Polymers zero shear viscosity

Polymer solutions are often characterized by their high viscosities compared to solutions of nonpolymeric solutes at similar mass concentrations. This is due to the mechanical entanglements formed between polymer chains. In fact, where entanglements dominate flow, the (zero-shear) viscosity of polymer melts and solutions varies with the 3.4 power of weight-average molecular weight. 

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to Mw whereas above this critical value r 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

Taking into account the relevance of the range of semi-dilute solutions (in which intermolecular interactions and entanglements are of increasing importance) for industrial applications, a more detailed picture of the interrelationships between the solution structure and the rheological properties of these solutions was needed. The nature of entanglements at concentrations above the critical value c leads to the viscoelastic properties observable in shear flow experiments. The viscous part of the flow behaviour of a polymer in solution is usually represented by the zero-shear viscosity, rj0, which depends on the con-... 

The viscosity level in the range of the Newtonian viscosity r 0 of the flow curve can be determined on the basis of molecular models. For this, just a single point measurement in the zero-shear viscosity range is necessary, when applying the Mark-Houwink relationship. This zero-shear viscosity, q0, depends on the concentration and molar mass of the dissolved polymer for a given solvent, pressure, temperature, molar mass distribution Mw/Mn, i.e. 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, q0. Flow curves (plots of log q vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

Galgali and his colleagues [46] have also shown that the typical rheological response in nanocomposites arises from frictional interactions between the silicate layers and not from the immobilization of confined polymer chains between the silicate layers. They have also shown a dramatic decrease in the creep compliance for the PP-based nanocomposite with 9 wt% MMT. They showed a dramatic three orders of magnitude drop in the zero shear viscosity beyond the apparent yield stress, suggesting that the solid-like behavior in the quiescent state is a result of the percolated structure of the layered silicate. 

Linear phosphonitrilic chlorides (LPNCs), silicone fluids and, 22 573 Linear photodiode arrays, 19 153 Linear polyesters, 14 116 Linear polyethylene fibers, 20 398 Linear polyimides, synthesis of, 20 273 Linear polymers, 20 391 25 455 high molecular weight, 23 733 zero-shear viscosity of, 19 839 Linear poly(thioarylene)s, 23 705 Linear PPS, 23 704. See also... 

Figure 14.7 Dependence of the zero-shear viscosity, uo, on molecular weight, M, for different dendrimer systems. (1) Dendrimers of different chemical composition but in the same state (i.e. PAMAM, PPI and PBzE dendrimers in bulk D, C and E, respectively). (2) Compositionally identical dendrimers (i.e. PAMAMs) in solutions and in the bulk state (A, B and D, respectively). (3) Compositionally identical dendrimers and linear polymers of comparable molecular weights (i.e. PAMAMs in the bulk state D and F, respectively)...

The limiting low shear or zero-shear viscosity r 0 of the molten polymer can be related to its weight-average molecular weight, M 9 by the same relations noted for concentrated solutions rj0 = KMW for low molecular weight and rj0 = Kfor high molecular weight. 

Controlled stress viscometers are useful for determining the presence and the value of a yield stress. The structure can be established from creep measurements, and the elasticity from the amount of recovery after creep. The viscosity can be determined at very low shear rates, often in a Newtonian region. This zero-shear viscosity, T 0, is related direcdy to the molecular weight of polymer melts and concentrated polymer solutions. 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

The zero-shear viscosity r 0 has been measured for isotropic solutions of various liquid-crystalline polymers over wide ranges of polymer concentration and molecular weight [70,128,132-139]. This quantity is convenient for studying the stiff-chain dynamics in concentrated solution, because its measurement is relatively easy and it is less sensitive to the molecular weight distribution (see below). Here we deal with four stiff-chain polymers well characterized molecu-larly schizophyllan (a triple-helical polysaccharide), xanthan (double-helical ionic polysaccharide), PBLG, and poly (p-phenylene terephthalamide) (PPTA Kevlar). The wormlike chain parameters of these polymers are listed in Tables... 

In Sect. 6.3, we have neglected the intermolecular hydrodynamic interaction in formulating the diffusion coefficients of stiff-chain polymers. Here we use the same approximation by neglecting the concentration dependence of qoV), and apply Eq. (73) even at finite concentrations. Then, the total zero-shear viscosity t 0 is represented by [19]... 

This article builds upon an earlier review on the same subject by Porter and Johnson in 1966 (14), and on the recent treatise on viscoelasticity in polymers by Ferry (15). We have generally tried to maintain the same nomenclature as the latter. Recent reviews on the relation between the zero-shear viscosity and molecular structure (16), crosslinked networks (17), and flow birefringence (18) in this same journal cover portions of the subject. We have tried to minimise redundancy with these works while at the same time making the review reasonably self-contained. 

In this way, the relaxation times can be calculated when the weight average molecular weight of the polymer and its (zero shear) viscosity are known (Mt and p are the free parameters of this equation). 

At this point it seems of interest to include a graph obtained on a quite different polymer, viz. cellulose tricarbanilate. Results from a series of ten sharp fractions of this polymer will be discussed in Chapter 5 in connection with the limits of validity of the present theory. In Fig. 3.5 a double logarithmic plot of FR vs. is given for a molecular weight of 720000. This figure refers to a 0.1 wt. per cent solution in benzophenone. It appears that the temperature reduction is perfect. Moreover, the JeR-value for fiN smaller than one is very close to the JeR value obtained from Figure 3.1 for anionic polystyrenes in bromo-benzene. As in the case of Fig. 3.1, pN is calculated from zero shear viscosity. The correspondence of Figs. 3.1 and 3.5 shows that also the molecules of cellulose tricarbanilate behave like flexible linear chain molecules. For more details on this subject reference is made to Chapter 5. 

Originally, Fox and Flory (121) found that the zero shear viscosity of polymer melts increases with the 3.4-th power of the molecular weight. Bueche (122) has shown that this relationship holds only above a certain critical molecular weight Mc which depends on the structure of the polymer chain. Below Mc the zero shear viscosity is found to depend on a significantly lower power of molecular weight. A theoretical interpretation of these facts has been given by the latter author on the basis of the free-draining model (Section 3.4.1.). 

This was in contrast to a glycerine solution having the same zero-shear-viscosity which was dispersed after a few pipe diameters at a relatively low Reynolds number. For injection concentrations of 0.25% and 0.3%, the polymer thread did not remain... 

Kulicke W-M, Kehler H, Bouldin M A consideration of the state of solution in the molecular modeling of the zero-shear viscosity for polymeric liquids Colloid Polym Sci (submitted)... 

For concentrated solutions of amorphous polymers, Bueche s mathematical model shows the ratio of zero shear viscosities of branched and linear polymer above the critical molecular weight in the entanglement region to be (28) ... 

Figure 3.3 Evolution of physical properties of the thermosetting polymer as a function of conversion of reactive groups (a) zero-shear viscosity and elastic modulus, (b) sol fraction.

The polyethenes prepared with catalyst 2 (Fig. 3a) have greatly elevated elastic modulus G values due to LCB compared to the linear polymers shown in Fig. 3b. LCB also shifts the crossover point to lower frequencies and modulus values. The measured complex viscosities of branched polymers (see also Table 2) are more than an order of magnitude higher than calculated zero shear viscosities of polymers having the same molecular weight but a linear structure. The linear polymers have, in turn, t] (0.02 radvs)... 

Getmanova and coworkers attached polar groups to the periphery of the hyperbranched poly(carbosilane) obtained from the polymerization of methyldiallylsilane (Scheme 25)78. This modified hyperbranched polymer possessed certain properties, such as zero shear viscosity, 7 g and surface tension, that were similar to comparably modified carbosilane dendrimers81. 

If rj is independent of the shear rate y a liquid is called Newtonian. Water and other low molecular weight liquids typically are Newtonian. If rj decreases with increasing y, a liquid is termed shear thinning. Examples for shear thinning liquids are entangled polymer solutions or surfactant solutions with long rod-like micelles. The zero shear viscosity is the value of the viscosity for small shear rates ij0 = lim,> o tj y). The inverse case is also sometimes observed rj increases with increasing shear rate. This can be found for suspensions and sometimes for surfactant solutions. In surfactant solutions the viscosity can be a function of time. In this case one speaks of shear induced structures. 

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