Polymer coil overlap

However, polymer coils overlap and dominate most of the physical properties of semidilute solutions (such as viscosity). Thus, adding a very small amount of polymer to a solvent can create a liquid with drastically different properties than the solvent. This unique feature of polymer overlap is due to their open conformations. Linear polymers in solution are fractals with fractal dimension I) < 3. In semidilute solutions, both solvent and other chains are found in the pervaded volume of a given coil. The overlap parameter P is the average number of chains in a pervaded volume that is randomly placed in the solution ... 

The use of high-molecular-weight thickeners As discussed above, high-molecular-weight materials such as HEC or xanthan gum, when added above a critical concentration (at which polymer coil overlap occurs), will produce very high viscosity at low stresses or shear rates (usually in excess of several hundred Pas), and this wiU prevent sedimentation of the particles. 

For the coil-overlap region (where polymer coils overlap) the viscosity is... 

Solutions with a-7 PMMA revealed pseudoplastic behavior at concentrations above 0.1 g/dl, as shown in Figure 3. At concentrations above 1 g/dl the solutions are jelly. These elastic properties may be attributed to entanglements. The values of tn] indicate that at concentrations exceeding 0.05-0.1 g/dl l the polymer coils overlap. From these measurements there is no evidence for a specific association. 

Critical concentration of polymer coil overlap Drag coefficient... 

Polymer-polymer interactions are neglected in the foregoing sections. This is justified for very dilute solutions. Because a polymer molecule contains mostly solvent in its coil, the molecules start to overlap at a relatively low concentration. At that concentration, the separation distances between the coils approach the dimensions of the coils themselves. A solution in which the polymer coils overlap is referred to as a nonconnected network. Taking 4/3( / g) as the coil volume, the concentration c (in mol monomer dm" ), at which the nonconnected network starts to form is given by... 

Another method of reducing creaming or sedimentation is to induce weak flocculation in the emulsion system. This may be achieved by controlling some parameters of the system, such as electrolyte concentration, adsorbed layer thickness and droplet size. These weakly flocculated emulsions are discussed in the next section. Alternatively, weak flocculation may be produced by addition of a free (non-adsorbing) polymer. Above a critical concentration of the added polymer, polymer-polymer interaction becomes favourable as a result of polymer coil overlap and the polymer chains are squeezed out from between the droplets. This results in a polymer-free zone between the droplets, and weak attraction occurs as a result of the higher osmotic pressure of the polymer solution outside the droplets. This phenomenon is usually referred to as depletion flocculation [59] and can be applied for structuring emulsions and hence reduction of creaming or sedimentation. 

Samples of protonated PDMS-h with Mw 22,500 47,700, and 79,900 and polydispersity Mw/Mn 1.03 were synthesized and characterized at Max Planck Institut fClr Polymerforschung, Germany, and PDMS-d (M = 27600 and 75600, M, M < 1.11) was purchased from Polymer Standards Service GmbH, Mainz, Germany. All solutions were prepared at the volume fraction of the polymer equal to the concentration of polymer coil overlap (< ) 0.1372, 0.0941, and 0.0727 for M > 22,500 47,700, and 79,900, respectively) which is practically indistinguishable from the critical concentration of phase demixing [3]. 

A distinction must be drawn between gels which are diluted first and then cross-linked and those which are cross-linked first and then swollen (Section F below). In the former, the network strands have their average random configurations in a more or less unstrained state (except perhaps for gels linked at quite high dilution, where the polymer coils overlap each others domains only to a limited extent). In the latter, however, the strands are all extended beyond their normal root-mean-square lengths, in proportion to the cube root of the swelling factor p/c = uj. ... 

We consider the approach of two polymer coils in dilute solution until the two coil domains overlap to some extent, as shown in Fig. 8.11. According to... 

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... 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The interpretation of A becomes clearer when two plates, originally at very small distance from each other, are separated. At a certain separation, equal to 2A, polymer penetrates into the gap. In dilute solutions, where the chains behave as individual coils, A is expected to be of the order or r, the radius of gyration. However, at concentrations where t e coils overlap, the osmotic pressure of the solution becomes so high that narrower gaps can be entered, and A becomes smaller than Tg. 

Figure 3. Depletion thickness A for four chain lengths as a function of polymer concentration. The arrows indicate the solution concentration where the polymer coils begin to overlap, x = 0.5, hexagonal lattice.

At concentrations greater than this value there will be polymer chains at any position in the solution but there will still be large fluctuations in local concentration with position because the density of each polymer coil is greatest at the centre of mass of the coil. The interactions between segments of neighbouring coils due to overlap progressively screen out... 

Consider two particles with adsorbed layers approaching each other. The adsorbed layers on the core particles first begin to overlap at the outermost extreme of the fringe, at which the surface exerts the least influence. As a first approximation, then, the initial encounter between two approaching core particles is comparable to the approach of two polymer coils in solution. In Chapter 3, Section 3.4a, we saw that the concept of excluded volume could be... 

Boss, et al., fitted Gq. (17) to their data vs. vdi enabling them to determine fp and D . At solvent concentration approaching vdiI = 0.95, the data revealed an enhancement above the value predicted by Eq. (17) as fitted to the lower-concentration data. The authors argued that under these circumstances macroscopic inhomogeneities in concentration (and hence in the free-volume distribution) should exist and enhance the diffusivity. Above v > 0.99 the polymer coils no longer overlapped substantially, depriving the solvent molecules of a set of obstacles fixed with respect to the laboratory, and solvent diffusion became related principally to intrinsic viscosity. 

PGSE measurements on polyethylene oxide) in aqueous dextran solutions were performed by Brown and Stilbs A2) as function of the concentrations of both polymers. The results for D(PEO) depend on the product of the concentration and the intrinsic viscosity of the dextran (host) component, and suggest that coil overlap in the concentrated host solution is the principal impediment to PEO diffusion. 

Since this chapter is concerned with block copolymers in dilute solution, it is useful to include a definition of the dilute regime for polymer solutions in the Introduction. This regime extends up to a volume fraction above which swollen coils overlap (de Gennes 1979) ... 

What is the size of the overlapping volume The complete independence of the constant in Equation 1 from the degree of polymerization shows that the overlapping volume always consists of the same portion of the volume of the polymer coil (12). This can be easily understood by assuming that two polymer coils are able to migrate nearly unhindered through each other. Then the mean depth of permeation and, therefore, the time of overlapping is determined only by the statistics of the free Brownian motion. Equation 1 is based on this assumption. 

The calculation proceeds as follows (13). First, we determine the frequency of encounter of two polymer chains following Smoluchowskis treatment. After two chains have come into contact with each other, we allow them to move freely in all directions so we obtain a mean volume and a mean lifetime of the overlapping. This mean volume is given by about 1/10 of the diameter of the polymer coil. This treatment is illustrated in Figure 4. 

Sufficiently dilute polymer solutions may be viewed as systems in which islands of polymer coils scattered in the sea of a liquid solvent occasionally impinge and interpenetrate. By this way, the spatial distribution of chain segments in them is quite heterogeneous and undergoes appreciable fluctuations from time to time. As the polymer concentration increases, the collision of the islands becomes more frequent and causes the chains to overlap and entangle in a complex fashion. 

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