Basic lattice cluster theory

The phase diagrams of PO-type random copolymers may be more complex than those for homopolymers with different, but uniform, molecular structure. Thus, similar to SLCT, but simplified further, is the basic lattice cluster theory (BLCT). The version was developed for the random copolymers or random and block copolymer systems. 

BLCT Basic lattice cluster theory (LCT) bPET Branched polyethylene terephthalate BR Polybutadiene, butadiene rubber... 

The remainder of Section I is devoted to a rather brief review of earlier work in the field in order to gain a little perspective. In Sections II to IV the basic results of the cluster method are derived. In Section V a very brief account of the application of the formal equations to some systems with short-range forces is given. Section VI is devoted to a review of the application to systems with Coulomb forces between defects, where the cluster formalism is particularly advantageous for bringing the discussion to the level of modern ionic-solution theory.86 Finally, in Section VII a brief account is given of Mayer s formalism for lattice defects69 since it is in certain respects complementary to that principally discussed here. We would like to emphasize that the material in Sections V and VI is illustrative of the method. This is not meant to be an exhaustive review of results obtainable. 

Basically, the process of tablet compression starts with the rearrangement of particles within the die cavity and initial elimination of voids. As tablet formulation is a multicomponent system, its ability to form a good compact is dictated by the compressibility and compactibility characteristics of each component. Compressibility of a powder is defined as its ability to decrease in volume under pressure, and compactibility is the ability of the powdered material to be compressed into a tablet of specific tensile strength [1,2], One emerging approach to understand the mechanism of powder consolidation and compression is known as percolation theory. In a simple way, the process of compaction can be considered a combination of site and bond percolation phenomena [5]. Percolation theory is based on the formation of clusters and the existence of a site or bond percolation phenomenon. It is possible to apply percolation theory if a system can be sufficiently well described by a lattice in which the spaces are occupied at random or all sites are already occupied and bonds between neighboring sites are formed at random. 

To introduce the basic concepts of percolation theory [1-3], let us consider the very simple model of a two-dimensional square lattice as illustrated in Figure 5.1 [11,12]. In this system, we assume a probability, p, that a lattice site is occupied and we denote the occupied sites by closed circles. In the present lattice case, two sites are considered connected only if they are nearest neighbors and if they are both occupied. We define a (connected) cluster as an ensemble of occupied sites such that each of them is connected to at least another occupied site of the ensemble. In... 

In this chapter, techniques for describing pore connectivity and conductivity were reviewed. These methods provide an framework understanding problems of transport in porous polymers. For example, although the basic concepts in percolation theory are relatively simple, they provide a powerful tool for understanding cluster behavior in porous systems. Likewise, simple simulation lattice walk techniques can provide considerable insight into the microscopic determinants of fluid or solute transport in constricted pores. 

---