Rouse theory

However, there is no interaction between beads other than the spring force. The restoring force on each of the beads is given by 

When the polymer is emerged in a solvent medium, an additional force, frictional drag, acts on the bead  

This set of linear first-order differential equations can be summarized in the form of a matrix equation 

Final results obtained for the viscoelastic properties of dilute solutions of coiling polymers are given in a series of equations. The real and imaginary components of a complex viscosity rj = rjj — iri2 are 

Each relaxation time makes a contribution of r kTxj to t q. The subscript runs from 1 to N. The longer relaxation time, ti, accounts for nearly all of the viscosity. The short relaxation time, accounts for only a small part of the viscosity. The two quantities may be expressed as 

In an undiluted polymer, the number of polymer molecules per cubic centimeter, n, is pNo/M rather than cNo/M, where p is the density. Equations 18 and 19 of Chapter 9 and the corresponding equation for G then become  

The characteristic length a will correspond to that in dilute solution in a 0-solvent. Since the low-frequency limit of G is ojt/o, the friction coefficient fo can be expressed in terms of i/o by a calculation analogous to the derivation of equation 27 of Chapter 9  

In the terminal zone of the frequency scale cf. Chapter 2, Section A3) the properties are dominated by the terminal relaxation time. 

The terminal segments of G and G plotted against to are described by the equations (in which 1.08 = 2 for the Rouse theory) 

Other features of the viscoelastic behavior in the terminal zone are represented by the constants rjo and Rearrangement of equations 7 and 9, together in the equation 51 of Chapter 9 with omission of r)s as appropriate for undiluted polymer, gives for these quantities 

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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 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. 

Table 3.5 Rouse Theory Expressions for the Modulus (entries labeled 1) and Compliances (entries labeled 2) for Tension and Shear Under Different Conditions ...

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

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 calculation of g for Gaussian uniform star chains was carried out by Zimm and Kilb (ZK) [83]. They used a modified version of the dynamic Rouse theory including preaveraged HI (in the non-draining limit) that considers the particular connectivity of units consistently with the star architecture. This ap-... 

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. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

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 ... 

These results make it clear that the forms of t]0 — rjs and Je° are completely independent of model details. Only the numerical coefficient of Je° contains information on the properties of the model, and even then the result depends on both molecular asymmetry and flexibility. Furthermore, polydispersity effects are the same in all such free-draining models. The forms from the Rouse theory cany over directly, so that t]0 - t]s, translated to macroscopic terms, is proportional to Mw and Je° is proportional to the factor A/2M2+, /A/w. Unfortunately, no such general analysis has been made for models with intramolecular hydrodynamic interaction, and of course these results apply in principle only to cases where intermolecular interactions are negligible. 

The failure of the Rouse theory was attributed to the pathological nature of medium motions in entangled systems, and not any special defect in the Rouse representation of the polymer chain itself. For Rouse chains in a deforming continuous medium, the frictional force depends on the systematic velocity of the bead relative to the medium. The frictional force on a bead is therefore a smootly... 

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) ... 

Ferry, J.D., Landel,R.F Williams, M.L. Extensions of the Rouse theory of viscoelastic properties to undiluted linear polymers. J. Appl. Phys. 26,359-362 (1955). 

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. 

G -G" co relationship) in Figure 1.20 for Zdoll at 10°C with the following remarks. The simple Rouse theory in Eq. (1.11) implies that... 

Fatkullin NF, Kimmich R, Kroutieva M (2000) The twice-renormalised Rouse formalism of polymer dynamics Segment diffusion, terminal relaxation, and nuclear spin-lattice relaxation. J Exp Theor Phys 91(1) 150-166 Ferry JD (1980) Viscoelastic properties of polymers, 3rd edn. Wiley, London Ferry JD (1990) Some reflections on the early development of polymer dynamics Viscoelasticity, dielectric dispersion, and self-diffusion. Macromolecules 24 5237-5245 Ferry JD, Landel RF, Williams ML (1955) Extensions of the Rouse theory of viscoelastic properties to undilute linear polymers. J Appl Phys 26 359-362 Fikhman VD, Radushkevich BV, Vinogradov GV (1970) Reological properties of polymers under extension at constant deformation rate and at constant extension rate. In Vinogradov GV (ed) Uspekhi reologii polimerov (Advances in polymer rheology, in Russian). Khimija, Moscow, pp 9-23... 

Hess W (1988) Generalised Rouse theory of entangled polymeric liquids. Macromolecules... 

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]. 

Figure 4-13 contains the predictions of the Rouse theory on the left and of the Zimm theory on the right. As is to be expected, the predictions of the Zimm theory that takes the hydrodynamic interactions into account predicts well experimental data. 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

Freely Draining Gaussian Chain (Rouse Theory) Dominant HI Theta Solvent (Zimm Theory) Dominant HI Good Solvent... 

Figure 3.13 Linear viscoelastic data (symbols) for polystyrene in two theta solvents, decalin and diocty Iphthalate, compared to the predictions (lines) of the Zimm theory with dominant hydrodynamic interaction, h = oo. The reduced storage and loss moduli and G are defined by = [G ]M/NAksT and G s [G"]M/A /cbT, where the brackets denote intrinsic values extrapolated to zero concentration, [G jj] = limc o(G /c) and [G j ] = limc +o[(G" — cor)s]/c), and c is the mass of polymer per unit volume of solution. The characteristic relaxation time to is given by to = [rj]oMrjs/NAkBT. For frequencies ro

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