Influence of particle arrangement on the permittivity of an elastomeric composite

Due to their low cost, lighter weight, and superior hydrophobicity, electrically insulating rubber-like materials, such as silicone rubber, are often used as the dielectric medium in capacitors. A dielectric medium is characterized by its permittivity, which is measured using a capacitor. At steady state, the relative permittivity, i.e., the ratio of the capacitance for a dielectric medium to that for a vacuum,¹ is also called the static relative permittivity.² The properties of its individual constituents influence the bulk properties of a composite material. The addition of conductive filler particles into an elastomer matrix improves the bulk relative permittivity of the composite, predictions of which can be based on mixing rules or effective medium theory.^3–5 However, since predictive methods only consider the volume fractions of the constituents, the geometrical arrangement of the filler particles, which also influences the bulk properties of the composite,^6,7 is not appropriately represented.

There are several means to organize the filler, e.g., using optical tweezers,⁸ electron beams,⁹ or field-assisted self-assemblies.^10–12 The former two methods require expensive tools, whereas the latter method uses simpler tools,¹³ such as application of an external magnetic field on a liquid polymer that is infused with suspended particles in ferrofluid.^{10,11,14–17} This creates a basis for a ‘material printer’. For instance, starting with a liquid polymer-particle mixture, desired bulk material attributes can be imparted to a material by a priori specifying the required particle arrangements. Enhancing the electric permittivity by organizing the filler particles in an elastomeric composite is expected to be beneficial for flexible electronics,¹⁸ 3D printed circuit boards^19,20 and sensors,^21,22 data storage,²³ optics,²⁴ and artificial muscles.²⁵ 

The simplest and most common particle organization using field-assisted control is a chainlike structure.^12,16,26 Upon application of an external field, filler particles within the liquid prepolymer align along the field direction to form straight¹² or zigzag chains.^12,27–29 Therefore, we arrange conductive particles in chainlike structures that lie perpendicular to the conductive plates within polydimethylsiloxane (PDMS), as shown in Figure 1. Numerical simulations are employed to investigate the effects of different particle arrangements and sizes on the static relative permittivity of the composite using the COMSOL Multiphysics® package along with its electrostatics physics module. Our simulations assume uniform spherical particles, a steady electrostatic condition, and the absence of defect, fatigue and hysteresis. Despite these assumptions, the results are useful for designing and understanding how the electric permittivity of polymer-particle composites can be improved.

The relative permittivity of a composite (ε) is predicted by simulating a parallel capacitor at steady state without any current flow,

where V denotes the applied voltage, $∫υρυdυ$ total charge, C overall capacitance, $ε0$ absolute permittivity of vacuum, A projected surface area of the plates, and L distance between the plates. The PDMS matrix has relative permittivity of 2.69.³⁰ Since the relative permittivity of metal is infinite,^31,32 we assume a large value, $2.8×105$⁠, for the conductive particles as an appropriate idealization. Our models are verified for pure PDMS and for PDMS-particle composites with randomly distributed particles using various approaches, as explained in the supplementary material. We also ensure that the applied voltages do not lead to electric field strengths inside the composites that exceed the breakdown strength for the PDMS matrix.³³ When particle chains are oriented parallel to the plates, ε lies within 3% of that obtained with randomly distributed particles. Therefore, only the results for chains oriented perpendicular to the plates are discussed.

We examine the influence of the following parameters on ε: (1) filler particle volume fraction ψ, (2) interparticle spacing d, (3) interchain spacing a, (4) clearance h between the outermost particle and its adjacent conductive plate, (5) zigzag angle θ (the acute angle between the center-to-center direction of adjacent particles and the overall chain direction), and (6) size of the filler particles r, all of which are illustrated in the schematic of Figure 1.

Organizing filler particles into a chainlike structure within a dielectric composite enhances ε. For the same ψ, this ε enhancement diminishes rapidly as the interparticle spacing $d/2r$ increases and ε approaches the corresponding value for a composite that contains only randomly distributed particles, as shown in Figure 2a. This result is explained through the interparticle long-range dipolar interaction that decays proportionally to $m−3$⁠, where m denotes the average distance between neighboring particles.³⁴ A composite with higher ψ has larger ε, which agrees with the mixing rule concept.⁵ The ε enhancement is most pronounced for $d/2r=0$ when the particles form continuous chains and it is negligible when $d/2r>0$⁠, i.e., when discontinuous chains are formed. Thus, when the composite contains continuous filler particle chains, ε enhancement is more significant than when discontinuous chains are included in it. Hence, results for composites that include continuous particle chains are discussed below.

Consider that the two clearances between each plate of a capacitor and their nearest particles form two smaller capacitors that are connected in series by a continuous conductive particle chain, i.e.,

where $V1$ and $V2$ denote the electric potentials across each of the two capacitors. No matter how $V1$ and $V2$ change, the electric potential $V$ across the two plates is constant. Therefore, when the entire filler organization is placed off-center, changing h does not significantly influence ε, as shown in Figure 2b. Thus, for both continuous and discontinuous chains, ε enhancement is independent of h.

Reducing a diminshes ε enhancement, as shown in Figure 3a. This result is counterintuitive with the general understanding that the capacitance, hence ε, should increase with decreasing distance between two conductive objects. However, the statement is only true for two conductive objects with equal and opposite charges.³⁵ Since each conductive chain herein has zero net charge, this leads to the different observed behavior.

Essentially, for the same applied voltage, as a decreases, charge interactions between adjacent particle chains reduce the electric charges at the chain ends. This results in a lower overall capacitance and therefore smaller $ε$ for the composite. Consider two continuous particle chains in a dielectric composite that are placed far enough to avoid long-range dipolar interactions.³⁴ The electric charges induced by an applied voltage V at the two ends of each chain are $±Q0$⁠, which can be visualized by examining the distribution of surface charge density (Figure 3b–d). When the chains are moved closer, i.e., a is decreased, like charges located at the ends of the chains repel each other and migrate towards the middle of these chains. Here, charges from opposite ends of the chain intersperse and neutralize their polarity (green region in Figure 3c and d). The remaining charges at the two ends of each chain are now $±Q1$ where $Q1<Q0$⁠, shown as the smaller and asymmetrical blue and purple (higher surface charge densities) regions in Figure 3d. For a capacitor of specific overall geometry and applied voltage, there is a proportional relationship between its electric charges $∫vρvdv=Q$ and $ε$⁠, cf. Equation (1). Therefore, with decreasing $a$⁠, charge interactions within the chains diminish the overall charge at their ends, diminishing $ε$ enhancement, as shown in Figure 3a. Since a has the maximum value $ax$ for any $ψ$ (see the supplementary material for details), Figure 3a also implies that when the particle chains are placed evenly with $a/ax=1$⁠, $ε$ enhancement is largest.

Increasing $ψ$ increases $ε$⁠. The $ε$ enhancement is more significant for a composite containing continuous particle chains than one that includes randomly distributed particles (⁠$εz$⁠), as shown in Figure 4a. When $ψ<$ 9%, a composite containing smaller filler particles has higher $ε$ than one containing larger particles, whereas when $ψ>$ 9% the behavior is opposite because chains formed with smaller particles have a greater overall surface area.³⁶ In addition, the surface morphologies of the interfaces between the matrix and smaller filler particles are more intricate, leading to a higher interfacial polarization,^37,38 a size advantage that diminishes as $ψ$ increases. Due to the competing influences of increasing $ψ$ and charge interactions discussed above, a composite containing smaller particles has an inherently smaller $ax$ when $ψ>$ 9%, resulting in more vigorous charge interactions that decrease the $ε$ enhancement.

For smaller ψ, $ax$ is larger so that interchain charge interactions are negligible. These interactions become stronger with increasing ψ for continuous particle chains regardless of particle size. Therefore, for organized particles, ε increases with ψ and $ax$ but the higher ψ is, the smaller $ax$ becomes whereas $εz$ depends on ψ only. Consequently, there exists an optimal value of $ψ$ for $ε$ enhancement that maximizes the benefit of using organized particles as compared to simply distributing them randomly. The permittivity reaches a maximum value $ε/εz≈1.85$ for any particle size at an optimal ψ value, as shown in Figure 4b. For instance, when $r=0.68$ $μm$⁠, the maximum value of $ε/εz$ occurs at $ψ=$ 11% while the maxima for $r=0.4$ $μm$ and 0.5 $μm$ lie at $ψ=$ 7% and 9%, respectively.

The particle chains discussed above are straight and aligned (⁠$θ=0°$⁠) with overall chain directions perpendicular to the conductive plates. In Figure 5, we demonstrate the influence of $θ=0°$ to $60°$ and ψ on ε for $r=0.5$ $μm$⁠. For continuous particle chains (Figure 5a), ε decreases with increasing θ and even more rapidly with increasing ψ. Without varying the number of particles in a chain, the larger θ is, the closer are the particles within a chain. Since the electric potential over the entire surface of the chain is the same, for a chain that is oriented perpendicular to the equipotential surfaces, the closer the particles are, the lower is the distortion of the equipotential surfaces. This leads to smaller electric field strengths and therefore smaller ε for the composite. For discontinuous chains, ε is more or less independent of θ, as shown in Figure 5b, the largest difference in ε being 0.265. Hence, through various combinations of ψ and θ for the same matrix and filler material, it is possible to design elastomer-particle composites with ε in the range from 3 to 8 (Figure 5c).

In summary, the manner in which conductive filler particles contained within a PDMS elastomeric matrix are arranged influences the electric permittivity of the composite material. Organizing the particles into chainlike structures enhances the bulk permittivity by 85% in comparison with that for a composite containing the same filler, but which is randomly distributed. The value of ψ that is most beneficial when organized particles are used is 9% α 2%. The bulk permittivity increases with increasing ψ and $a(x)$⁠, but decreases with increasing $d$ and ψ. When $ψ<$ 9%, a composite containing smaller particles has higher ε, while for $ψ>$ 9%, introduction of larger particles increases ε. The enhancement in ε is more gradual beyond $ψ≈$ 9% due to the competition between effects induced by increasing ψ, which increases ε, and decreasing $ax$⁠, which decreases ε. When adjacent particles are interconnected, ε enhancement is more pronounced because continuous particle chains that are oriented perpendicular to the equipotential surfaces increase the induced charge of the composite, hence its bulk permittivity. These results are relevant for the design of soft dielectric materials with different permittivities when identical material compositions are used but their filler particles morphologies are varied, such as for additive manufacturing flexible electronics and 3D printed circuits.