# The Impact of the Anisotropy of the Media between Parallel Plates on the Casimir Force

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}will have different values in different directions. Accordingly, this makes the condition for repulsive force more complicated. One of our recent works shows that it is still possible to have repulsive force in this case and that the force can be significantly affected by the anisotropy of the media between the plates [21]. As we know liquid like nitrobenzene can be anisotropic under an external field [22,23,24]. This means that the external field can affect the Casimir force by affecting the anisotropy of the intervening media. This is important for both theoretical and experimental investigations on the Casimir effect. In order to supplement our previous work [18], in this paper, we calculate the fully relativistic Casimir force between two isotropic plates separated by a gap filled with a third uniaxial material, and then analyze the exact impact of the anisotropy of the intervening media and dispersion on the Casimir force. Furthermore, we discuss the influence of the external electric field on the Casimir effect by affecting the anisotropy of the intervening media though the Kerr electro-optical effect.

## 2. Calculation of the Casimir Force between Parallel Slabs Separated by Uniaxial Material

_{q}and ${a}_{q}^{+}$ in (4) and (5) are the usual creation and annihilation operators, respectively.

**k**= (k

_{x}, k

_{y}, k

_{z}) is the wave vector. We can choose k

_{y}= 0 without loss of generality, and we can have

**k**= (k

_{x}, 0, k

_{z}) = K

_{0}(α, 0, γ), with K

_{0}= ωc

^{−1}. In order to determine the mode of the electric and magnetic field

**e**

_{q}and

**h**

_{q}with frequency ω, we can use the classical Maxwell equations without sources.

_{i}(i = 1, 2, 3) is the dielectric permittivity in region I, II, and III, which could be expressed by Equations (1)–(3). The solution of the electromagnetic fields can be expressed in the plane wave form as:

_{x}, e

_{y}, and e

_{z}indicate the elements of the electric field (

**E**) polarization vectors, and b

_{x}, b

_{y}, and b

_{z}indicate the elements of the magnetic field (B) polarization vectors. As all of the materials that are considered in this paper are assumed to be nonmagnetic, we can have

**B**= μ

_{0}

**H**.

_{x}, e

_{y}b

_{x}, and b

_{y}in region: I

_{1})

^{2}= α

^{2}− ε

_{1}. These four eigenvalues correspond to four independent mode solutions whose linear superposition should be the general solution of the electromagnetic field. Further, the superposition coefficients should be the amplitudes of each mode. Therefore, the transverse elements of the electromagnetic field modes in (4) and (5) in region I can be expressed as:

_{2})

^{2}= α

^{2}− ε

_{2}.

_{3x})

^{2}= α

^{2}− ε

_{3x}and (t

_{3z})

^{2}= (α

^{2}− ε

_{3z})(ε

_{3x}/ε

_{3z}).

_{I,3}= A

_{I,4}= A

_{II,1}= A

_{II,2}= 0. As the boundary conditions require the transverse elements of the electromagnetic field to be continuous at z = a and z = 0, we can get a set of eight linear homogeneous equations that are relating the unknown coefficients A

_{I,1}, A

_{I,2}, A

_{II,3}, A

_{II,4}, A

_{III,1}, A

_{III,2}, A

_{III,3}, and A

_{III,4}. It has nontrivial solutions if the determinant of its coefficients is equal to zero, which is, accordingly, the equation to determine the proper frequency ω of the modes. This equation can be expressed as:

**k**

**= (k**

_{||}_{x}, k

_{y}) and z (according to previous assumption, this plane is x-z plane), respectively. We introduce δ = (ε

_{3z}/ε

_{3x}) − 1 to express the degree of anisotropy of the intervening material between the plates, and δ = 0 refers to isotropic media. The detailed expression of the functions D

_{1}and D

_{2}can be found in Appendix A, and the functions G

_{1}and G

_{2}can be expressed as (ω = iξ),

^{2}= 1 − α

^{2}/ε

_{3x}, (s

_{1})

^{2}= −(t

_{1})

^{2}/ε

_{3x}, and (s

_{2})

^{2}= −(t

_{2})

^{2}/ε

_{3x}have been introduced.

_{1}does not depend on δ, which means that the anisotropy affects the Casimir force only though the second term in the bracket in (23). When δ = 0, (23) becomes:

_{3x}, ε

_{3z}, ε

_{1}, and ε

_{2}with their values at ξ = 0 (ε

_{3x0}, ε

_{3z0}, ε

_{10}, and ε

_{20}) [3]. Also, (23) can be written with considerable accuracy in the form:

## 3. The Impact of the Anisotropy of the Intervening Material between the Plates on Casimir Force

_{10}> ε

_{20}. We need to emphasize that the case in which ε

_{10}> ε

_{3x0}> ε

_{20}and ε

_{10}> ε

_{3z0}> ε

_{20}are satisfied at the same time is not within our interests. This is because in this case, all the possible values of ε

_{30}are greater than ε

_{10}and smaller than ε

_{20}, and this will make the force repulsive, which is almost the same as Lifshitz’s result [3]. For the same reason, we also do not have interests in the case when both ε

_{3x0}and ε

_{3z0}are greater than ε

_{10}or when both ε

_{3x0}and ε

_{3z0}are smaller than ε

_{20}. Therefore, we only need to focus on two cases: 1. only one of ε

_{3x0}and ε

_{3z0}is between ε

_{10}and ε

_{20}; and, 2. one of ε

_{3x0}and ε

_{3z0}is smaller than ε

_{20}, while the other one is greater than ε

_{10}.

_{10}> ε

_{3x0}> ε

_{20}> ε

_{3z0}(or ε

_{3z0}> ε

_{10}> ε

_{3x0}> ε

_{20}). The value of Φ for different δ, the degree of anisotropy, is shown in Figure 3. The positive Φ indicates that the force is attractive, while the negative Φ indicates that the force is repulsive. In Figure 3a, where ε

_{10}> ε

_{3x0}> ε

_{20}> ε

_{3z0}, we can see that the Casimir force is attractive at beginning and the magnitude decreases as the degree of anisotropy δ changes. Finally, it switches into repulsive. In Figure 3b, where ε

_{3z0}> ε

_{10}> ε

_{3x0}> ε

_{20}, the Casimir force is repulsive at beginning and then switches into attractive as the degree of anisotropy δ changes. Therefore, we can conclude that the magnitude, as well as the direction, of the Casimir force could be significantly affected by the anisotropy of the intervening material in case 1. For a case where ε

_{3z0}is between ε

_{10}and ε

_{20}, but ε

_{3x0}is not, the result is similar.

_{3z0}> ε

_{10}> ε

_{20}> ε

_{3x0}. The value of Φ for different δ, the degree of anisotropy, is shown in Figure 4. In Figure 4, we can see that the Casimir force is repulsive at the very beginning and increases as δ increases. Then, as δ increases, the repulsive force starts to decrease, before finally switching into attractive. Therefore, in this case, still, the magnitude as well as the direction of the Casimir force could be significantly affected by the anisotropy of the intervening material.

_{0}represents the contribution of isotropy to the total force, while δΦ

_{δ}represents the contribution of anisotropy. As Φ

_{0}does not depend on δ, it will always be the same value and sign when δ changes. Therefore, the changes of the magnitude and direction of the Casmir force are caused by δΦ

_{δ}, which depends on the anisotropy of the intervening material. In case 1, Φ

_{0}is always negative and produces a repulsive force. In Figure 3a,b, it is clear that when the absolute value of δ increases, the repulsive force becomes smaller and finally changes into an attractive force. This means that the anisotropy can produce an attractive force against the repulsive force that is produced by Φ

_{0}, the isotropic term. When the contribution from anisotropy is greater than the contribution from isotropy, the force switches its direction. In case 2, Φ

_{0}is always positive and produces an attractive force. However, the total force is not always attractive, because the anisotropy can produce a repulsive force against the attractive force produced by Φ

_{0}, the isotropic term. One should notice that the curve in Figure 3a represents a good linear relationship, while the curves in Figure 3b and Figure 4 do not represent a linear relationship. This is because the expansion in (28)–(30) is only valid for small δ, but the values of δ are not so small in Figure 3b and Figure 4. The higher order terms of δ produce the nonlinear relationship between Φ and δ. However, in general, we can draw the conclusion that, in both case 1 and case 2, if the isotropy of the material between the plates contributes a force in a certain direction, the anisotropy could produce a force in the opposite direction. The direction of the Casimir force depends on which one has greater magnitude, which reflects the competition between the contribution from anisotropy and isotropy. The change in the direction of the Casimir force is caused by the change of the degree of anisotropy of the material between the plates in this limiting case. This confirms our conclusion in [21], where we roughly divided the force into two parts.

## 4. The Impact of the Dispersion

_{3}is the average value of ε

_{3}

_{z}and ε

_{3}

_{x}. The models of the dispersion relations and the parameters used are described in Appendix B. The Casimir forces at different separations for different values of δ are calculated with (23) and the result is shown in Figure 5b. The positive value indicates that the force is attractive, while the negative value indicates that the force is repulsive. The force is repulsive in wide distance range, and can be attractive only at very small separations. This behavior can be explained by the inequality between ε

_{1}, ε

_{2}and ε

_{3}. ε

_{3}is the average value of ε

_{3}

_{z}and ε

_{3}

_{x}. The inequality ε

_{1}> ε

_{3}> ε

_{2}is satisfied for the frequencies lower than ~2 × 10

^{15}rad/s, and the modes with these low frequencies will contribute to a repulsive force. For the frequencies that are higher than ~2 × 10

^{15}rad/s, the inequality ε

_{1}> ε

_{3}> ε

_{2}is no longer valid, and the modes with these high frequencies will contribute to an attractive force. The direction of the total force depends on which contribution is greater. At small separation, the contribution from high frequency modes is dominant and the force is attractive; while at large separation, the contribution from low frequency modes is dominant and the force switches into repulsive.

_{1}, ε

_{2}and ε

_{3}. The negative δ will make ε

_{3}, the average value of ε

_{3z}and ε

_{3x}, smaller, and the inequality ε

_{1}> ε

_{3}> ε

_{2}will invalidate at even lower frequency. That is why negative δ can help to produce an attractive force.

^{isotropic}(a) to express the impact of anisotropy of the intervening media. As shown in Figure 5c, at the separation greater than few nanometers, negative δ always produces an attractive force, which is coincident with the result in the paragraph above. One may also notice that ε

_{1}is much greater tha n ε

_{2}and ε

_{3}in wide frequency range. Therefore, for small positive δ, it is not possible to make ε

_{3}, the average value of ε

_{3}

_{z}and ε

_{3}

_{x}, go beyond ε

_{1}at lower frequency. That is why the impact of positive δ is always negative in Figure 5c.

## 5. The Impact of the External Electric Field on the Casimir Force

_{ext}is applied across the material in z direction, then the intervening material will become birefringent, and the relationship between permittivity in z direction and x direction should be: [24]

_{k}is associated with the property of the material. Here, we only consider the Kerr electro-optical effect for the material between the plates and do not consider that for the plates, since this effect in solid is often not as strong as that in liquid. As we have introduced δ = (ε

_{3z}/ε

_{3x}) − 1 to express the degree of anisotropy of the intervening material, we can have:

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

_{1}and G

_{2}can be found in (20) and (21) and are transformed into another form with the relations mentioned in Section 2.

_{1}and ∆

_{2}are analytic and have no poles along the closed path in (A7) and (A8), argument theorem is capable here. If we assume that along any radial direction in complex plane we can have:

^{2}= 1 − α

^{2}/ε

_{3x}, (s

_{1})

^{2}= −(t

_{1})

^{2}/ε

_{3x}, and (s

_{2})

^{2}= −(t

_{2})

^{2}/ε

_{3x}.

## Appendix B

_{1}), we use the Drude model:

_{2}) and the intervening material between the plates (ε

_{3x}), we use the following model:

_{2}, we use the parameters for quartz [9]: C

_{IR}= 1.920, C

_{UV}= 1.350, ω

_{IR}= 2.09 × 10

^{14}rad/s and ω

_{UV}= 2.04 × 10

^{16}rad/s. For ε

_{3x}, we use the parameters for bromobenzene [26]: C

_{IR}= 2.967, C

_{UV}= 1.335, ω

_{IR}= 5.47 × 10

^{14}rad/s and ω

_{UV}= 1.28 × 10

^{16}rad/s. In Figure 5a, δ = −0.1 is used.

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**Figure 1.**Two parallel isotropic plates interact across an anisotropic media. The surfaces of the plates are parallel to the x-y plane. The optical axis of the intervening anisotropic media between the plates is in the z direction.

**Figure 2.**The schematic of the system under investigation. The space can be considered to be divided into three regions with corresponding permittivities.

**Figure 3.**Φ and Φ

_{0}vs. δ. ε

_{10}/ε

_{3x0}= 1.5 and ε

_{20}/ε

_{3x0}= 0.8. Φ is shown with the solid line and Φ

_{0}is shown with the red dash line. The positive value indicates that the force is attractive, while the negative value indicates that the force is repulsive. (

**a**) ε

_{10}> ε

_{3x0}> ε

_{20}> ε

_{3z0}; (

**b**) ε

_{3z0}> ε

_{10}> ε

_{3x0}> ε

_{20}.

**Figure 4.**Φ and Φ

_{0}vs. δ. ε

_{10}/ε

_{3x0}= 1.5, ε

_{20}/ε

_{3x0}= 1.1 and ε

_{3z0}> ε

_{10}> ε

_{20}> ε

_{3x0}. Φ is shown with the solid line and Φ

_{0}is shown with the red dash line. The positive value indicates that the force is attractive, while the negative value indicates that the force is repulsive.

**Figure 5.**The Casimir force when dispersion is considered. The positive value indicates that the force is attractive, while the negative value indicates that the force is repulsive. The horizontal dash-dot line indicates F = 0 and the intersections between this line and other curves indicate the reverses of the direction of the force. (

**a**) The frequency dependent pemittivities (The models of dispersion relation are described in Appendix B); (

**b**) F(a) vs. a; (

**c**) The impact of anisotropy of intervening media; (

**d**) F(a) vs. δ.

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**MDPI and ACS Style**

Deng, G.; Pei, L.; Hu, N.; Liu, Y.; Zhu, J.-R.
The Impact of the Anisotropy of the Media between Parallel Plates on the Casimir Force. *Symmetry* **2018**, *10*, 61.
https://doi.org/10.3390/sym10030061

**AMA Style**

Deng G, Pei L, Hu N, Liu Y, Zhu J-R.
The Impact of the Anisotropy of the Media between Parallel Plates on the Casimir Force. *Symmetry*. 2018; 10(3):61.
https://doi.org/10.3390/sym10030061

**Chicago/Turabian Style**

Deng, Gang, Ling Pei, Ni Hu, Yang Liu, and Jin-Rong Zhu.
2018. "The Impact of the Anisotropy of the Media between Parallel Plates on the Casimir Force" *Symmetry* 10, no. 3: 61.
https://doi.org/10.3390/sym10030061