The structures of various systems, such as colloidal suspensions including electrorheological fluids,^[1–3] magnetorheological suspensions,^[4–7] ferrofluids,^[8–11] living cells suspensions,^[12] and so on, are very sensitive to the change of interparticle forces. Unfortunately, some shearing flow that is inevitable in experiments and natural environments can cause the rotation of the colloidal particles,^[13–23] thus reducing the interparticle force to some extent. An interaction between a rotating electric field and suspended particles or cells can also lead to a rotational motion of the particles. This phenomenon, known as electrorotation, has been increasingly employed as a sensitive tool for noninvasive studies of a broad variety of microparticles, ranging from living cells to nanowires, seeds as well as synthetic materials.^[24–29] In addition, insulating particles suspended in a liquid with low conductivity can also rotate in the presence of an electric field, which leads to the so-called negative effective viscosity effect.^[30,31]

The dielectric relaxation can affect the inter-particle force, and even the property of the semi-conducting materials.^[32–34] So far, the relaxation of polarized charges in dielectric particles has been investigated within the well-known Maxwell–Wagner relaxation theory. However, it was shown that the Maxwell–Wagner relaxation time is valid for an isolated stationary dielectric sphere or infinite large plank, whose dielectric responses are represented by a dielectric constant and electric responses by a conductivity.^[35]

When a dielectric particle starts to rotate, the polarization charges tend to deviate from the original position due to the rotation of the dielectric particle. On the other hand, the charges tend to return to the original position due to the directed external field. Thus, at dynamic equilibrium, the two competitive effects can change the polarization charge distribution. As a result, this change affects the interaction between the two particles one of which is rotating. The same process also happens when a metallic particle substitutes for the dielectric one. However, for the extremely high conductivity of the metallic material, it is believed that free charges on the surface of the metallic particle will not cause an obvious force decrease in a low rotation speed. On the contrary, Tao and Lan^[36] experimentally reported that the rotation of a metallic particle can reduce significantly the attracting interparticle force between the rotating metallic particle and a stationary dielectric or metallic one in argon gas. To the best of our knowledge, this was the first direct experiment conducted on the interparticle force between two touching rotating particles. Tan et al.^[37] found that the local electric field between two metallic particles weakens as one rotates faster, which can be used to qualitatively explain the phenomenon of Tao and Lan’s work.^[36] However, a quantitative explanation is still lacking for explaining Tao and Lan’s observations,^[36] which is important for practice. To solve this, we believe that the relaxation processes in dynamic metallic particles should be understood carefully, which are also expected to receive a broad interest in various fields related to dynamics in (soft) condensed matter. Reference [38] described a method to solve similar systems. However, only dielectric particles were under their consideration. In this work, we improve the method in Ref. [38]. Based on a first-principles approach, we establish a relaxation theory for dynamic metallic particles in the presence of an external AC field, thus being called AC relaxation theory.

This paper is organized as follows. In Section 2, we establish an AC relaxation theory for dynamic metallic particles on the basis of a first-principles approach. In Section 3, the comparison between theory and experiment is presented, which is followed by a conclusion in Section 4.

In the experiment,^[36] two spherical particles of diameter 19.06 mm were arranged vertically inside a capacitor composed of two horizontal electrodes. One of the particles, which could move up and down to adjust the gap between particles, was connected to a motor and was able to rotate around the horizontal axis. Another one rested on a microbalance. The whole device was placed inside a glove box, which was filled with dry argon gas. When there was no electric field, the reading on the microbalance was just the weight of the stationary particle. When an electric field was applied, the induced attracting force between the rotating metallic particle (cooper) and the stationary metallic particle (cooper) or the dielectric one (polyamide) reduced the reading on the microbalance. Thus, the attracting force between the two particles was able to be determined.

Now we are in a position to compare theory and experiment. For this purpose, consistent with the experiment,^[36] the diameters of particles are also set to be 19.06 mm. The distance between the centers of the two particles, R, which is used in our calculations, can be got by R = gap + 19.06 mm, where the gap is the surface-to-surface separation between the particles (see Fig. 1). The values of gap are acquired from the experiment.^[36] By using Eq. (20), we obtain a fitting of Tao and Lan’s experimental data of a rotating metallic particle (made of copper of dielectric constant 6) and a stationary metallic one (Fig. 2) or a stationary dielectric one (Fig. 3).^[36] For the fitting, we first fit the case of R = 19.441 mm (or gap = 0.381 mm) by choosing appropriate relaxation times, τ_R = 4.342 × 10⁻⁴ s and τ_I = 7.3 × 10⁻⁸ s for the metallic–metallic system (Fig. 2) and τ_R = 4.352 × 10⁻⁴ s and τ_I = 3.6 × 10⁻¹¹ s for the metallic–dielectric system (Fig. 3), according to which the other groups of experimental data are also fitted well. As ω increases, the attracting force between the rotating particle and the stationary one decreases. However, such a reduction of the force weakens with gap enlarging.^[36] Here it should be remarked that the relaxation time τ_R used for this fitting is reasonably comparable to that of the particles suspended in electrorheological fluids.^[41]

Fig. 2.

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Fig. 2. Fitting of the experimental data of a rotating metallic particle (copper) and a stationary metallic one (copper) in an AC electric field with frequency ω0 = 300 Hz. The symbols denote the experimental data which are extracted from Ref. [36]. From upper to bottom: R = 19.314 mm (or gap = 0.254 mm) and E0 = 100 V/mm; R = 19.441 mm (or gap = 0.381 mm) and E0 = 100 V/mm; R = 19.314 mm (or gap = 0.254 mm) and E0 = 80 V/mm; R = 19.568 mm (or gap = 0.508 mm) and E0 = 100 V/mm; R = 19.441 mm (or gap = 0.381 mm) and E0 = 80 V/mm; R = 19.568 mm (or gap = 0.508 mm) and E0 = 80 V/mm. Similarly, according to our theory [Eq. (20)], the dotted line is plotted by choosing appropriate relaxation times, τR = 4.342 × 10−4 s and τI = 7.3 × 10−8 s, for the fitting of the case of R = 19.441 mm (or gap = 0.381 mm) and E0 = 80 V/mm. The two values, τR = 4.342 × 10−4 s and τI = 7.3 × 10−8 s, are then used for fitting the other five groups of the experimental data (dashed lines). In this figure, the unit of ω, “rpm”, stands for “rotations per minute”.

Fig. 3.

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Fig. 3. Fitting of the experimental data of a rotating metallic particle (copper) and a stationary dielectric one (polyamide) in an AC electric field with frequency ω0 = 300 Hz. The symbols denote the experimental data which are extracted from Ref. [36]. From upper to bottom: R = 19.314 mm (or gap = 0.254 mm) and E0 = 100 V/mm; R = 19.441 mm (or gap = 0.381 mm) and E0 = 100 V/mm; R = 19.441 mm (or gap = 0.381 mm) and E0 = 80 V/mm; R = 19.568 mm (or gap = 0.508 mm) and E0 = 80 V/mm. According to our theory [Eq. (20)], the dotted line is plotted by choosing appropriate relaxation times, τR = 4.352 × 10−4 s and τI = 3.6 × 10−11 s, for the fitting of the case of R = 19.441 mm (or gap = 0.381 mm) and E0 = 80 V/mm. The two relaxation times, τR = 4.352 × 10−4 s and τI = 3.6 × 10−11 s, are then used for fitting the other three groups of the experimental data (dashed lines).

For the rotating metallic sphere, the well-known Maxwell–Wagner relaxation time^[35,42] predicts , where ε_h (or σ_h) means the real part of the complex dielectric constant (or conductivity) of the host argon gas. According to the experiments by Tao and Lan,^[36] we have and σ_h ≈ 0, thus yielding τ_MW = 1.22 × 10⁻¹⁸ s, which is much smaller than both of the above-mentioned fitting parameters, τ_R and τ_I. It is worth noting that charge imbalances could relax in a period of time, which is comparable to the characteristic collision time (τ′) of electrons in conductors,^[43] . Here m, n, and e are the electron mass, average electron density, and electron charge, respectively. It is because the time dependence is so rapid that the inertia of electrons must be taken into account. For copper, one obtained τ′ ≈ 2 × 10⁻¹⁴ s.^[43] In view of the precision of the experiments reported in Ref. [36], we expect that the collision time (τ′) should be comparable to the relaxation time τ_I. Then we can conclude that neither the collision process described by τ′ nor the leaky dielectrics process characterized by τ_MW^[35,42] serves as the source of the relaxation time τ_R.

According to our results, the relaxation time τ_R is generally much longer than τ_I. In fact, the τ_R sources from polarized charges (described by ) under the local field of touching particles, as mentioned in Subsection 2.2. Because of the short separation between the particles, a strong local field is formed, and shifts the property of electrons in metallic particles. When a particle is rotating, the local field should change greatly.^[37] Thus, the interaction force decreases.^[36] In general, the change of local field of touching rotating metallic particles is likely to bring such a long relaxation time ∼ 10⁻⁴ s, due to the effect of multipolar interactions. As far as we know, theories of this process based on the first principles have not yet been found for predicting τ_R directly. In this situation, our method is a possible way to obtain τ_R of metallic particles by comparing with experiment data. On the other hand, the relaxation time τ_I, which originates from free charges (corresponding to ), is much shorter. The collision time as mentioned above can be used to understand the general vanishing value of τ_I. We can conclude that τ_I refers to the inertia of electrons in conductors. Since τ_R is much longer, we believe that the dielectric relaxation process plays a key role in the force reduction in the experiment.

When displaying the theoretical results in Figs. 2 and 3, we have assumed that the relaxation times of the touching rotating particle are unchanged, even if the separation between the two particles or the external electric field is changed. In fact, these factors also have an effect on the relaxation times, τ_R and τ_I. Therefore, we try to use Eq. (20) to fit the experimental data, by using more accurate relaxation times, as listed in Tables 1 and 2 explicitly. It is found that the relaxation times τ_R always take the values with the same order of magnitude for the current systems. The stability of the values of τ_R is an evidence of the validity of our fitting presented in Figs. 2 and 3. For the relaxation time τ_I ranges over several orders of magnitude, we think that either the limit of experimental precision or the inaccuracy of the fitting process is a possible reason. As mentioned above, the relaxation time τ_I is vanishing (compared with τ_R). Thus, tiny fluctuations of experimental data may lead to a great change of the fitting values of τ_I.

Table 1.

Table 1.

Table 1. Relaxation times, τR and τI, for obtaining good fitting with experimental data of Fig. 2. . F0/10−5 N E0/V·mm−1 R/mm gap/mm τR/10−4 s τI/10−4 s 76.0 100 19.314 0.254 10.159 7.1 × 10−6 53.5 100 19.441 0.381 3.863 3.9 × 10−7 37.1 100 19.568 0.508 2.266 1.7 × 10−4 42.0 80 19.314 0.254 4.405 2 × 10−10 33.0 80 19.441 0.381 4.342 7.3 × 10−4 22.0 80 19.568 0.508 2.646 2.1 × 10−6

Table 1. Relaxation times, τR and τI, for obtaining good fitting with experimental data of Fig. 2. .

Table 2.

Table 2.

Table 2. Relaxation times, τR and τI, for obtaining good fitting with experimental data of Fig. 3. . F0/10−5 N E0/V·mm−1 R/mm gap/mm τR/10−4 s τI/10−4 s 4.3 100 19.314 0.254 6.171 2 × 10−10 3.9 100 19.441 0.381 4.855 1.7 × 10−3 2.4 80 19.441 0.381 4.352 3.6 × 10−7 2.2 80 19.568 0.508 3.710 2 × 10−10

Table 2. Relaxation times, τR and τI, for obtaining good fitting with experimental data of Fig. 3. .

Here some comments are in order. Our method introduced the conductivity () to metallic particles. This can be understood from the charge conservation law

Here, the current can be written as Ohm’s law , while, according to the Maxwell equations, free charge . Such relations naturally give birth to Eq. (6), which contains both and .

To sum up, based on a first-principles approach, we have established an AC relaxation theory for dynamic metallic particles. An imaginary part has been introduced to the method established in Ref. [38] because the particle is conducting. The relaxation times were obtained according to the fitting with the experimental data. Good agreement between theory and experiment has shown that the relaxation processes of surface-polarized and free charges yield the reduction in the attracting interparticle forces between a rotating metallic particle and a stationary metallic or dielectric one. Our theory can help to determine the relaxation time of dynamic metallic particles in many fields.