Kinetic model monodisperse

Simplifications in kinetic modeling with monodisperse particle sizes and separations, and the detailed characterization offered via [4] more systematic studies of the influence of these parameters can be made, and the associated kinetic modeling is then simplified and becomes more accurate. Furthermore, the active catalyst phase is in direct contact with the reacting gas [e.g., no pore diffusion needs to be accounted for, which simplifies mass transport calculations (when needed)]. 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

The BGK-like kinetic models for monodisperse particles have the linearized form... 

Eor inelastic collisions, the coefficient of restitution will appear explicitly in the kinetic model as seen above in the monodisperse case. We will consider a binary case with inelastic... 

The polydisperse kinetic model can be compared with the monodisperse case by setting 02 = 0. It is straightforward to show that... 

A more rigorous approach to the description of the colloid surfactant diffusion to the interfaee was proposed by Noskov [133]. The reduced diffusion equations for micelles and monomers, which take into account the multistep nature of micellisation and the polydispersity of micelles, were derived for time intervals corresponding to the fast and slow processes using the method applied initially by Aniansson and Wall to uniform systems. Analogous equations have been derived later by Johner and Joanny [135] and also by Dushkin et al. [137]. Recently Dushkin has studied also the adsorption kinetics in the framework of a simplified model of quasi-monodisperse micelles. In this case the assumption of the existence of two kinds of micelles permits to study the main features of the surface tension relaxation in real micellar solution [138]. The main steps of the derivation of surfactant diffusion equations in micellar solutions are presented below [133, 134]. 

Similar to the FR measurements of sorbates in monodispersed micro-porous materials, the major difficulty in the application of the FR method is to identify the true kinetic mechanisms occurring in the systems from the rate spectra or the FR spectra as there are generally several combinations of parameters, i.e., several theoretical models, which could produce virtually the same FR curves. This difficulty has been tackled by investigating the systems over a range of reasonable or possible parameter values, e.g., variation of particle size or temperatures. 

Alternatively, the kinetics considerations may be treated in terms of a two-state model, where the probe is assumed to be free in the continuous phase or uniformly bound with monodisperse aggregates, either micelles or microemulsion droplets. For instance, according to the so-called pseudophase model, the following relationship is obtained [5] ... 

In 1944 Kramers [1] published a phase-space kinetic theory for the steady-state potential flow of monodisperse dilute polymer systems in which the polymer molecule is modeled as a freely jointed bead-rod chain. Subsequent scholars developed kinetic theories for shearing flows of monodisperse dilute polymer solutions Kirkwood [2] for freely rotating bead-rod chains with equilibnum-averaged hydrodynamic interaction. Rouse [3] and Zimm [4] for freely jointed bead-spring chains, and others. These theories were all formulated m the configuration space of a single polymer chain. 

In 1975 Curtiss, Bird, and Hassager [S] developed a more general kinetic theory formulated in the phase space of the entire polymenc liquid, the theory being patterned after the Irving and Kirkwood [6] theory for monatomic liquids. The theory includes the possibility of more than one polymer species, polydispersity, models with or without internal constramts, and both dilute and concentrated systems. The Curtiss-Bird-Hassager theory was summarized later in textbook chapters [7,8] and also applied to polymer melts, both monodisperse [9] and polydisperse [10]. 

Selection of solid-form and appropriate particle attributes is one of the foundation elements of drng product design. One challenge is to deliver consistent dissolution rates of API particles used in formulations. Additionally as the chemistry route is optimised and the final step isolation is refined, there is a need to be able to define the potential impact of changes of API particle size and shape on product efficacy. The model used to predict the dissolution kinetics of the monodisperse crystalline... 

The first meaningfnl stndies on the kinetics of dynamic processes within micellar solutions, which looked at dissociation rates via temperature-jump (T-jump) and related techniqnes, were carried out in the 1960s. The first notable attempt at a complete theory of micellization kinetics was the work of Kresheck et al. [77], who proposed a stepwise surfactant aggregation model based on a monodisperse system, where all micelles have the same aggregation nnmber, n. However, this model found... 

The kinetically controlled stereoselection depends on very small increments of free activation enthalpy. It is therefore an excellent sensitive probe for dendrimer effects. As monodispersed macromolecules, chiral dendrimer catalysts provide ideal model systems for less regularly structured but commercially more viable supports such as hyperbranched polymers. However, the results obtained by a considerable number of research groups in the field have also established that the structural characteristics of the established dendrimer systems, such as the absence of a well-defined secondary structure, have limited the development of efficient abiotic enzyme mimics based on dendrimers. To achieve this ambitious goal, more effort in dendrimer synthesis will be necessary. The use of dendritic... 

The approach used in this thesis to study the chain-length dependence of termination reactions can be characterized as empirical and model-independent. The basic thought behind this study is that when using single-pulse pulsed-laser initiation, the radical chain-length distribution is so narrow that a monodispersity assumption for this radical population seems to be justified. In this line of thought, two novel kinetic analyses have been presented which should enable the direct model-independent measurement of these microscopic chain-length... 

---