Rouse-Bueche theory

The Rouse-Bueche theory is useful especially below 1% concentration. However, only poor agreement is obtained on studies of the bulk melt. The theory describes the relaxation of deformed polymer chains, leading to advances in creep and stress relaxation. While it does not speak about the center-of-mass diffusional motions of the polymer chains, the theory is important because it serves as a precursor to the de Gennes reptation theory, described next. 

Compare the Rouse-Bueche theory with the de Gennes theory. How do they model molecular motion ... 

Combining this with the prediction of the Rouse-Bueche theory for Hnear, imentangled melts (Eq. 6.16), we obtain ... 

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. 

Although theories of the Rouse-Bueche-Zimm type have been very successful in rationalizing the behavior of polymeric systems from a molecular point of view, another class of theories is presently commanding the most attention. These theories treat the motion of polymer molecules in terms of reptation, a reptile-like diffusive motion of each polymer molecule through a matrix formed by its neighbors. To a considerable extent, this new approach has overcome some of the most important shortcomings of the normal-mode theories, which... 

These theories are associated with the names of Rouse, Bueche and Zimm [26-28] and are based on the idea of representing the motion of polymer chains in a viscous liquid by a series of linear differential equations. They are essentially dilute solution theories, but we shall see that, rather unexpectedly perhaps, they can be extended to predict the behaviour of the pure polymer. Because of its simplicity, we will give an account of the theory due to Rouse [26]. 

The Rouse, Zimm and Bueche theories are satisfaetory for the longer relaxation times, which involve movement of submolecules. This has been eonfirmed for dilute polymer solutions, where the theory would be expeeted to be most appropriate [29,30]. More remarkably, it also holds for solid amorphous polymers (Referenee 11, Chapter 13), provided that the friction coefficient is suitably modified. 

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

The well-known theory of Zimm (80) for an infinitely dilute solution of chain molecules will be discussed in more detail. Other theories are considered as special cases [Rouse (16), Bueche (53)] or refinements [Ptitsyn and Eizner (84), Hearst (85) and Tschoegl (36)]. If a linear polymer chain is built up of IV subchains, there are IV — 1 junction-points and two end-points. At these points beads of equal diameter are assumed. In this way the hydrodynamic interaction with the solvent is accomplished. The positions of the said IV + 1 beads are described by 3 (IV + 1) Cartesian coordinates. This system is considered as a visco-elastic system in the sense of the previous section. 

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... 

Molecular theories describing the dielectric relaxation behavior of polymers have been developed and are summarized in references 5, 12 and 13. Again, if the dipoles are rigidly attached along the chain contour, normal mode theories such as those of Rouse and Bueche described in Chapter 3 for the mechanical case might be expected to be applicable. In addition, the time-temperature superposition principle also generally applies. 

The molecular theory of polymer viscoelasticity rests on the work of Bueche (101-104), Rouse (105), and Zimm (106,107), investigating behavior of diluted solutions of linear polymers. The molecular theory of viscoelasticity has not been very successful in describing viscoelasticity of solid polymers over the whole temperature interval, and thus modified theory of rubber elasticity has to be used... 

The first molecular theories concerned with polymer chain motion were developed by Rouse (57) and Bueche (58), and modified by Peticolas (59). This theory begins with the notion that a polymer chain may be considered as a succession of equal submolecules, each long enough to obey the Gaussian distribution function that is, they are random coils in their own right. These submolecules are replaced by a series of beads of mass M connected by springs... 

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. ... 

Chapter 6 treats mean-field theories of melt behavior. We begin with the Rouse model for molecules in dilute solution and its modification by Bueche to deal with unentangled melts. The longest Rouse relaxation time emerges from this treatment and plays an important role in all molecular models. The tube model is introduced, in which the basic relaxation... 

As in the case of the relaxation modulus, the dependence on temperature is quite weak, because within the experimentally accessible temperature range, the variation of p T is small. According to the Bueche-Rouse theory, over some range of times or frequencies away from the terminal zone, where the two or three longest relaxation times can be neglected, the relaxation modulus can be approximated by Eq. 6.17 ... 

Horizontal and vertical shifts comply equally well with Arrhenius equations as shown in Figures 7.18-7.19. The vertical shifts can be explained by the density and temperature product changes as assumed in the Bueche-Rouse theories of the linear viscoelaslicity of unentangled polymer melts and solutions [157]. Moreover, if the Delay and Plazek [157] definition is... 

---