Critical overlap

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 )... 

As shown in Fig. 10, the shear strain is proportional to the relative displacement of two parallel aligned adjacent chains. Therefore, it seems plausible to assume that the maximum shear strain value /3=( -8b) at which fracture of the fibre is initiated will be related to a critical overlap length between adjacent... 

Figure 3.1.3a shows a normal urine organic acid profile. Panel represents an expansion of the same profile (5% of original abundance) to emphasize the underlying complexity of the urine organic acid fraction and the need to appreciate how informative findings could be hidden within a background of comparable intensity (see section 2.1.8.3). For this reason, exclusive reliance on computer-driven identification is not a safe option because visual evaluation is required, for example in the case of the critical overlaps discussed later in this chapter. 

Table 3.1.12 Critical overlaps, on species included in this table correspond to the most informative ion chromatograms to be displayed for visual inspection and resolution of the overlapping compounds ... 

The way to remove entanglements, viz. the manner in which topological constraints limit the drawability, is seemingly well understood and crystallization from semi-dilute solution is an effective and simple route to make disentangled precursors for subsequent drawing into fibers and tapes [ 17,18]. A simple 2D model visualizing the entanglement density is shown in Fig. 3. Here 0 is the polymer concentration in solution and 0 is the critical overlap concentration for polymer chains. 

Fig. 3 A simple 2D model envisaging how the entanglement density varies upon crystallization at decreasing polymer concentration, . is the critical overlap concentration for polymer chains...

Fig.9 SAN scattering curves of aqueous solutions at three different PDADMAC concentrations above the critical overlap concentration, c, without additional salt. The polymer concentration is given in monomol IT1 MW(PDADMAC)=100,000 g mol1 (from Ref. [42])...

Terpolymer solution behavior is dependent on compositional microstructure, polymer concentration, ionic strength, pH, temperature, and shear history. The viscosities of these low charge density terpolymers would be expected to be insensitive to changes in ionic strength below the critical overlap concentration, C above C intermolecular attractive forces might be expected to cause enhanced viscosities similar to those of hydrophobically associating copolymers. 

Table 6.1 Critical overlap concentration, C, for polystyrene coils in solution measured by energy transfer between carbazole- and anthracene-labeled polystyrene chains...

Fig. 4.2 Illustration of different concentrations of polymer solutions with respect to the critical overlap concentration C the dilute solutions (C < C, left) and the concentrated solutions (C > C, right)...

The territory of an expanded single coil is much larger than the volume the monomers actually occupy. Therefore, in comparison to the collisions of small molecules, the probability of mutual collisions between two coils is significantly enhanced. When such fluffy coils diffuse, the mutual friction yields a solution with a high viscosity. In the early history of polymer science, the high viscous polymer solution was misimderstood as a colloidal gel. However, polymer solutions are actually the molecular dispersions of long chains in the solvent molecules. With the increase of the polymer concentration, the coils start to interpenetrate into each other. We can define an illusive critical overlap concentration C, as illustrated in Fig. 4.2. Then, polymer solutions with the concentrations beyraid C are called concentrated solutions. 

However, we wish to note that near the critical overlap concentration, the coil size is actually dependent on the polymer concentration in the concentrated polymer solutions. 

At the critical overlap concentration, assuming that chain units are homogeneously distributed within each coil,... 

We know that the higher concentration C causes more interpenetration, leading to the decrease of With the reference of the critical overlap concentration C, we... 

The hydrophobic polyelectrolyte chain appears rigid due to the electrostatic repulsion along the chain. In the concentrated solutions, the electrostatic repulsion will be gradually screened due to interpenetration of polyelectrolyte chains (Dobrynin and Rubinstein 2001). According to (4.68), the critical overlap concentration for the transition from the dilute solution to the concentrated solution is... 

When C > C, similar to the treatment of neutral polymers in semi-dilute solutions, we can use the blob model to describe the polyelectrolyte segment holding the electrostatic repulsion in semi-dilute solutions. Assuming the blob size as a characteristic length to maintain the chain rigidity, with reference to the critical overlap concentration C, ... 

Observations from this viscosity study and previous studies with other gel systems suggest that there is a minimum polymer concentration for gelation for a particular polymer. Below this concentration, IMC may be overwhelming. For the Aldrich polyacrylamide, this minimum concentration appears to fall somewhere between 2000 and 4000 ppm. The minimum concentration is suspected to be related to the critical overlap concentration commonly used to describe the transition between a dilute solution and a semi-dilute solution. The critical overlap concentration for the Aldrich PAAM was estimated from its intrinsic viscosity (5.0 g/dl)... 

Figure 38 Proposed interaction of HPMC with increasing concentrations of SDS in water where CAC is the critical aggregation concentration, Cp is the concentration of the polysaccharide, c is the critical overlap concentration for the polysaccharide, and CMC is the critical micelle concentration. (Reprinted with permission from Ref. 163. Copyright 1995 American Chemical Society.)...

For combinations of surfactants and HPMC, Figure 38 shows that surfactant micelles create more intermolecular chain entanglement as the concentration of SDS increases than was seen in Figure 37 for SDS and HPC. In Figure 38, entanglement requires the polymer concentration, Cp, to exceed a critical overlap concentration, c (i.e., the polymer must be in the concentrated regime), or the polysaccharide tends to collapse and the viscosity drops. It is currently felt that at sufficiently high surfactant concentrations, both HPC and HPMC become fully extended and the lipophilic domains become enshrouded in surfactant micelles. 

Consider the case xas = Xbs = 1/2 where the dilute A and B chains are nearly ideal. Here the critical overlap concentration d> becomes NcP / o Below d> the chains still repel each other strongly (because of the three-body interaction of Fig. IV.9), and segregation cannot take place. However, as soon as d> > d>, we have full overlap and phase separation occurs. Thus,... 

Using the block copolymer information in Section 12.3.6.2, compare the interfacial concentration with the critical overlap concentration. Section... 

Figure 8 The zero-shear-rate viscosity of a polymer solution versus polymer concentration, showing the critical overlap concentration c. ...

---