The effect of agglomeration of magnetic nanoparticles on the Casimir pressure through a ferrofluid

TL;DR

This study examines how nanoparticle agglomeration in ferrofluids influences Casimir pressure between plates, revealing conditions under which the force's sign can change, with implications for microdevice design and Casimir force theory.

Contribution

It demonstrates that nanoparticle clustering can alter the Casimir force's magnitude and sign, depending on the optical model used and nanoparticle clustering parameters.

Findings

Agglomeration causes quantitative changes in Casimir pressure for dielectric plates.

Using the plasma model, nanoparticle clustering can reverse the force's sign.

The effect varies with nanoparticle size, clustering degree, and plate separation.

Abstract

The impact of agglomeration of magnetic nanoparticles on the Casimir pressure is investigated in the configuration of two material plates and a layer of ferrofluid confined between them. Both cases of similar and dissimilar plates are considered in the framework of the Lifshitz theory of dispersion forces. It is shown that for two dielectric (SiO_2) plates, as well as for one dielectric (SiO_2) and another one metallic (Au) plates, an agglomeration of magnetite nanoparticles results in only quantitative differences in the values of the Casimir pressure if the optical data for Au are extrapolated to low frequencies by means of the Drude model. If, however, an extrapolation by means of the plasma model is used in computations, which is confirmed in experiments on measuring the Casimir force, one finds that the pressure changes its sign when some share of magnetic nanoparticles of sufficiently large diameter is merged into clusters by two or three items. The revealed effect of sign change is investigated in detail at different separations between the plates, diameters of magnetic nanoparticles and shares of particles merged into clusters of different sizes. The obtained results may be useful when developing ferrofluid-based microdevices and for resolution of outstanding problems in the theory of Casimir forces.

Equations

$$P(a)=-\frac{k_{B}T}{8\pi a^3}\sum _{l=0}^{\infty }{}^{{}^{\prime }}{\int }_{\sqrt{{\epsilon }_{\mathrm{ff},l}{\mu }_{\mathrm{ff},l}}{\zeta }_l}^{\infty }{y}^{2}dy$$

$$\times \sum _{\alpha }{\left[\frac{e^y}{r_{\alpha }^{(1)}(i{\zeta }_l,y)r_{\alpha }^{(2)}(i{\zeta }_l,y)}-1\right]}^{-1}.$$

$${\zeta }_l=\frac{{\xi }_l}{{\omega }_{\mathrm{cr}}}=\frac{2a{\xi }_l}{c}=\frac{4\pi a{k}_{B}Tl}{\hslash c},l=0,1,2,\dots ,$$

$${\epsilon }_{\mathrm{ff},l}\equiv {\epsilon }_{\mathrm{ff}}(i{\xi }_l)={\epsilon }_{\mathrm{ff}}(i{\omega }_c{\zeta }_l),$$

$${\epsilon }_l^{(k)}\equiv {\epsilon }^{(k)}(i{\xi }_l)={\epsilon }^{(k)}(i{\omega }_c{\zeta }_l),k=1,2,$$

$${\mu }_{\mathrm{ff},l}\equiv {\mu }_{\mathrm{ff}}(i{\xi }_l)={\mu }_{\mathrm{ff}}(i{\omega }_c{\zeta }_l).$$

$$r_{\mathrm{TM}}^{(k)}(i{\zeta }_l,y)=\frac{{\epsilon }_l^{(k)}y-{\epsilon }_{\mathrm{ff},l}\sqrt{y^2+({\epsilon }_l^{(k)}-{\epsilon }_{\mathrm{ff},l}{\mu }_{\mathrm{ff},l}){\zeta }_l^2}}{{\epsilon }_l^{(k)}y+{\epsilon }_{\mathrm{ff},l}\sqrt{y^2+({\epsilon }_l^{(k)}-{\epsilon }_{\mathrm{ff},l}{\mu }_{\mathrm{ff},l}){\zeta }_l^2}},$$

$$r_{\mathrm{TE}}^{(k)}(i{\zeta }_l,y)=\frac{y-{\mu }_{\mathrm{ff},l}\sqrt{y^2+({\epsilon }_l^{(k)}-{\epsilon }_{\mathrm{ff},l}{\mu }_{\mathrm{ff},l}){\zeta }_l^2}}{y+{\mu }_{\mathrm{ff},l}\sqrt{y^2+({\epsilon }_l^{(k)}-{\epsilon }_{\mathrm{ff},l}{\mu }_{\mathrm{ff},l}){\zeta }_l^2}}.$$

$${\chi }_{\mathrm{ff}}^{(1)}(0)=N\frac{M_1^2}{3k_{B}T},$$

$${\chi }_{\mathrm{ff}}^{(k)}(0)$$

$${\mu }_{\mathrm{ff}}^{(k)}(0)=1+4\pi {\chi }_{\mathrm{ff}}^{(k)}(0).$$

$${\mathcal{P}}_k=\frac{P({\kappa }_k)}{P({\kappa }_1)},$$

Full text

I Introduction

During the last few years much attention was given to ferrofluids which are colloidal liquids consisting of magnetic nanoparticles suspended in some carrier liquid (see, e.g., the monograph 1 and Refs. 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ). Ferrofluids are used in optical switches, optoelectronic communications, mechanical and medical applications 4 ; 6 ; 10 ; 11 ; 12 , and also in microdevices playing a broad spectrum of roles 13 ; 14 ; 15 ; 16 ; 17 ; 18 . In the latter case, ferrofluids may be confined in a narrow, submicrometer, gap between two material plates. Under these conditions, the plates are subjected to the Casimir force caused by the zero-point and thermal fluctuations of the electromagnetic field 19 . In the presence of a fluid in the gap, the Casimir force may be both attractive and repulsive depending on materials of the plates.

Nowadays the Casimir force is under active theoretical and experimental studies (see Refs. 20 ; 21 ; 22 ; 23 for a review). Specifically, there is abundant evidence that the Casimir force can be used in micro- and nanoelectromechanical devices, such as Casimir oscillators, silicon chips, switches, optical choppers etc. 24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 . This raises the question of whether the Casimir force should be taken into account in ferrofluid-based microdevices. In Ref. 33 the Casimir pressure was investigated in the case of magnetite nanoparticles suspended in kerosene or water between two SiO2 plates. It was shown that an addition of a 5% volume fraction of magnetite nanoparticles leads to significantly different Casimir pressures as compared to the case of nonmagnetic intervening liquid. It was found also that at a fixed separation between the plates an addition to carrier liquid of magnetite nanoparticles of some definite diameter does not influence the Casimir pressure.

It has been known that magnetic particles suspended in a ferrofluid undergo agglomeration which tends to diminish their surface energy 1 ; 6 . To decrease an extent of agglomeration, magnetic nanoparticles are usually coated with a surfactant which makes lower the surface tension between a nanoparticle and a carrier liquid. However, even with a surfactant, some share of magnetic nanoparticles merge into clusters composed of two, three or more particles. The question arises: What is the effect of agglomeration on the Casimir pressure between two material plates separated by a ferrofluid?

In this paper, we investigate the Casimir pressure through a ferrofluid under different assumptions about nanoparticle diameter, materials of the plates, and the extent of agglomeration of magnetite nanoparticles. It is shown that the extent of agglomeration affects the pressure in a nontrivial way depending on the other parameters of the problem and, what is more important, on the theoretical approach used in calculations. Specifically, for two similar dielectric plates separated by a ferrofluid the effect of agglomeration of magnetite nanoparticles leads to only relatively small quantitative differences in the Casimir pressure. The most interesting results are found for a ferrofluid confined between two dissimilar plates, one metallic (Au) and another one dielectric (SiO2). This is an example of three-layer systems much studied in the case of nonmagnetic intervening layer 19 ; 22 , where the pressure can be repulsive. According to our results, for magnetite nanoparticles of 10 nm diameter suspended in water as a carrier liquid agglomeration leads to only relatively small variations in the values of repulsive Casimir pressure. However, for nanoparticles of 20 nm diameter the effect of agglomeration on the pressure appears essentially dependent on the used model of the low-frequency dielectric response of Au.

It is the subject of considerable literature that the Lifshitz theory of the van der Waals and Casimir forces agrees with the measurement results only if the available optical data of a plate metal are extrapolated down to zero frequency by means of the lossless plasma model. If the lossy Drude model is used for extrapolation, the theoretical predictions are excluded by the experimental data at the highest confidence level (see Refs. 19 ; 20 ; 23 ; 34 ; 34a ; 34b for a review and more recent experiments 35 ; 36 ; 37 ; 38 ; 39 ; 40 ). This result received the name Casimir puzzle because it implies that the dielectric response of a metal to the low-frequency fluctuating field is not that which is normally expected to an ordinary electromagnetic field.

Specifically, we demonstrate that for magnetite nanoparticles of 20 nm diameter the effect of agglomeration on the Casimir pressure remains again entirely quantitative and relatively small if the optical data of Au are extrapolated by means of the lossy Drude model. If, however, the experimentally consistent lossless plasma model is used for extrapolation, the agglomeration of magnetite nanoparticles results in the change of sign of the Casimir pressure from repulsion to attraction at some separation distance between the plates. The physical explanation to this effect is provided. The transition conditions of the Casimir pressure from repulsion to attraction under an impact of agglomeration are investigated as functions of the share of nanoparticles merged into clusters and of nanoparticle diameter. Possible applications of the obtained results are discussed.

The paper is organized as follows. In Sec. II, we present the Lifshitz formula for the Casimir pressure adapted for a configuration of two dissimilar plated separated by a ferrofluid. Section III contains the computational results on the impact of agglomeration of magnetite nanoparticles on the Casimir pressure for both dissimilar and similar plates obtained using different theoretical approaches. In Sec. IV, the conditions for a change of sign of the Casimir pressure are investigated for different shares of nanoparticles merged into clusters and different nanoparticle diameters. Section V contains our conclusions and a discussion.

II The Lifshitz formula for two dissimilar plates separated

by a ferrofluid

We consider the configuration of two parallel nonmagnetic plates described by the dielectric permittivities $\varepsilon^{(1)}(\omega)$ and $\varepsilon^{(2)}(\omega)$. The gap between the plates of thickness $a$ is filled with a ferrofluid described by the dielectric permittivity $\varepsilon_{\rm ff}(\omega)$ and magnetic permeability $\mu_{\rm ff}(\omega)$. Material plates can be considered as semispaces if they are thicker than 100 nm 19 and $2~{}\mu$m 41 in the case of metallic and dielectric materials, respectively. The Casimir pressure at temperature $T$ can be conveniently expressed in terms of the dimensionless variables by the following Lifshitz formula:

[TABLE]

Here, $k_{B}$ is the Boltzmann constant, the dimensionless Matsubara frequencies $\zeta_{l}$ are expressed via the dimensional ones $\xi_{l}$ by

[TABLE]

and all dielectric permittivities and magnetic permeability are calculated at these frequencies along the imaginary frequency axis

[TABLE]

The prime on the first summation sign in Eq. (1) means that the term with $l=0$ is divided by $2$, and the summation in $\alpha$ is over two independent polarizations of the electromagnetic field, transverse magnetic ($\alpha={\rm TM}$) and transverse electric ($\alpha={\rm TE}$). Finally, the reflection coefficients on the first and second plates are given by

[TABLE]

Now calculation of the Casimir pressure through a ferrofluid can be performed by Eqs. (1)–(4) if one knows the values of dielectric permittivities of the plates and ferrofluid at the pure imaginary frequencies, $\varepsilon_{l}^{(k)}$, $\varepsilon_{{\rm ff},l}$, and of ferrofluid magnetic permeability $\mu_{{\rm ff},l}$. Note, that the magnetic permeability quickly decreases with increasing frequency and at room temperature becomes equal to unity at $\xi\ll\xi_{1}$. For this reason, the magnetic properties of a ferrofluid, as well as of any other magnetic body, influence the Casimir force only through the term of Eq. (1) with $l=0$ 42 .

Below we consider Au and SiO2 plates with an intervening layer of water-based ferrofluid containing $\Phi=0.05$ volume fraction of magnetite nanoparticles. The dielectric permittivity of Au along the imaginary frequency axis was obtained using the optical data of Ref. 43 extrapolated to lower frequencies by either the plasma or the Drude model and repeatedly used in the literature 19 ; 20 ; 34 ; 34a ; 34b ; 35 ; 36 ; 39 ; 40 . At low frequencies the respective permittivities behave as $\varepsilon_{p}^{(1)}\sim\omega_{p}^{2}/\xi^{2}$ and $\varepsilon_{D}^{(1)}\sim\omega_{p}^{2}/(\gamma\xi)$, where $\omega_{p}$ is the plasma frequency and $\gamma$ is the relaxation parameter of Au.

The permittivities of SiO2 and water are taken from Refs. 44 ; 45 , respectively. Specifically, for SiO2 one has $\varepsilon^{(2)}(0)=3.801$. An analytic expression for the dielectric permittivity of magnetite was found in Ref. 33 using the measured optical data of Ref. 46 and the Kramers-Kronig relations. Combining the permittivities of water and magnetite with the help of Rayleidh’s mixing formula, the permittivity of ferrofluid $\varepsilon_{{\rm ff},l}$ with a given volume fraction of nanoparticles $\Phi=0.05$ was obtained 33 . With omitted conductivity of magnetite at low frequencies (see Refs. 19 ; 20 ; 48 ; 49 ; 50 for the reasons why this option is more realistic in computations of the Casimir force), one arrives at $\varepsilon_{{\rm ff}}(0)=77.89$.

The resulting permittivities of Au (in two variants), SiO2 and of a ferrofluid are used in Secs, III and IV to investigate an impact of agglomeration of nanoparticles on the Casimir pressure. The static magnetic permeability of a ferrofluid, which depends on the extent of agglomeration, is determined in the next section.

III Impact of agglomeration on the sign of Casimir pressure

According to the results of Ref. 33 , the initial susceptibility of a ferrofluid containing single nanoparticles is given by

[TABLE]

where $N=\Phi/V_{1}$, the volume of each nanoparticle $V_{1}=\pi d^{3}/6$ is expressed via its diameter $d$, and $M_{1}$ is the magnitude of a nanoparticle magnetic moment. The later is related to the saturation magnetization per unit volume $M_{1}=M_{S}V_{1}$, where $M_{S}$ may take different values for a bulk material and for its parts. Specifically, for single-domain magnetic nanoparticles of spherical shape considered here one has $M_{S}\approx 300~{}\mbox{emu/cm}^{3}=3\times 10^{5}~{}$A/m 47 .

Let us now assume that as a result of agglomeration the share $\kappa_{k}$ of all nanoparticles is merged into clusters containing $k$ particles each. Taking into account that the size of clusters is far less than the size of magnetic domain, it would appear reasonable to put the magnitude of the magnetic moment of a cluster equal to $M_{k}=kM_{1}$. In doing so, the ferrofluid contains $N(1-\kappa_{k})$ single magnetic nanoparticles having the magnetic moments of magnitude $M_{1}$ and $N\kappa_{k}/k$ clusters with magnetic moments of magnitude $M_{k}$. Then, the initial susceptibility of a ferrofluid of this type takes the form

[TABLE]

where $\chi_{\rm ff}^{(1)}(0)$ is defined in Eq. (5). As a result, the static magnetic permeability of a ferrofluid allowing for the effect of agglomeration of nanoparticles is equal to

[TABLE]

Now we are in a position to compute the Casimir pressure between two parallel plates separated by a ferrofluid with due account for the effect of agglomeration of magnetic nanoparticles. We begin with the case of an Au plate described using an extrapolation of the optical data to low frequencies by means of the plasma model and a SiO2 plate. Computations are performed by substituting the dielectric permittivities $\varepsilon^{(1)}$ of Au, $\varepsilon^{(2)}$ of SiO2 and $\varepsilon_{\rm ff}$ of ferrofluid discussed in Sec. II to Eqs. (1)–(4).

The magnetic permeability of a ferrofluid with account of the effect of agglomeration is found from Eqs. (6) and (7). Thus, if magnetite nanoparticles have $d=10~{}$nm diameter, one obtains $\mu_{\rm ff}^{(1)}(0)=1.24$ if all nanoparticles are single and $\mu_{\rm ff}^{(2)}(0)=1.36$ and $\mu_{\rm ff}^{(3)}(0)=1.48$ if half of all nanoparticles are merged into clusters by two and three particles, respectively. In a similar way, if the nanoparticle diameter is $d=20~{}$nm, one obtains from Eqs. (6) and (7) $\mu_{\rm ff}^{(1)}(0)=2.9$, $\mu_{\rm ff}^{(2)}(0)=3.85$, and $\mu_{\rm ff}^{(3)}(0)=4.8$ for the cases when all nanoparticles are single and when one-half of them are merged into clusters by two and three, respectively.

Here and below all computations are performed at room temperature $T=300~{}$K for a water-based ferrofluid containing $\Phi=0.05$ volume fraction of magnetite nanoparticles.

In Fig. 1(a) the computational results for the Casimir pressure are shown as functions of separation between the plates for nanoparticles of $d=10~{}$nm diameter by the solid and dashed lines obtained with no agglomeration and when half of nanoparticles is merged into clusters by three ($\kappa_{3}=0.5$), respectively. As is seen in Fig. 1(a), for nanoparticles of $d=10~{}$nm diameter the Casimir pressure is repulsive and agglomeration leads to only minor quantitative defferences in the pressure values. Computations show that if one-half of particles were merged into clusters by two, the respective line in Fig. 1(a) would be sandwiched between the solid and dashed lines.

In Fig. 1(b) the computational results for the magnitude of the Casimir pressure are shown for magnetite nanoparticles of $d=20~{}$nm diameter. The solid, short-dashed and long-dashed lines are plotted for the cases when all nanoparticles are single, and when half of them is merged into clusters by two and by three, respectively. As is seen in Fig. 1(b), for larger nanoparticles the agglomeration not only changes the magnitude of the Casimir pressure significantly, but also leads to a qualitatively different picture by replacing a repulsion with an attraction when separation between the plates decreases. Thus, for $\kappa_{2}=0.5$ the Casimir force becomes attractive at $a<228~{}$nm (the short-dashed line) and for $\kappa_{3}=0.5$ at $a<640~{}$nm (the long-dashed line).

A profound effect of agglomeration on the Casimir pressure for sufficiently large nanoparticles finds simple physical explanation. The point is that the agglomeration does not influence on the dielectric permittivity of a ferrofluid, but makes an impact only on $\mu_{\rm ff}(0)$. For small $d$, this impact is also rather small and increases with increasing $d$ (see above). In the Lifshitz formula (1), all terms with $l\geq 1$ lead to an attraction, as well as the TE contribution to the term with $l=0$ (in the system under consideration the latter is not equal to zero only in the presence of magnetic properties of intervening liquid). As to repulsion, it is produced by the TM contribution to the term with $l=0$. With increasing $d$, the permeability $\mu_{\rm ff}(0)$ quickly increases, and the combined effect of the terms with $l\neq 0$ and the TE contribution to the term with $l=0$ causes a transition from repulsion to attraction at sufficiently short separations between the plates.

In computations of Fig. 1, an extrapolation of the optical data for Au to low frequencies by means of the plasma model was used. Now we repeat the same computations but using an extrapolation of the same data by means of the Drude model. The computational results for the Casimir pressure through a ferrofluid as functions of separation between the plates are shown in Fig. 2 by the top and bottom pairs of solid and dashed lines obtained for nanoparticles with $d=10~{}$nm and $d=20~{}$nm diameter, respectively. In each pair, the solid line is for single nanoparticles and the dashed line is for the case when half of them is merged into clusters by three. By contrast to Fig. 1, in Fig. 2 no qualitative effect caused by the agglomeration of nanoparticles is observed even for nanoparticles with $d=20~{}$nm diameter. This is physically explained by the fact that for typical values of $\mu_{\rm ff}(0)$ calculated above the TE contribution to the term of the Lifshitz formula with $l=0$ obtained using the Drude model is much less than that obtained using the plasma model.

For comparison purposes, we also consider the role of agglomeration of nanoparticles when the ferrofluid is confined between two similar SiO2 plates. The computational results for the magnitude of the (negative) Casimir pressure are presented in Fig. 3 by the top and bottom pairs of solid and dashed lines obtained for nanoparticles with $d=20~{}$nm and $d=10~{}$nm diameter, respectively. As above, the solid lines are for the case of single nanoparticles and the dashed lines refer to the case when half of them is merged into clusters by three. As is seen in Fig. 3, for two similar plates agglomeration of nanoparticles results in only quantitative differences in the magnitudes of the Casimir pressure. This is explained by the fact that for similar materials of the plates all contributions to the Lifshitz formula (1) add to the effect of attraction.

IV Dependence on the extent of agglomeration and diameter of magnetic

nanoparticles

As found in the previous section, the Casimir pressure between metallic and dielectric plates is subject to change of sign from repulsion to attraction under an impact of agglomeration of nanoparticles. This effect occurs when the low-frequency dielectric response of a metal (Au in our case) is described by the experimentally consistent plasma model (see Sec. I). Here, we investigate the effect of sign change in relation to the share of nanoparticles, which are merged into clusters, and nanoparticle diameter.

We again consider the water-based ferrofluid containing 5% volume fraction of magnetite nanoparticles sandwiched between Au and SiO2 plates. For convenience in graphical displays, we compute the ratio of the Casimir pressures

[TABLE]

where ${P(\kappa_{k})}$ is the Casimir pressure through a ferrofluid where the share $\kappa_{k}$ of all particles is merged into clusters by $k$ particles. Below, computations of the quantity (8) are made for $k=2$ and 3.

The computational results for ${\cal P}$ as a function of the share of particles merged into clusters $\kappa=\kappa_{2}$ or $\kappa_{3}$ are shown in Fig. 4(a) for the plates at $a=200~{}$nm separation and in Fig. 4(b) for $a=500~{}$nm. In each of these figures, the top and bottom pairs of lines are computed for nanoparticle diameters $d=10$ and 20 nm, respectively. The short- and long-dashed lines label the cases when the share $\kappa$ of all nanoparticles is merged into clusters by two and three particles, respectively.

As is seen in Figs. 4(a) and 4(b), for nanoparticles of $d=10~{}$nm diameter the effect of sign change does not occur no matter what is the share of particles merged into clusters. This generalizes the respective results obtained in Sec. III. From Figs. 4(a) and 4(b) it is also seen that if clusters contain lesser number of particles the effect of sign change occurs when larger share of all particles is merged into clusters. Thus, in Fig. 4(a) the sign change takes place for $\kappa_{2}=0.43$ (the short-dashed line) and $\kappa_{3}=0.23$ (the long-dashed line). Furthermore, from the comparison of Fig. 4(a) and Fig. 4(b), one can conclude that at larger separation between the plates the effect of sign change comes for larger shares of particles merged into respective clusters (at $a=500~{}$nm we have $\kappa_{2}=0.87$ and $\kappa_{3}=0.42$ for clusters consisting of two and three particles, respectively).

Finally, we consider how the quantity ${\cal P}$, defined in Eq. (8), depends on the diameter $d$ of single nanoparticles. For this purpose, we compute ${\cal P}$ as a function of $d$ under different assumptions concerning the share of merged particles, separation between the plates and the character of clusters. The computational results for ${\cal P}$ as a function of $d$ at $a=200~{}$nm are shown in Figs. 5(a) and 5(b) for the cases when some share of magnetic nanoparticles is merged into clusters by two and three, respectively. In each figure, the solid lines counted from top to bottom are computed for the share of merged particles $\kappa_{2}$ and $\kappa_{3}$ equal to 0.1, 0.3, 0.5, 0.7, and 0.9, respectively.

As is seen in Fig. 5(a), the effect of sign change occurs only for nanoparticles of sufficiently large diameter. According to this figure, the sign of the Casimir pressure changes for $d>18.1$, 18.7, and 19.6 nm if the shares $\kappa_{2}=0.9$, 0.7, and 0.5 of all nanoparticles are merged into clusters by two. Note that we do not consider nanoparticles with more than 20 nm diameter because otherwise it would be necessary to increase the minimum separation distance between the plates. Comparing Figs. 5(a) and 5(b), one can conclude that if some share of magnetic nanoparticles is merged into the larger clusters by three particles each the effect of sign change occurs starting from lesser nanoparticle diameters. Thus, from Fig. 5(b) we find that the Casimir pressure changes its sign from repulsion to attraction for $d>15.9$, 16.7, 17.8, and 19.2 nm if the following respective shares of all nanoparticles are merged into clusters by three: $\kappa_{2}=0.9$, 0.7, 0.5, and 0.3. This opens opportunities to control the sign of the Casimir pressure through a ferrofluid by choosing an appropriate nanoparticle diameter.

V Conclusions and discussion

In the foregoing, we have investigated an impact of agglomeration of magnetic nanoparticles on the Casimir pressure in the configuration of a ferrofluid sandwiched between two material plates. The nanoparticles should be ferromagnetic and their size is restricted by the size of one domain for the material under consideration. To determine the role of agglomeration, one needs to know the specific magnetic properties of nanoparticles. The most important one is the initial magnetic susceptibility of a single nanoparticle which is usually different from that determined for bulk material. Both cases of two similar (dielectric) and dissimilar (one dielectric and another one metallic) plates were considered. Computations of the Casimir pressure through a ferrofluid have been performed at room temperature using the Lifshitz theory for Au and SiO2 plates and the water-based ferrofluid containing 5% fraction of magnetite nanoparticles with different diameters. It was assumed that some share of this nanoparticles is merged into clusters containing two or three particles. The dielectric response of Au was described using the measured optical data extrapolated to low frequencies by means of either the lossless plasma or the lossy Drude model.

According to our results, for a ferrofluid sandwiched between two dielectric plates, as well as between dielectric and metallic plates if the optical data of the latter are extrapolated by means of the Drude model, an agglomeration of magnetic nanoparticles makes only a quantitative impact on the Casimir pressure depending on nanoparticle diameter, but retaining the pressure sign unchanged.

A completely different type of situation occurs for a ferrofluid sandwiched between one metallic and one dielectric plates when the low-frequency response of a metal (Au) is described by the plasma model. In this case, for a sufficiently large nanoparticle diameter, the agglomeration results in the sign change of the Casimir pressure from repulsive to attractive. As an example, for magnetite nanoparticles of 20 nm diameter, half of which is merged into clusters by three, the pressure becomes attractive at separations between the plates exceeding 640 nm. It should be taken into account that numerous experiments of Casimir physics are consistent with the theoretical predictions using an extrapolation by means of the plasma model and exclude with certainty the theoretical results obtained with the help of the Drude model (see Sec. I). Because of this, the change of the pressure sign as a result of agglomeration of nanoparticles can be considered as a quantitative effect which merits detailed consideration.

Based on this conclusion, we have investigated an impact of agglomeration of nanoparticles on the change of sign of the Casimir pressure when the share of nanoparticles merged into clusters of different size and nanoparticle diameter vary continuously. It was found that the effect of sign change under an impact of agglomeration becomes more pronounced at shorher separations between the plates, for larger clusters and arises only for sufficiently large nanoparticle diameter. To take one example, if 70% of all nanoparticles are merged into clusters by three, the Casimir pressure between Au and SiO2 plates at a distance 200 nm changes its sign if nanoparticle diameter exceeds 16.8 nm. The proposed effects can be observed experimentally in microdevices exploiting ferrofluids 14 ; 15 ; 16 ; 17 and in measurements of the Casimir force through a liquid layer liq1 ; liq2 when the latter possesses magnetic properties. In doing so, it is simple to generalize all the above results for the plates made of any materials if the dielectric properties of these materials are available.

To conclude, the obtained results can be used to predict the effect of agglomeration of magnetic nanoparticles on the Casimir pressure in microdevices exploiting ferrofluids for their functionality. The revealed difference regarding the predicted effect of sign change when using two alternative extrapolations of the optical data of metals to low frequencies may be of interest for further investigation of the Casimir puzzle which as yet awaits for its resolution.

Acknowledgments

V. M. M. was partially funded by the Russian Foundation for Basic Research, Grant No. 19-02-00453 A. His work was also partically supported by the Russian Government Program of Competitive Growth of Kazan Federal University.