Coil overlap parameter

Fig. 8.9. Power law exponent d as a function of the coil overlap parameter c[ ] at low concentrations. The filled circles are narrow distribution polystyrene solutions (1 77, 316, 318), the open circles are poly(a-methyl styrene) (198, 318). Solvents are chlorinated di-phenyls except the intrinsic viscosity data which were obtained in toluene. Symbols are for polystyrene M= 13.6 x 106, 4 1-8 x 10 , and 0.86 x 106 for poly(a-methyl styrene) O M = 7.5 x 10 , 6 3.3 xlO6, Cr 1.82 xlO6, O- 1.14x10 , a. 0.694x10 , and...

The dimensionless product c[k]] is defined as the coil overlap parameter it provides information about the changing nature of the interactions in a dispersion (Blanshard and Mitchell, 1979 Morris et al., 1981). For dilute dispersions, i.e., below c, the slope of log( qsp/cI) vs log(c[T ]) universally approximates 1.4. At the upper practical extreme, with exceptions (especially the galactomannans Morris et al., 1981), the slope increases sharply to 3.3, illustrating wide deviations from Newtonian flow in the segment approaching elasticity. The deviations are significant when 5 < < 10 (Barnes... 

FIG. 16.17 Power law exponent — ft — 1 — n as a function of the coil overlap parameter c[ 7], for solutions of polystyrene (filled symbols) and poly(a-methyl styrene) in chlorinated biphenyls (open symbols). The values of [77] were obtained in toluene. Molecular weights range from 860 to 13,600 kg/mol for polystyrene and from 440 to 7500 kg/mol for poly(a-methyl styrene). From Graessley (1974). Courtesy Springer Verlag. 

The effect of concentration on the zero-shear viscosity of biopolymer dispersions can be expressed in terms of the coil overlap parameter, c[ j], and the zero-shear specific viscosity as described in Chapter 4 in connection with food gum dispersions. 

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.

It is obvious from these results that the parameter essentially describes the frictional contribution of the contacts between the chains and the frictional resistance of single chains. This can be seen even more clearly by plotting the ratio of the entanglement contributionri j at zero shear rate and ofrij, . versus the coil overlap parameter hie C2 in Fig. 9... 

F. 9. Ratio of the entanglement part of the shear viscosity (cf. Eq. (14)) at zwero shear rate and the frictional part of the viscosity as a function of the coil overlap parameter... 

Equation (5) shows that the left term, which is called the specific viscosity (nsp), is directly proportional to the product of intrinsic viscosity and concentration. This product is referred to as the "coil overlap parameter = C[n ], Polymer molecules with high ratios of hydrodynamic volume to molecular weight (large intrinsic viscosities)... 

Equation (5) applies only to very dilute solutions in which the coil overlap parameter is less than about 0.1. At higher overlap parameters, interaction between polymer coils becomes significant and equation (5) must be expanded to account for coil-coil interactions. 

So we see that a good rule of thumb for predicting when concentration effects will become important is when the coil overlap parameter c[rj] (or cM ) is near unity. 

Simha and Zakin (126), Onogi et al (127), and Comet (128) develop overlap criteria of the same form but with different numerical coefficients. Accordingly, flow properties which depend on concentration and molecular weight principally through their effects on coil overlap should correlate through the Simha parameter c[ /], or cM , in which a is the Mark-Houwink viscosity exponent (0.5 < a < 0.8). If coil shrinkage, caused by the loss of excluded volume in good... 

Further examination of the Williams approach seems called for, both to improve the method for estimating parameters such as the relaxation time, and to clarify the relationship between the intramolecular potential form and non-thermodynamic frictional forces. The method might provide a fairly unified description of non-linear flow porperties if a suitable potential function for large scale molecular friction were found. Aside from the Williams work, there have been no theoretical studies dealing with t] vs. y at low to moderate concentrations. The systematic changes in the master curve /(/ ) with coil overlap c[ij] are thus without explanation at the present time. 

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 ... 

A second major problem that causes error in EOR polymer intrinsic viscosity measurements concerns the uncertainty that dilute solution conditions exist during viscosity measurements. As shown by Figure 1, dilute solution conditions exist only when the overlap parameter is less than one. Extrapolation to "zero concentration reduced viscosity is linear only if viscosity data is taken in the dilute region. However, a polymer solution having an intrinsic viscosity of 30 dl/g is dilute only if the concentration is less than 1/30 g/dl (33 ppm). At concentrations higher than this, excessive coil overlap may exist. Solution reduced viscosities determined above the dilute solution region can not be easily extrapolated to zero concentration. When linear extrapolation is done, the apparent intrinsic viscosities which are obtained are larger than the true intrinsic viscosity. 

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. 

The parameter of prime importance m SIA is the degree of penetration or overlap of adjacent zones. This is dependent on the relative volumes, m addition to the usual parameters of tubing size and length, reaction coil geometry, and so forth. The degree of penetration and the dispersion determine the relative signal that is recorded. We may define the zone penetration... 

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