Vector percolation model

This model is excellent when applied to very weak interfaces (Gic 1 J m ) where most of the fracture events occur in a well-defined two-dimensional plane and chain pullout dominated the fracture process. However, it is not applicable to strong interfaces (Gic 100-1000 J m ) where the fracture events occur in a three-dimensional deformation zone at the crack tip. Here, the vector percolation model is appropriate. 

A theory of fracture was developed for such entangled polymers that was based on the vector percolation model of Kantor and Webman. The percolation model is used to describe connectivity between the chains and to relate the interfacial structure to the breakdown process of the deformation zone at the crack tip. Vector percolation involves the transmission of forces (vectors) through a two- or three-dimensional lattice where a certain fraction of the bonds are missing or broken. Thus, we can examine how the stiffness or strength of a lattice changes with bond fracture or disentanglement. 

The nail and vector percolation models of polymer-polymer adhesion have been successfully applied to many problems.Some of these are treated in articles on Polymer-polymer adhesion weld strength, Polymer-polymer adhesion molecular weight dependence and Polymer-polymer adhesion incompatible interfaces. 

Polymer-polymer adhesion models R P WOOL Nail and vector percolation models... 

We examined the role of vector percolation in the fracture of model nets at constant strain and subjected to random bond scission, as shown in Fig. 11 [1,2]. In this experiment, a metal net of modulus Eo containing No = 10" bonds was stressed and held at constant strain (ca. 2%) on a tensile tester. A computer randomly selected a bond, which was manually cut, and the relaxation of the net modulus was measured. The initial relaxation process as a function of the number of bonds cut N, could be well described by the effective medium theory (EMT) via... 

Fig. I I, The role of vector percolation in the fracture of model nets at constant strain and subjected to random bond scission.

To obtain the effective Heuniltoniein, describing the behavior of SAWs on the percolation cluster, we make use of results from [91,22,101]. Let us consider the m-vector spin model on a diluted lattice with bond reindomness, described by the Hamiltonian ... 

The self-adhesion (autohesion) of polymers is of considerable practical importance as many moulding operations involve the need for streams of merging polymer melt to self-adhere to form a strong bond necessary to the integrity of the moulded article. Mechanisms of polymer diffusion are discussed in Polymer diffusion reptation and interdigitation and theories that relate interfacial structure to strength in Polymer-polymer adhesion models. In the latter article, the vector percolation (VP) model was described, which is here applied to welding, and the practical consequences of its predictions are drawn out. - ... 

Fracture in this region can be treated by the vector percolation theory, which models the polymer as a three-dimensional network, the links of which are progressively broken (here by chain disentanglement) until a critical value, the percolation threshold, is reached, and the whole net fractures (Wool 2005). 

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