Phenomenological Dynamical Model

Tensile tests on different HDPE resins indicate that the stress-strain behaviour below the yield point can generally be described by the following empirical relationship (Fig. 4.2)  

Be EoiTref, fgf) and D2 Tref, ) the values of these moduli that pa- 

By looking at the pair of values (T, f ) for a specified reference temperature Tr(f, and reference deformation rate satisfy Eq. 4.9, one can 

Using this relation one can now determine the stress-strain curve at a certain reference temperature even for deformation rates that are too low to be achievable in practice in the tensile test experiment. For this purpose the tensile test is carried out at a deformation rate as low as technically possible and at a corresponding high temperature. Deformation rates and temperatures will be selected in such a way that Eq. 4.10 is fulfilled. 

So far the deformation rate served as a parameter to characterize the deformation behaviour. However, time can also be used for it. Let us consider the triple (T, ,t) attributed to each point cr, , f) in the family of curves in Fig. 4.2 over the relationship  

---

Another possibility to quantify the response of a stochastic system to periodic signals is to generalize the notion of synchronization, which is known from deterministic nonlinear oscillators. We will pursue this idea in what follows. To this end we review in section 2.2 the notion of effective synchronization in stochastic systems. The mean number of synchronized system cycles turns out to be an appropriate quantity to characterize the synchronization properties of the system to the periodic signal. However the task remains to calculate this quantity. This calculation will be based on discrete renewal models for bistable and excitable dynamics. These discrete models are introduced in section 2.3. We first recapitulate the well known two state model for the stochastic dynamics of an overdamped particle in a doublewell system [10] and afterwards introduce a phenomenological discrete model for excitable dynamics. In section 2.4 a theory to calculate the mean frequency and effective diffusion coefficient in periodically driven renewal processes is presented. These two quantities allow to calculate the mean number of synchronized cycles. Finally in section 2.5 we apply this theory to investigate synchronization in bistable and excitable systems. 

One possibility to simplify a continuous stochastic system is the reduction to a description in terms of a few discrete states. The system s behavior is then specified by the transition times between these discrete states. For example when investigating a neurons behavior, the important aspect are often only the times when a spike is emitted and not the complex evolution of the membrane potential [16]. In a doublewell potential system, depending on the questions asked, it may be sufficient to know in which of the two wells the system is located, neglecting the fluctuations in the wells as well as the actual dynamics when crossing from one well to the other. In these cases a reduction to a discrete description can be considered as an appropriate simplification. We first review the two state description of bistable systems [10] and then introduce a phenomenological discrete model for excitable dynamics. 

The phenomenological disengagement dynamics models [46-48] assumed that the solvent concentration at the solvent-polymer interface is independent of the solvent concentration history. There are parameters used that cannot be obtained from experiments. So the physical origin of such parameters is not very clear. In addition, there is a lot of empiricism in the approach. 

Some heat flows in connection with entropy production are associated with other thermodynamic variables. Typical single fluxes and forces are summarized above. It may be noted that steady fluxes are considered. Kinetic theory provides theoretical justification of some of these flux force relations (J = LX). Here, L is called phenomenological coefficient. But kinetic theory has limitation in the sense that first approximation to distribution function corresponds to local equilibrium hypothesis. It may be noted that non-equilibrium molecular dynamics (model and simulation) provides justification of these laws for a wide range. Nevertheless, justification has to be provided by experiments (Table 2.1). 

An alternative way of solving this difficulty is the use of dynamical models, based on combinations of first principles and neural networks (NN), called grey-box neural models (GNM). A GNM normally consists of a phenomenological part (heat and/or mass balances differential equations) and an empirical part (a neural network in this work). Due to the inherent flexibility of NN, models based on this structure are well suited to represent complex functions such as those encountered in chemical reaction processes. This work proposes incorporate in the RTO system a dynamical GNM of the... 

To describe the detailed kinetics at interphase boundary within the LG or UGAL models one has to know a large niunber of the probabilities and rate constants of elementary and kinetic processes. Depending on the information available they can be taken from classical and quantum dynamical models, thermodynamics and phenomenology, and from experiment. We are going to consider here some models of elementary processes in adsorbed layer and corresponding approximations for the probabilities and kinetic coefficients. 

The crossover between these two extremes of behavior is found to be quite extensive. These results involve many approximations, which are not always well controlled, but they represent one of the first microscopic approaches to dynamics in semi-dilute solutions and they seem to confirm the more phenomenological tube model. It should also be noticed that the confinement of the chain in what would be the tube is not due to topological entanglements, as in Edwards approach, but rather to the excluded volume interactions with the other chains. The relative role of these two effects has not yet been studied. 

Numerous computer simulations have been carried out in order to examine the transition from Rouse to reptation dynamics [70, 78-88]. Entanglement effects on chain dynamics clearly showed up. However, the discussion as to what extent the characteristic featmes of the reptation model for concentrated polymer liquids are verified by these simulations still remains controversial. It also should be mentioned that a series of phenomenological nonreptative models were published recently [89-94]. They mainly focus on viscoelastic properties of entangled polymer systems. 

For modelling conformational transitions and nonlinear dynamics of NA a phenomenological approach is often used. This allows one not just to describe a phenomenon but also to understand the relationships between the basic physical properties of the system. There is a general algorithm for modelling in the frame of the phenomenological approach determine the dominant motions of the system in the time interval of the process treated and theti write... 

An alternative framework to Newtonian dynamics, namely Langevin dynamics, can be used to mask mild instabilities of certain long-timestep approaches. The Langevin model is phenomenological [21] — adding friction and random... 

The model is able to predict the influence of mixing on particle properties and kinetic rates on different scales for a continuously operated reactor and a semibatch reactor with different types of impellers and under a wide range of operational conditions. From laboratory-scale experiments, the precipitation kinetics for nucleation, growth, agglomeration and disruption have to be determined (Zauner and Jones, 2000a). The fluid dynamic parameters, i.e. the local specific energy dissipation around the feed point, can be obtained either from CFD or from FDA measurements. In the compartmental SFM, the population balance is solved and the particle properties of the final product are predicted. As the model contains only physical and no phenomenological parameters, it can be used for scale-up. 

Models of a second type (Sec. IV) restrict themselves to a few very basic ingredients, e.g., the repulsion between oil and water and the orientation of the amphiphiles. They are less versatile than chain models and have to be specified in view of the particular problem one has in mind. On the other hand, they allow an efficient study of structures on intermediate length and time scales, while still establishing a connection with microscopic properties of the materials. Hence, they bridge between the microscopic approaches and the more phenomenological treatments which will be described below. Various microscopic models of this type have been constructed and used to study phase transitions in the bulk of amphiphihc systems, internal phase transitions in monolayers and bilayers, interfacial properties, and dynamical aspects such as the kinetics of phase separation between water and oil in the presence of amphiphiles. 

---