Zimm model viscoelasticity

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

These predictions of the Zimm model are compared with experimental data on dilute polystyrene solutions in two -solvents in Fig. 8.7. The Zimm model gives an excellent description of the viscoelasticity of dilute solutions of linear polymers. 

The time-dependent viscoelastic response of polymers is broken down into individual modes that relax on the scale of subsections of the chain with Njp monomers. The Rouse and Zimm models have different structure of their mode spectra, which translates into different power law exponents for the stress relaxation modulus G t) ... 

The Rouse-Zimm Model Describes the Dynamics of Viscoelastic Fluids... 

Figure 33.9 The Rouse-Zimm model for viscoelasticity. Applying a shear flow to a polymer solution stretches out the chains, indicated here in terms of a beads-and-spring model, leading to an elastic entropic retractive force.

Fig. 16.9 shows the low frequency slopes of 2 and 1, respectively, as expected for viscoelastic liquids and the high frequency slopes Vi and 2/3 for Rouse s and Zimm s models, respectively. Experimentally it appears that in general Zimm s model is in agreement with very dilute polymer solutions, and Rouse s model at moderately concentrated polymer solutions to polymer melts. An example is presented in Fig. 16.10. The solution of the high molecular weight polystyrene (III) behaves Rouse-like (free-draining), whereas the low molecular weight polystyrene with approximately the same concentration behaves Zimm-like (non-draining). The higher concentrated solution of this polymer illustrates a transition from Zimm-like to Rouse-like behaviour (non-draining nor free-draining, hence with intermediate hydrodynamic interaction).

This latter model was employed by Rouse (27) and by Bueche (28) in the calculation of viscoelasticity and is sometimes called the Rouse model. It was used later by Zimm (29) in a more general calculation which may be regarded as an application of the Kirkwood theory. As illustrated in Fig. 2.1, the Rouse model is composed of N + 1 frictional elements represented by beads connected in a linear array with N elastic elements or springs, hence the bead-spring model designation. The frictional element is assumed to represent the translational friction... 

The zero mode is the self-diffusion of the center of mass whose diffusion coefficient is given by the Stokes-Einstein relation D = k TIN. The time Tj will be proportional to the time required for a chain to diffuse an end-to-end distance, that is, R )/D = t N b lk T. This means that for time scales longer than Tj the motion of the chain will be purely diffusive. On timescales shorter than Tj, it will exhibit viscoelastic modes. However, the dynamics of a single chain in a dilute solution is more complex due to long-range forces hydrodynamic interactions between distant monomers through the solvent are present and, in good solvents, excluded volume interactions also have to be taken into account. The correction of the Rouse model for hydrodynamic interaction was done by Zimm [79]. Erom a mathematical point of view, the problem becomes harder and requires approximations to arrive at some useful results. In this case, the translational diffusion coefficient obtained is... 

In the theories for dilute solutions of flexible molecules based on the bead-spring model, the contribution of the solute to the storage shear modulus, loss modulus, or relaxation modulus is given by a series of terms the magnitude of each of which is proportional to nkT, i.e., to cRTjM, as in equation 18 of Chapter 9 alternatively, the definition of [C ]y as the zero-concentration limit of G M/cRT (equations 1 and 6 of Chapter 9) implies that all contributions are proportional to nkT. Each contribution is associated with a relaxation time which is proportional to [ri Ti)sM/RT-, the proportionality constant (= for r i) depends on which theory applies (Rouse, Zimm, etc.) but is independent of temperature, as is evident, for example, in equation 27 of Chapter 9. Thus the temperature dependence of viscoelastic properties enters in four variables [r ], t/j, T explicitly, and c (which decreases slightly with increasing temperature because of thermal expansion). 

In the case of polymers, viscosity is considered an important property to explain the viscoelastic behavior of polymers under stress and strain [154]. At this point, two theories are considered which deal with the flow behavior of polymer mixtures the first, which was proposed by Rouse [155] and is based on the studies of Kargin and Slonimsky, is KSR model the second, as proposed by Zimm [156] and based on the studies of Kirkwood and Risemann, is the KRZ model. 

Since molecular theories of viscoelasticity are available only to describe the behavior of isolated polymer molecules at infinite dilution, efforts have been made over the years for measurements at progressively lower concentrations and it has been finally possible to extrapolate data to zero concentration. The behavior of linear flexible macromolecules is well described by the Rouse-Zimm theory based on a bead-spring model, except at high frequencies . Effects of branching can be taken into account, at least for starshaped molecules. At low and intermediate frequencies, the molec-... 

The independence of the model prediction from the choice of the subchain, reflecting the independence of the stress from this choice explained earlier (for the internally equilibrated subchains ), demonstrates the reliability of the bead-spring models. In fact, the slow viscoelastic responses of dilute chains are well described by the bead-spring model considering the hydrodynamic interaction, as noted from the agreement between the Zimm-Kilb prediction (solid curves) and the data for the dilute star-branched chain shown in Figure 7. 

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