Viscosity Rouse theory

Although the Rouse theory is the source of numerous additional relationships, Eq. (3.98) is a highpoint for us, because it demonstrates that the viscosity we are dealing with in the Rouse theory for viscoelasticity is the same quantity that we would obtain in a flow experiment. Several aspects of this statement deserve amplification ... 

Inspection of Fig. 3.9 suggests that for polyisobutylene at 25°C, Ti is about lO hr. Use Eq. (3.101) to estimate the viscosity of this polymer, remembering that M = 1.56 X 10. As a check on the value obtained, use the Debye viscosity equation, as modified here, to evaluate M., the threshold for entanglements, if it is known that f = 4.47 X 10 kg sec at this temperature. Both the Debye theory and the Rouse theory assume the absence of entanglements. As a semi-empirical correction, multiply f by (M/M. ) to account for entanglements. Since the Debye equation predicts a first-power dependence of r) on M, inclusion of this factor brings the total dependence of 77 on M to the 3.4 power as observed. 

In semi-dilute solutions, the Rouse theory fails to predict the relaxation time behaviour of the polymeric fluids. This fact is shown in Fig. 11 where the reduced viscosity is plotted against the product (y-AR). For correctly calculated values of A0 a satisfactory standardisation should be obtained independently of the molar mass and concentration of the sample. 

At the high polymer concentration used in plasticized systems the viscosity of amorphous polymer is given by the modified Rouse theory at low molecular weight, M - 2Mr [from equation (47)] and by the modified Doi-Edwards equation at high molecular weight. In the first case... 

The incorporation of non-Gaussian effects in the Rouse theory can only be accomplished in an approximate way. For instance, the optimized Rouse-Zimm local dynamics approach has been applied by Guenza et al. [55] for linear and star chains. They were able to obtain correlation times and results related to dynamic light scattering experiments as the dynamic structure factor and its first cumulant [88]. A similar approach has also been applied by Ganazzoli et al. [87] for viscosity calculations. They obtained the generalized ZK results for ratio g already discussed. 

The number of beads in the model macromolecule is n, and is the Stokes law friction coefficient of each bead. The are to be evaluated for each macromolecule in its own internal coordinate system, with origin at the molecular center of gravity and axes (k = 1,2,3) lying along the principal axes of the macromolecule. The coordinates of the ith bead in this frame of reference are (x ]),-, (x2)i, and (x3)f. The averaging indicated by < > is performed over all macromolecules in the system. Thus, < i + 2 + 3) is simply S2 for the macromolecules. The viscosity is therefore identical, for all free-draining models with the same molecular frictional coefficient n and the same radius of gyration, to the expression from the Rouse theory ... 

Theories based on the uniformly effective medium have the practical advantage that they can be extended quite easily to polydisperse systems (227). Viscosity master curves can be predicted from the molecular weight distribution, for example. The only new assumption is that the entanglement time at equilibrium for a chain of molecular weight M in a polydisperse system has the form suggested by the Rouse theory (15) ... 

Due to difficulties in measuring the zero-shear viscosity of such high molecular weight polymers, and thus deducing the monomeric friction coefficient from Graessley s uncorrelated drag model [43], the following equation adapted from the modified Rouse theory has been applied [8]. 

The theory of Zimm (7) uses the assumptions of the Rouse theory and in addition considers hydrodynamic interactions between the moving submolecules and the solvent. The theory also makes use of the method formulated by Kirkwood and Riseman for the evaluation of the viscosity of dilute polymer solutions. A parameter h = where r is the... 

The Doi-Edwards theory provides expressions for G%, r[, and D that contain two adjustable parameters the friction coefficient and the primitive path step length L. The friction coefficient can be obtained from the relationship between the viscosity and molecular weight in the Rouse theory [Eq. (11.36)] or from the relaxation spectrum discussed below. Moreover, the step length a can be determined from the plateau modulus G%. Actually, according to the Doi-Edwards theory... 

Figure 3.66 shows the steady-shear viscosity for a polymer system at three molar masses. Note the plateau in viscosity at low shear rates (or the zero-shear viscosity). Also note how the zero-shear viscosity scales with to the power 3.4. (This is predicted by Rouse theory (Rouse, 1953).) Figure 3.67 shows the viscosity and first normal-stress difference for a high-density polyethylene at 200 C. Note the decrease in steady-shear viscosity with increasing shear rate. This is termed shear-thinning behaviour and is typical of polymer-melt flow, in which it is believed to be due to the polymer chain orientation and non-affine motion of polymer chains. Note also that the normal-stress difference increases with shear rate. This is also common for polymer melts, and is related to an increase in elasticity as the polymer chain motion becomes more restricted normal to flow at higher shearing rates. 

Another interesting result obtained for solutions of PIB in cyclohexane by Tanaka et al. (2,92) is shown in Fig. 3.4. The reduced storage modulus at infinite dilution is close to the prediction of the Rouse theory while the loss modulus is not. This result indicates the existence of an additional contribution to the intrinsic viscosity at high frequency which might be a frequency-independent high frequency limit [in contrast with the results for PMS (94,99) and for PIB in toluene (2,92)]. We will return to this subject in the following chapter. 

A relaxation spectrum similar to that of Fig. 4.2 is obtained for the diffusional motion of a local-jump stochastic model of IV+ 1 beads joined by N links each of length b, if a weak correlation in the direction of nearest neighbor links is taken into account for the probability of jumps (US). On the other hand, relaxation spectra similar to that of the Rouse theory (27) are obtained for the above mentioned model or for stochastic models of lattice chain type (i 14-116) without the correlation. Iwata examined the Brownian motion of more realistic models for vinyl polymers and obtained detailed spectra of relaxation times of the diffusional motion 117-119). However, this type of theory has not gone so far as to predict stationary values of the dynamic viscosity at high frequencies. 

Since the viscosity is the integration of G t) (Eq. (4.30)), the relaxation modulus G t) should contain more detailed information than the single viscosity value r/o. A wrong conclusion can be made if the conclusion is simply based on the viscosity result. On the basis of the viscosity data (Fig. 4.7), the onset of entanglement was traditionally believed to occm at Me 2.4Me. Shown in Chapters 10 and 11, detailed studies of the viscoelastic spectra in terms of the extended reptation theory as well as the Rouse theory have indicated that the onset occurs at Me-... 

Fig. 10.9 Comparison of the measured viscosity values ( without the so-called Tg correction) of nearly monodisperse polystyrene samples and those calculated from Ekj. (9.24) (solid line 1), from the Doi-Edwards theory (solid line 2), from the Rouse theory (solid line 3), and from integrating numerically Eq. (9.19) with K JK = 1 (the bottom dashed line), K fK = 3 (the middle dashed line), and K /K = 5.5 (the top dashed line).

Six theoretical viscosity curves calculated with the same K value are shown in Fig. 10.9 curve 1 is calculated from Eq. (9.24) curve 2 from the Doi-Edwards theory (Eq. (8.57)) curve 3 from the Rouse theory (Eq. (7.61)) and three dashed lines calculated numerically from integrating Eq. (9.19) with K /K = 1, 3.3, and 5.5, respectively. The ratio of K /K = 5.5 was obtained by multiplying K /K = 3.3 by a factor which was estimated from comparing the areas obtained from integrating... 

Shortly after the Doi-Edwards theory was published, Graessley proposed using the viscosity value at Me to calculate the K value on the basis of the Rouse theory... 

Without the entropy-driven modes of motion — as described by either the ERT or the Rouse theory — in OTP, the viscosity can be expressed by... 

A quite different type of observation which leads also to the concept of an entanglement network is the dependence of viscosity on molecular weight in undiluted polymers or at constant concentration in concentrated solutions, as advanced by Bueche. This is illustrated in Fig. 10-10 for fractions of polystyrene. At low molecular weights, rjo increases only slightly more rapidly than directly proportional to Af, and its magnitude is actually predicted by the Rouse theory, in accordance with the principle of Bueche. Thus, from equations 4 and 6, rjo is given by... 

According to the modified Rouse theory of Section A1 of Chapter 10, for uncross-linked polymers of sufficiently low molecular weight to avoid entanglements, the monomeric friction coefficient is related to the steady-flow viscosity by the following equation, which can be obtained from equations 4 and 10 of Chapter 10 and was originally derived by Bueche. ... 

---