Dilute regime, definition

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. 

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. 

Since this chapter is concerned with block copolymers in dilute solution, it is useful to include a definition of the dilute regime for polymer solutions in the Introduction. This regime extends up to a volume fraction above which swollen coils overlap (de Gennes 1979) ... 

In the dilute domain, the polymers Remain far apart and they repel one another weakly their size is then about the same as the size of an isolated polymer. The concentration effects really appear when the concentration C of the polymers becomes large enough so that they begin to overlap. By letting the size of the chain grow, we finally obtain the semi-dilute regime in which the chains strongly overlap. The transition occurs in the vicinity of a concentration C. Various rather equivalent definitions of C are possible. The theoretician likes to refer to the mean square end-to-end distance of the isolated polymer. This leads to the definition... 

The study of the dependence of the screening length on temperature is also very significant. We know that in the semi-dilute regime, the screening length e is proportional to the Kuhnian overlap length. By definition... 

The definition of this apparent correlation length can be extended to dilute regimes, in which case app gives g/V3 (see eqn (2.72)). The behaviour of app is entirely different in dilute solutions and concentrated solutions. In dilute solutions, lapp dincreases with the molecular weight and the excluded volume, while in concentrated solutions app — I is independent of the molecular weight and decreases as a function of concentration and excluded volume (see eqn (5.36)). The reason can be easily understood from Fig. 5.1 once polymers overlap each other, the excluded volume interaction tends to make the concentration homogeneous. 

Starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (u, = /xRe/dpp). The following is an interpolation formula that can be applied over all three settling regimes ... 

There are four to six concentration regimes (Dautzenberg et al., 1994). Edwards (1966) classified them into three broad types on the basis of the number of polymer chains present, the number and length of the monomer constituting them, and the volume of the monomer relative to the volume Vt. At use levels as low as parts per thousand in food, no one definition of dilute, semidilute, or concentrated encompasses the range of weight-volume or volume-volume concentrations necessary to elicit a par-... 

According to the preceding definition, the relationship between the saturation carrying capacity K and fluid velocity Ut can be calculated, as shown in Fig. 5, which defines the transition from the PFC regime to the FD regime, that is, from fast fluidization to dilute transport, as corroborated by the experimental points (the data at high velocities was transported from Fig. 7). 

However, a fluidized bed operating at a high gas velocity alone (> [7, ) does not make it a fast fluidized bed. A generally accepted definition of a fast fluidized bed is the coexistence of a dilute phase and dense phase regimes (see review by Rhodes and Wang, 1998). 

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