ω – k_xV, and

$$\begin{gathered} {{Q}_{e}} = (q + {{q}_{1}}{\text{/}}{{\varepsilon }_{1}})(q + {{{\tilde {q}}}_{2}}{\text{/}}{{{\tilde {\varepsilon }}}_{2}})\exp (ql) \\ - \;(q - {{q}_{1}}{\text{/}}{{\varepsilon }_{1}})(q - {{{\tilde {q}}}_{2}}{\text{/}}{{{\tilde {\varepsilon }}}_{2}})\exp ( - ql) \\ \end{gathered} $$

and the Q_μ value is determined by the same expression with the replacement ε → μ. Formula (A.1) describes the frictional force acting on plate 1, moving in the direction of the x axis with a constant speed V. Its important features are the absence of the Fresnel reflection coefficients in an explicit form and the absence of the separation of the contributions of homogeneous and inhomogeneous modes on the axis of wave vectors. From (A.1), in the linear approximation in velocity at t₁ = T₂ = T, it follows (see (23) in [20])

$$\begin{gathered} {{F}_{x}} = \frac{{\hbar V}}{{2{{\pi }^{2}}}}\int\limits_0^\infty {d\omega \frac{{dn(\omega )}}{{d\omega }}} \\ \times \;\int {{{d}^{2}}kk_{x}^{2}\left[ {\operatorname{Im} \left( {\frac{{{{q}_{1}}}}{{{{\varepsilon }_{1}}}}} \right)\operatorname{Im} \left( {\frac{{{{q}_{2}}}}{{{{\varepsilon }_{2}}}}} \right)\frac{{{{{\left| q \right|}}^{2}}}}{{{{{\left| {{{Q}_{e}}} \right|}}^{2}}}}} \right.} \\ \left. { + \;\operatorname{Im} \left( {\frac{{{{q}_{1}}}}{{{{\mu }_{1}}}}} \right)\operatorname{Im} \left( {\frac{{{{q}_{2}}}}{{{{\mu }_{2}}}}} \right)\frac{{{{{\left| q \right|}}^{2}}}}{{{{{\left| {{{Q}_{\mu }}} \right|}}^{2}}}}} \right], \\ \end{gathered} $$

(A.2)

and for T₁ = T₂ = 0, respectively, is the expression for the quantum friction force [18, 20]

$$\begin{gathered} {{F}_{x}} = \frac{\hbar }{{4{{\pi }^{3}}}}\int\limits_{ - \infty }^{ + \infty } {d{{k}_{y}}\int\limits_0^\infty {d{{k}_{x}}{{k}_{x}}} } \\ \times \;\int\limits_0^{{{k}_{x}}V} {\exp ( - 2kl)\operatorname{Im} {{\Delta }_{{1\varepsilon }}}\operatorname{Im} {{{\tilde {\Delta }}}_{{2\varepsilon }}}{{{\left| {{{D}_{\varepsilon }}} \right|}}^{{ - 2}}} + (\varepsilon \to \mu ).} \\ \end{gathered} $$

(A.3)

Here Δ_iε = (ε_iq_i – q_i)/(ε_iq_i + q_i), i = 1, 2, D_ε = 1 – ${{\Delta }_{{1\varepsilon }}}{{\Delta }_{{2\varepsilon }}}\exp ( - 2kl)$, and for the corresponding magnetic contributions we should replace ε → μ. We note that in Section 3, the case of nonmagnetic materials when μ_1, 2 = 1, is considered.