Overlap concentration definition

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. 

Really, there is no established consensus about the definition of the overlap concentration. In fact, Hager and Berry [56] replace the factor 3/4tt by 1/5, and Graessley [57] by 1/8, while Ying and Chu [58] drop this factor and use (0)/4 for (0). Thus we have to be careful in comparing reported overlap concentrations. 

On the contrary, when the concentration is larger than the overlap concentration, the solution becomes more homogeneous and the study of the function H(q) more revealing. In fact, let us consider the definition... 

At low concentrations, these spheres or cubes are separated from each other. As the concentration c (mass concentration expressed in g/mL, for instance) increases, they become congested and eventually touch each other. At the so-called overlap concentration (c ), the whole volume of the solution is packed with these spheres. The overall concentration of the solution is equal to the concentration in the sphere given by Eq. 1.107 at the chain overlap. Thus c More quantitative definitions of c are usually used ... 

Secondly, we tried to avoid the crossover region (C - C ) for the purpose of our present studies. Since the overlap concentration C is arbitrarily defined as the concentration at which polymer coils begin to overlap. There are several definitions based upon different viewpoints, such as C = 1/[q] with [q] being the intrinsic viscosity of the polymer... 

The concept of screening survives in semidilute conditions. At the overlap concentration, by definition, the average distance between the chains is proportional to the Flory radius (Rp) in a good solvent. Therefore we can guess the following form for ... 

Sigma (a) bonds Sigma bonds have the orbital overlap on a line drawn between the two nuclei, simple cubic unit cell The simple cubic unit cell has particles located at the corners of a simple cube, single displacement (replacement) reactions Single displacement reactions are reactions in which atoms of an element replace the atoms of another element in a compound, solid A solid is a state of matter that has both a definite shape and a definite volume, solubility product constant (/ p) The solubility product constant is the equilibrium constant associated with sparingly soluble salts and is the product of the ionic concentrations, each one raised to the power of the coefficient in the balanced chemical equation, solute The solute is the component of the solution that is there in smallest amount, solution A solution is defined as a homogeneous mixture composed of solvent and one or more solutes. 

A large amount of work has been reported on potentially bis- or trishomocycloprope-nium ions. Many reviews exist including those bv Winstein " , Story and Clark Williams and Kurtz, Koptyug Hogeveen and Kwant, Lenoir and Siehl and a particularly full account by Barkhash The topic also overlaps that of the pyramidal cations and related boranes In this present section we will again not give an exhaustive account but concentrate on a few key systems where there has been some recent definitive work reported. 

In the definition of the overlap volume fraction 0, the pervaded volume V is taken at, since polymer size and hence its pervaded volume may change with concentration. 

As no specific standard methods cover the full range C4-C44 for TPH, a combination of GRO and DRO can be used to assess TPH. The concentration of C4-C10 from the GRO method and that of C12-C44 from the DRO method can be summed. The overlap in ranges of products can be accounted for by averaging the concentration of C10+ determined in the GRO method with the Cl2- in the DRO method. These two values should correlate within 10% and this average value can be added to the GRO and DRO to provide a TPH value. Simply adding the GRO and DRO measurements is only a guide but not a definitive measure of TPH. 

In fact, the decisive experimental proof of the macromolecular hypothesis has been given by osmotic pressure measurements in very dilute solutions. A second fundamental contribution is more recent. It deals with the study of systems of strongly overlapping chains. The observations made by Noda and his collaborators (1980) definitely showed that, in this case, the osmotic pressure dependence of the solute concentration follows a universal law the crossover with the dilute state is also universal. These results gave an experimental proof of the principles which constitute the basis of the modern theory introduced by de Gennes and des Cloizeaux. 

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... 

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