Spring model, bead-and

The bead and spring model is clearly based on mechanical elements just as the Maxwell and Voigt models were. There is a difference, however. The latter merely describe a mechanical system which behaves the same as a polymer sample, while the former relates these elements to actual polymer chains. As a mechanical system, the differential equations represented by Eq. (3.89) have been thoroughly investigated. The results are somewhat complicated, so we shall not go into the method of solution, except for the following observations ... 

Fig. 8a,b. Off-lattice representations of a three-functional star a Bead and Rod model b Bead and Spring model... 

Figure 5.1 Schematic illustration of bead-and-spring model of atomic force between atoms. Reprinted, by permission, from M. F. Ashby and D. R. H. Jones, Engineering Materials 1, 2nd ed., p. 44. Copyright 1996 by Michael F. Ashby and David R. H. Jones.

Figure 7.5 Bead-and-spring model of a polymer chain.

Figure 3.5 (a, b) Illustration of beads-and-springs models. Part (b) shows a two-bead flexible dumbbell. (From Larson 1988.)... 

The Bead-and-Spring Model in Bad and Good Solvents Dynamics of the Collapsed Chain... 

Having proceeded this far, we are now in a position to write the equation of motion for our linear bead-and-spring model. To do this we divide the polymer molecule into z submolecules so that there are z springs and z + 1 beads. We now introduce a unidirectional deformation. The restoring force on each of the beads is given by ... 

Now one may associate these relaxation times with the relaxation times of a generalized Maxwell model. Thus the stress relaxation behavior for the bead-and-spring model is given as... 

Quite disappointing, however, is the comparison of the curves in this figure with those presented in Chapters 2, 4 and 5 that show the observed experimental response of bulk polymers. Experimentally, high-molecular-weight polymers exhibit two major transitions while the bead-and-spring model predicts only one, albeit with a broadened time response relative to a Maxwell model. 

Bueche6 has shown that the flexible bead-and-spring model leads to the following expression for the creep compliance of an undiluted polymer ... 

Is such a deformable chain model inconsistent with the non-Newtonian intrinsic viscosity Finding an answer to this question is the goal of this paper. To this end, the viscosity of xanthan solutions was measured over a broad range of shear stress, including especially the low-shear Newtonian limit which has not been measured by Whitcomb and Macosko. The intrinsic viscosity at various shear stresses was then determined and the resultant experimental curve was compared to theoretical expectations for a flexible chain (bead-and-spring) model. 

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. 

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