We apply an exact analytical model to demonstrate that multiple particle systems can bind in nontouching, minimum energy equilibrium configurations. It is predicted that such systems may self-assemble due entirely to electrostatic interactions without the aid of external forces. We demonstrate the potential energy equilibria for ternary systems where the effective dipoles cancel and the anomalous electrostatic forces are described by interactions between higher order multipoles.

Mixtures of charged microscopic particles can self-assemble into larger material systems. Like-charged particle interactions in colloidal suspensions have been experimentally studied for two decades and can exhibit nonlinear behavior and anomalous forces.1–7 Theoretical models have been advanced to explain the phenomenon of like charge attraction.8–15 However, anomalous repulsive forces in oppositely charged particle systems, which can occur in inverted dielectric systems (i.e., when the submerging fluid dielectric constant is larger than that of the particles), have not received such attention.16 Understanding such forces is critical to the design of novel, microscale materials and tunable surfaces. In particular, attractive van der Waals forces could be mitigated and lattice tunability achieved by nontouching minimum energy particle systems dominated by electrostatic forces.

Critical to engineering advanced tunable materials using electrostatic interactions is to determine if nontouching, stable equilibrium between more than two particles can be achieved in inverted systems. Previously, an approximate description of binary systems attributed the anomalous forces to the monopole-dipole interactions in inverted dielectric systems.16 Oppositely charged monopoles and the induced dipoles are attractive in nature. However, the monopole-dipole interactions produce repulsive forces in inverted dielectric systems, which can dominate at short distances. The balance between the attractive and repulsive forces can produce nontouching equilibrium configurations for binary systems of dielectric particles in fluids with stronger dielectric response. In ternary and larger systems, it is expected that the induced dipoles will effectively cancel, and the possibility of nontouching equilibrium comes into question.

In this correspondence, we apply an analytical solution of Maxwell’s equations to demonstrate that multiple spherical particles can bind in nontouching, minimum energy equilibrium configurations even when the induced dipoles are canceled by the interactions between multiple particles. In many such systems, the submerging fluid often includes charge carriers due to electrolytes or dissociation of the submerging fluid. Such fluids will screen electrostatic interactions. Even pure water has a Debye length (i.e., the length at which Coulombic effects dissipate to e1) on the order of 1μm.17,18 As with previous similar modeling studies of long-range forces, we limit our systems to length scales much smaller than the Debye length of weakly polarizable materials.19 Although the total electromagnetic and induced material stresses in the surrounding fluid is responsible for the observable motion and equilibria of colloidal spheres,20 the nonlinear force phenomena presented herein can be equivalently modeled in terms of effective particle multipoles. Such nonlinear interactions can produce stable potential wells for multiple particles in the vicinity of a larger, dissimilar particle.

The general model that we consider for demonstration is depicted in Fig. 1. Particle 1 at (x,y,z) and particle 3 at (x,y,z) are identical and negatively charged, and particle 2 at (0,0,0) is positively charged. The submerging background fluid has a larger dielectric constant than the particles (ϵb>ϵj) providing an inverted dielectric system.

FIG. 1.

Three particle equilibrium where particle 1 (top) and particle 3 (bottom) are negatively charged, and particle 2 (middle) is positively charged. The induced dipoles are reversed when the background fluid has a higher dielectric constant than the particles ϵb>ϵj, where j{1,2,3}.

FIG. 1.

Three particle equilibrium where particle 1 (top) and particle 3 (bottom) are negatively charged, and particle 2 (middle) is positively charged. The induced dipoles are reversed when the background fluid has a higher dielectric constant than the particles ϵb>ϵj, where j{1,2,3}.

Close modal

The field solution is obtained by analytical solution in spherical coordinates.11,14,16 The electrostatic field solution to the Laplace equation near each particle j is re-expanded near every other particle. External to particle j, the real electrostatic potential is

ψext(rj,θj,ϕj)=n=0Nm=nn[Bnm(j)rj(n+1)+Wnm(j)rjn]Pn|m|(cosθj)eimϕj,
(1)

where the complex mode coefficients give the magnitudes of the spherical modes near the jth particle due to the particle (Bnm(j)) and due to all other sources besides particle j (Wnm(j)).

The unique solution is determined by re-expansion of the static potential about each particle and application of the boundary conditions at the surface of each particle. The mode coefficient Wnm(j) includes contribution from all other particles via

Wnm(j)=kLnm(jk).
(2)

The coefficient Lnm(jk) provides the magnitude of mode {n,m}, which is scattered to particle j from particle k and can be written in the form of a transfer function. For a particle system far from interacting surfaces,

Lnm(jk)=ν=0Nμ=ννTnmνμ(jk)Bνμ(k),
(3)

where the Tnmνμ(jk) is actually a 6th-rank tensor representing the fields from particle k interacting with particle j. It is computed by re-expansion,

Tnmνμ(jk)=(1)nΓnmνμrkj(n+ν+1)Pn+ν|mμ|(cosθkj)ei(mμ)ϕkj.
(4)

Here, rkj,θkj,ϕkj represents the coordinates from the origin at particle j to the particle k. Finally, the mode interaction coefficient is given by

Γnmνμ=i|μm||μ||m|(n+m+νμ)!(nm+ν+μ)(n+m)!(νμ)!(nm)!(ν+μ)!.
(5)

The electric field in terms of the potential is

E¯=ψ=r^ψrθ^1rψθϕ^1rsinθψϕr^Er+θ^Eθ+ϕ^Eϕ.
(6)

The electric field components inside particle j are

Erint=n=0Nm=nnnAnm(j)rjn1Pn|m|(cosθj)eimϕj,
(7a)
Eθint=n=0Nm=nnAnm(j)rjn1θjPn|m|(cosθj)eimϕj,
(7b)
Eϕint=n=0Nm=nnimAnm(j)rjn1Pn|m|(cosθj)eimϕj.
(7c)

Likewise, the electric field components outside particle j are

Erext=n=0Nm=nn[(n+1)Bnm(j)rj(n+2)nWnm(j)rjn1]Pn|m|(cosθj)eimϕj,
(8a)
Eθext=n=0Nm=nn[Bnm(j)rj(n+2)+Wnm(j)rjn1]θjPn|m|(cosθj)eimϕj,
(8b)
Eϕext=n=0Nm=nnim[Bnm(j)rj(n+2)+Wnm(j)rjn1]Pn|m|(cosθj)eimϕj.
(8c)

The force components of the electrostatic field acting on each particle are computed by a summation, which includes the mode coefficients Bnm(j) and Wnm(j).21 The forces are verified via numerical integration of the Minkowski stress tensor around each particle.22 Further discussion of the force computation and distribution is subsequently provided herein.

The solution is exact as N. However, convergence to N=10 modes is generally sufficient for comparison with experimental results involving closely packed spheres.20 The modal expansion in (1) can be considered as a multipole expansion with the lowest modes representing the monopoles and dipoles of the particle. In reference to Eq. (1), the B00(j) term yields the well-known monopole potential for a uniform charge proportional to rj1. Likewise, the B10(j) term gives the dipole potential term proportional to rj2. Because of system symmetries, it is expected that higher multipoles must be responsible for any stable, nontouching equilibria.

For demonstration, we consider nonpolar particles with dielectric constant ϵ1,2,3=2.0 submerged in a weakly polar dielectric background with ϵb=6.0. Particles 1 and 3 have uniform surface charges of Q1,3=1fC and radii R1,3=80nm, and particle 2 at the origin has a uniform surface charge of Q2=+1fC and radius R2=320nm.

A one-dimensional system, where all three particles are on the z-axis as shown in Fig. 1, is considered first. Figure 2 illustrates the z-directed force and potential energy vs surface-to-surface separation distance for particle 1. Note that the force on particle 3 is equal and opposite to that on particle 1, and the force on particle 2 is zero due to symmetry. A nontouching equilibrium at the surface-to-surface separation distance of 46 nm is revealed as the solution converges for increasing N. As expected, the N=1 solution expansion is inadequate for predicting the equilibrium position indicating that the ternary, unlike the binary system,16 is dominated by higher order multipoles. Additionally, Fig. 2 demonstrates excellent convergence with as few as N=10 modes, in spite of recent concerns over the convergence of the re-expansion method applied herein.19 

FIG. 2.

(a) Force on the top particle 1 and (b) corresponding potential energy are plotted vs separation distance for the three particle configuration. The top particle 1 spans z, and the bottom particle 3 (not shown) spans z. The middle particle 2 is maintained at the origin. The forces and potentials are rotationally symmetric about the z axis, and the system is electrically inverted with dielectric constant of the particles (ϵ1=ϵ2=ϵ3=2) and the background fluid (ϵb=6). The uniformly distributed surface charges are qj=(1,1,1)fC, and the radii are Rj=(80,320,80)nm.

FIG. 2.

(a) Force on the top particle 1 and (b) corresponding potential energy are plotted vs separation distance for the three particle configuration. The top particle 1 spans z, and the bottom particle 3 (not shown) spans z. The middle particle 2 is maintained at the origin. The forces and potentials are rotationally symmetric about the z axis, and the system is electrically inverted with dielectric constant of the particles (ϵ1=ϵ2=ϵ3=2) and the background fluid (ϵb=6). The uniformly distributed surface charges are qj=(1,1,1)fC, and the radii are Rj=(80,320,80)nm.

Close modal

To study the three-dimensional equilibria behavior, the force and potential energy are computed for the two smaller particles in the xz plane. Figure 3 shows the force field and resulting potential energy for particle 1 at various points (x,z). Particle 2 is at the origin, and particle 3, not shown, is at (x,z). Because the three particles define a plane, the configuration considered is sufficient for visualizing the three-dimensional trapping potential of particle 1. The one-dimensional potential energy well depicted for particle 1 in Fig. 2 is reinforced by the three-dimensional well shown in Fig. 3, though the energy well is broader in the xy plane than in the z-direction. It should be pointed out that, because of symmetry, the potential well and associated force field are rotationally symmetric about the z-axis. The results demonstrate that a ternary system will form a chain with the solid angle of 180°. This results because the two smaller, like-charged particles repel as the solid angle decreases from this equilibrium position. Such preferred, self-assembled arrangements have been previously observed experimentally for similar, contacting chains of spherical particles.17,23 It should be noted that, while Demirörs et al. experimentally considered similar particle systems of comparable dielectric constants, the smaller size ratios of 2:1 to 3:1 and significantly larger overall scale would, in our calculations, result in touching ternary clusters as experimentally observed.17 

FIG. 3.

Force (arrows) and resulting potential energy (surface) in nJ on the top particle 1 is plotted as a function of position in the y=0 plane. The top particle 1 spans (x,z), and the bottom particle 3 (not shown) spans (x,z). The middle particle 2 is maintained at the origin. The system is electrically inverted with the dielectric constant of the particles (ϵ1=ϵ2=ϵ3=2) and the background fluid (ϵb=6). The uniformly distributed surface charges are qj=(1,1,1)fC, and the radii are Rj=(80,320,80)nm.

FIG. 3.

Force (arrows) and resulting potential energy (surface) in nJ on the top particle 1 is plotted as a function of position in the y=0 plane. The top particle 1 spans (x,z), and the bottom particle 3 (not shown) spans (x,z). The middle particle 2 is maintained at the origin. The system is electrically inverted with the dielectric constant of the particles (ϵ1=ϵ2=ϵ3=2) and the background fluid (ϵb=6). The uniformly distributed surface charges are qj=(1,1,1)fC, and the radii are Rj=(80,320,80)nm.

Close modal

The analytical results from the present study are presented in terms of interactions between effective particle multipoles (i.e., monopoles, dipoles, and higher order terms). The particle forces are calculated by closed surface integration of the Minkowski stress tensor T¯¯=12(D¯E¯)I¯¯D¯E¯ applied in the background submerging fluid region just outside of a particle where D¯=ϵbϵ0E¯. However, the Minkowski stress-energy-momentum (SEM) tensor is known to include some contributions from the materials.

In order to understand the physics of this system, it is necessary to consider the field-kinetic subsystem of electromagnetics, which separates the stress, energy, and momentum within the fields from that carried by the matter.24,25 For stationary electrostatic systems, the field-kinetic force density can be written as

f¯Fk=(P¯)E¯+ρE¯,
(9)

where ρ is the local free charge density, P¯=(ϵ1)ϵ0E¯ is the local polarization density, and ϵ is the local dielectric constant. Because the background fluid and particle materials are considered to be homogeneous, the force density in Eq. (9) is zero everywhere except at each particle surface. This is due to media discontinuities and the uniformly distributed surface charges. Equation (9) reduces to pressure distributions at each particle surface due to forces on bound charges and free charges given by the first and second terms, respectively. Therefore, a pressure difference exists between the submerging background fluid and the particle. Due to the nonzero polarization density P¯ of both the submerging background fluid P¯=(ϵb1)ϵ0E¯ and the particle P¯=(ϵp1)ϵ0E¯, the electrostatic field exerts pressure upon both the submerging background fluid and the particle, which can be individually identified by stress tensor integration approaches.26 The sum of these pressures integrated over the boundaries of the background-particle media interfaces results in the observable forces presented herein. For the system presented here, there is an equivalence in the observable force, which can be calculated as the total field-kinetic force exerted on the particles’ free and bound charges plus the electromagnetic force exerted on the bound charges of the submerging background fluid surrounding the particle. In other words, the force consists of a force on both the particle and the submerging fluid.27 

Several electromagnetic force equivalencies have been presented in the literature.28 For example, the sum of kinetic-field surface forces exerted on the particles and the background dielectric fluid can be calculated as the divergence of the Minkowski stress tensor as applied herein or, equivalently, by summing the force on bound and free charges resulting from an “effective” polarization density. The effective polarization density is P¯eff=ϵ0(ϵϵb)E¯, where ϵ=ϵb in the background submerging fluid yields P¯eff=0 and ϵ=ϵj in particle j yields P¯eff=ϵ0(ϵpϵb)<0. In this model description, the force on the background fluid is zero and the anomalous forces exerted on the particles result because of an inverted polarization density as depicted in Fig. 1 for the dipole terms. However, while this effective polarization density description is useful in modeling, it does not reasonably describe the physics of the system.

Multiple charged dielectric particles can bind in nontouching, minimum energy equilibrium configurations. This can be modeled by the anomalous interactions between the dielectric multipoles in inverted dielectric systems. The potential energy wells can exist for nontouching particle systems even when the induced dipoles are canceled by the interactions of multiple particles. The physics of the systems is best described by the combination of forces on the particles and the submerging fluid, which, in the case of homogenous materials, result in surface forces on both media. This distribution of surface pressures on the surfaces of the particles and the fluid boundaries with the particles can result in stable, nontouching equilibria for ternary systems. Such systems are proposed as the building blocks for the microscopic assembly of photonic surfaces.

Funding for this research was provided by the Center for Advanced Surface Engineering, under the National Science Foundation Grant No. IIA-1457888 and the Arkansas EPSCoR Program, ASSET III.

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