Concentration regimes dilute regime

Fiber suspensions are typically classified into three concentration regimes dilute, semidilute, and concentrated, which are based on their volume fraction, unit volume. The dilute regime is such that the fibers within the suspension are free to both rotate and translate without hydrodynamic interaction or direct contact. Theoretically, this occurs when the average distance between the center of mass of two fibers is greater than L leading to the constraint of n < 1/L or (p < a. The transition to the semidilute region occurs just above the dilute upper limit. Here hydrodynamic interaction is the predominant phenomenon with little fiber contact. However, the suspension orientation state is not subject to... 

Similarly, contaminant concentrations in rivers or streams can be roughly assessed based on rate of contaminant introduction and dilution volumes. Estuary or impoundment concentration regimes are highly dependent on the transport mechanisms enumerated. Contaminants may be localized and remain concentrated or may disperse rapidly and become diluted to insignificant levels. The conservative approach is to conduct a more in-depth assessment and use model results or survey data as a basis for determining contaminant concentration levels. 

In the semi-dilute regime, the rate of shear degradation was found to decrease with the polymer concentration [132, 170]. By extrapolation to the dilute regime, it is frequently argued that chain scission should be nonexistent in the absence of entanglements under laminar conditions. No definite proof for this statement has been reported yet and the problem of isolated polymer chain degradation in simple shear flow remains open to further investigation. 

Let us remark that relation (6) is given for polymer concentration c lower than the critical overlapping concentration c above which higher terms in c must be considered. In fact, the concentration practically used ( around 10 3 g/cm3) corresponds to the semi-dilute regim for which the behavior is not well known in the case of polyelectrolytes. We have however kept relation (6) by introducing for K a mean apparent value determined from our experiments ( K - 1 )... 

The relevant part of the phase diagram (x > 0) is shown in Fig. 38. The c-x-plane is divided into four areas. The dilute regime I and I are separated from the semi-dilute regimes III and II, where the different polymer coils interpenetrate each other, by the so-called overlap concentration... 

All the measurements reported in this paper were carried out at a particle concentration of 1 weight %. Only one concentration was investigated since it has been observed previously (6) that there is little variation in the UCFT as a function of particle concentration in the dilute concentration regime. The dispersion being investigated was added to the pressure bomb which was sealed and -200 bars pressure applied to the system. 

Finally, much work remains to be done on concentrated micellar solutions, which have been poorly investigated compared to the dilute regime. Interesting properties, such as lyotropic mesophase behavior, are expected to be observed for these concentrated micellar solutions. 

Dendrimers remain discrete objects in dilute solution, avoiding interpenetration. As the concentration increases above overlap, the dendrimers preferentially shrink in size rather than interpenetrating. When dried to a solvent-free condition, the dendrimers must either deform from their spherical shape into polyhedrons, or must interpenetrate. The solvent-free condition would require deuterium labeled dendrimers, and experiments are under way to probe this last concentration regime. 

We can express our transformation in the semi-dilute and concentrated regime as given by Equation (5.113) but with concentration included. So we obtain... 

Branched polymers can also be dissolved at fairly high concentrations. Because of the higher segment density in the isolated macromolecules the overlap concentration will also be increased. For this reason the semi-dilute regime of branched polymers may in some cases be larger than for linear chains, say about 20% or more. Clearly, however, a full interpenetration, as was assumed for flex-... 

Most important, however, was the discovery by Simha et al. [152, 153], de Gennes [4] and des Cloizeaux [154] that the overlap concentration is a suitable parameter for the formulation of universal laws by which semi-dilute solutions can be described. Semi-dilute solutions have already many similarities to polymers in the melt. Their understanding has to be considered as the first essential step for an interpretation of materials properties in terms of molecular parameters. Here now the necessity of the dilute solution properties becomes evident. These molecular solution parameters are not universal, but they allow a definition of the overlap concentration, and with this a universal picture of behavior can be designed. This approach was very successful in the field of linear macromolecules. The following outline will demonstrate the utility of this approach also for branched polymers in the semi-dilute regime. 

By the definition of T (f), is the volume of the cavity in which the particular solvent is replaced by the solute. Therefore Vp depends on the particular solvent chosen for the SANS-experiment and also on concentration. For the dilute regime under consideration here the latter dependence can safely be dismissed. 

Fd h = 0) should increase as when the chain concentration increases. A very different picture is predicted in the case of adsorbing polymers [49]. The layer of adsorbed chains may be partially interpenetrated by free chains in the bulk and therefore the range and strength of the attraction are not determined by the solution concentration. Instead, they are rather sensitive to the coverage and thickness of the adsorbed chains which depend essentially on the solvent quality and on the mean chain length in the dilute regime. 

Here, A is the depletion layer thickness (assumed equal to the radius of gyration of the polymer, RG). H = r - 2a is the surface-to-surface particle separation, V ° is the molar volume of the solvent, and ji and ji are the solvent chemical potentials for the polymer solution and the pure solvent. It appears that the assumption A = RG is generally acceptable providing that the polymer solution is in the dilute concentration regime. At higher polymer concentrations, however, the value of A is reduced according to the relationship (Vincent, 1990) ... 

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