# Finite-Size Effects of Casimir–van der Waals Forces in the Self-Assembly of Nanoparticles

## Abstract

**:**

## 1. Introduction

## 2. Lifshitz Theory and the Hamaker Constant

## 3. Finite-Size Effects

## 4. Results

## 5. Discussion

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The variation of ${E}_{r}$ as a function of the separation, L, of the two plates for different values of the thickness c of each as indicated, and $a=b=2000$ nm. Even for large enough values of c, the energy ratio is less than unity.

**Figure 3.**For two fixed separations between the plates, $L=50$ nm and $L=100$ nm, the ratio ${E}_{r}$ increases with increasing the value of the thickness c, leveling-off asymptotically to the value expected for half-spaces. The dimensions of the plate are $a=b=2000$ nm.

**Figure 4.**Energy ratio, ${E}_{r}$ between two cubes facing each other and for two tilted cubes, as a function of the separation L. The nanocubes have dimensions $a=b=c=80$ nm.

**Figure 5.**Energy ratio, ${E}_{r}$, between two tilted nanocubes of different sizes. The sizes considered are 20 nm (blue line), 80 nm (red line), and 150 nm (orange line). The maximum in each curve happens when the separation equals the size of the cube.

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**MDPI and ACS Style**

Esquivel-Sirvent, R.
Finite-Size Effects of Casimir–van der Waals Forces in the Self-Assembly of Nanoparticles. *Physics* **2023**, *5*, 322-330.
https://doi.org/10.3390/physics5010024

**AMA Style**

Esquivel-Sirvent R.
Finite-Size Effects of Casimir–van der Waals Forces in the Self-Assembly of Nanoparticles. *Physics*. 2023; 5(1):322-330.
https://doi.org/10.3390/physics5010024

**Chicago/Turabian Style**

Esquivel-Sirvent, Raul.
2023. "Finite-Size Effects of Casimir–van der Waals Forces in the Self-Assembly of Nanoparticles" *Physics* 5, no. 1: 322-330.
https://doi.org/10.3390/physics5010024