Conductivity within a Single Particle

In conventional silver flake filled ICAs, the conductivity of the flakes can be considered to be equivalent to the bulk conductivity of silver (Li et al. 1995; Li et al. 1997; Lu et al.

1999). However, in the case of ICAs formulated with silver coated polymer spheres fillers, the filler conductivity is less than that of bulk silver. This is because the silver coating only accounts for a small proportion of the volume of the filler particles compared to the highly insulating polymer core. Therefore, the thickness and morphology of the silver coating is expected to have a significant influence on the filler particle conductivity. Metal coated polymer spheres (especially Nickel (Ni) and Gold

(Au) coated) are already widely used in ACA applications (Dou et al. 2004). However, both heat and pressure are simultaneously applied to ensure the formation of electrical connections in ACAs which significantly deforms the MPS. The deformation of MPS in ACA assembly is illustrated in Figure 4.4. Williams et al. (1993), Hu et al. (1997) and Shi et al. (1999) presented models to calculate the resistance of an individual solid metal particle in an ACA assembly. Unfortunately, their models are not applicable to the metal-coated polymer spheres used in the present study. Määttänen (2003) first presented an analytical model to calculate the resistance of a single metal-coated polymer sphere, Rmps. Määttänen calculated the resistance as a function of the degree of deformation and coating thickness by treating the MPS as a series of metal rings of thickness tc around the polymer particle. When external pressure is applied an MPS is deformed by depth d to height hd as shown in Figure 4.4.

Figure 4.4 Deformation of MPS in ACA application

Määttänen calculated the average cross-sectional area, Avr, of the thin metal rings

Chapter 4: Theoretical Model of Electrical Conductivity of ICAs Filled with Silver Metalised Polymer Spheres

Assuming that the average cross sectional area through which the current passes is the average area of the thin metal rings around the polymer particle, he calculated the MPS resistance as:

𝑹_𝒎𝒑𝒔= (𝑹𝒆𝒔𝒊𝒔𝒕𝒊𝒗𝒊𝒕𝒚 𝒐𝒇 𝒎𝒆𝒕𝒂𝒍 𝒄𝒐𝒂𝒕𝒊𝒏𝒈) ∗(𝑳𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒅𝒆𝒇𝒐𝒓𝒎𝒆𝒅 𝑴𝑷𝑺)

𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝒄𝒓𝒐𝒔𝒔−𝒔𝒆𝒄𝒕𝒊𝒐𝒏𝒂𝒍 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒎𝒆𝒕𝒂𝒍 𝒓𝒊𝒏𝒈𝒔 4.15 Combining equations 4.13 and 4.14 in equation 4.15 gives:

𝑹_𝒎𝒑𝒔= _𝝅.𝒕^𝝆𝒎

𝒄 𝒍𝒏 𝒕𝒂𝒏 [^𝝅_𝟒 (𝟏 +^𝒉_𝟐𝒓^𝒅)] 4.16 where ρm is the resistivity of metal coating.

By also assuming that the MPS metallisation is made up of metal rings of thickness tc

around the polymer particle Dou et al. (2003) derived another model to calculate the MPS resistance. They also obtained the particle resistance function based on the degree of deformation and coating thickness. However, their particle resistance function was more difficult to integrate, and required numerical solution.

However, in the case of ICAs, the degree of deformation would be negligible as compared to ACAs. This is because no external pressure is applied and deformation is due to any adhesive shrinkage upon curing. In this study resistance model of a single MPS for ICA applications is developed in terms of on contact radius instead of the degree of deformation of the sphere and is presented in the next section.

4.1.3.1 Conductivity Model of a Single Metalised Polymer Sphere

In this model the MPS is considered as a spherical shell of radius r and coating thickness tc , made up of annular rings of radius rsinθ and length rdθ , as shown in Figure 4.5. The following assumptions are made in this model;

(i) The metal coating is smooth and homogenous; and

(ii) Current flows uniformly in the metal coating parallel to the surface from one pole to the other.

The resistance of each annular ring can be calculated using Ohm’s law and then integrated to calculate the resistance of whole MPS. The resistance contribution, dR, of a ring is given by:

𝒅𝑹 = 𝝆_𝒎. 𝒓𝒅𝜽

𝟐𝝅. 𝒓𝒔𝒊𝒏𝜽. 𝒕_𝒄 𝟒. 𝟏𝟕 where ρm is the metal coating resistivity (Ωm); and 2πrsinθtc is the cross-sectional area of this annular ring (m²).

Figure 4.5 Metal coated polymer sphere shown as made up of small annular rings (for clarity only two rings are shown)

As the geometry is symmetrical the resistance of a half spherical shell is calculated, which is then multiplied by two to calculate the total resistance. The resistance Rhs of the half spherical shell can be obtained by integrating Equation 4.17 between θ values of 0 and π/2.

𝑹_𝒉𝒔 = ∫ ^{𝝆𝒎. 𝒓𝒅𝜽}

𝟐𝝅. 𝒓𝒔𝒊𝒏𝜽. 𝒕_𝒄 = 𝝆_𝒎

𝟐𝝅. 𝒕_𝒄 ∫ ^𝒅𝜽 𝒔𝒊𝒏𝜽

𝜽=𝝅 𝟐⁄

𝜽=𝟎

𝟒. 𝟏𝟖

𝜽=𝝅 𝟐⁄

𝜽=𝟎

Iout

rsinθ

r θ dθ

rc

θc

I I

Iin

Chapter 4: Theoretical Model of Electrical Conductivity of ICAs Filled with Silver Metalised Polymer Spheres Therefore equation 4.18 can be restated as:

𝑹_𝒉𝒔=_{𝟐𝝅.𝒕}^𝝆𝒎 infinite. This is because the model effectively assumes that the particle contact area is an infinitesimal. In reality the current will not flow through a point but through an area of contact between two particles. Assuming a small contact area between adjacent particles, a small value of θ1 is taken instead of zero, where this angle of contact θc = sin^-1(rc/r) and rc is the contact radius as shown in Figure 4.5. The resistance Rhs of the half spherical shell can now be calculated by integrating equation 4.18 from θ1 = θc to θ2 = π/2: For small angle, sinθ ≈ θ so equation 4.26 reduces to:

𝑹_𝒉𝒔 = ^𝝆𝒎

𝟐𝝅.𝒕𝒄 {– 𝒍𝒏 [𝒕𝒂𝒏^𝒓𝒄

Ø]} 4.27

And as the resistance of the whole MPS, Rmps, is twice the resistance of a half spherical shell:

𝑹_𝒎𝒑𝒔 = ^𝝆𝒎

𝝅.𝒕_𝒄 {– 𝒍𝒏 [𝒕𝒂𝒏^𝒓𝒄

Ø]} 4.28

Equation 4.28 shows that the resistance of a metal coated polymer sphere depends on the (i) metal coating resistivity, (ii) metal coating thickness, (iii) contact radius, and (iv) MPS diameter (Ø). Figure 4.6 plots the variation of Rmps for different metallic coatings, MPS Ø and coating thickness, as a function of contact radius. Figure 4.6 (a) plots the variation of resistance of a 4.8μm MPS when coated with 120nm of the noble metals silver and gold. Figure 4.6 (a) shows that the silver coated MPS has lower resistance as compared to one gold coated, however gold has other advantages such as it does not oxidise and its ions are less susceptible to migration. Figure 4.6 (b) plots the variation of the MPS resistance with diameter and coated with 100nm of Ag. It shows that as the MPS diameter increase its resistance increase. Figure 4.6 (c) plots the variation in resistance of a 4.8μm MPS coated with Ag of different thickness. It shows that as the thickness of the Ag increases the resistance decreases. All of the plots in Figure 4.6 show that resistance decreases with increasing contact radius. It can be further observed from Equation 4.5 that contact radius depends upon the stiffness of the polymer core and the normal force between adjacent MPS (due to shrinkage). Thus a MPS with a soft core are preferred. Therefore, experiments were carried out to investigate the effect of coating material, MPS Ø, coating thickness, contact radius, polymer core material and applied normal force on MPS conductivity. The details of these experiments are presented in Chapter 5.

Chapter 4: Theoretical Model of Electrical Conductivity of ICAs Filled with Silver Metalised Polymer Spheres

Figure 4.6 Variation of Rmps with (a) different metallic coatings, (b) MPS diameter (Ø) and (c) coating thickness

4.2 Conductivity Model for an ICA formulated with Metalised Polymer