# The Casimir Interaction between Spheres Immersed in Electrolytes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Two Spheres Immersed in an Electrolyte

#### 2.1. Basic Formalism

#### 2.2. The n = 0 Contribution

_{1}(ψ

_{2}) may be understood as the potential of the sphere 1(2) in the absence of the other (although such interpretation would hold if κ

_{D}L ≫ 1), as it is evidenced by the fact that all coefficients a

_{ℓm}, b

_{ℓm}, c

_{ℓm}, d

_{ℓm}depend on the boundary conditions imposed by the two spheres. Despite its relative simplicity, expression (7) is not convenient as it makes reference to two different (but interdependent) coordinate systems. In order to write the solution (7) in terms of a single coordinate system, we use an important spherical addition theorem [62] and arrive at

_{n}(x) and y

_{n}(x) in [62], and it differs from our formula for modified Bessel functions i

_{n}(x) and k

_{n}(x) by the pre-factor ${(-1)}^{\nu}$.

#### 2.3. The $n\ge 1$ Contributions

#### 2.3.1. Plane-Wave Representation

#### 2.3.2. Numerical Application

## 3. Results and Discussion

#### 3.1. Numerical Considerations

#### 3.2. The Screening Effect

#### 3.3. Zero-Frequency Contribution for Very Small Spheres.

#### 3.4. Charge Fluctuations

#### 3.5. Very Large Spheres and Comparison with PFA

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Description of the geometry with two solid spheres of radii ${R}_{1}$ and ${R}_{2}$ separated by a center-to-center distance L along the z-axis. We also indicate the surface-to-surface distance $a=L-({R}_{1}+{R}_{2})$. A generic point can be described using either sphere center: ${r}_{1}$ (${r}_{2}$) is a vector from the center of the sphere 1 (2) to the point.

**Figure 2.**Variation with the surface-to-surface distance of the (

**a**) Casimir free energy, (

**b**) Casimir force, (

**c**) relative zero frequency contribution to the free energy, and (

**d**) relative zero frequency contribution to the force. We consider two PS spheres of radius $1\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$.

**Figure 3.**Zero-frequency contribution to the Casimir energy for two silica spheres of radius R as a function of $R/L$ for a fixed $L=1\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ and different Debye lengths: numerical (solid) and small-sphere approximation (46) (dashed).

**Figure 4.**Attractive electrostatic interaction between two fluctuating dipoles in an electrolyte solution for two configurations: (

**a**) Dipoles perpendicular to the line connecting them and (

**b**) Dipoles parallel to the line connecting them. In situation (

**a**), the electrostatic potential generated by each dipole does not induce a charge, while, in situation (

**b**), it does, with the signs indicated in the figure.

**Figure 5.**Relative discrepancy between the PFA and the exact result as a function of the Debye length for the total free energy ${\mathcal{F}}_{\mathrm{cas}}$ (red squares) and the zero-frequency contribution ${\mathcal{F}}_{\mathrm{cas}}^{0}$ (blue dots). The horizontal dashed line indicates the discrepancy when considering positive frequencies only, which do not depend on ${\lambda}_{D}.$ We consider two PS spheres of radius 1 μm separated by a distance of 20 nm.

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**MDPI and ACS Style**

Nunes, R.O.; Spreng, B.; de Melo e Souza, R.; Ingold, G.-L.; Maia Neto, P.A.; Rosa, F.S.S.
The Casimir Interaction between Spheres Immersed in Electrolytes. *Universe* **2021**, *7*, 156.
https://doi.org/10.3390/universe7050156

**AMA Style**

Nunes RO, Spreng B, de Melo e Souza R, Ingold G-L, Maia Neto PA, Rosa FSS.
The Casimir Interaction between Spheres Immersed in Electrolytes. *Universe*. 2021; 7(5):156.
https://doi.org/10.3390/universe7050156

**Chicago/Turabian Style**

Nunes, Renan O., Benjamin Spreng, Reinaldo de Melo e Souza, Gert-Ludwig Ingold, Paulo A. Maia Neto, and Felipe S. S. Rosa.
2021. "The Casimir Interaction between Spheres Immersed in Electrolytes" *Universe* 7, no. 5: 156.
https://doi.org/10.3390/universe7050156