Dilute Solution Zero-shear Viscosity

A unique characteristic of polymers is that even a small concentration (c) of a high molecular veight polymer can significantly enhance the viscosity of a solution (t ) as compared to the viscosity of the solvent This is because the expanded polymer coils in slo vly deforming dilute solutions behave as rigid spheres vith a radius of the order of resulting in a large polymer volume fraction (f [Eq. (9)] 

A polymer s capacity to enhance solution viscosity is described in terms of its intrinsic viscosity ([ ])  

Combining Eqs. (9)-(ll) vith Eq. 2.1, and realizing that n M, one finds that [tf has an dependence on polymer molecular veight. In reality, the factor a [Eq. (5)] depends to a small extent on M, resulting in the Mark-Hou vink relation [Eq. (12)]. 

The constants K and a are obtained by fitting experimental data for a given polymer-solvent system. The intrinsic viscosity [p] is conveniently measured by observing the time for fio v of a certain volume of solution (and solvent) through a capillary, and thus provides an easy indication of the molecular veight from the tabulated Mark-Hou vink constants [57]. 

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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 intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is... 

Figure 6.20 (a) Viscosity of solution of poly(y-benzyl-L-glutamate) (PBLG molecular weight 231,000), in m-cresol as a function of shear rate at 29 C for various mass percentages ranging from dilute ( 0.31%) to semidilute (0.31-2.5%) to concentrated isotropic (5-9.5%). The different symbols and lines refer to data taken on different instruments, (b) Zero-shear viscosity as a function of concentration, (reprinted with permission from Mead and Larson, Macromolecules 23 2524. Copyright 1990 American Chemical Society.)... 

The zero-shear viscosity of a concentrated polymer solution can be treated by a modified version of the method used to calculate the zero-shear viscosity of a polymer melt. The modifications take the two effects of the solvent (plasticization and true dilution of the polymer) into account. Approximations are involved, however, in determining the appropriate mixing rules for the plasticization effect and the magnitude of the true dilution effect. The zero-shear viscosity of concentrated polymer solutions will be discussed briefly in Section 13.G. 

An especially appealing aspect of the use of equations 11.25 and 13.8 to estimate Mcr is that Equation 11.25 was developed [11] by using data on rubbery polymers. It therefore relates more directly to the zero-shear viscosity of a polymer melt than does an extrapolation from data on the viscosities of dilute solutions under theta conditions. We have found that the use of equations 11.25 and 13.8 is generally preferable to the use of Equation 13.7 in calculating Mcr. The correlation developed for Vw in Section 3.B has allowed the much more general use of Equation 11.25 than was previously possible by using group contributions. 

Heo, Y., and Larson, R. G., The scaling of zero-shear viscosities of semi-dilute polymer solutions with concentration, J. RheoL, 49, 1117-1128 (2(X)5). 

McKerma, G. B., Hadziioarmou, G., Lutz, P., Hild, G., Strazielle, C., Straupe, C., Rempp, P., and Kovacs, A. J., 1987. Dilute-solution characterization of cyclic polystyrene molecules and their zero-shear viscosity in the melt. Macromolecules, 20 498-512. 

Relationships between the viscosity and concentration in semidilute regimes of linear homopolymers of PMMA in DMF was investigated for electrospun nanofibers (Fig. 1.7]. The plot of the zero shear viscosity with the C/C distinctly separated into different solution regimes, viz. dilute C/C < 1], semidilute unentangled 1 3]. 

The diffusion constant of the rods in semi-dilute solution decreases with the seventh power of molecular weight, which is a phenomenal difference from dependence for flexible entangled polymers [20], Consequently, the zero shear viscosity of rigid rods depends strongly on the molecular weight [2] ... 

The zero shear viscosity scales with Nf" to contrast Af dependence for isotropic polymers [20] So far, we have examined the dynamics of rod-Uke macromolecules in isotropic semi-dilute solution. For anisotropic LCP solutions in which the rods are oriented in a certain direction, the diffusion constant increases, and the viscosity decreases, but their scaling behavior with the molecular weight is expected to be unchanged [2,17], Little experimental work has been reported on this subject. The dynamics of thermotropic liquid crystalline polymer melts may be considered as a special case of the concentrated solution with no solvent. Many experimental results [16-18] showed the strong molecular weight dependence of the melt viscosity as predicted by the Doi-Edwards theory. However, the complex rheological behaviors of TLCPs have not been well theorized. 

The zero shear viscosity of flexible linear polymers varies experimentally with and theoretically with [20]. Due to the highly restricted rotational diffusion, the viscosity of TLCPs is much more sensitive to the molecular weight than that of ordinary thermoplastics as discussed in section 3. Doi and Edwards predicted that the viscosity of rod-like polymers in semi-dilute solutions scales with A/ [see Equation (12)] [2]. Such a high power dependence of viscosity on the molecular weight has been experimentally observed both for lyotropic LCPs [14,15] and for TLCPs [16-18]. The experimental values of the exponent range from 4 to 7 depending on the chemical structure, the chain stiffness, and the domain or defect structure of the liquid crystalline solution or melt. The anisotropicity of the liquid seems to have little effect on the exponent. A slightly smaller exponent for the nematic phase than for the isotropic phase (6 in the nematic phase versus 6.5 in the isotropic... 

The Rouse model was initially designed to treat the dynamics of polymers in very dilute solutions [1]. Ironically, however, it turned out that dilute solutions are not appropriate systems for it. Indeed, in the Rouse model the maximal relaxation time, Tchain> and the diffusion coefficient, I>chain> scale with the molecular weight, M, as and M respectively (see Eqs. 57 and 61). Furthermore, it is a straightforward matter to demonstrate that for the Rouse model at = 0 the zero shear viscosity [ ] (0)] is proportional to M, see Eq. 22. All these theoretical findings disagree with the experimental data... 

Zimm molecular mass dependencies of the maximal relaxation time, of the diffusion coefficient, and of the zero shear viscosity are all consistent with the experimental findings [3]. The Zimm model also agrees with the experiment with respect to the frequency dependence of the storage and loss moduli of dilute polymer solutions under 0-conditions both G (co) and G" co) show... 

Measurements of the steady shear viscosity in the Newtonian or linear regime yields valuable information on molecular interactions. The molecular weight (Mw) and polymer concentration (C) dependence of the zero-shear viscosity, rjo, of cellulose solutions, exhibits two distinct regions. The dilute regime shows a linear increase of zero-shear viscosity with respect to CMw In the semi-dilute region, the CM is no longer linearly proportional to r o-It is well established for linear, flexible polymer chains that rjo is proportional to [13]. This proportionality for cellulose in the NH3/NH4SCN... 

In this part, we are dealing with the shear-thickening transition in dilute surfactant solutions. Only solutions with zero-shear viscosity close to that of the solvent and for which no apparent viscoelasticity is observed at rest will be considered. Systems showing both viscoelasticity and shear-thickening have also been found and wiU be evoked in the part devoted to the semidilute regime. 

For more concentrated suspensions, other parameters should be taken into consideration, such as the bulk (elastic) modulus. Clearly, the stress exerted by the particles depends not only on the particle size but on the density difference between the partide and the medium. Many suspension concentrates have particles with radii up to 10 pm and a density difference of more than 1 g cm . However, the stress exerted by such partides will seldom exceed 10 Pa and most polymer solutions will reach their limiting viscosity value at higher stresses than this. Thus, in most cases the correlation between setfling velocity and zero shear viscosity is justified, at least for relatively dilute systems. For more concentrated suspensions, an elastic network is produced in the system which encompasses the suspension particles as well as the polymer chains. Here, settling of individual partides may be prevented. However, in this case the elastic network may collapse under its own weight and some liquid be squeezed out from between the partides. This is manifested in a dear liquid layer at the top of the suspension, a phenomenon usually... 

Equations 15.27 and 15.28 require that the G, all be the same and that rjo = Srj, (with rji = 2,Gj). They reduce the number of necessary parameters to three rjo, the steady-state zero-shear viscosity Ig, a maximum relaxation time and a, an empirical constant. The Rouse theory [7] for dilute polymer solutions predicts a =2, but for concentrated solutions and melts, better fits are obtained with a s between 2 and 4 [6]. 

System relaxation times have been determined from the relaxation of the stress after abrupt cessation of shear flow. Representative applications of the approach are found in Takahashi, etal. 9), who examined 355-3840 kDa polystyrenes in benzyl- -butylphthalate, at concentrations identified as showing dilute-solution behavior for the steady-state compliance Je and semidilute behavior for the zero shear viscosity. The stress relaxation after shear cessation, identified as the transient viscosity, decreased exponentially with time except at the shortest times studied, leading to an identification of an observed longest relaxation time Xm, whose c and M dependences were determined. 

The oldest, simplest and most widely used method for obtaining information about the molecular weight of a polymer is based on the measurement of the viscosity of dilute solutions. We will see that this quantity is less sensitive to molecular weight than the zero-shear viscosity of the melt. However, the apparatus required is much simpler and can be used in combination with GPC to determine the molecular weight distribution. Furthermore, it is often impossible using a commercial rheometer to determine the zero-shear viscosity of a melt. 

The behavior of harmonic dumbbells in dilute solution has been studied in detail analytically [230]. These results can be used to predict the zero-shear viscosity t] and the storage and loss moduli, G (co) and G"((o) in oscillatory shear with frequency m, of the MPC dumbbell fluid. This requires the solvent viscosity and diffusion constant of monomers in the solvent. Since the viscoelastic MPC fluid consists of dumbbells only, the natural assumption is to employ the viscosity tjmpc and diffusion constant D of an MPC point-particle fluid of the same density. The zero-shear viscosity is then found to be [229]... 

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