Polymer Particle Morphogenesis

Let us first consider the catalyst/polyolefin particle in the early stage of its evolution. The particle consists of the solid catalyst carrier with catalyst sites immobilized on its surface, polymer phase, and pores. The first-principle-based meso-scopic model of particle evolution has to be capable of describing the formation of polymer at catalyst sites, transport of monomer(s) and other re-actants/diluents through the polymer and pore space, and deformation of the polymer and catalyst carrier (including its fragmentation). Similar discrete element modeling techniques have been applied previously to different problems (Heyes et al., 2004 Mikami et al., 1998 Tsuji et al., 1993). 

The z th micro-element is characterized by the position of its center xh velocity vh radius rh monomer concentration ch and by its type tt. The translational movement of each micro-element is governed by kinematic equation and Newton s equation of momentum 

The rate of the growth of the /th micro-element of type P depends on the monomer concentration in this micro-element 

The mathematical model of catalyst/polymer particle evolution consists of the set of differential-algebraic equations (61)-(64). The constitutive equations describing the force interactions, transport of monomer, phase equilibria at the interface between polymer and pore phase as well as the rules for connectivity of micro-elements have to be specified (Grof and Kosek, 2005 Grof et al., 2005a). 

Magnitudes of binary and ternary force interactions required in Eq. (62) are calculated by simple elastic or visco-elastic constitutive equations and then projected into force vectors. Elastic model is the direct implementation of the Hook s law with the stress between micro-elements A and B dependent linearly on the strain eAB  

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The methodology is demonstrated via several examples including phase transition and structure evolution in porous and granular media, the morphogenesis of polymer particles, and heterogeneous catalysis. Several future potential applications of the methodology are identified. 

It is noteworthy that the same polymer backbone (for example PEO-fo-PEI in Eig. 24) can not only lead to entirely different crystal morphologies as mentioned above, but also to a single crystal in the case of the sulfonate functions, or to polycrystalUne particles in the case of the weaker acidic carboxyl functions. Thus, it is evident that different morphogenesis mechanisms are active for the different DHBCs and the important role of the interacting fimctional polymer block is highhghted. 

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