Crossover concentration

The crossover concentration calculated for polystyrene with this expression is in the range of the reported values(12,13) Finally, the above discussion points out that c is not a sharp dividing line It is therefore not physically meaningful to scale the concentration axis in a plot of D/Dq versus concentration divided by the crossover concentration Equation 13 demonstrates that it is better to plot the relative diffusion coefficient vs the weight concentration of the polymer, as was done by Munch et al (12) ... 

In subsequent work, ordering in solutions of the same matched diblock and triblock spanning a broader range of volume fractions, 0.1 < < 0.4, was explored (Hamley et al. 1997). For liquid-like and SAXS showed that there was no inter-micellar order in the liquid. Above a crossover concentration 0.2, ordering of micelles was shown by the presence of a structure factor peak. The ordered micellar structure, identified as hexagonal-packed cylinders for more concentrated solutions, persisted up to an order-disorder transition located from a discontinuity in the... 

To determine the concentration at the "crossover" point (where stable and unstable regions of the front intersect), the method of Leavitt (4) can be used, provided the carrier gas is essentially unadsorbed, and axial diffusion forces and thermal conductivities are negligible (see the aforementioned reference for the corrections for these latter effects). For such a case, an energy balance across the stable and unstable portions of the front, along with relations to describe the equilibrium loadings at the extrema of these fronts (for example, the LRC model) can be combined to solve for the crossover concentration. The energy balance across each front takes the form ... 

For solutions in which C > C, overlapping of domains occurs. Solutions whose concentration lies in the range C < C < C are said to be semidilute, while concentrated solutions are those for which C > C. The crossover concentration from semidilute to concentrated is estimated as (3)... 

The crossover concentration remained constant or decreased as the concentration of the toxic cation increased. 

Recall from Chapter 5 that the crossover concentration (p Ki jb [Eq. (5.36)] denotes the boundary between semidilute and concentrated solutions. For 0 > 0 chains are nearly ideal in concentrated solutions, whereas for 0 < 0 chains are swollen on intermediate scales. Network modulus and equilibrium swelling depend on the relative value of preparation and fully swollen concentrations (0o and l/Q) with respect to the crossover concentration 0. Since the swollen concentration is always lower than the preparation concentration (l/Q < 0o) there are three... 

In Fig. 7.17, the dry modulus of various PDMS networks is plotted as a function of their equilibrium swelling in toluene. A single curve results for both model networks (open symbols) made by end-linking linear chains with two reactive ends and networks with intentionally introduced defects in the form of dangling ends (filled symbols) made by end-linking mixtures of chains with one and two reactive ends. The data are fit to Eq. (7.91) as the solid lines in Fig. 7.17 and their intersection determines the crossover concentration 0 = 0.2, which is typical for good solvents. 

Fig. 15 Normalized diffusion coefficients D(q 0)/Do as functions of the volume fraction for PMMA-grafted silica particles DP 150 open square) and DP 760 (open circle) in CCL4 (good solvent for PMMA) at 2(FC [208]. The different crossover concentrations in the dynamics are indicated by solid and vertical arrows. The inset shows in magnification the area bounded by the rectangle around a normalized diffusion coefficient value of 1 in the main plot, using the same colors and symbols (solid ) for the two systems. Lines are drawn to guide the eye...

Figure 8. Results of a geochemical model investigating the effects of pH buffering by acetic acid-acetate on the response of the carbonate system to changes in pCO. Increases in the pC02 of calcite-saturated solutions will promote calcite undersaturation for initial concentrations of acetic acid up to approximately 0.06 m (3600 mg/I). This is called the crossover concentration. (Reproduced with permission from Ref. 7. Copyright 1989 Elsevier).

Figure 9. Temperature dependence of the acetic acid crossover concentration for calcite, 0.5 m NaCl concentration, and a pC02 change from 0.1 to 1.0 atm. Crossover concentration (see Fig. 8) decreases with increasing temperature as a result of the decrease in the pK of acetic acid. (Reproduced with permission from Ref. 7. Copyright 1989 Elsevier).

To determine the critical solvent concentration, o, at which the diffusivity changes to a reptative mode, the following argument was applied. The characteristic crossover concentration for a change from a concentrated to a semi-dilute solution for polymer melts [63] is used as an initial estimate for uf ... 

Long-chain molecules in dilute solution influence a region of space large with respect to the sum of the volumes of the individual mers. This cc ept was expressed in terms of the pervaded volume, Vp = (4jt/3) R, in Chapter 5. When the sum of the pervaded volumes of the solute macromolecules equals the macroscopic solution volume, the solution can no longer be considered dilute. It is convenient to define a crossover concentration (c ) to the semidilute regime as ... 

It will be shown that many of the properties of polymer solutions in the semidilute regime depend in a detailed way on the value of the crossover concentration, c. ... 

The crossover concentration c scales with the total number of monomer units N in the system as the individual polymer chains are indistinguishable from each other at these higher concentrations. Thus, measures such as (r ) and even M become irrelevant for the system and are in effect replaced by the mesh parameter... 

Adsorption of polymers on adsorbing particles of varying sizes is concentration specific. The analysis of such adsorption is very difficult, and to describe the experimental data, the first crossover concentration, where macromolecular coils begin to overlap and interact, can be used. Therefore we can use the simple criterion of transition from dilute to semi-dilute solutions7 ... 

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