Propagation-Controlled Model

Development of the propagation-controlled model for the absorption of the initiator-derived free radicals is based on the following major assumptions [22]  

generation of free radicals with z monomeric units via the propagation reaction of free radicals with (z - 1) monomeric units with monomer molecules in the continuous aqueous phase is the rate-limiting step for the absorption of free radicals by the latex particles. The rate of absorption of free radicals by a single latex particle (p = pJNp) can be written as follows  

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One of the most important parameters in the S-E theory is the rate coefficient for radical entry. When a water-soluble initiator such as potassium persulfate (KPS) is used in emulsion polymerization, the initiating free radicals are generated entirely in the aqueous phase. Since the polymerization proceeds exclusively inside the polymer particles, the free radical activity must be transferred from the aqueous phase into the interiors of the polymer particles, which are the major loci of polymerization. Radical entry is defined as the transfer of free radical activity from the aqueous phase into the interiors of the polymer particles, whatever the mechanism is. It is beheved that the radical entry event consists of several chemical and physical steps. In order for an initiator-derived radical to enter a particle, it must first become hydrophobic by the addition of several monomer units in the aqueous phase. The hydrophobic ohgomer radical produced in this way arrives at the surface of a polymer particle by molecular diffusion. It can then diffuse (enter) into the polymer particle, or its radical activity can be transferred into the polymer particle via a propagation reaction at its penetrated active site with monomer in the particle surface layer, while it stays adsorbed on the particle surface. A number of entry models have been proposed (1) the surfactant displacement model (2) the colhsional model (3) the diffusion-controlled model (4) the colloidal entry model, and (5) the propagation-controlled model. The dependence of each entry model on particle diameter is shown in Table 1 [12]. 

Two major entry models - the diffusion-controlled and propagation-controlled models - are widely used at present. However, Liotta et al. [28] claim that the collision entry is more probable. They developed a dynamic competitive growth model to understand the particle growth process and used it to simulate the growth of two monodisperse polystyrene populations (bidisperse system) at 50 °C. Validation of the model with on-line density and on-line particle diameter measurements demonstrated that radical entry into polymer particles is more likely to occur by a collision mechanism than by either a propagation or diffusion mechanism. 

Marek and coworkers simulated the experiments using an abstract two-variable propagator-controller model (Sevcikova and Marek, 1986). To include electric field effects, they added another term to the reaction-diffusion equation ... 

A number of models dealing with absorption of free radicals by the latex particles were proposed. They are (a) the collision-controlled model [1,17,18], (b) the diffusion-controlled model [19], (c) the surfactant displacement model [20], (d) the colloidal model [21], and (e) the propagation-controlled model [22, 23]. The dependence of the rate constant for absorption of free radicals by the latex particles on the particle diameter d ) predicted by these models is summarized in Table 4.1 [24]. At present, the most widely accepted models... 

As mentioned above, the two most popular reaction mechanisms involved in the absorption of free radicals by the monomer-swoUen micelles and polymer particles are the diffusion- and propagation-controlled models. Nevertheless, liotta et al. [39] were inclined to support the colUsion-controlled model. A dynamic competitive particle growth model was developed to study the emulsion polymerization of styrene in the presence of two distinct populations of latex particles (i.e., bimodal particle size distribution). Comparing the on-line density and particle size data with model predictions suggests that absorption of free radicals by the latex particles follows the collision-controlled mechanism. 

More recent work has shown that the observed variation in propagation rate constants with composition is not sufficient to define the polymerization rates.5" 161,1152 There remains some dependence of the termination rate constant on the composition of the propagating chain. Thus, the chemical control (Section 7.4.1) and the various diffusion control models (Section 7.4.2) have seen new life and have been adapted by substituting the terminal model propagation rate constants (ApXv) with implicit penultimate model propagation rate constants (kpKY -Section 7.3.1.2.2). 

In the classical diffusion control model it is assumed that propagation occurs according to the terminal model (Scheme 7.1). The rate of the termination step is limited only by the rates of diffusion of the polymer chains. This rate may be dependent on the overall polymer chain composition. However, it does not depend solely on the chain end.166,16... 

The driving force for isoselective propagation results from steric and electrostatic interactions between the substituent of the incoming monomer and the ligands of the transition metal. The chirality of the active site dictates that monomer coordinate to the transition metal vacancy primarily through one of the two enantiofaces. Actives sites XXI and XXII each yield isotactic polymer molecules through nearly exclusive coordination with the re and si monomer enantioface, respectively, or vice versa. That is, we may not know which enantio-face will coordinate with XXI and which enantioface with XXII, but it is clear that only one of the enantiofaces will coordinate with XXI while the opposite enantioface will coordinate with XXn. This is the catalyst (initiator) site control or enantiomorphic site control model for isoselective polymerization. 

The enantiomorphic site control model attributes stereocontrol in isoselective polymerization to the initiator active site with no influence of the structure of the propagating chain end. The mechanism is supported by several observations ... 

The polymer stereosequence distributions obtained by NMR analysis are often analyzed by statistical propagation models to gain insight into the propagation mechanism [Bovey, 1972, 1982 Doi, 1979a,b, 1982 Ewen, 1984 Farina, 1987 Inoue et al., 1984 Le Borgne et al., 1988 Randall, 1977 Resconi et al., 2000 Shelden et al., 1965, 1969]. Propagation models exist for both catalyst (initiator) site control (also referred to as enantiomorphic site control) and polymer chain end control. The Bemoullian and Markov models describe polymerizations where stereochemistry is determined by polymer chain end control. The catalyst site control model describes polymerizations where stereochemistry is determined by the initiator. 

Equations (7.58) and (7.68) yield the rate of copolymerization, and may be taken from previous studies of the chemical control model, or from an empirical correlation between this parameter and the r T2 product [27] which is based on the fact that cross-termination over homo-termination. Direct measurements of have been obtained [26] by measuring the absolute values of the rates of propagation and termination in pure monomers and in mixtures of various compositions. In the case of styrene-/7-metho)q styrene, = 1, indicating that no polar or other influences favor cross-termination. In most cases, however, cross-termination is... 

The benefit of such a model is that better understanding of the wave propagation process may be gained. Also, it is possible to make controlled parameter studies in order to understand the influence of e.g. defect orientation, probe angle and frequency on the test results. Results may be presented as A-, B- or C-scans. 

Overview Reconciliation adjusts the measurements to close constraints subject to their uncertainty. The numerical methods for reconciliation are based on the restriction that the measurements are only subject to random errors. Since all measurements have some unknown bias, this restriction is violated. The resultant adjusted measurements propagate these biases. Since troubleshooting, model development, ana parameter estimation will ultimately be based on these adjusted measurements, the biases will be incorporated into the conclusions, models, and parameter estimates. This potentially leads to errors in operation, control, and design. 

The model is a straightforward extension of a pool-fire model developed by Steward (1964), and is, of course, a drastic simplification of reality. Figure 5.4 illustrates the model, consisting of a two-dimensional, turbulent-flame front propagating at a given, constant velocity S into a stagnant mixture of depth d. The flame base of width W is dependent on the combustion process in the buoyant plume above the flame base. This fire plume is fed by an unbumt mixture that flows in with velocity Mq. The model assumes that the combustion process is fully convection-controlled, and therefore, fully determined by entrainment of air into the buoyant fire plume. 

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

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