Semi-dilute concentration regime

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... 

The constant in Equation (5.112) cannot be readily evaluated using scaling theory. Our transformation applies equally well to the radius of gyration or the root mean square end-to-end length, only the numerical constant changes. We would like to be able to apply this idea to the role of concentration in semi-dilute and concentrated polymer regimes. In order to do this we need to define a new parameter s, the number of links or segments per unit volume ... 

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. 

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 ... 

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]. 

Although motional averaging might occur in ways other than that envisioned by Cates, temperature-jump experiments have yielded values of Tbr that indicate Tbr < in the region where the relaxation is nearly monoexponential, in agreement with Cates theory. In addition, Cates theory offers distinctive predictions for the concentration-dependencies of the viscoelastic behavior these allow the theory to be tested rather stringently. To obtain these predictions, we note that in the semi-dilute regime, the mean-field reptation time is L 4>, where 0 is the volume fraction of surfactant. Hence, from Eqs. (12-31) and... 

Finally, we may say that polymers in solution interact with one another and that this interaction which is observed by osmotic pressure in the dilute concentration domain, in good solvents, is related to the chain size. We shall later examine this relation in detail, and we shall also see that it extends to the semi-dilute regime (see Chapter 13). 

The membranes used to measure the osmotic pressure are impermeable to polyions but permeable to counter-ions. In spite of this, the osmotic pressure of polyelectrolytes in pure water and for rather small concentrations (but in the semi-dilute regime), is huge. Let us assume that the polyelectrolyte has been put in cell I (see Fig. 5.1). It partially ionizes but both cells must remain practically neutral. The counter-ions which, theoretically, can cross the membrane, are retained in cell I and a contact-potential difference appears at the boundary between the cells. Actually, it looks as if the counter-ions contributed like polyions to the osmotic pressure. Let C be the polyion concentration. We may write approximately... 

This broader correspondence led to the discovery of new scaling laws in particular, for the osmotic pressure. Then, Daoud, de Gennes, and Jannink who wanted to know how, in the semi-dilute regime, the radius of gyration and the screening length decrease with concentration, proposed a coherent set of simple laws to describe the situation. Experiments were systematically performed in order to test these laws and they produced favourable results. 

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 semi-dilute regime corresponds to the domain CXd 1. In this domain Ilfi is expected to depend only on the monomer concentration. As was pointed out in Section 2.1, the number of monomers of a chain is proportional to X11", and consequently, in this Kuhnian limit, the monomer concentration must be defined as the Kuhnian concentration. 

The threshold region at which overlap commences, signalling the transition from the dilute to the semi-dilute regime, would be expected to occur when the total volume of the chains Just fills the available volume. This occurs at a polymer concentration Ci (whose value must be comparable to that in the polymer coil), which is given by... 

This demonstrates that decreases slowly with the polymer concentration in the semi-dilute regime. Of course, in a 0-solvent, the corresponding value of y leads to i=0, i.e. Rg N. ... 

In the semi-dilute regime, the mesh size depends only upon the concentration of polymer, at least for polymers of high molecular weight. This seems intuitively reasonable because the mesh size should be determined solely by the total segmental concentration, being independent of chain length if that... 

Further experimental proof of the occurrence of interpenetration resides in the verification of the scaling laws for concentrated polymers, reviewed in Section 4.6.3.5. Scaling law theories assume that polymer chains interpenetrate in the semi-dilute regime and their predictions appear to be confirmed experimentally. 

The steric layer theory of Vincent, Luckham and Waite (1980) also provides reasonable semi-quantitative predictions of the onset of flocculation. According to this theory, the onset of flocculation marks the entrance into the semi-dilute polymer concentration regime (i.e. V2 = C2 ). Flocculation arises because there is a net decrease in free energy when two particles come together and displace into the bulk solution some of the free polymer molecules that are interpenetrating the stabilizing moieties. Restabilization is said to be associated with the onset of the so-called concentrated polymer regime (i.e. 

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