Cross-over concentration

Note 2 Cross-over concentration is defined as a range because different measurement techniques give different values. The symbol c refers usually to amount concentration, but in polymer science it is generally used for mass concentration. 

Thus, JcR is expressed as a universal function of CM. Moreover, it was reported that the cross-over concentration from dilute to entangled regions for J is about 5 times higher than that from dilute to semidilute regions for 7 at the constant molecular weight[5]. 

Figure 5 indicates that the higher the molecular weight, or the lower the added-salt concentration, the lower the cross-over polymer concentration between the two regions is. Although the polymer concentration and molecular weight dependences of Jo are different from those of non-ionic polymer solutions, the order of cross-over concentrations is reasonable. 

If the concentration becomes suffidently large, the fluctuations become small and can be treated by a simple mean field theory. Such a solution is called concentrated. As we shall show later, the cross-over concentration... 

It may be noted that this classification is conceptual usually, the crossover between various regimes is not sharp, and experimentally it is often difficult to identify the cross-over concentration. 

Note that condition (35) implies that for sufficiently large concentrations, the blobs desorb. Using equations (10) and (35), we find the cross-over concentration C for desorption of blobs... 

Theoretical analysis of the process of adsorption and desorption in the framework of the concept of phase transition was already discussed. The phase diagram in reduced coordinates e/Ecrit = eN and( )/( ) =( )N have been obtained, where is the energy of the chain attraction to the surface, cr is the critical energy, N is the number of monomeric units in the chain, and ( ) and ( ) are concentrations of polymer in solution and cross-over, respectively. The latter work is important because cross-over concentration is considered — essential for adsorption from semi-diluted solution (see section 1.6). 

A generalized approach gives the possibility to consider the transition conditions from semi-dilute to concentrated solutions. For statistical coils, the critical concentration of overlapping (cross-over concentration) is expressed as... 

Detailed osmotic studies by Mandel et al. [140] on NaPSS yield an exponent of 9/8 at low concentration and 9/4 at high concentration of the osmotic pressure-concentration power law (n c ) which again was interpreted as a dilute-semidilute concentration transition. A recent literature study [66] confirmed the experimental scaling exponents but clearly demonstrated that the cross-over concentration does not depend on the molar mass of the polyions. 

Comparison to the MC-simulations and theory indicates that the osmotic pressure is most likely governed by intermolecular interactions and not by changes in the polyion conformation. Intermolecular interactions at low concentration correspond to an excluded volume of k at higher concentration compared to the hard-core diameter d. The cross-over concentration, as stated earlier, corresponds to k = d or, equivalently, to = 1. 

C cross-over concentration between di- r modulus of the vector position... 

The reason for the ultramicrochemical test was to establish whether the bismuth phosphate would carry the plutonium at the concentrations that would exist at the Hanford extraction plant. This test was necessary because it did not seem logical that tripositive bismuth should be so efficient in carrying tetrapositive plutonium. In subsequent months there was much skepticism on this point and the ultramicrochemists were forced to make repeated tests to prove this point. Thompson soon showed that Pu(Vl) was not carried by bismuth phosphate, thus establishing that an oxidation-reduction cycle would be feasible. All the various parts of the bismuth-phosphate oxidation-reduction procedure, bulk reduction via cross-over to a rare earth fluoride oxidation-reduction step and final isolation by precipitation of plutonium (IV) peroxide were tested at the Hanford concentrations of... 

In a traditional, two-way cross-over study, blood samples are taken at predetermined times (e.g., 0.5, 1, 1.5, 3, 6, 9, 12, 24, and 36 hours after dosing). The samples are assayed for drug (and if necessary metabolites). Fortunately, analytical methods, especially HPLC, are now available that make the quantification of many drugs in blood or serum convenient, rapid, and relatively inexpensive. The method selected should normally have an acceptance precision so that concentrations of drug of one tenth Cmax can be reliably quantified. 

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