Dynamic Inversion Maps

There are many apparent contradictions to be found in the phase inversion literature. Some of these contradictions have been explained by Salager s observa- 

Shinoda et u/. s PIT work and Marzairs - - EIP work are studies of dynamic inversion. PIT inversions (induced by change in temperature altering the surfactant affinity) can now be seen to be transitional inversions, while EIP inversions (induced by adding a dispersed water phase to a continuous oil phase) are catastrophic inversions. 

WOR maps are limited to examining one surfactant concentration per map. Phase behaviour changes with composition in these systems are sometimes represented on triangular diagrams (these have been used for surfactant-oil-water systems by Smith et However, triangular diagrams can only 

---

Dynamic inversion is studied by producing a change that moves the point that represents the formulation and composition of an emulsion on the map from one side of the inversion line to the other side. In practice the system is first equilibrated and then emulsified, to produce the initial emulsion. Then its formulation or composition is altered continuously or by small increments, while a low-... 

The two different ways of crossing the inversion line are associated with quite different behaviors. The first one, which is known as transitional inversion, is produced by changing formulation at a constant water-to-oil ratio, i.e., along a vertical path in the bidimensional map. Such a crossing takes place in the A region in the central zone of the map. The experimental evidence indicates that, in this kind of dynamic process, the inversion takes place at the very moment the standard inversion line is crossed, i.e., essentially at SAD = 0, whatever the direction of change [from A to A or vice versa as indicated with white arrows in Fig. 12 (left)]. The horizontal branches of the standard and dynamic inversion lines are thus identical. The term... 

The dynamical inversion problem consists in determining the unique control trajectory u(t) associated to a given input-output realization [d(t), y(t)] over [0, tp] [13]. For this purpose, take the time derivative of the output map z = g(x) (Eq. 9h), substitute dx/dt by (Eq. 1), and obtain the 2m algebraic equation set... 

Apparent equilibrium time and dynamic inversion. Salager discussed two experiments which concern dynamic inversion. The first considered the minimum contact time between phases before emulsification so that the resulting emulsion was indiscernible from an equilibrium system. He terms this contact time apparent equilibrium time and concluded that the equilibrium time decreases as SAD —> 0 and that it is essentially zero for some near optimum formulations. The second study tried to mimic actual processes by starting with an emulsion produced from a pre-equilibrated mixture and shifting its position on the WOR map, e.g. by changing temperature, or by changing its WOR at constant SAD. A shift across the inversion locus is a dynamic inversion. The results are shown in Figure 6.5. 

Such changes may be taken into account by shifting the representative point of the emulsion on the formulation-composition map. In some cases this point could trespass on the standard inversion line and emulsion inversion could take place in a dynamic fashion. Recent studies have shown that there are two kinds of dynamic inversions (1) the vertical crossing of the horizontal branch, which is produced by changing a formulation variable from A to A" region or vice versa, and (2) the horizontal crossing of one of the vertical branches, which takes place by changing somehow the water-to-oil ratio. The first type has been called transitional inversion because it happens smoothly in some reversible way. The second one was termed catastrophic dynamic inversion because it develops as a sudden instability and exhibits several characteristics of the cusp catastrophe model, such as hysteresis and metastability [55]. 

FIG. 4 Transitional and catastrophic dynamic inversions on a formulation-composition map. 

An essential property of A is the existence of the inverse transformation A . This allows us to go back and forth between Hamiltonian dynamics and Markovian dynamics. In other words, A maps deterministic reversible dynamics to irreversible stochastic dynamics. 

Fig. 1.3 Relaxation map of polyisoprene results from dielectric spectroscopy (inverse of maximum loss frequency/w// symbols), rheological shift factors (solid line) [7], and neutron scattering pair correlation ((r(Q=1.44 A )) empty square) [8] and self correlation ((t(Q=0.88 A" )) empty circle) [9],methyl group rotation (empty triangle) [10]. The shadowed area indicates the time scales corresponding to the so-called fast dynamics [11]...

The two variables change their role with respect to their dependent versus independent, intensive versus extensive nature. This is also true of e.g. calorimetric, conductometric and spectrophotometric titrations using UV-, IR- or NMR-spectrosco-py We additionally have to consider that in the titration of the catalytic process only the external dynamics are measured a direct comparison with the actual metal fraction of the related intermediate complexes is generally not possible We call this analysis of homogeneous catalytic systems by a metal-ligand titration the method of inverse titration and for the resulting diagrams we use the term li nd-concentration control maps ([L]-control maps) . 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

R. W. Field Prof. Rabitz, I like the idea of sending out a scout to map a local region of the potential-energy surface. But I get the impression that the inversion scheme you are proposing would make no use of what is known from frequency-domain spectroscopy or even from nonstandard dynamical models based on multiresonance effective Hamiltonian models. Your inversion scheme may be mathematically rigorous, unbiased, and carefully filtered against a too detailed model of the local potential, but I think it is naive to think that a play-and-leam scheme could assemble a sufficient quantity of information to usefully control the dynamics of even a small polyatomic molecule. 

Theorem C.6. A compact limit set of a monotone dynamical system in 0 can be deformed by a Lipschitz homeomorphism (with a Lipschitz inverse) to a compact invariant set of a Lipschitz system in IR"" in such a way that trajectories are mapped to trajectories and such that the parameterization of solutions is respected. 

The Rouse model is the simplest molecular model of polymer dynamics. The chain is mapped onto a system of beads connected by springs. There are no hydrodynamic interactions between beads. The surrounding medium only affects the motion of the chain through the friction coefficient of the beads. In polymer melts, hydrodynamic interactions are screened by the presence of other chains. Unentangled chains in a polymer melt relax by Rouse motion, with monomer friction coefficient C- The friction coefficient of the whole chain is NQ, making tha diffusion coefficient inversely proportional to chain length ... 

The search for a phenomenological alternative to RRKM inversion distribution mapping does not represent a novel idea. The first step in the RRKM modeling procedure for a chemically activated species involves the a priori characterization of its initial excitation energy distribution (70,89,90). For species produced from exoergic reactions this information is normally obtained from thermochemical data. A correspondingly simple direct method has not yet emerged for hot atom activation processes, because the associated dynamics are incompletely imderstood. 

Feedback error learning (FEL) is a hybrid technique [113] using the mapping to replace the estimation of parameters within the feedback loop in a closed-loop control scheme. FEL is a feed-forward neural network structure, under training, learning the inverse dynamics of the controlled object. This method is based on contemporary physiological studies of the human cortex [114], and is shown in Figure 15.6. 

---