Polymer overlap concentration

The upper solid curve in Fig. 3a is a plot of Eq. (4) with /2//1 =0.11. The equation for the non-adsorption mixture fits the data well. Our previous study of the same system also suggests that the PEP polymer does not adsorb onto the colloidal spheres. The two end-functionalized polymers are found to be partially adsorbed onto the colloidal surfaces. The lower solid curve in Fig. 3a is a fit to Eq. (8) with ao = 0.21. In the range 0 < a < 24, ao is found to be independent of the molar ratio u. In this range of a , the overall polymer concentration P2 is below the polymer overlap concentration. The zwitterion-PEP data can also be fitted to Eq. (8) with a constant ao = 0.18. The dashed curve in Fig. 3b is a plot of Eq. (8) with a = 1 (complete adsorption). One can immediately see from Fig. 3 that the measured y(u ,ao) for our end-functionalized polymers lies in-between the non-adsorption and complete adsorption curves. 

Experimental work on the determination of the depletion layer thickness commenced in this period. The depletion thickness S of polystyrene at a nonadsorbing glass plate was measured using an evanescent wave technique by Aflain et al. [120]. The value found for S was indeed close to the radius of gyration of the polymer. Ausserre et al. [121] measured the depletion thickness of xanthan (a polysaccharide) at a quartz waU below and above the polymer overlap concentration. In dilute solutions, below overlap, S was close to the radius of gyration of... 

In this section we consider the phase behaviour of dispersions containing spherical colloids and interacting polymer chains in a common solvent. For small polymer-to-colloid size ratios, q < 0.4, the relevant part of the phase diagram hes below the polymer overlap concentration ((j>p < 1). Then accounting for interactions between the polymers is not essential to properly describe the phase diagram and it is still sufficient to approximate the polymer-induced osmotic pressure by the ideal gas law as assumed within free volume theory [18]. However, for >0.4, the polymer concentrations where phase transitions occur are of the order of and above the polymer overlap concentration in that case interactions between the polymer segments should be accounted for. 

Above the polymer overlap concentration we enter the semi-dilute regime (Fig. 4.7c). The length scale over which the polymer segments are correlated is denoted as correlation length Below overlap (p<(p ) this quantity is the coil size Rg which depends only on M (and solvency). Above overlap

q> ) we have the famous De Gennes scaling law [40]... 

Solubility characteristics of high molecular weight polymers can be problematic. Long dissolution times and fish eyes in the final solution are often observed. Associating polymers have additional complexity. The solution history of the polymer can affect inter- and intramolecular associations which influence dissolution rates. Preparing solutions above the polymer overlap concentration, C, requires patience and care. The design of the polymer structure can greatly affect solubility and dissolution characteristics. 

In HOPC, a concentrated solution of polymer is injected. The concentration needs to be sufficiently higher than the overlap concentration c at which congestion of polymer chains occurs. The c is approximately equal to the reciprocal of the intrinsic viscosity of the polymer. In terms of mass concentration, c is quite low. For monodisperse polystyrene, c is given as (4)... 

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 )... 

The relevant part of the phase diagram (x > 0) is shown in Fig. 38. The c-x-plane is divided into four areas. The dilute regime I and I are separated from the semi-dilute regimes III and II, where the different polymer coils interpenetrate each other, by the so-called overlap concentration... 

The s.a.n.s. experiments were carried out using the D17 camera at the I.L.L., Grenoble. Data were collected at two wavelengths, 0.8 and 1.4 nm at a sample to detector distance of 1.8 m. The overlapping spectra were combined to give a sufficiently wide Q range to enable the data to be numerically inverted to obtain the density distributions. The latex dispersions were prepared at a solids concentration of 4% and polymer solution concentrations between 200 and 300 ppm. 

A rough measure for the overlap concentration 4>ov is that volume fraction of polymer at which close-packed spheres with radius rg just touch. Then 4> =0.74 rl /(4Trr /3). Taking r — A((J> - 0) ... 

This means that the expansion factor depends only upon the end-to-end length R2(Ms) of chains of molecular weight Ms relative to the chain in 6 conditions. We can derive an effective concentration of the chains since this is related to the overlap concentration of the polymer of molecular weight Ms. ... 

Non-dilute solutions also allow for theoretical descriptions based on scaling theory [16, 21]. When the number of polymer chains in the solution is high enough, the different chains overlap. At the overlapping concentration c , the long-scale density of polymer beads becomes uniform over the solution. Consequently c can be evaluated as... 

Branched polymers can also be dissolved at fairly high concentrations. Because of the higher segment density in the isolated macromolecules the overlap concentration will also be increased. For this reason the semi-dilute regime of branched polymers may in some cases be larger than for linear chains, say about 20% or more. Clearly, however, a full interpenetration, as was assumed for flex-... 

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. 

In order to resolve these challenges, it is essential to account for chain connectivity, hydrodynamic interactions, electrostatic interactions, and distribution of counterions and their dynamics. It is possible to identify three distinct scenarios (a) polyelectrolyte solutions with high concentrations of added salt, (b) dilute polyelectrolyte solutions without added salt, and (c) polyelectrolyte solutions above overlap concentration and without added salt. If the salt concentration is high and if there is no macrophase separation, the polyelectrolyte solution behaves as a solution of neutral polymers in a good solvent, due to the screening of electrostatic interaction. Therefore for scenario... 

Surface pressure was measured as a function of surface concentration for monolayers of linear and cyclic polydimethylsiloxanes. In addition, the comparisons of linear and cyclic polymer above the overlap concentration c lead to the surprising conclusion that even for three-dimensional semi-dilution solutions, the ratio c/c was not a universal reduced concentration [174]. 

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...

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