Critically branched state

Peniche-Covas, C. A. L., Dev, S. B., Gordon, M., Judd, M. Kajiwara, K. (1974). The critically branched state in covalent synthetic systems and the reversible gelation of gelatin. In Gels and Gelling Processes. Faraday Discussions of the Chemical Society, No. 57, pp. 165-80. 

Starburst/cascade dendrimer dense-packing critical branching state, nanoscopic steric effects. 

Starburst (de Gennes) Dense Packing — A Critical Branched State ... 

Clearly, this proposed congestion at the critical branched state M, should exhibit (1) sterically-inhibited reaction rates, (2) sterically-induced stoichiometry, and, quite possibly, (3) a critical phase change due to surface cooperativity (association). These phenomena have been observed experimentally for the Starburst PAMAM dendrimers and may prove to be diagnostic probes for this critical branched state in other dendrimer families. 

The cause of the Buna fiasco was never absolutely determined, but its effect was immediate. The infantry s confidence in the flame thrower was shattered. In January 1943 Colonel Copthorne, Chief Chemical Officer, USAFFE, informed General Porter that "the way the flame throwers let the infantry down at a critical point brought them into such ill-repute that I am afraid that they may never want to use them again. In Washington Colonel Benner, chief of the CWS Field Requirements Branch, stated that a weapon such as the flame thrower with its "temperamental nature has no place in modern warfare where ruggedness and reliability are essential. ... 

The combinations of failures and non-failed conditions define the state of the pJani at the right branches. The damage associated with these plant damage states are calculated using thermal-hydraulic analyses to determine temperature profiles that are related to critical chemical reactions, explosions and high pressure. These end-states serve as initiators fot breaking confinement that leads to release in the plant and aquatic and atmospheric release outside ol the plant,... 

As the tangent plane rolls on the primitive surface, it may happen that the two branches of the connodal curve traced out by its motion ultimately coincide. The point of ultimate coincidence is called a plait point, and the corresponding homogeneous state, the critical state. 

Conditions (30) and (31) are sufficient to discuss the principal properties of the critical state of a one-component system. We observe that the existence of a critical state for such a system cannot be inferred from a j)riori considerations, because it is not necessary that the two branches of the connodal curve should ultimately coalesce that such is the case must be regarded as established for systems containing liquid and vapour by the experiments of Andrews ( 86), and the following discussion is limited to such systems (cf. 103). 

The two conditions stated above do not assure the occurrence of gelation. The final and sufficient condition may be expressed in several ways not unrelated to one another. First, let structural elements be defined in an appropriate manner. These elements may consist of primary molecules or of chains as defined above or they may consist of the structural units themselves. The necessary and sufficient condition for infinite network formation may then be stated as follows The expected number of elements united to a given element selected at random must exceed two. Stated alternatively in a manner which recalls the method used in deriving the critical conditions expressed by Eqs. (7) and (11), the expected number of additional connections for an element known to be joined to a previously established sequence of elements must exceed unity. However the condition is stated, the issue is decided by the frequency of occurrence and functionality of branching units (i.e., units which are joined to more than two other units) in the system, on the one hand, as against terminal chain units (joined to only one unit), on the other. 

Gas-phase ion chemistry is a broad field which has many applications and which encompasses various branches of chemistry and physics. An application that draws together many of these branches is the synthesis of molecules in interstellar clouds (Herbst). This was part of the motivation for studies on the neutralization of ions by electrons (Johnsen and Mitchell) and on isomerization in ion-neutral associations (Adams and Fisher). The results of investigations of particular aspects of ion dynamics are presented in these association studies, in studies of the intermediates of binary ion-molecule Sn2 reactions (Hase, Wang, and Peslherbe), and in those of excited states of ions and their associated neutrals (Richard, Lu, Walker, and Weisshaar). Solvation in ion-molecule reactions is discussed (Castleman) and extended to include multiply charged ions by the application of electrospray techniques (Klassen, Ho, Blades, and Kebarle). These studies also provide a wealth of information on reaction thermodynamics which is critical in determining reaction spontaneity and availability of reaction channels. More focused studies relating to the ionization process and its nature are presented in the final chapter (Harland and Vallance). 

In Chapter 3, the conditions for a chain branching explosion were developed on the basis of a steady-state analysis. It was shown that when the chain branching factor a at a given temperature and pressure was greater than some critical value acrit, the reacting system exploded. Obviously, in that development no induction period or critical chain ignition time rc evolved. 

Furthermore, a related and common criticism of the MFT method is that a mean-field approach cannot correctly describe the branching of wave packets at crossings of electronic states [67, 70, 82]. This is true for a single mean-field trajectory, but is not true for an ensemble of trajectories. In this context it may be stressed that an individual trajectory of an ensemble does not even possess a physical meaning—only the average does. 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

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