Here, we use the standard formulation of the problem, in which the plates are assumed to be made of homogeneous and isotropic materials with permittivities ${\epsilon }_1$, ${\epsilon }_2$ and permeabilities ${\mu }_1,{\mu }_2$, depending on the frequency $\omega$ and local temperatures $T_1$ and $T_2$ (Figure 1).

In line with [26,27], the power $F_{x}V$ of the friction force $F_x$ per unit surface area applied to plate 2 in the laboratory coordinate system associated with plate 1, is given by

Here $P_1$and $P_2$ are the heat fluxes of the plates from a unit surface area per unit time, and $\gamma ={\left(1-V^2/c^2\right)}^{-1/2}$. For all quantities, indices 1 and 2 here and in what follows correspond to numbering in Figure 1. Moreover, $P_1$and $P_2$ are calculated in the rest frames of the plates. General relativistic expressions for $P_1$and $P_2$were obtained in [26]. In the nonrelativistic case $V/c\ll 1$, but taking retardation into account, a more compact form of $P_1$and $P_2$reads [27]

Here ${\omega }^{-}=\omega -k_{x}V$, $q=\sqrt{k^2-{\omega }^2/{\mathrm{с}}^2},$ $q_{\mathrm{1,2}}=\sqrt{k^2-{\epsilon }_{\mathrm{1,2}}{\mu }_{\mathrm{1,2}}{\omega }^2/{\mathrm{с}}^2}$, and $a$ is the gap width in Figure 1. Variables with a tilde, such as ${\tilde{q}}_2,$ should be used replacing $\omega \to {\omega }^{-}$ . The terms $\left({\mu }_{\mathrm{1,2}}\leftrightarrow {\epsilon }_{\mathrm{1,2}}\right)$ are defined by the same expressions with appropriate replacements. In the general case, the expressions depending on ${\epsilon }_{\mathrm{1,2}}\mathrm{a}\mathrm{n}\mathrm{d}$ ${\mu }_{\mathrm{1,2}}$ correspond to the contributions of electromagnetic modes with P and S polarizations. The quantities $P_1$ and $P_2$ are directly related to the heating (cooling) rates of the plates: $d{Q}_1/dt=-P_1$ and $d{Q}_2/dt=-P_2$.

Using (1)–(4), the power of the friction force $F_{x}V=P_1+P_2$takes the form

Formula (5) can also be recast into a more familiar form in terms of the Fresnel reflection coefficients [9,23].

At $T_1=T_2=T$, due to the symmetry of the system, the heating rates of identical plates are equal. We then have $F_{x}V=2P_{\mathrm{1,2}}$, and the friction force can be determined using the heating rate of any plate. For $T_1\ne T_2$, it follows $P_1\ne P_2$, but $P_1\left(T_1,T_2\right)=P_2\left(T_2,T_1\right)$ and, correspondingly, $P_1\left(T_1,T_2\right)+P_2\left(T_1,T_2\right)=P_1\left(T_1,T_2\right)+P_1\left(T_2,T_1\right)=P_2\left(T_1,T_2\right)+P_2\left(T_2,T_1\right)$. This means that, when measuring the CL friction force, it is sufficient to control the temperature of only one plate.

In order to treat the problem of temperature-dependent CL friction force between ordinary metals, we model them by the Drude model in terms of plasma frequency ${\omega }_p$and damping parameter $\nu \left(T\right)={\omega }_p^2\rho \left(T\right)/4\pi$, with $\rho \left(T\right)$ being the resistivity:

Figure 2 plots the dependences $\rho \left(T\right)$ corresponding to the Bloch–Grüneisen (BG) model [28] and the modified Bloch–Grüneisen (MBG) model [29]. In the former case, the residual resistance is zero or can be specified by indicating the effective temperature, below which it is constant. In the MBG model, the residual resistivity is ${\rho }_0=2.3·{10}^{-10}\mathsf{\Omega }\bullet \mathrm{m}$ (see Figure 2).

Hereinafter, for simplicity, we assume that the plates are made of the similar nonmagnetic metal (${\mu }_1={\mu }_2=1$) with the same plasma frequency ${\omega }_p$, but different dependence $\nu \left(T\right)$.

Because $\epsilon \left(\omega \right)\gg 1$ for good conductors, and the inequality gets stronger as $T\to 0,$the terms with ${\epsilon }_{\mathrm{1,2}}$ in (2) , (3), (5), corresponding to $P$ modes are negligible compared to the terms with ${\mu }_{\mathrm{1,2}}$, corresponding to $S$ modes. So, in what follows, the contributions of P modes are omitted.

When calculating the integrals in (2), (3), (5), it is convenient to introduce a new frequency variable $\omega ={\nu }_m\left(T_1,T_2\right)t$, with ${\nu }_m\left(T_1,T_2\right)=\mathrm{m}\mathrm{a}\mathrm{x}({\nu }_1\left(T_1\right),{\nu }_2\left(T_2\right))$ and ${\nu }_i\left(T_i\right)$ being the damping parameters of plates 1 and 2 depending on their temperatures $T_1$ and $T_2$. The absolute value $k$ of the two-dimensional wave-vector (using the polar coordinates $k,\varphi$ in the plane ${(k}_x,k_y\left)\right)$ is expressed as $k=\left({\omega }_p/c\right)\sqrt{y^2+{\beta }_m^2{t}^2}$ in the evanescent sector $k>\omega /c$ ($0\le y<\infty )$ and $k=\left({\omega }_p/c\right)\sqrt{{\beta }_m^2{t}^2-y^2}$ in the radiation sector $k<\omega /c$ ($0\le y\le {\beta }_{m}t)$). Moreover, we introduce additional parameters ${\beta }_m={\nu }_m/{\omega }_p$ , ${\alpha }_i=\hslash {\nu }_i/T_i$ , ${\gamma }_i={\nu }_i/{\nu }_m$, $\lambda ={\omega }_{p}a/c$ , $\zeta =(V/c){{\beta }_m}^{-1}$, and $K=ћ{\nu }_m^2{\left({\omega }_p/c\right)}^2/2{\pi }^3$. With these definitions, for $k>\omega /c$, Eqs. (2), (3) and (5) take the form

In the sector $k<\omega /c$, formulas (14), (15) should be modified by replacing $y\to \mathrm{i}y$ and substituting ${\beta }_{m}t$ for $\infty$ in (7)–(9) in the integrals over $y$. The expressions for $\mathrm{I}\mathrm{m}{w}_{\mathrm{1,2}}$ can be additionally simplified. For example, it follows

The $\mathrm{I}\mathrm{m}{w}_2$ is defined by the same expression (16), substituting ${\gamma }_2$ for ${\gamma }_1$and $t^{-}$ for $t$. For two identical plates at quasithermal equilibrium, it follows ${\gamma }_1={\gamma }_2=1,$and a simpler useful expression is obtained by expanding the square root in (16) and leaving the expansion terms up to the second order:

In this case, an approximate analytical consideration can be carried out.

In the case $T_1=T_2=0$, corresponding to the conditions of quantum friction, the main role is played by the evanescent modes $k>\omega /c$. At finite temperatures, the evanescent modes make the dominant contribution at $a<1$ µm. This range of distances is very promising experimentally. For this reason, I consider hereinafter only evanescent modes, omitting the small term ${\beta }_m^2{t}^2$ in (9), (14) and other formulas. So, at zero temperature, substituting the identity $Z\left(t,y,\varphi \right)=\mathrm{s}\mathrm{g}\mathrm{n}\left(t\right)-\mathrm{s}\mathrm{g}\mathrm{n}\left(t-\zeta y\mathrm{c}\mathrm{o}\mathrm{s}\left(\varphi \right)\right)$ into Eq.$\left(9\right)$yields

The simplest asymptotics of (18) can be worked out for two identical plates in the limit of low velocities, $\zeta \ll 1$. Using (15) and (17), we get

Inserting (19) into (18) yields where ${\rho }_0$ is the residual resistivity corresponding to the zero-temperature damping factor ${\nu }_0=\nu \left(0\right).$

The limit of large velocities, $\zeta \gg 1$ is more laborious. A reasonable representation of the triple integral in (9) can be worked out using an approximate expression for $\mathrm{I}\mathrm{m}{w}_1\mathrm{I}\mathrm{m}{w}_2$, based on (17): where $b=\zeta y\mathrm{c}\mathrm{o}\mathrm{s}\varphi$. The product $\mathrm{I}\mathrm{m}{w}_1\mathrm{I}\mathrm{m}{w}_2$, as a function of $t$ in the range $0\le t\le b,$reaches its maximum close to the point $t=b/2$, with zeroing at the end points $t=0,t=b$ of the integration domain of the inner integral in (9). At the same time, the dependence on $t$ in ${\left|D\right|}^2$ is much weaker. By virtue of this, we insert $t=b/2$ into the denominator of (21) and into ${\left|D\right|}^2$(in the latter case, we also put $\mathrm{c}\mathrm{o}\mathrm{s}\varphi \approx 1)$. Expression (21) then takes the form

With these transformations, it follows (see Appendix A) where ${\left|D\right|}^2$ and $\psi \left(y,\varphi \right)$are given by (A2) and (A3). The integrals over $t$ and $\varphi$ are calculated explicitly, and finally we get (see (A4) and (A6))

As follows from (24), in this approximation, the power of the friction force does not depend on the velocity at $\zeta \gg 1$. However, for $\zeta \ll 1$, this formula also agrees rather well with numerical calculations and approximation (20) (see Sec. 3.2).

In the quasiequilibrium thermal regime, $T_1=T_1=T$, for two identical metal plates in the linear in velocity approximation, Eqs. (5) and (9) can be recast into the form [19,20]

In this limit, the friction parameter $\eta =F_x/V$ does not depend on $V$. It is the dependence $F_x\propto {\alpha }^{-1}$ in (25) that leads to a large enhancement of friction at low temperatures, when $\alpha =\hslash \nu \left(T\right)/T\to 0,$because the function $Y_1\left(\lambda ,\alpha \right)$ weakly depends on $\alpha .$ The main contribution to $Y_1\left(\lambda ,\alpha \right)$ in this case make the values $t<1$, $y~1/2\lambda ~1$, and we can again use (17) for $\mathrm{I}\mathrm{m}{w}_1$. At the same time, ${\alpha }^2{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-2}(\alpha t/2)\approx 4/t^2$ (this is a good approximation at $\alpha <0.3)$ $\mathrm{a}\mathrm{n}\mathrm{d}{\left|D\right|}^2\approx 16y^4\exp \left(2\lambda y\right).$ Making use of these simplifications in (26), we arrive at (see Appendix B) where ${\mathsf{H}}_1\left(x\right)$ and $N_1\left(x\right)$ are the Struve and Neumann functions [30]. Using their series representations yields

An even simpler and physically clearer low-temperature representation of (25) is obtained by using the relation $\nu \left(T\right)={\omega }_p^2\rho \left(T\right)/4\pi$ between the damping factor and resistivity, yielding

Combining the relation $\alpha \ll 1$, which implies $\hslash \nu \left(T\right)\ll T$, and $\zeta \ll 1$, which implies the limit of low-velocities $V/c\ll \nu \left(T\right)/{\omega }_p,$we conclude that formula (29) holds at

As a result, the conditions of low-temperature increase in friction and the applicability of the low-speed approximation are met at$V/c\ll T/{\omega }_p\hslash$. For gold, at $T=1K$, this implies $V/c\ll {1.5\bullet 10}^{-5}$.

According to [19,20], dependence (29) is associated with a growing penetration depth of S-polarized electromagnetic modes and an increase in their density at low temperatures. A significant low-temperature increase in the friction parameter was also noted in the case of movement of a metal particle above the metal surface [21].

The limit $\zeta \gg 1$at finite but low temperatures ($\alpha \ll 1)$ can be analyzed similarly to the case of zero temperatures using the properties of function (21). When substituting (21) into (9) with allowance for (13), the first exponential term in (13) makes the dominant contribution at $t~1\ll b=\zeta y\mathrm{c}\mathrm{o}\mathrm{s}\varphi$. Because of this, we can take advantage of the substitution $\left|t-b\right|\to b$ in the denominator of (21). For the second term in (13), we introduce a new variable $t^{\text{'}}=t-b$, and make the substitution $\left|t^{\text{'}}+b\right|\to b$ in the denominator of (21), while the integral (9) is then determined by the large exponential factor${\left(\exp \left(\alpha t^{\text{'}}\right)-1\right)}^{-1}$at $t^{\text{'}}~1\ll b$. Then, taking into account these transformations in (9), and summing both contributions, the triple integral in (9) finally takes the form (see Appendix C) where and

To proceed further, we replace the function $t/\left(e^{\alpha t}-1\right)$ by $1/\alpha$ in the inner integral (31), which is again a good approximation for $\alpha <0.3$. The remaining integral yields where $\mathrm{K}\left(q\right)$ is the complete elliptic integral [24]. Taking this into account, the $\varphi$-integral in (31) can be evaluated as the arithmetic mean between the integrals calculated with the limit functions on the left and right sides of the inequality (see Appendix C)

Substituting (C9) into (31) and (9) finally yields where $Y_2(\lambda ,\zeta )$ is given by

The heat transfer of plates is described by the equations with ${с}_i\left(T_i\right)$being the specific heat capacities, $h_i$ and ${\rho }_i$are the thicknesses and densities of materials, $P_1\left(T_1,T_2\right)$ and $P_2\left(T_1,T_2\right)$are defined by Eqs. (2) and (3), and the temperature gains ${∆T}_i$ correspond to the interval of time $∆t$. The dependences $T_2\left(T_1\right)$ and $T_1\left(T_2\right)$ can be determined from the equation

For identical plates, in (39), one can use the replacements $P_2(T_1,T_2)\to P_1(T_2,T_1)$, $P_1(T_1,T_2)\to P_2(T_2,T_1)$. Further on, we will consider only this case.

When writing Eqs. (38), (39), it is also assumed that the heat exchange due to radiative heat transfer occurs much slower than under thermal diffusion, and the plates acquire equal temperature at all points because of high thermal conductivity. Really, using the thermal diffusion equation along the normal to the plates, $\partial T/\partial t=a^2{\partial }^{2}T/\partial z^2$, the characteristic time of the heat diffusion necessary to reach thermal quasiequilibrium, is $\tau =h^2/a^2$(where $a^2=\kappa /c\rho$, and $\kappa$ is the thermal conductivity). Then it follows ${\tau =h}^{2}c\rho /\kappa$ and in the case of gold at $T=10\mathrm{K}$ and $h_{\mathrm{1,2}}=h=500\mu m$ $c=2.2$ J/(kg·K), $\kappa =3200$ W/(m·K), $\rho =19.8·{10}^3$kg/m³ [31]) we obtain $\tau \simeq 3$ $\mathsf{\mu }\mathrm{s}$. In turn, the kinetics of heating induced by friction takes dozens of seconds or minutes (see Sec. 3.3), depending on the velocity and other parameters. Assuming that $P_i\left(T,T\right)=-0.5\eta (T,V)V^2$, from Eq. (38) we obtain where $t$ is the heating time from the initial temperature $T_0$ to the final temperature $T$. In the simplest case $\eta =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ and $с\left(T\right)=a_{1}T+a_2{T}^3$(this is a typical low-temperature dependence for metals) it follows from Eq. (40) that where $\beta =a_1/a_2$. At $T_1\ne T_2$ and relatively low velocities of plate 2, as follows from numerical calculations (see Sec. 3.1), the heating/cooling rates of metal plates differ only in sign, i.e. $P_1\left(T_1,T_2\right)=-P_2\left(T_1,T_2\right)$. This is the normal mode of heat transfer, when a hotter body cools down, and a colder one heats up. Then the left sides of equations (38) can be equated, and the corresponding quasistationary temperature of the plates is given by where $T_1$ and $T_2$ are their initial temperatures. After establishing quasithermal equilibrium, the temperature of the plates will increase according to Eqs. (40), (41).

Figure 3 shows the velocity-dependent quantum friction force between the plates of gold, calculated using formulas (20) (green line), (9) (red line), and (24) (blue line).

The curves on panels (a) and (b) were calculated at residual resistances of $2.13\bullet {10}^{-13}$ Ω·m and $2.3\bullet {10}^{-10}$ Ω·m, which correspond to BG model (43) at$T=5$K and MBG model at $T=0$K. Note that in the latter case, the residual resistance coincides with that defined by formula (43) at T = 20.9 K.

Figure 4 shows the plots of the friction parameters $\eta =F_x/V$ depending on the temperature $T$ of gold plates, corresponding to the BG and MBG models. The curves with symbols were calculated using Eq. (9) for $V=1$ m/s. Solid curves were plotted using approximation (25) along with (26) (green lines) or with (28) at $\lambda \gg 1$(blue lines). On panel (a), both the solid lines merge. The presence of maxima and their positions on the curves agree with (29) and (30). These results show that the linear in velocity approximation is valid only to the right of the maxima of the dependences $\eta \left(T\right)$.

Figure 5 demonstrates the velocity dependences of $\eta$ in the BG model (a) and MBG model (b)). Red, blue and green lines correspond to quasiequilibrium temperatures of 5, 10, and 77 K. The different order of lines 1–3 in panels (a) and (b) is explained by the high residual resistance of gold in the MBG model: the condition $\hslash \nu \left(T\right)<T$, which is necessary for the law-temperature increase in friction, is violated at $T=5$ and 10 K.

Table 1 shows the calculated friction parameters $\eta$ of gold plates at $V=1$ m/s, depending on the temperature $T$ and separation distance a. As in Figure 4, one can note the effect of increasing friction (up to a maximum) with decreasing temperature at $T<{\theta }_D$, which is more expressed in the BG model. The height of this maximum depends on the velocity-to-resistivity ratio. When the temperature becomes sufficiently low, the condition (30) is violated and the coefficient of friction decreases.

The dependence of $\eta$on the separation distance a in all the cases is close to inverse proportionality ($\eta \propto a^{-1}$). This is clearly seen from the data of the table and agrees with our previous results [19,20,27].

Figure 6 and Figure 7 show the calculated heating rates of plate 1 (panels (a)) and friction parameters (panels (b)) depending on the velocity$V$ of plate 2 for various thermal configurations.

One can see that at at $V<10$ m/s (Figure 6, panel (a)) and $V<{10}^3$ m/s (Figure 7, panel (a)), the heating rates of plates 1, 2 are equal in absolute value, differing in the sign. According to their temperatures, $T_1=4$K and $T_2=6$K, plate 1 heats up and plate 2 cools down, realizing the “normal” heat exchange regime. At the same time, the friction parameters weakly depend on the temperature (panels (b)). When the speed of plate 2 becomes higher, both plates heat up faster. Then we can see the effect of “anomalous” heating of plate 2 for some time, when it continues to heat up despite the higher temperature. This is similar to the case of heating a hotter metal particle moving above a cold surface [21]. However, due to different magnitudes of the heating rate (cf. the upper and lower lines with □ on panels (a)), the temperature of plate 1 “catches up” the temperature of plate 2, and further on, both plates heat up with the same rate.

Drop in friction parameters for high velocities of plate 2 (panels (b) in Figures 6,7) is explained by the change in sign of the Doppler-shifted frequency ${\omega }^{-}=\omega -k_{x}V=\omega -kV\mathrm{c}\mathrm{o}\mathrm{s}\varphi$ in Eq. (5). This occurs at $V>\nu \left(T\right)a$, because the characteristic absorption frequency is $\omega ~\nu \left(T\right)$ and the characteristic wave-vector is $k~1/a$. The positions of the "kinks" on the curves $\eta \left(V\right)$ in Figures 6, 7 correlate with resistivity, since $\nu \left(T\right)~\rho \left(T\right)$. Indeed, it follows from Figure 2 that ${\rho }_{MBG}/{\rho }_{BG}={10}^2÷{10}^3$ at $T=4÷6\mathrm{K}$. At the same time, the ratio ${\eta }_{MBG}/{\eta }_{BG}$ in this case is in the inverse proportion to resistivities (see (29) and Table 1).

In general, as follows from the calculations for all considered temperatures and velocities (Figure 5, Figure 6 and Figure 7, Table 1), the maximum friction parameter in the BG and MBG models (at $a=10$nm) is ${10}^{-6}-{10}^{-3}$ kg/(m²· s).

Figure 8 shows the heating time of plates vs. velocity of plate 2, calculated by numerical integration of (39) from 4 to 5 K and from 4 to 8 K. In these calculations, the fitting parameters $a_1=0.0035\mathrm{J}/(\mathrm{k}\mathrm{g}\bullet {\mathrm{K}}^2)$, $a_2=0.0023\mathrm{J}/(\mathrm{k}\mathrm{g}\bullet {\mathrm{K}}^4)$ of the dependence $с\left(T\right)=a_{1}T+a_2{T}^3$ were determined using the data [31] for gold at $T<20\mathrm{K}$.

As follows from Figure 8, quite comfortable (from the experimental point of view) values of the plate heating times (1–100 s) can be obtained in the velocity range $1–{10}^3$m/s. On the contrary, heating by 1 K at $T_0=300$K, $a=10$ nm, and $V={10}^3$ m/s will take about 2 h. Thus, low-temperature thermal measurements have great advantages over measurements under normal conditions due to a significant reduction in measurement time, elimination of noise and other undesirable effects.

Substituting $t=b/2=\zeta y\mathrm{c}\mathrm{o}\mathrm{s}\varphi /2$ into (22), we take into account that, typically, $\zeta y\mathrm{c}\mathrm{o}\mathrm{s}\varphi /2\gg 1$(since $\zeta \gg 1$, $y~1/2\lambda ~1$) and $\left|w_{\mathrm{1,2}}\right|\approx \sqrt{y^2+1}$. Then ${\left|D\right|}^2$ takes the form

With these simplifications, formula (18) reads

The $t$ integral in (A2) is simply ${\zeta }^3{y}^3{\mathrm{c}\mathrm{o}\mathrm{s}}^3\varphi /6$, while the integral over $\varphi$ is where $u=2/\zeta$. The integral in (A4) is calculated explicitly using the table integral [30]

Using (A5) yields

Substituting (A6) into (A2), yields (24).

In the case $\alpha \ll 1$, the main contribution to (26) make the values $t<1$, $y~1/2\lambda ~1$. Then from (14) it follows $\left|w_{\mathrm{1,2}}\right|=\left|{\left(y^2+t/\left(t+i\right)\right)}^{1/2}\right|\approx y$. Using this, we find