Neck radius, between particles

Figure 11.5 Evolution of the structure of two planar arrays of copper spheres during sintering. Two different starting densities, expressed as x/a, are shown, where x is the neck radius between particles of radius a. (From Ref. 16.)...

German gives several geometric relations between the particle radius, r the neck radius, p the movement of the center of the particle toward the plane of contact, h the area of contact, A and the volume of material that must be transported to form the neck, V (Figure 14.3). 

Two cases of sintering Transport of material from the spherical surfaces to the neck (top) does not contribute to densifi-cation. Transport of material from the interface between the particles to the neck (bottom) does contribute to densification. p is the neck s radius of curvature, r is the particle radius, 2h is the decrease of distance between particle centers, and x is the radius of contact. 

Neck formation between spherical particles (a) without shrinkage but with a decrease in particle radius, r (b) with shrinkage [34]. 

Fig. 8.12. Schematic picture of contact points (a) and neck formation and sintering between particles (b), with radius R and neck width x. Path 1 represents matter transport from the grain boimdary to the neck surface with curvature radius p.

Fig. 5.48 Model of the sintering processes that can occur between two particles [B.61]. (a) Depicts schematically bonding. X, Y, a, and p are bridge radius, penetration depth, particle radius, and neck radius, respectively, (b) Is the bonding area in more detail showing the diffusion paths (Tab. 5.7).

The dominant electrical parameter of our model is the capacitance C. The connection between C and geometrical and/or structural properties can be established by an assumption that the contact region can be viewed as a planar micro-capacitance. Fig. 10. shows a geometric model of two spheres in contact, where u is the radius of the spherical particles, and x is the neck radius. 

Figure 5.15. Determining the coalescence factor by partial dissolution and r different degrees of reinforcement fa radius of neck between particles r - radius of...

This result is the Kelvin equation. In this approximation, we have further assumed that the particle is essentially a flat particle (with a vapor pressure of Pq) compared to the radius of curvature of the neck. We can calculate the rate at which the neck increases by equating the rate of material transfer to the surface of the lens between the spheres with the increase in its volume. The rate of condensation, m, is proportional to the difference in equilibrium vapor pressure, ZiP, as given by... 

The particles formed, usually oxide, form a packing with contact points, the number of which per unit of volume is determined by their coordination number and particle radius (see e.g. Ref. [24]). On these contact points a "physical reaction" takes place which results in the formation of necks between the particles as shown in Fig. 8.12. This process is called the initial state of sintering. 

For short times, the radius of the neck between two solid particles grows as when lattice diffusion dominates (Kingery and Berg, 1955). After a short initial period the rate of decrease in the surface area for coalescing solid particles approaches the linear rate law 12.4) with Zf given by (Friedlander and Wu. 1994)... 

As shown in Fig. 5.44, a large pore remains unfilled because of the preferential wetting of the necks between the particles [101]. As the grains continue to grow, the liquid phase reaches a favorable condition for filling the pore as determined by the curvature of the liquid-vapor meniscus [61, 76, 101]. From Fig. 5.45, the radius of curvature of the meniscus is given by [101]... 

It is possible to estimate a sensible particle size for an ideal support microstructure consisting of spherical spheres. Let us consider an alloy containing 22 wt % Cr and a critical thickness of the Cr203 scale around 1 pm, which is assumed to fully cover the sphere surface ( 0.5 pm of the original sphere radius). It is assumed that the necks between metal particles should have a thickness of least 4-5 pm to avoid complete corrosion of these conduction paths. Cr concentration needed to form this scale is calculated using the density of Cr203 scale. The remaining Cr concentration in the bulk can then be deduced, as illustrated in (Fig. 3). 

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