Conductivity of Metal-Polymer Composites

Colloidal and particle-based coatings and slurries can be described as polymer composites in which the properties of the bulk coating are determined by the properties of a dispersed material within a dispersion medium [78]. These composites are typically cured at tempera-tures significantly lower than those required for ceramic or metal sintering and thus exhibit different properties and phenomena. A number of models to describe such composites have been proposed since their inception, but such systems are most often characterized as dis-ordered systems in which there is no regular or dis-ordered structure to the dispersant in the medium. The framework of percolation theory thus emerged to describe the electrophysical properties of disordered systems [78].

Accompanying percolation theory itself is typically a discussion of fractal geometry, which emerged when it was observed that individual particles would self-assemble into similar agglomerates, which then was adopted as the long-range unit structure of study [68]. Fractal geometry provides a framework to quantitatively describe the disordered structure of the aggregate particles by their fractal parameters [94, 27]. An in-depth discussion of fractal geometry is beyond the scope of this work, but the geometry of the aggregates affects the long-range percolation behavior of the composite [78].

Percolation theory provides a framework to describe the change of a composite material from an insulator to a conductor as the amount of conductive material loaded into an in-sulating matrix is increased. As the volume fraction of conductive material increases, more percolated linkages form, and the probability of a conductive pathway being formed increases accordingly [81]. This is shown in Figure 3.1. Once a critical number of linkages allowing for a conductive pathway is formed, the conductivity of the composite increases significantly and continues to increase as the volume fraction of conductive particles increases. The volume fraction at which this occurs is called the percolation threshold, Vc [67].

Bueche [12] proposed a model that simplifies the composite to a mixture of two phases, where the overall resistivity is a weighted sum proportional to the amount of each phase, and accounting for the fact that not all conductive particles present may be in contact and contributing to the conductive pathway. This is given by Equation 3.1, where ⇢ is the resistivity of the composite, ⇢m is the resistivity of the nonconductive matrix, ⇢p is the resistivity of the conductive particles, Vp is the volume fraction of the conductive particles, and wg is the fraction of contributing conductive particles.

1

⇢ = 1 Vp

⇢m

+Vpwg

⇢p (3.1)

Further refinements to this model were made by Springett [90], which provide more re-alistic behavior at the percolation threshold and, though spherical particles are assumed, include a lattice packing fraction term which helps further account for the disordered posi-tions of the conductive particles. This average model of conductance is given by Equation 3.2, where ↵ is the insulator conductivity, m is the metallic conductivity, vm is the volume fraction of metal, and f is the lattice packing fraction.

Figure 3.1: Dependence of conductivity on conductive particle loading as described by per-colation theory. Vc indicates the volume loading corresponding to the percolation threshold of the composite.

4 = ⁰_m ⁰_↵+⇥

( _m + ⁰↵)²+ 8 _{m ↵}⇤1/2

(3.2)

m0 =

✓3vm

f 1

◆

m

↵0 =

✓

2 3vm

f

◆

↵

However, despite the formation of a continuous conductive pathway, it is more accurate to consider the pathway as a series of resistors made up of each percolated linkage, with the resistance of each linkage and the interfacial contact resistance between linkages contributing to the overall resistance of the entire pathway [81]. The contact resistance, Rc, can be further broken down as the sum of the constriction resistance, Rcr, and tunneling resistance, Rt

(Equation 3.3) [67].

R_c = R_cr+ R_t (3.3)

Constriction resistance results from a constriction of electron flow from each particle to the contact area between two particles. This resistance has been derived and shown to be inversely proportional to the diameter of the contact spot by Equation 3.4, where ⇢i is the resistivity of the particle material and d is the diameter of the contact spot. [81]. The denominator of this relationship has been defined as the diameter of the contact spot rather than the contact spot area because the ratio of the particle diameter, D, and the the contact spot diameter determines the upper limit of validity of this relationship (when D/d > ⇠10).

Rcr = ⇢i

d (3.4)

Tunneling resistance derives from resistances associated with any insulating film or layer that may be coating each particle [81]. In the case of metals in a polymer matrix, these layers can be oxide layers formed on each metal particle or even the polymer itself. Depending on the manufacturing and processing methods performed on the particles, there may be residual organic films remaining as well. Based on the physics of quantum-mechanical tunneling, an electron may ”tunnel through” instead of overcome a potential barrier based on a probability proportional to the conductor’s work function, the film thickness, and the film’s relative dielectric permittivity. The resistivity of the film is absent from this proportionality, so for primarily organic and polymer films, the thickness of the film becomes the primary differentiating factor. This tunneling resistance is inversely proportional to the contact spot area, as in Equation 3.5, where ⇢i is the resistivity associated with tunneling and a is the contact spot area.

R_t = ⇢t

a (3.5)

The degree of contact resistance is also dependent on any internal stresses that may be present in the composite. These stresses may result from differences in thermal expansion during the curing process, capillary action from solvent evaporation in the slurry, and shrink-age from polymer crosslinking during the curing process [67]. The degree to which these play a role depends on the compositions of the polymer, conductive particles, and slurry, but the resulting internal stresses impart elastic and plastic deformation on the conductive particles, which assists in maintaining good contact.

In general, the magnitude of the tunneling resistance outweighs the magnitude of the constriction resistance and is impacted by the thickness of any insulating layers between conductive particles and the contact area, which are both consequently a result of internal stresses between particles, particle morphology, and packing fractions. Depending on the relationship between these factors, the overall resistivity of the composite may vary wildly [81].