Semi-dilute regime

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... 

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... 

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. 

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... 

Analysis of polyelectrolytes in the semi-dilute regime is even more complicated as a result of inter-molecular interactions. It has been established, via dynamic light-scattering and time-dependent electric birefringence measurements, that the behavior of polyelectrolytes is qualitatively different in dilute and semi-dilute regimes. The qualitative behavior of osmotic pressure has been described by a power-law relationship, but no theory approaching quantitative description is available. 

In the case of polymer solutions in the semi-dilute regime, the elastic scattered intensity is given by Omstein-Zernike (OZ) type equation [10, 73, 74]... 

In dilute solutions, the chains are far apart on average. When the polymer concentration c increases, there exists a concentration c at which the chains begin to overlap. This is the onset of the semi-dilute regime. It may write ... 

The semi-dilute regime exists in the limits of infinite chains, since the singularity is independent of chain size [61]. 

In Ref. [76] we showed that the necklace conformations can exist also in the presence of counterions and that they exhibit a variety of conformational transitions as a function of density. The end-to-end distance was found to be a non-monotonic function of concentration and showed a strong minimum in the semi-dilute regime. Here we have found for short chains a collapse of each chain into a globular stable state which repel each other due to their remaining net charge. The focus of a more recent work was to analyze, by extensive computer simulations in detail, three possible experimental observables, namely the form factor, the structure factor and the force-extension relation, which can be probed by scattering and AFM techniques [77]. The details of the simulation techniques can be found in Refs. [76, 77]. 

Experimentally, good solvent conditions have been observed [22,23,27,28, 34,35]. On the other hand, none has been reported for the prediction of the theta condition, y = 101, whereas the prediction of poor solvent conditions giving rise to y > 3 has been reported. These all have y < 20 except for two they are poly(methyl acrylate) at lower temperatures [34] and poly(dimethyl siloxane) [24]. Others have failed to reproduce them since. A caveat needs to be raised with these results. Since the semi-dilute regime is so narrow in r before the collapse state sets in whereby the power exponent is commonly deduced for a r range less than one full decade hence, the r scaling is at best qualitative in the static characterization. 

Fig. 25 Loss tangent (tan 8) for PXcMA polymers on 0.01 M HCl at 25 °C. The symbols are the same as Fig. 2. The dashed lines are guidelines for each polymer and the solid line shows the slope of tan8 at high 77 or at small A tan8 A 10 — 772. The values represent averages over three wave vectors with error bars amounting to about 30% omitted for clarity. In both plots, PHcMA data in the dilute regime are not included since they are scattered much more than the rest, whereas the conclusion drawn for the semi-dilute regime is not affected by the omission...

Fig. 26 Conjecture of chain conformation is illustrated for PXcMA polymer monolayers at AIW (a) in the dilute regime and (b) in the semi-dilute regime. Gray balls represent hydrophilic carboxyl groups and black lines represent hydrocarbon chains...

We close this section by examining the status of applications of these methods to polymer monolayers. Initially, ISR was used to probe the 2D nematic state of phthalocyaninatopolysiloxane, descriptively called a hairy rod , dispersed in eicosanol [ 149], and subsequently applied to a set of poly(f-butyl methacrylate) in the semi-dilute regime and beyond [150]. In the semi-dilute regime, the surface viscosity is found to scale linearly with molecular weight, which is in good accord with the results of Sacchetti et al. [134]... 

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