# Puzzling Low-Temperature Behavior of the Van Der Waals Friction Force between Metallic Plates in Relative Motion

## Abstract

**:**

## 1. Introduction

^{−}at $T=0$.

## 2. Theoretical Background

## 3. Numerical Results

## 4. Discussion

^{2}, whereas the corresponding friction force was $3.2\times {10}^{-5}$ N/m

^{2}(at $V=1$ m/s, $T={T}_{0}=3$ K), i.e., much higher.

## 5. Conclusions

- (1)
- For metals without defects and impurities, van der Waals friction parameter $\gamma ={F}_{x}/V$ increases with decreasing temperature as ${T}^{-4}$ at temperatures $T<100$ K.
- (2)
- In the presence of residual resistance ${\rho}_{0}$, the temperature dependence of the friction force reaches saturation ($\gamma \propto {\rho}_{0}^{-0.8}$) at $T<{T}_{0}$, where ${T}_{0}$depends on the magnitude of ${\rho}_{0}$.
- (3)
- Dependence of the friction parameter on distance is weak: $\gamma ~{a}^{-q}$ with $q\le 1$up to $a=3$µm.
- (4)
- Absolute values of van der Waals friction at $T<1\xf75$ K are large enough to be measured with existing AFM techniques.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Dependence $p\left(a\right)\mathrm{in}\left(22\right)\left(\mathrm{circles}\right)$ as a function of the gap width $a,$obtained when equalizing (16) and (22) for $T=1\mathrm{K}$ and ${\mathsf{\nu}}_{0}=0$ in (7). Solid line extrapolation (23).

**Figure 3.**Friction force between two Au plates as a function of gap width $a$ for $T={T}_{0}=1\mathrm{K},V=1\mathrm{m}/\mathrm{s}$. The black line shows the result of numerical calculation. The red, green, and blue lines correspond to analytic fits $5{a}^{-1},1.37{a}^{-0.75}$, and $0.41{a}^{-0.25}$.

**Figure 4.**Friction force between Au plates as a function of temperature $T$ and parameter ${T}_{0}$ for a gap width $a=10$ nm and $V=1$m/s. The top (red) to bottom (light blue) curves correspond to ${T}_{0}=0.1,0.2,0.5,1,3\mathrm{K}$, respectively.

**Figure 5.**Friction force between Au plates as a function of temperature $T$ and parameter ${T}_{0}$ for a gap width $a=100$ nm and $V=1$m/s. The top (red) to bottom (light blue) curves correspond to ${T}_{0}=0.1,0.2,0.5,1,3\mathrm{K}$, respectively.

$\mathit{a}=10\mathbf{nm}$ | ||||
---|---|---|---|---|

T K | ${\mathit{I}}_{\mathit{m}\mathit{n}}$ | ${\mathit{I}}_{\mathit{e}\mathit{n}}$ | ${\mathit{I}}_{\mathit{m}\mathit{r}}$ | ${\mathit{I}}_{\mathit{e}\mathit{r}}$ |

1 | 8.08 × 10^{4} | 4.91 × 10^{−34} | 8.43 × 10^{−29} | 8.94 × 10^{−29} |

5 | 129.3 | 7.49 × 10^{−23} | 4.12 × 10^{−21} | 4.36 × 10^{−21} |

10 | 8.087 | 4.84 × 10^{−18} | 8.02 × 10^{−18} | 8.89 × 10^{−18} |

50 | 0.0785 | 1.46 × 10^{−9} | 2.91 × 10^{−12} | 3.77 × 10^{−12} |

$a=50$nm | ||||

T K | ${I}_{mn}$ | ${I}_{en}$ | ${I}_{mr}$ | ${I}_{er}$ |

1 | 3.031 × 10^{4} | 9.77 × 10^{−35} | 2.91 × 10^{−29} | 7.14 × 10^{−29} |

5 | 48.5 | 1.49 × 10^{−23} | 1.42 × 10^{−21} | 3.49 × 10^{−21} |

10 | 3.031 | 9.77 × 10^{−19} | 2.78 × 10^{−18} | 7.1 × 10^{−18} |

50 | 0.0296 | 6.54 × 10^{−11} | 1.14 × 10^{−12} | 3.07 × 10^{−12} |

**Table 2.**Van der Waals friction force between two Au plates (N/m

^{2}), $V=1$m/s. The upper-rows of data correspond to Equation (15) with (16), while the lower-rows data correspond to Equation (24) with (23).

a, nm | T = 0.5 K | T = 1 K | T = 5 K | T = 10 K | T = 50 K |
---|---|---|---|---|---|

1 | 13.1 13.0 | 0.818 0.814 | 1.309 × 10^{−3}1.302 × 10 ^{−3} | 8.18 × 10^{−5}8.14 × 10 ^{−5} | 7.90 × 10^{−7}7.95 × 10 ^{−7} |

5 | 9.55 9.72 | 0.597 0.608 | 9.55 × 10^{−4}9.72 × 10 ^{−4} | 5.97 × 10^{−5}6.08 × 10 ^{−5} | 5.78 × 10^{−7}5.94 × 10 ^{−7} |

10 | 7.48 7.57 | 0.468 0.473 | 7.48 × 10^{−4}7.57 × 10 ^{−4} | 4.68 × 10^{−5}4.73 × 10 ^{−5} | 4.54 × 10^{−7}4.62 × 10 ^{−7} |

20 | 5.29 5.24 | 0.330 0.328 | 5.29 × 10^{−4}5.24 × 10 ^{−4} | 3.31 × 10^{−5}3.28 × 10 ^{−5} | 3.22 × 10^{−7}3.20 × 10 ^{−7} |

30 | 4.09 3.98 | 0.256 0.248 | 4.09 × 10^{−4}3.98 × 10 ^{−4} | 2.56 × 10^{−5}2.49 × 10 ^{−5} | 2.49 × 10^{−7}2.43 × 10 ^{−7} |

40 | 3.33 3.18 | 0.208 0.198 | 3.33 × 10^{−4}3.18 × 10 ^{−4} | 2.08 × 10^{−5}1.99 × 10 ^{−5} | 2.03 × 10^{−7}1.94 × 10 ^{−7} |

50 | 2.81 2.63 | 0.175 0.164 | 2.81 × 10^{−4}2.63 × 10 ^{−4} | 1.75 × 10^{−5}1.64 × 10 ^{−5} | 1.71 × 10^{−7}1.61 × 10 ^{−7} |

100 | 1.55 1.39 | 0.0969 0.0868 | 1.55 × 10^{−4}1.39 × 10 ^{−4} | 9.69 × 10^{−6}8.68 × 10 ^{−6} | 9.48 × 10^{−8}8.48 × 10 ^{−8} |

150 | 1.063 0.965 | 0.0665 0.0603 | 1.063 × 10^{−4}9.65 × 10 ^{−5} | 6.65 × 10^{−6}6.03 × 10 ^{−6} | 6.53 × 10^{−8}5.90 × 10 ^{−8} |

200 | 0.404 0.773 | 0.0504 0.0483 | 8.07 × 10^{−5}7.73 × 10 ^{−5} | 5.05 × 10^{−6}4.83 × 10 ^{−6} | 4.91 × 10^{−8}4.72 × 10 ^{−8} |

a, nm | 1 | 2 | 3 | 5 | 7 | 10 | 15 | 20 |
---|---|---|---|---|---|---|---|---|

p(a) | 0.352 | 0.385 | 0.360 | 0.334 | 0.318 | 0.297 | 0.272 | 0.254 |

a | 30 | 40 | 50 | 60 | 80 | 100 | 150 | 200 |

p(a) | 0.225 | 0.204 | 0.188 | 0.175 | 0.154 | 0.139 | 0.113 | 0.0955 |

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Dedkov, G.
Puzzling Low-Temperature Behavior of the Van Der Waals Friction Force between Metallic Plates in Relative Motion. *Universe* **2021**, *7*, 427.
https://doi.org/10.3390/universe7110427

**AMA Style**

Dedkov G.
Puzzling Low-Temperature Behavior of the Van Der Waals Friction Force between Metallic Plates in Relative Motion. *Universe*. 2021; 7(11):427.
https://doi.org/10.3390/universe7110427

**Chicago/Turabian Style**

Dedkov, George.
2021. "Puzzling Low-Temperature Behavior of the Van Der Waals Friction Force between Metallic Plates in Relative Motion" *Universe* 7, no. 11: 427.
https://doi.org/10.3390/universe7110427