Dilute regime

Mattoussi H ef a/1998 Properties of CdSe nanocrystal dispersions in the dilute regime structure and interpartiole interactions Phys. Rev. B 58 7850... 

The scaling analysis, mentioned in Sec. IIIB, predicts that one should observe a different kind of distribution in the dilute regime for chains which are smaller than the blob size [32] and thus behave essentially as isolated chains. These chains, which are fully swollen and may slip through the network made up from the chains of average size (L) without being seriously... 

Since the power 7 is easier to detect in two than in three dimensions, the first MC study [62] sampled a two-dimensional MWD in a range of temperatures (that is, of (L)), so that a change in the degree of interpenetration should trigger a crossover from dilute to semi-dilute regime at some density 0. Evidently, indeed, from Fig. 4, the MWD follows the form of Eq. (16). At 0 one observes a power 7eff 1.300 0.005 which comes closely to the expected one. Above 0 one finds 7eff —> 1, and the distribution (11) becomes relevant. 

An important quantity whieh has been frequently studied is the mean ehain length, (L), and the variation of (L) with the energy J, following Eq. (12), has been neatly eonfirmed [58,65] for dense solutions (melts), whereas at small density the deviations from Eq. (12) are signifieant. This is demonstrated in Fig. 6, where the slopes and nieely eonfirm the expeeted behavior from Eq. (17) in the dilute and semi-dilute regimes. The predieted exponents 0.46 0.01 and 0.50 0.005 ean be reeovered with high preeision. Also, the variation of (L) at the threshold (p, denoted by L, shows a slope equal to... 

The simulation data, represented in Figs. 27(a-c), confirms Eq. (57) in the region of weak /tot BN < 4) for A = 8, 16, 32 at dilute regimes whereby the values for Dq have been measured for the same system in equilibrium [20] (cf. Sec, VIIB). Substantial discrepancies emerge only for high density of the medium. 

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... 

Figure 3 for four chain lengths, expressed as the number of segments per chain, r. As expected, in dilute solutions A is independent of In this dilute regime, A is approximately proportional to the square root of the chain length for not too short chains A/l = 0.56 (Sr - 2), which is of the order of rg/1. Note that for a random flight chain rg/1 = /r/6 = 0.41 Sr, and a lattice chain is expected to be slightly more expanded. 

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. 

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. 

At present, there are only a few experimental results known on the osmotic modulus of randomly branched macromolecules or randomly cross-Hnked chains in the semi-dilute regime. One possible explanation for this lack of data may be based on the prejudice that the universaHty predicted by de Geimes [4] for Hnear chains will hold in the same maimer also for branched materials. In particular it is expected that the individual characteristics of the macromolecules are lost due to the strong overlap of the segments from different macromolecules. The following data, mainly from the author s own research group, revealed however, that the characteristics of the special architectures are not lost. 

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. 

Following the pioneering experiments of Lyklema and van Vliet [29], Mondain-Monval et al. [30,31] have measured the repulsive forces between polymer-covered liquid interfaces in the low interaction regime (force/radius < 10 " N/m). They used two different force measurement techniques, i.e., MCT and TFB techniques. All the experiments were performed in the dilute regime (pb where (pl is the dilute... 

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. 

Attempts to measure the depletion force in nonadsorbing polymer medium with an SEA have failed essentially because measurements are hindered by the slow exclusion of the polymer from the narrow gap due to the large viscosity of the polymer solutions. However, depletion forces have been measured in solutions of living polymers in a semi-dilute regime by Kdkicheff et al. [50]. The... 

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