Curvature-Driven Forces Based on Natural Exponential Pair Potential at Micro/Nanoscales

Dan Wang; Yajun Yin; Zheng Zhong; Zhili Hu

doi:10.1007/s10338-019-00079-6

Acta Mechanica Solida Sinica

pp 1–15 | Cite as

Curvature-Driven Forces Based on Natural Exponential Pair Potential at Micro/Nanoscales

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Dan Wang
Yajun Yin
Zheng Zhong
Zhili Hu

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First Online: 09 February 2019

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Abstract

The natural exponential potential ($C\hbox {e}^{-R/{\lambda _0 }})$ widely exists at micro/nanoscales; this paper studies the interaction potential between a curved-surface body and an outside particle base on the natural exponential potential. Mathematical derivation proves that the interaction potential can be expressed as a function of curvatures. Then, idealized numerical experiments are designed to verify the accuracy of the curvature-based potential. The driving forces exerted on the particle are discussed and confirmed to be a function of curvatures and the gradient of curvatures, which may explain some abnormal movements at micro/nanoscales.

Keywords

Micro/nanoscales Curvature-driven forces Natural exponential pair potential Curvature-based potential

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Notes

Acknowledgements

This research is supported by the Natural Science Foundation of Jiangsu Province (Nos. BK20180411, BK20180416) and the start-up funding awarded by Nanjing University of Aeronautics and Astronautics (Nos. 56SYAH17065, 90YAH17065).

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Conflict of interest

The authors declare that they have no conflict of interest.

Appendix

The Uniform Expression of the Intersection Area

To derive the intersection area A, Eq. (3) is substituted into Eq. (1):

$$\begin{aligned} z=\frac{1}{2}\left( {c_1 \cos ^{2}\theta +c_2 \sin ^{2}\theta } \right) r^{2}=\frac{1}{2}k_n r^{2} \end{aligned}$$

(A1)

where $k_n $ is the normal curvature.

For the convex and concave bodies (Figs. 3b and 4b), we obtain the below relation in triangle ABP:

$$\begin{aligned} R_k ^{2}+(z+h)^{2}=R^{2} \end{aligned}$$

(A2)

where $R_k =\left| {\mathop {BA}\limits ^\rightarrow } \right| $. The intersection area A(R) is:

$$\begin{aligned} A(R)= & {} \int _0^{2\pi } {\hbox {d}\theta \int _0^{R_k (\theta ,c_1 ,c_2 )} {\sqrt{1+\left( {\frac{\partial z}{\partial r}} \right) ^{2}}} } r\hbox {d}r \nonumber \\= & {} \int _0^{2\pi } {\hbox {d}\theta \int _0^{R_k (\theta ,c_1 ,c_2 )} {\sqrt{1+\left( {\frac{-r}{\sqrt{R^{2}-r^{2}}}} \right) ^{2}}} } r\hbox {d}r \nonumber \\= & {} \int _0^{2\pi } {R\left( {R-\sqrt{R^{2}-R_k ^{2}}} \right) \hbox {d}\theta } \end{aligned}$$

(A3)

By combining Eqs. (A1) and (A2), we have the calculation formula for $R_k $:

$$\begin{aligned} R_k ^{2}=\frac{2\left[ {\sqrt{\left( {Rk_n } \right) ^{2}+2hk_n +1}-\left( {hk_n +1} \right) } \right] }{k_n ^{2}} \end{aligned}$$

(A4)

Substitution of Eq. (A4) into Eq. (A3) yields:

$$\begin{aligned} A(R)= & {} \int _0^{2\pi } R \left( {R-\sqrt{R^{2}-R_k ^{2}}} \right) \hbox {d}\theta \nonumber \\= & {} \int _0^{2\pi } R \left( {R-\sqrt{\frac{\left( {Rk_n } \right) ^{2}+2hk_n +1-2\sqrt{\left( {Rk_n } \right) ^{2}+2hk_n +1}+1}{k_n ^{2}}}} \right) \hbox {d}\theta \nonumber \\= & {} \int _0^{2\pi } R \left( {R+\frac{1}{k_n }-\frac{1}{k_n }\sqrt{\left( {Rk_n } \right) ^{2}+2hk_n +1}} \right) \hbox {d}\theta \end{aligned}$$

(A5)

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Authors and Affiliations

Dan Wang
- 1
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Yajun Yin
- 2
Zheng Zhong
- 3
Zhili Hu
- 1

1.State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
2.Department of Engineering Mechanics, School of AerospaceTsinghua UniversityBeijingChina
3.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina

Curvature-Driven Forces Based on Natural Exponential Pair Potential at Micro/Nanoscales

Abstract

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Notes

Acknowledgements

Compliance with Ethical Standards

Conflict of interest

References

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