Spring-Bead Model Zimm Theory

For the bead-spring model, a/b has to be smaller than 1/2 to avoid the interpenetration of the neighboring spheres. The value of ft described by Eq. (3.1) satisfies this criterion and so is consistent with the model on which the theory is based However, it should be noted that this favorable result is obtained within the framework of the Zimm theory. The value of ft at the non-free draining limit is 1/4 for Gaussian chains but it is different from 1/4 for chains of other distribution. Moreover,... 

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-... 

We refer to this model as the bead-spring model and to its theoretical development as the Rouse theory, although Rouse, Bueche, and Zimm have all been associated with its development. 

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... 

Only recently has the theory of chain dynamics been extended by Peterlin (J [) and by Fixman (12) to encompass the known non-Newtonian intrinsic viscosity ofTlexible polymers. This theory, which is an extension of the Rouse-Zimm bead-and-spring model but which includes excluded volume effects, is much more complex than that for undeformable ellipsoids, and approximations are needed to make the problem tractable. Nevertheless, this theory agrees remarkably well (J2) with observations on polystyrene, which is surely a flexible chain. In particular, the theory predicts quite well the characteristic shear stress at which the intrinsic viscosity of polystyrene begins to drop from its low-shear Newtonian plateau. 

In 1944 Kramers [1] published a phase-space kinetic theory for the steady-state potential flow of monodisperse dilute polymer systems in which the polymer molecule is modeled as a freely jointed bead-rod chain. Subsequent scholars developed kinetic theories for shearing flows of monodisperse dilute polymer solutions Kirkwood [2] for freely rotating bead-rod chains with equilibnum-averaged hydrodynamic interaction. Rouse [3] and Zimm [4] for freely jointed bead-spring chains, and others. These theories were all formulated m the configuration space of a single polymer chain. 

The relaxation time 1 and the pol3mer contribution to the viscosity rjp depend strongly on the polymer molecular weight, concentration, and equihbrium conformation. Kinetic theory can be used to obtain scaling behavior for these quantities. At dilute concentrations, for example, the Zimm bead-spring model predicts the relaxation time as a function of the drag on polymer chain segments. 

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). 

The theory of cyclization dynamics was first presented by Wileaski and Fixman [WF] (5). A number of curious features of the theory prompted detailed attention by Doi (11), by Perico and Cuniberti (12), and by others (13). The theory is developed in terms of the bead-and-spring Rouse-Zimm [RZ] model (14). Unrealistic in detail, this model is quite useful for describing low frequency, large flmq[>litude chain motions. The RZ model, figure 2, treats the chain as a series of n beads connected by (n-1)harmonic springs of root-mean-squared length b. 

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