Trapping of a Conducting Nanoparticle by Long-Range Surface Forces

We study theoretically the trapping of a nanoparticle by a rotational motion around another nanoparticle caused by surface forces comprising electrostatic and Casimir forces. We introduce a local power exponent of the surface force as a function of the separation distance and show that it strongly depends on the electric charges carried by the nanoparticles. The stability of circular motions depends mainly on the local power exponent, and if the electric charges are not zero and the mass of the nanoparticle is small, then the stable tapping is possible in the Earth’s gravity field. [DOI: 10.1380/ejssnt.2011.301]


I. INTRODUCTION
Many scientific insights covering a wide scale range have been brought to light by investigating circular motions.We focus on circular motions on the mesoscopic scale, in which quantum long-range forces play an important role [1].In particular, we consider the Casimir force [2,3] as a quantum long-range force acting on conducting nanoparticles.In contrast to elemental interactions such as Coulomb interactions between point charges and gravitational interactions, surface interactions do not obey an inverse-square law in most cases.In addition, the strength of the surface force interaction often diverges as the surfaces approach.Thus, the presence of stable motions caused by surface forces is not obvious.
The dependence of surface forces on the separation distance is an essential factor for the stability of circular motion.The electrostatic force and the Casimir force between conducting objects depends on their shapes.We consider the interactions between two perfectly conductive spheres.The Casimir force is an entirely quantum force, and it acts on the sphere even if the electric charges of spheres are zero.For small surface separations, the strength of the Casimir force between perfectly conductive spheres is inversely proportional to the cube of the surface separation distance [3], and it is the major contributor to the force between spheres.On the other hand, for large separations, the dominant force is the electrostatic force.
We have used the coexistence state of the Casimir force with an electrostatic force to measure the Casimir force precisely [4,5].In addition, it was used also in actuators [6] and sensors [7] in micro electro mechanical systems [8], and in these applications, the study of the motions induced by an electrostatic force and the Casimir force is indispensable.We show that the Casimir effect on the interactions between nanoparticles is significantly large at small separations if the electric charge of the nanoparticle is small, and it prevents the stable circular motion of the nanoparticle.However, the stable circular motion is possible, and it changes drastically when even a single electron is transported into the nanoparticle.

II. SURFACE FORCES BETWEEN CONDUCTING NANOPARTICLES
We consider the motion of the conducting sphere labeled 1 of radius R and mass m around a conducting sphere labeled 2 of the same radius.The center of sphere 2 is fixed at the origin and the position of sphere 1 is indicated by the distance from the origin to the center of the sphere r and the angle φ as shown in Fig. 1.The separation distance between the surfaces, a, is given by r − 2R.Spheres 1 and 2 carry electric charges Q 1 and Q 2 , respectively, and they are fixed if the transfer of charges to sphere 2 is blocked.
We assume that the spheres attract each other by the surface force F s (a), which consists of electrostatic force F E (a) and the Casimir force F C (a).We derive the electrostatic force between conducting spheres as follows: The electric potentials of spheres 1 and 2 are considered to be ϕ 1 and ϕ 2 , respectively.The relation between the electric charges and the electric potentials are expressed using the capacitance coefficients matrix C = [c i,j ] i=1,2,j=1,2 as where C −1 is the inverse of matrix C. All possible electrostatic configurations of the two spheres are described by eq. ( 1), and the electrostatic force F E is calculated by −∂U E /∂a.If the radii of the spheres are the same, C 11 and C 12 are equal to C 22 and C 21 , respectively.The exact expressions of the capacitance coefficients matrix [9] are given by where u ≡ cosh −1 [(r/2R) 2 − 1] and ϵ 0 is the dielectric constant of vacuum.If the spheres carry equal and opposite charges +Q and −Q, then the asymptotic forms of the electrostatic force are ( Thus, if the Casimir force did not exist, the local power exponent of the surface force which is defined by β = −d ln(−F S (a))/d ln a changes from −1 to −2, which corresponds to the exponent of the Coulomb force.We note that the local power exponent is independent of the electric charge in the absence of the Casimir force.
Although the Casimir force between the two perfectly conductive spheres has not been calculated exactly, the asymptotic forms are given by where the coefficients c n are given by Emig et al. up to n = 9 [10][11][12].To express the Casimir force between spheres using a compact function, we applied the Padé approximant [13] of order [7/8] to the series expansion for long distances so as not to contradict the asymptotic behavior occurring at short distances.If the electrostatic force does not exist, the local power exponent changes from −3 to −8 as the separation increases, which corresponds to the exponent of the Casimir-Polder force [14].
Figure 2 shows the dependence of the local power exponent of the surface force between 10-nm-radius spheres that carry equal and opposite charges Q and −Q on the separation distance.We use the absolute value of electric charge of an electron |e| = 1.602 × 10 −19 C as the unit of Q.For small separations, the Casimir force contributes dominantly to the surface force.Thus, the local power exponent decreases from −3 as the separation increases at small separations, but it increases as the separation increases further because of decrease in contributions of the Casimir force.For large separations, the electrostatic force is major contributor to the surface force and the power exponent decreases again and approaches −2.The inset in Fig. 3 shows the results of the Padé approximant for the ratio of the Casimir energy between two perfectly conductive spheres to the energy calculated using the proximity force approximation E PFA = −(π 3 /1440)ℏcR/a 2 , which is good agreement with the numerical results calculated by Eming et al. [10].

III. STABLE CONDITIONS OF ROTATIONAL MOTIONS
To determine the stable rotational motion conditions, we consider the following effective potential: ( Here the first term is the approximate surface potential at radius r with the local power exponent −α = β + 1 and the second term is the centrifugal potential where L denotes the angular momentum of sphere 1.We consider the effect of gravity on the motion later.is given by W ′ (r) = 0 and W ′′ (r) > 0. These conditions yield constraints α < 2 and From these constraints, we find that the stable rotational motion cannot be realized only by the Casimir force in the considered system, because the local power exponent of the Casimir energy between perfectly conductive spheres is always smaller than −2.The minimum radius of the stable circular motion is given by the solution of the equation This equation only includes the radius of the sphere and the local power exponent as the parameters.Therefore, it is independent of the angular momentum or coefficient of potential C s .The region of stable circular motion of a sphere of a radius with R = 10 nm is the shaded region in Fig. 2. The effective surface potential differs significantly from the effective potential comprising Coulomb potential between point charges the centrifugal potential, as it negatively diverges at r = 2R.Thus, the maximum value of the effective surface potential is finite.In contrast, if α < 2, the maximum of the effective Coulomb potential is positively infinite because the centrifugal potential diverges at r = 0. We consider the depth of the effective potential ∆U , which is defined as the minimum of the difference between the local maximum and the global minimum of the effective potential.Figure 3 shows the depth of the effective potential normalized by C s /(2R) α as a function of ratio γ ≡ (|L| 2 /2m)/(C s /(2R) α ).
Stable circular motion is allowed if the depth of the effective potential is positive.Thus, the value of γ must be larger than the critical values at which the effective potential is zero.Near the critical value, the local maximum of the effective potential is smaller than zero, as shown by the bottom line in the inset of Fig. 3, and the difference between the maximum and the minimum values of the effective potential is small.When the value of γ increases, the maximum value increases more rapidly than the minimum value.Thus, the depth of the effective potential increases.However, when values of γ increase further, the local maximum value is larger than zero, leading to a decrease in the depth of the effective potential.

IV. CIRCULAR MOTIONS OF A NANOPARTICLE
We now consider the dynamics of sphere 1.The equation of motion is given by where g is the gravitational acceleration on Earth.In contrast to an electron in an atom, the mass of a nanoparticle is large.For instance, the mass of a 10-nm-radius gold sphere is approximately 8 × 10 −20 kg, which corresponds to 9 × 10 10 electron mass.Thus, we examine whether the circular motion considered above is stable on Earth.Let us assume that sphere 1 rotates along a circle of radius R 0 with an angular velocity Ω 0 in the absence of gravity.If the effect of gravity on the motion is sufficiently small, the radius and the rotation angle can be written as r(t) = R 0 + R(t) and φ(t) = Ω 0 t + Ω(t), where R(t) ≪ R 0 and Ω(t) ≪ Ω 0 .If the surface force at radius r can be expressed by F s = −αC s /(r − 2R) α+1 and the second order in the perturbation is negligibly small, the differential equation of R(t) and φ(t) is given by The solution for the radius under the initial condition that R = Ṙ = 0 and Ω = Ω = 0 is given by where Here, the value of Ω 1 must be a real number.Thus, we have the first constraint R 0 > 6R/(2 − α).On the other hand, the condition The surface forces decrease as the separation increases.Thus, the second constraint is not satisfied at large separations.In addition, Ω 1 converges to zero in the limit R 0 → 6R/(2 − α), therefore, the second constraint is not satisfied near R 0 = 6R/(2 − α).Accordingly, the stable circular motion under the gravitational field is allowed only for a limited range of radius satisfying these two conditions.This is possible.For instance, the righthand side of the last constraint is evaluated as 9.1×10 −12 N for R = 10 nm, Q 1 = −Q 2 = 10, α = 0.377, and R 0 = 60 nm.On the other hand, the gravity force acting on a 10-nm-radius gold nanoparticle is approximately mg = 7.8×10 −19 N and it is much smaller than the upper limit.
Unlike an atom, the electric charge of sphere 2 can be changed by attaching an electrode to it, and the change in the electric charge may be measured precisely using nonelectric devices such as a single electron transistor [15].Hence we simulate the change of the orbital motion by decreasing the electric charge of sphere 2 from −10 over a time of 10µs. Figure 4 shows the change in orbital motion and the fluctuation of the radius for Q = −10, −9 and −8 for the initial radius R 0 = 60 nm and the initial angular velocity Ω 0 = 3.67 × 10 7 rad/s.The electric charge of sphere 1 is fixed to Q 1 = +10.By decreasing the electric charge, the local minimum point of the effective potential shifts to the surface of sphere 2 and the depth of the potential barrier decreases.Accordingly, the radius of the rotation decreases and the fluctuation of the radius increases.When the electrical charge changes from −8 to −7, sphere 1 overcomes the potential barrier and contacts sphere 2.
We here consider how to generate highly charged par-ticles.We introduce two methods: the first is based on photoionization, and the second is realized by the attachment of a highly charged ion such as O 5+ and Ar 8+ to a neutral particle.Näher et al. have succeed to generate highly charged sodium clusters Na z+ n with 2 ≤ z ≤ 14 by using two different lasers [16].In this method, Na clusters is transported by a cold helium gas stream into a high vacuum chamber and is photoionized with a dye laser pulse at a wavelength of 400 nm.By the first laser, Na clusters are singly charged and can be accelerated in an electrical field.Then Na cluster ion beam is multiply ionized by an excimer laser.Finally, the ionized clusters are selected by using time-of-flight mass spectrometer.Chandezon et al. have succeed to generate charged sodium clusters up to 6 by bombarding free sodium clusters with atomic ion beams [17].The difference between two methods is the temperature of the highly charged clusters.The temperature of the cluster formed by the bombardment is lower than that by the photoionization.
Although the above-mentioned methods was applied to Na clusters, whose size is smaller than the nanoparticle considerd in the present study, both methods may be available to generate highly charged nanoparticles.We note that if the excess electric charge is much large, the charged cluster is unstable due to disrupting effect of Coulomb repulsion.However, for a cluster size n, allowed charge excess increases by a power law n 2/3 .Thus, large nanoparticles can hold many excess charges than small clusters.Consequently, the charged particles in the ion beams can be trapped by the other nanoparticle with the opposite electric charge if the impact parameter and the projectile velocity are adjusted to satisfy the condition (6).

V. CONCLUSIONS
Our numerical results show that stable traping of a perfectly conductive nanoparticle can be realized on Earth.The examined nanoparticle system is similar to an atom, hence, we regard the combined state of the nanoparticle as an artificial atom.However, the most distinguishing property is that the Casimir force prevents the stable circular motion of the nanoparticle.Since the Casimir force increases rapidly as the separation distance decreases, the Casimir force cannot be balanced by the centrifugal force.This implies that the stabilized circular motion by the presence of the electrostatic force is still sensitive to small changes in electric charges if the electrostatic force is weak.Even though the quantum fluctuation of the nanoparticle is small because of its large mass, if the electric charge of sphere 2 changes probabilistically by quantum effects, the stability of the motion of sphere 1 must be considered in the framework of quantum mechanics.

FIG. 1 :
FIG.1: Configuration of the conducting spheres.Sphere 1 rotates around sphere 2 that is fixed at the origin in the y − z plane.Spheres 1 and 2 carry electric charges Q1 and Q2, respectively.

FIG. 2 :
FIG.2: Dependence of the local power exponent of the surface forces (electrostatic force and the Casimir forces) on the electric charge Q1 = −Q2 = Q for perfectly conductive spheres of 10 nm radius.The shaded region indicates the region where the stable circular motions are allowed.Inset: Normalized Casimir energy between perfectly conductive spheres as a function of the ratio of the sphere radius to the distance between the centers of the spheres.

FIG. 4 :
FIG.4: Change in the orbital motion and the radius of sphere 1 (radius 10 nm) by decreasing the electric charge of sphere 2 from Q2 = −10 to −8 in the period 10µs.Solid circles indicate sphere 2. The time is reset to zero each time the electric charge is decreased.