Polymers chain overlap parameter

There have been many studies of hydrophobic crosslinking. For example, Flynn40 produced a series of poly (acrylamides) (PAM) and recorded the low shear rate viscosity as a function of the chain overlap parameter. This was performed for a range of molecular weights and concentrations. This procedure was then repeated with the same polymer backbone but with the addition of differing concentrations of alkyl side chains which give rise to hydrophobic association (HPAM). A comparison between hydrophobe and non-hydrophobe polymers is shown in Figure 5.30. 

The chain overlap parameter has been very successful at superimposing the data from the systems without hydrophobic modification, producing the continuous curve. However, it is clear from Flynn s work that once the hydrophobes are introduced into the polymer the viscosity rapidly increases at lower values of the chain overlap parameter. Increasing the mole percentage of hydrophobes also increases the viscosity at lower values of the chain overlap parameter. The position and number of the hydrophobes on a chain are important in determining the structure that forms and the onset of the increase in viscosity. The addition of side chains to hydroxyethyl cellulose modifies the network modulus as a function of concentration. This is discussed further in Section 2.3.4. 

The non-aqueous system of spherical micelles of poly(styrene)(PS)-poly-(isoprene)(PI) in decane has been investigated by Farago et al. and Kanaya et al. [298,299]. The data were interpreted in terms of corona brush fluctuations that are described by a differential equation formulated by de Gennes for the breathing mode of tethered polymer chains on a surface [300]. A fair description of S(Q,t) with a minimum number of parameters could be achieved. Kanaya et al. [299] extended the investigation to a concentrated (30%, PI volume fraction) PS-PI micelle system and found a significant slowing down of the relaxation. The latter is explained by a reduction of osmotic compressibihty in the corona due to chain overlap. 

In the electronic Hamiltonian t +i, is the transfer integral, i.e. the re-electron wavefunction overlap between nearest neighbour sites in the polymer chain, and is equivalent to the parameter /3 in Equation (4.20), and c 1+,s. and clhS are creation and annihilation operators that create an electron of spin s ( 1/2) on the carbon atom at site n-f 1 and destroy an electron of spin, s at the carbon atom on site n, i.e. in effect transfer an electron between adjacent carbon atoms in the polymer chain. The elastic term is just the energy of a spring of force constant k extended by an amount ( +1— u ), where the u are the displacements along the chain axis of the carbon atoms from their positions in the equal bond length structure, as indicated in Fig. 9.8(b). The extent of the overlap of 7i-electron wavefunction will depend on the separation of nearest neighbour carbon atoms and is approximated by ... 

In earlier studies on solutions of synthetic polymers (Ferry, 1980), the zero-shear viscosity was found to be related to the molecular weight of the polymers. Plots of log r] versus log M often resulted in two straight lines with the lower M section having a slope of about one and the upper M section having a slope of about 3.4. Because the apparent viscosity also increases with concentration of a specific polymer, the roles of both molecular size and concentration of polymer need to be understood. In polymer dispersions of moderate concentration, the viscosity is controlled primarily by the extent to which the polymer chains interpenetrate that is characterized by the coil overlap parameter c[r] (Graessley, 1980). Determination of intrinsic viscosity [r]] and its relation to molecular weight were discussed in Chapter 1. The product c[jj] is dimensionless and indicates the volume occupied by the polymer molecule in the solution. 

Figure 4-6 Illustration of Dilute and Concentrated Regimes in Terms of Log c[tj (coil overlap parameter) against Log t)sp = [( o >ls)l>ls] (Vsp = specific viscosity) slope of 3.3 for entangled polysaccharide chains dissolved in good solvents and 4.1 for polymers with specific intermolecular associations.

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 ... 

Estimate the overlap parameter P for polymers with fractal dimension D in the melt (4>=l) if the degree of polymerization is N. Estimate the overlap parameter for an ideal chain (with Z) = 2) in a melt with A = 10 monomer segments. 

Polymeric thickening agents used in foods typically are well soluble polysaccharides, with an excluded volume parameter [j clearly above zero. This implies that especially the dilute and semidilute regimes are often of importance. Viscosity of very dilute solutions has been discussed in Sections 6.2.2 and 6.3.2. For higher concentrations, the reduced viscosity (j sp/c) is higher, as is true for any system (see Figure 5.5), but for polymer solutions the viscosity increases far stronger with concentration as soon as the chain overlap concentration is reached. 

This number appears to be constant for flexible polymers, with the average value = 20.6( 8%). Table 25.1 shows data for polyolefin melts listing density p, plateau modulus Ge, melt chain dimensions from SANS o/M, entanglement molar mass Me calculated from Eq. (25.6), Kuhn length b, packing length p, tube diameter a, and the overlap parameter for entanglement P, all at temperature T. 

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. 

Fujita and coworkers [79] smdied fluorescently labeled polyoxyethylene chains and found a good correlation between the concentration dependence of the friction coefficient evaluated from the anisotropy measurements and from the macroscopic viscosity. Fujita developed the fi ee-volume theory which describes reasonably well the concentration dependence of in the whole concentration region, [80] but it does not enable prediction of the parameters at a molecular level. Hyde et al. [81] used the Fujita theory for fairly successful interpretation of the experimental data. An interesting paper has been published by Viovy and Moimerie [82]. The authors studied concentrated solutions of anthracene-labeled polystyrene in toluene. They found good correlation of the local dynamics with the viscosity in the range of high concentrations and made one very important observation the local dynamics are unaffected by the overlap of the polymer chains that occurs at concentrations higher than c (concentration of the first overlap—see chapter Conformational and Dynamic Behavior of Polymer and Polyelectrolyte Chains in Dilute Solutions ). 

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