Qualitative description of the model

We consider that the powder consists of a certain number N of spherical grains all with idenhcal radius R and molar volume F. Each grain is in contact with a certain number of close neighbors called the coordination number N. This number depends on arrangement of the powder and its compactness and can vary from 4 to approximately 7. 

We define the fractional extent of the transfomiation as the ratio of the volume of matter transferred to the initial volmne. If we consider Ft ns as the volume of matter transferred by grain through contact with a neighbor and if o is the amount of matter that constimtes a grain, the fractional extent is defined by  

the total fractional extent of the powder is equal to the fractional extent a of a grain. 

We assume here that the phenomenon of coalescence uses only one process with several steps in a pseudo-steady state and that, in the majority of the cases, there is a rate-determining step for the kinetics of coalescence. Thus, we will be able to admit the separable rate in the form  

We are introducing a reactivity (independent of time with constant external constraints and expressed in mole per square meter per second) and a space function E (function of time and expressed in square meters per mole) of the rate-determining step. 

---

A. General Phenomenological Theory 1. Qualitative Description of the Model... 

The qualitative description of the model is broadly supported by experimental evidence. The later theoretical explanation of this qualitative model is experimentally proven, insofar as the necessary energy input ( excess interfacial tension ) can be measured during dispersion of a given phase in a given polymer matrix. The non-equilibrium phase diagram, with the phase separation and flocculation description, and especially the non-equilibrium energy level, is experimentally well supported. On the other hand, the theoretical description is... 

In practice, such a fractionation experiment could be carried out by either lowering the temperature or adding a poor solvent. In either case good temperature control during the experiment is important. Note that the addition of a poor solvent converts the system to one containing three components, so it is apparent that the two-component Flory-Huggins model is at best only qualitatively descriptive of the situation. A more accurate description would require a... 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

The difference between the static or equilibrium and dynamic surface tension is often observed in the compression/expansion hysteresis present in most monolayer Yl/A isotherms (Fig. 8). In such cases, the compression isotherm is not coincident with the expansion one. For an insoluble monolayer, hysteresis may result from very rapid compression, collapse of the film to a surfactant bulk phase during compression, or compression of the film through a first or second order monolayer phase transition. In addition, any combination of these effects may be responsible for the observed hysteresis. Perhaps understandably, there has been no firm quantitative model for time-dependent relaxation effects in monolayers. However, if the basic monolayer properties such as ESP, stability limit, and composition are known, a qualitative description of the dynamic surface tension, or hysteresis, may be obtained. 

The major part of the reports discussed above provides only a qualitative description of the catalytic response, but the LbL method provides a unique opportunity to quantify this response in terms of enzyme kinetics and electron-hopping diffusion models. For example, Hodak et al. [77[ demonstrated that only a fraction of the enzymes are wired by the polymer. A study comprising films with only one GOx and one PAH-Os layer assembled in different order on cysteamine, MPS and MPS/PAH substrates [184[ has shown a maximum fraction of wired enzymes of 30% for the maximum ratio of mediator-to-enzyme, [Os[/[GOx[ fs 100, while the bimolecular FADH2 oxidation rate constant remained almost the same, about 5-8 x 10 s ... 

In the mass action model the micellar system can be described by only one parameter, and despite this simplicity, a good qualitative description of the main physical properties is obtained, for example the onset of cmc (critical micelle concentration), as shown in Figure 9.7. Notice that the formation of micelles becomes appreciable only at the cmc, and after that, by increasing further the surfactant concentration, all added surfactant is transformed directly into micelles, so that the surfactant concentration in solution remains constant at the level of cmc. 

A detailed analysis of the dynamics of polyatomic photodissociation presents a formidable problem. For this reason a number of simplified models have been introduced. Based on these models, one can obtain a qualitative description of the phenomenon, and of some of the important aspects. 

The oscillatory behavior of product-inhibited cultures cannot simply be described by a common inhibition term in the equation for the biomass growth. A better description must include an indirect or delayed effect of the product ethanol on the biomass growth rate as indicated in experiments. The decay rate pmaa was introduced to account for the accumulation of the inhibitory product pyruvic acid. Other more mechanistic, structured models can be formed that relate to the internal key-compound e. In these, the inhibitory action of ethanol is accounted for in the inhibition of the key-compound e formation. Mathematically, however, these two model descriptions are equivalent, except that the key-compound e is washed out as a part of the biomass in continuous cultures and the rate constant //ma55 does not vary. Our proposed indirect inhibition model provides a good qualitative description of the experimental results shown in Figure 7.25. 

Our canvas here is to provide a qualitative description of current models for the NM-M transition, developed for both metal-ammonia and metal-methylamine solutions. For this purpose we also draw upon interpretations from other systems in which the transition from localized to itinerant electron regimes is well recognized (78). 

Simulation Results. A onc-dimensional simulation model based on the Nernst-Planck and Poisson equations [14, in which all the acid-base reactions occurring in the membrane are taken into account, has been used to give a qualitative description of the pH step titration process. In these simulations, a pH step is applied outside a 2 mm thick stagnant layer, which is assumed to be present in front of an 8 mm thick membrane. Diffusion coefficients in the membrane are assumed to be 4/10 of those in water (this value is based on experience with ion step experiments). Lysozyme, used as a model protein, is assumed to contain 11 carboxylic groups (pKa = 4.4), 2 imidazole groups (pKa = 6.0), and 9 amino groups (pKa = 10.4) per molecule. Concern... 

Up to this point, the analysis is rigorously correct and general it is simply a restatement of the Born-Von Karman expansion of the energy in terms of relative displacements—.see Eq. (8-17). However, we shall now make a major approximation in taking the force constants from the very simple valence force field that we described in Chapter 8. This will give us a clear and correct qualitative description of the vibration spectra and will even give semiquantitativc estimates of the frequencies. Afterward, we shall consider the influence of the many terms that are omitted in this simple model. 

An important qualitative description of the spectral behavior of class II compounds was presented by Robin and Day. This simple model has found apphcabihty to the discussion of the spectra of numerous mixed valence compounds in which some delocalization occurs. In this model, it is assumed that the ground-state wave function contains the function, a, which describes mixing of the wave function for site A with the wave function of site B. 

Both computations provide a good qualitative description of the birefringence patterns, but the mPTT model captures more accurately the shape of the fringes in the downstream region. 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

Our calculations show that the smallest size quantum model (MOD-A) does not provide an adequate description for neither structural, nor electronic or dynamical properties. In contrast, a cluster model of the size of MOD-B is able to reproduce the structural properties of the real system quite accurately and provides also a qualitative description of the electronic and dynamic features ... 

Here, some general properties of the spectrum of the confined hydrogen atom are considered, leading to a simple expression for the energies. It may be noted that since the confining model potentials are qualitative simulations of complicated situations, a qualitative description of the spectrum is very useful. 

It was a century ago that researchers started to study the factors affecting the behaviour of water-oil-surfactant systems but it is only with the introduction of Winsor s R theory (1954) that the formulation effects could be interpreted. Winsor s R theory was the first qualitative description of the formulation, paving the way to an understanding of how intermolecular interactions among the different chemical species present in a system are related to its behaviour. Throughout the following decades, several empirical experimental correlations such as the phase inversion temperature (PIT), semiempirical ones such as the cohesive energy ratio (CER), and models based on thermodynamics such as the surfactant affinity difference (SAD) or the hydrophilic-lipophilic deviation (HLD) [15, 143, 144] led... 

A qualitative description of the structural and spectroscopic properties of H-bonded crystals requires the use of a simple two-mode linear model (9.1). 

Although the Weibull model allows qualitative description of the sensor reliability and thus the comparison Q15 to other variants, more specific experiments were needed to identify and understand the parameters of the failure mechanism itself. Thus, dedicated experiments, accompanied by simulations, were designed. With this setup, it became feasible to test several membranes differing in thickness, layer composition, and length/width ratio in a reasonable amount of time. The effects of particle size, velocity, and intensity Q16 on lifetime were also investigated. 

---