Order parameter aggregation

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

The order parameter values calculated from the data of Fig. 4 are illustrated in Fig. 5. The data there suggest the existence of two continuous transitions, one at a = 0.85 and another at a = 0.7. The first transition at a = 0.85, denoted by the arrow labeled a in Fig. 5, is assigned to the formation of percolating clusters and aggregates of reverse micelles. The onset of electrical percolation and the onset of water proton self-diffusion increase at this same value of a (0.85) as illustrated in Figs. 2 and 3, respectively, are qualitative markers for this transition. This order parameter allows one to quantify how much water is in these percolating clusters. As a decreases from 0.85 to 0.7, this quantity increases to about 2-3% of the water. 

Deuterium nmr describes in equally clear-cut and elegant a manner the static (order parameter) and the microdynamic (correlation time t ) properties of hydrocarbon chains in aggregated surfactants 13). The example of octanoate, in various phases, resembles the membrane behavior of phospholipids a decrease of the order parameter is found for deuterium relaxation rates,... 

The specific surface area, i.e. primary particle size, and the corresponding order of aggregation of primary particles, as well as the content of silanols, are key-parameters affecting many physicochemical properties of powders and aqueous suspensions of nanosilica. These key characteristics are responsible for the properties of Silics as a medicinal preparation acting as a sorbent. 

Also included in Table 1 are the average areas and Ay obtained from the entire chains (i.e. order parameters accounts for the fact that half thickness <(> [19,20,8]. The problems encountered when correlating chain order parameters and the average interfacial areas per chain in liquid crystalline aggregates have been recently discussed by Nagle [21]. 

By an order parameter S =Sqd thus the residual anisotropy, which is not averaged by fast local motions, is characterised. This is the main information obtained for partially ordered systems, and can be considered a characterisation of the mobility of the C-2H bond, which on the other hand describes the chain packing order in oriented aggregates. 

In practice, the situation can be more complicated, since motions around interdependent axes might lead to complex motional patterns, or the EFG may have no axial symmetry. The resulting 2H wide/line spectrum then can exhibit a modified shape, from which further information about the geometry of motions can be obtained. 2H wide-line spectra thus contain a wealth of information about the time scales and geometries of local bond motions in systems with partial motional averaging. Order parameters are established in characterising liquid crystalline phases, particularly for lipids. Typical values of S in aggregated alkyl chains of volume phases are on the order of 0.1- 0.2. 

Fig. 11 Left Relative order parameters in a series of volume aggregates, in which the aggregate curvature is decreasing with increasing surfactant concentration (micelles, hexagonal, lamellar phase). Right Relative order parameters in adsorption layers at different surface coverage. Figures taken from [38] with permission.

The analysis of the correlation time for motion provided information on the interactions of probes with the dendrimer surface, or the interactions of the dendrimers with other species, such as biomolecules, surfactant aggregates, and different types of liquid and sohd surfaces. In the case of interactions of the dendrimer with surfactant aggregates, the use of radical surfactants inserted in the aggregates also provided information about the type and the strength of the interactions. The order parameter is extracted from the analysis of the ESR spectra. 

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