Coil overlap

A second source of intermolecular interactions arises from segment-segment contacts between molecules. This mode of interaction has become popularly known as entanglement. 

The symbol signifies depends on and is meant to give a functional dependence, not an equality. Thus eq. 11.3.11 states that the 

Viscosity at various concentrations and molecular weights in the low to moderate concentration range. Polystrene-decalin and polymethyl methacrylate-xylene are theta or near-theta systems the remainder are good solvent systems. Note that the c[ j] reduction is somewhat better in theta solvents, and that the Martin equation, which would give a straight line in the figure, is a somewhat better representation for theta solvents. Adapted from Graessley (1974). 

A simple illustration is an estimation of the overlap threshold c. We expect c to be comparable with the local concentration within a single chain, thus 

We can determine how varies with concentration from scaling arguments and a few simple physical considerations. First, we expect that for c c, the solution structure on the length scale will not depend on molecular weight because we are looking only at rather small sections of the molecule. Second, at c = c, we expect I to be approximately the same as the isolated coil size I 2 Vv ). Thus we are led to the scaling form  

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Studies of the structure and molecular size of wheat AX [41] revealed that they are shear-thinning and exhibit two critical concentrations, which correspond to the onset of coil overlapping. The existence of three domains provided the evidence for the formerly suggested rigid, rod-Uke conformation of AX in solution. In a recent study [116], the previously reported conflicting suggestions on the conformation of AX were discussed. 

Simha [53] made the first attempts to model the transition from a dilute to a concentrated solution. He assumed that in the range from lscaling laws a theory has been developed which allows for the prediction of the influence of Mw c and the solvent power on the screening length [54,55]. This theory is founded on the presumption that above a critical concentration, c, the coils overlap and interpenetrate. Furthermore it is assumed that in a thermody-... 

Efforts at synthesis and studies of temperature-dependent solution behaviour of these chemically hydrophobized polyacrylamides are now in progress. However, it is reasonable to point out that in this case, contrary to the hydrophilization of the hydrophobic precursor, the problems associated with additional swelling of the globular core (as the modification proceeds) are absent however, the problem of the choice of working concentration for the precursor is still present since above the coil overlapping concentration the intermolecular aggregation processes at elevated temperatures can compete with the intramolecular formation of core-shell structures. 

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. 

In some case, however, only a flattening of the osmotic modulus curve is observed. Such a case is found with star-branched macromolecules. This observation has rather comprehensively been investigated by Roovers et al. with stars of 64 and 128 arms [172]. The authors give the following explanation. At the point of coil overlap and at somewhat higher concentrations the stars feel the interaction as a quasi colloidal particle. Hence, a steeper increase of the osmotic mod-... 

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. 

The number of molecules per unit volume v is 6.02 x 1023 c/M. If the molecular centers are distributed randomly, the product vV is the average number of other molecules with centers lying within the pervaded volume of any one molecule. Accordingly, v V is a measure of the potential degree of coil overlap, and with = 2.68 xlO23 ... 

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... 

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...

Although taking place over a somewhat wider range of c[ij], this transition parallels the variation of JeR from Zimm-like to Rouse-like behavior at low concentrations (15). It also supports the contention (Section 5) that coil overlap is the principal structural variable affecting viscoelastic behavior at low to moderate concentrations. 

Intcrmolecular Contributions. Increasing concentration reduces the effects of excluded volume and intramolecular, hydrodynamic on viscoelastic properties (Section 5). Internal viscosity and finite extensibilty have already been eliminated as primary causes of shear rate dependence in the viscosity. Thus, none of the intramolecular mechanisms, even abetted by an increased effective viscosity in the molecular environment, can account for the increase in shear rate dependence with concentration, e.g., the dependence of power-law exponent on coil overlap c[r/] (Fig. 8.9). Changes in intermolecular interaction with increased shear rate seems to be the only reasonable source of enhanced shear rate dependence, at least with respect to the early deviations from Newtonian behavior and through a substantial portion of the power law regime. 

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. 

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) ... 

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. 

When soluble polymers are attached by one end to a surface, the thickness of the resulting layer, L, depends on the surface density of chains a as well as n and the excluded volume v/l3 (de Gennes, 1980). At low densities risolated chains extend 1/2 into the solution, creating a layer with the density profile shown in Fig. 26a and a thickness of L = n1/2f for ideal chains and L n3/sl in good solvents. When 1 the coils overlap and the interactions will cause the chains to expand away from the surface into the bulk. The configurations of the individual molecules and the density profile within the layer (Fig. 26b) differ markedly from the dilute situation. When cl2 1 the molecules become fully stretched. 

Numerical results for a 10% solution of polymer [N - 1000) in Its own monomer are given in fig. 5.15 (solid curves). It is clear that long chains try to avoid the surface region because of the incurred loss of conformational entropy. The available space Is occupied by the monomer which does not suffer from these entroplcal restrictions. Hence, the polymer is depleted. It can be shown that at low (p (below coil overlap) the thickness of the depletion zone is proportional to Vw, see sec. 5.3e. At higher (as in fig. 5.15), the osmotic pressure pushes the chains closer to the surface, making the depletion layer thinner and its thickness more weakly dependent on chain length. 

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.

The concept of polymer entanglements represents intermolecular interaction different from that of coil overlap type interaction. However, it is difficult to define the exact topological character of entanglements. The entanglements concept was aimed at understanding the important nonlinear rheological properties, such as the shear rate dependence of viscosity. However, viscoelastic properties could not be defined quantitatively as is possible with the reptation model. Because an entanglement should be... 

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