Hookean spring models

The inclusion of internal viscosity raises considerably the free-energy storage capacity of a rapidly deforming macromolecule as compared to the idealized Hookean spring model and could play a decisive role in mechanochemical reactivity in transient elongational flow. 

Theories for polymer dynamics of dilute polymer solutions include the elastic (Hookean) spring model (Kuhn, 1934) which considers that the system is mechanically equivalent to a set of beads attached with a spring. The properties are then based on a spring constant between beads and the friction of beads through solvent. The viscosity of a Hookean system is then described by... 

In the study of Smith et al. [1999], a high temporal and spatial resolution microscopy is used to reveal features that previous studies could not capture [Alon et al., 1995]. They found that the measured dissociation constants for neutrophil tethering events at 250 pN/bond are lower than the values predicted by the Bell and Hookean spring models. The plateau observed in the graph of the shear stress vs. the reaction rate suggests that there is a force value above which the BeU and spring models are not valid. Since the model proposed so far considers the ceU as a rigid body, whether the plateau is due to molecular, mechanical or ceU deformation is not clear at this time. 

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

The results were compared to MD-simulations [317]. Whereas the scattering function of pure PEO could be well described, the dynamics of the salt-loaded samples deviates from the predictions obtained with various electrostatic interaction models. The best but still not perfect and - at least for longer times -unphysical model assumes Hookean springs between chains to simulate the Na-ion mediated transient cross-links [317]. 

Models are used to describe the behavior of materials. The fluid or liquid part of the behavior is described in terms of a Newtonian dashpot or shock absorber, while the elastic or solid part of the behavior is described in terms of a Hookean or ideal elastic spring. The Hookean spring represents bond flexing, while the Newtonian dashpot represents chain and local segmental movement. 

In this model, derived originally for star-shaped branched molecules, polymer molecules are represented by beads connected by identical Hookean springs, and the decrease in viscosity with branching is expressed by the g1/2 rule. 

Figure 14.19 Stress-strain plots for (a) a Hookean spring where E is the slope (6) a Newtonian dash pot where s is constant, (c) stresstime plot stress for relaxation in the Maxwell model, and (d) stresstime plot stress for a Voigt-Kelvin model.

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

This equation describes Hookean elasticity, and Po = G (G is the modulus of rigidity). In Fig. 9, the classical mechanical spring model representing Eq. (14) is illustrated. If, however, it is assumed that jSi is the only nonzero constant in Eq. (13), then ... 

The two extreme cases of mechanical behavior can be reproduced very well by mechanical models. A compressed Hookean spring can serve as a model for the energy-elastic body under load (Figure 11-11). On releasing the load, the compressed spring immediately returns to its original position. The relationship between the shear stress (021) = Oe, the shear modulus Ge, and the elastic deformation ye is given by Hooke s law [Equation (11-1)] ... 

Thus, the restoring force is proportional to the extension and the onedimensional chain behaves as a Hookean spring. This important result simplifies the analysis of the normal modes of motion of a polymer. Polymer chain models can be treated mathematically by the much simpler linear differential equations because second order effects are absent. (It should be noted diat, while the elastic equation for a polymer chain is identical in form with Hooke s law, the molecular origin of the restoring force is very different). 

This model consists of two identical beads with bead friction coefficient C joined by a Hookean spring with spring constant H. 

This model consists of identical beads each with friction coefiicient C joined together linearly by Hookean springs each with spring constant H. For this model the solution to Eq. (13.4) is ... 

Once the singlet distribution function has been found, we are in a position to evaluate the various contributions to the fluxes that depend on (see Table 1). In this section we discuss the contnbutions to the stress tensor, and in the next two sections the contnbutions to the mass and heat flux vectors. In these sections, for illustrative purposes, we restrict ourselves to the Rouse bead-spring chain and the Hookean dumbbell models, for which we can use the singlet distribution functions , given in Eqs. (13.5) and (13.8). 

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