Overlap threshold

Zero shear viscosities have been determined in solution over a wide range of concentrations with a cone-plate Rheometrics Stress Rheometer. For linear macromolecules, the viscosity is proportional to c below the so called "entanglement concentration", c above c, is proportional to c. However, the viscosity will rise steeply at some concentration below c in the case where particular interconnections are formed at the concentration at which the molecules come into contact with one another. Ideally this will be the overlap threshold c. Below c, the molecules may associate partially but cannot form a network continuous over the entire sample space. Above c, plastic flow will require separation and... 

The liquid-like order present in the pair correlation function manifests itself as a peak in the static structure factor (S(q)). The scaling of the position qm of this maximum with the density has attracted much attention in the literature [40, 51-53]. Scaling arguments suggest [35, 42, 49, 51] that qm obeys the relation qm p1/3 for dilute solutions and qm pv/(3v 1) for semidilute solutions. Here v is the scaling exponent for the end-to-end distance, i.e., RE hT. The overlap threshold concentration is estimated as p N1 3v. As a conse-... 

Fig. 2. Determination of the overlap threshold of dextran T 2000 as buffer additive in phosphate 50 mmol/1 (pH 2.0) with the CE instrument. Conditions capillary length 50-60 cm pressure 1.5 psi...

In Fig. 11 the influence of the molecular mass of the sieving medium is demonstrated. There is no difference between dextran T 500 and T 2000 at identical concentration. Less efficient separations are only observed with dextran T 70. The mass concentration here is just around the overlap threshold. For practical reasons it is advisable to use the sieving polymer which gives the lowest viscosity in the buffer solution at the required concentration. 

A wide variety of polymeric materials can be employed for size separations. It appears that the molecular weight and concentration of the polymer is more important that the polymer type itself. Once the polymer concentration is greater than the overlap threshold, the porous matrix is defined and reproducible. It is best if the polymer does not interact with the macromolecules being separated. 

According to polymer theory, with a semidilute polymer solution above an overlap threshold concentration, c, ... 

In the NP+LMWL system (a good solvent), the molecular branches are repelled from each other due to the long-range interactions (cf. subsection 3.1.1), but the crosslinks bring them into contact. The situation that arises is similar to the overlap threshold on the dilute/semidilute solution interface provided that the length of the macromolecules in them is equal to that of the gel branch in the same solvent. 

A fundamental distinction exists between dilute polymer solutions where the coils are separate (Hg. III.3a) and more concentrated solutions where the coils overlap (Hg. in.3c). At the overlap threshold (c = c ) the coils begin to be densely packed. Clearly this threshold is not sharp it is more properly defined as a region of crossover between regimes (a) and (c), but the scaling properties of c are essential. We expect c to be comparable with the local concenhation inside a single coil. In a ery good (athermal) solvent this implies ... 

What we have is a set of closely packed coils sealed together by the crosslinks. The situation is reminiscent of the overlap threshold in semi-dilute solutions (Chapter III). Thus, the gel automatically nuummns a concentration c proportional to c. ... 

The essential requisite of the scaling approach is an assumption that both below and above the overlap threshold, a polymer solution is characterized by a single characteristic length scale in dilute solution, this is the size of the polymer coil,... 

Regions Ig and correspond to dilute solution of Gaussian (in 0 solvent) and swollen (in good solvent) polymer coils, respectively. The polymer concentration is below the overlap threshold, c[Pg.57]

When the polymer concentration is beyond the overlap threshold the solution of the star polymers is envi-... 

In table 1 we show total energies, ionization potentials and TT-band widths as a function of the number of neighbors and the overlap threshold applied in order to der monstrate the internal accuracy of our crystal orbital calculations and therefore also the direct comparability to standard molecular calculations. The latter is fur-... 

Next we consider the other limit, x > 1, associated with an entangled semi-dilute solution far above the overlap threshold. Here, the degree of polymerization N must become irrelevant, as is also demonstrated by the data in... 

Viscoelustic propsrties usually occur at conditions, where the lengths of the rod-shaped particles are larger than their mean distance of separation. The overlap threshold of anisometric micelles can be very low, and it is often observed in the millimolar range. Semi-dilute solutions of entangled micelles show rather simple scaling laws the osmotic pressure n and the equifibrium shear modulus Gg increase as a function of the volume fraction > 2,i8o,i8i. 

A simple illustration is an estimation of the overlap threshold c. We expect c to be comparable with the local concentration within a single chain, thus... 

Here c denotes the overlap threshold of a semidilute solution of free chains of polymerization degree N and radius Rp i.e., c = NIR. The swollen... 

The lowest critical concentration, C, of polymer solutions is associated with the overlap threshold of the coils ... 

First it is important to note that the overlap threshold for a brush in a collapsed state is smaller than that found in a good solvent. Since the dimensions of an isolated collapsed globule scale as iZ a instead of a in a good... 

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