Pseudo-bulk kinetics

In this limit, termination within the particle is rate determining. An important implication of termination being diffusion controlled (and hence chain-length dependent) is that, in conventional free-radical polymerisation, termination events are dominated by termination between a mobile short chain (formed in an emulsion polymerisation by entry of a z-mer, by re-entry of an exited radical or by transfer to monomer) and a long, relatively immobile one. This is known as short-long termination . 

Under any of these circumstances, a pseudo-bulk emulsion polymerisation follows the same kinetics as the equivalent bulk system  

Here the average termination rate coeiiicient k ) is the average of the chain-length dependent termination rate coeiiicient over the distribution of radicals of each chain length, with concentrations Ri (Russell etal, 1992)  

What is important about Equation 3.7 is that, just as with Equation 3.6, it is an equation with only two parameters, which can thus be used for imambiguous data interpretation and prediction, without a plethora of adjustable parameters. An example of this is the extraction of kt) from y-relaxation rate data in the emulsion polymerisation of styrene (Clay et al, 1998). The data so obtained are in accord (within experimental scatter) of kt) values inferred from treatment of the molecular weight distributions, and also from a priori theory. This data reduction method has also been performed recently for a corresponding methyl methacrylate emulsion polymerisation (van Berkel et al., 2005). The information gained from these data is particularly useful it supports the supposition that termination is indeed controlled by short-long events. Moreover, for the methyl methacrylate system, the data show that radical loss is predominantly caused by the rapid diliusion of short radicals generated by transfer to monomer (i.e. the rate coefficient for termination is a function of those for transfer and primary radical termination). Such mechanistic information is clearly useful for the interpretation and design of emulsion polymerisation systems in both academia and industry. 

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The rate of dispersion (co)polymerization of PEO macromonomers passes through a maximum at a certain conversion. No constant rate interval was observed and it was attributed to the decreasing monomer concentration. At the beginning of polymerization, the abrupt increase in the rate was attributed to a certain compartmentalization of reaction loci, the diffusion controlled termination, gel effect, and pseudo-bulk kinetics. A dispersion copolymerization of PEO macromonomers in polar media is used to prepare monodisperse polymer particles in micron and submicron range as a result of the very short nucleation period, the high nucleation activity of macromonomer or its graft copolymer formed, and the location of surface active group of stabilizer at the particle surface (chemically bound at the particle surface). Under such conditions a small amount of stabilizer promotes the formation of stable and monodisperse polymer particles. 

In this case, the kinetic behavior is quite similar to that of suspension polymerization, except that the polymer particles are supphed with free radicals from the external water phase. When the polymerization proceeds according to Eq. 48, the system is sometimes referred to as obeying pseudo-bulk kinetics. 

The values predicted by Eq. 51 agree well with those predicted by Eq. 49 within less than 4%. This type of plot is called a Ugelstad plot and has been applied as a criterion to determine whether a system under consideration obeys either zero-one kinetics (n<0.5) or pseudo-bulk kinetics (n>0.5). 

A pseudo-bulk system is one in which the compartmentalized nature of the locus of polymerization has no effect on any kinetic property (rate, molar mass or particle size distributions). A system in which n is appreciably greater than 0.5 will always be pseudo-bulk there are so many radicals in a particle that the polymerization will be indistinguishable from the equivalent bulk one. However, a system with a low value of n can also be pseudo-bulk, if (for example) radical desorption results in the desorbed radical suffering no other fate except to re-enter another particle [1,3]. It is then apparent that the polymerization process will not see the walls between particles. Because pseudo-bulk kinetics can occur in systems where n 0.5, a pseudo-bulk system is different from the Smith-Ewart Case 3. 

In this system the number of radicals in a particle is relatively so high that the polymerization resembles bulk polymerization. The average number of radicals per particle, h, is, almost, always greater than 0.5. Compartmentalization has no effect on the kinetics of a pseudo-bulk system, and termination, which is rate determining, is always diffusion controlled. 

The quantification of termination kinetics is relatively straightforward in a pseudo-bulk system, as follows [51-55]. Define / ,- as the population of growing... 

A system obeying pseudo-bulk behaviour is one wherein the kinetics are such that the rate equations are the same as those for polymerisation in bulk. In these systems, n can take any value in a pseudo-bulk system. Common cases are (a) when the value of n is so high that each particle effectively behaves as a microreactor, and (b) when the value of n is low, exit is very rapid and the exited radical rapidly re-enters another particle and may grow to a significant degree of polymerisation before any termination event. (This case is not the same as Smith and Ewart s Case 3 kinetics, because these were applicable only to systems with n significantly above. )... 

Systems whose kinetics do not fall imambiguously into the zero-one or pseudo-bulk categories pose a problem for routine interpretation and prediction, let alone for obtaining useful mechanistic information such as that discussed in the preceding section. One can always use Monte Carlo modelling (Tobita, 1995), but the enormous amount of computer time this requires, and the plethora of imknown parameters, precludes its use for obtaining mechanistic information from experiment. 

Over 25 years ago the coking factor of the radiant coil was empirically correlated to operating conditions (48). It has been assumed that the mass transfer of coke precursors from the bulk of the gas to the walls was controlling the rate of deposition (39). Kinetic models (24,49,50) were developed based on the chemical reaction at the wall as a controlling step. Bench-scale data (51—53) appear to indicate that a chemical reaction controls. However, flow regimes of bench-scale reactors are so different from the commercial furnaces that scale-up of bench-scale results caimot be confidently appHed to commercial furnaces. For example. Figure 3 shows the coke deposited on a controlled cylindrical specimen in a continuous stirred tank reactor (CSTR) and the rate of coke deposition. The deposition rate decreases with time and attains a pseudo steady value. Though this is achieved in a matter of rninutes in bench-scale reactors, it takes a few days in a commercial furnace. 

The concentration of monomers in the aqueous phase is usually very low. This means that there is a greater chance that the initiator-derived radicals (I ) will undergo side reactions. Processes such as radical-radical reaction involving the initiator-derived and oligomeric species, primary radical termination, and transfer to initiator can be much more significant than in bulk, solution, or suspension polymerization and initiator efficiencies in emulsion polymerization are often very low. Initiation kinetics in emulsion polymerization are defined in terms of the entry coefficient (p) - a pseudo-first order rate coefficient for particle entry. 

The units on the rate constants reported by DeMaria et al. indicate that they are based on pseudo homogeneous rate expressions (i.e., the product of a catalyst bulk density and a reaction rate per unit mass of catalyst). It may be assumed that these relations pertain to the intrinsic reaction kinetics in the absence of any heat or mass transfer limitations. 

Equations 9.2-28 and -29, in general, are coupled through equation 9.2-30, and analytical solutions may not exist (numerical solution may be required). The equations can be uncoupled only if the reaction is first-order or pseudo-first-order with respect to A, and exact analytical solutions are possible for reaction occurring in bulk hquid and liquid fdm together and in the liquid film only. For second-order kinetics with reaction occurring only in the liquid film, an approximate analytical solution is available. We develop these three cases in the rest of this section. 

Kinetic analysis revealed that this reaction is pseudo-first order in dienophile at low conversions (200). At higher conversions, the rate deviates from a linear relationship, suggestive of product inhibition. Indeed, addition of product to the reaction at the start resulted in a decrease in rate by 18%. A number of competitive inhibitors were identified in this study. Particularly interesting was the observation that the matched chiral dienophile product was less effective as an inhibitor than the mismatched product. The authors suggest that the sterically matched complex (where the ligand bulk and imide chirality is on the same side of the complex) is thermodynamically less stable than the mismatched complex. 

Poly(acrylic acid) is not soluble in its monomer and in the course of the bulk polymerization of acrylic acid the polymer separates as a fine powder. The conversion curves exhibit an initial auto-acceleration followed by a long pseudo-stationary process ( 3). This behaviour is very similar to that observed earlier in the bulk polymerization of acrylonitrile. The non-ideal kinetic relationships determined experimentally in the polymerization of these two monomers are summarized in Table I. It clearly appears that the kinetic features observed in both systems are strikingly similar. In addition, the poly(acrylic acid) formed in bulk over a fairly broad range of temperatures (20 to 76°C) exhibits a high degree of syndiotacticity and can be crystallized readily (3). 

The most straightforward of the various models describing micellar kinetics is the Menger-Portnoy model for (pseudo) unimolecular reactions.The Menger-Portnoy model assumes rapid equilibration of the reactant of interest over bulk water and the micellar pseudophase with equilibrium constant K. The reaction then proceeds in both pseudophases with rate constants and in bulk water and the micellar pseudophase, respectively (Scheme 4). 

If the concentration of B CBL in the bulk liquid is much greater than CAi, a common case being where A reacts with a pure liquid B, the kinetics of the reaction become pseudo first-order, and the above equations can be solved analytically to give ... 

Such a form is more amenable to analysis. Spielman (2) used the data of Ciborowski (3) and assumed the concentration of oxygen in the system remained constant and also that the high temperature and presence of a cobalt naphthenate catalyst caused the hydroperoxide to decompose extremely rapidly so that it does not appear in the kinetic scheme. He then calculated the pseudo first order rate constants of this scheme. Assuming the bulk phase was not saturated with oxygen, these rate constants must be a function of the available interfacial area. 

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