Elastically effective cross-link

Efficiency of the cross-linking reaction can be defined as the ratio of effective cross-link density, px, to the theoretical cross-link density, p(. Theoretical cross-link density assumes that all of the cross-linker added to the formulation created elastically effective cross-links however, it cannot be defined for nonchemical methods. Thus the theoretical cross-link density, p is given as... 

Figure 5. Elastically effective cross-link density versus bake temperature for a high...

In 65% sugar jellies the hydrogen bridges alone are active. Such gels are easily deformed and are elastic. In calcium pectinate gels, the most effective cross links are calcium ion bonds between the carboxyl groups. The bond distances are short therefore an inelastic, rather brittle, gel results. 

Determination of cross-link density from compression experiments is perhaps the most effective means of determining cross-link density as long as samples of the appropriate geometry can be prepared. When a hydrogel is subjected to an external force, it undergoes elastic deformation which can be related to the effective cross-link density of the network [63,99], Here the measurements made to extract cross-link density from polymer deformation are briefly discussed. 

What happens upon equilibration with liquid water instead of water vapor According to Equation (6.13), the capillary radius would go to infinity for PVP —> 1. Thus, in terms of external conditions, swelling would be thermodynamically unlimited, corresponding to the formation of an infinitely dilute aqueous solution of ionomer. However, the self-organized polymer is an effectively cross-linked elastic medium. Under liquid-equilibrated conditions, swelling is not controlled by external vapor... 

What came before the reptation model Experimental data like the one in Figure 12.6 a used to be explained in the following way. A polymeric liquid was thought to contain some kind of effective cross-links. In contrast to the usual chemical cross-links (formed by chemical bonds), the effective ones do not live for long. They can only last for a period of the order of T. Then they break (or decay ), and new cross-links are created at other places, and so on. Thus, when t r, the cross-links do not have enough time to vanish. They hold the sample together, so it behaves like an elastic body. In contrast, when t r, the cross-links start decaying, and the sample flows. 

Let s go back to the simplest experiment shown in Figure 12.6, where we applied a small constant stress viscous flow) and t r (elasticity) From (12.6) and (12.2), we deduce that J t) Jit t/r] for t T. On the other hand, from (12.3) we have J t) l/E for t r. Can we tell what happens for t r Obviously, the two estimates should merge smoothly into one another. This idea leads us to a very important relationship between the viscosity rj, the longest relaxation time r, and Young s modulus E for a network of effective cross-links ... 

A network of effective cross-links behaves as a normal elastic network for t T. We discussed the classical theory of high elasticity in Chapter 7. The Young s modulus of a network, as you remember, is of the order of ksT multiplied by the density of cross-links. (As usual, ks is Boltzmann s constant, and T is the temperature.)... 

The rate at which coals swell is dictated by the rate at which the solvent diffuses into the coal. This is controlled by solvent properties, the size of the coal particles, and the average molecular weight between the cross-links of the coal matrix (Olivares and Peppas, 1992). Coal is a glassy solid at room temperature, but transitions to a flexible state as it absorbs solvent and the flexible nature of the swollen coal suggested lower effective cross-link density, and suggested that the elasticity of the solvent swollen coal may be predominantly rubber-like. The transition from the glassy to rubbery state is generally very sharp (Olivares and Peppas, 1992). 

In the statistical theory of elasticity for spatially structured polymers, the modulus of elasticity is a function of the number of effective cross-links ... 

The initial modulus is determined in the limit of small strain. The initial portion of the force-length curve is usually reversible. The deformation of the disordered interlamellar region is involved and the lamellar structure remains essentially intact. Interpreting the modulus, in terms of the basic structural and molecular parameters that define a semicrystalline polymer, is complex. In this region of very small strain, the primary effect is a rubber-like elastic deformation, whereby chain entanglements and other topological features act as effective cross-links. The total system is constrained by the bounding lamellae and their broad basal planes. 

Measurement of the rubbery plateau modulus ( ) provides a direct means by which to assess cross-link density in polymer networks. Classical rubber elasticity theory (Treloar, 1975 Mark, 1982 Hill, 1997) relates the elastic modulus in the rubbery region to effective cross-link density (v ) as shown, for example, in the equation suggested by Hill (1997) ... 

Usually two methods of estimating the effective cross-linking density are used. The first is based on elasticity theory. It computes the concentration of the cross-links in the network from the value of the elastic modulus and the polymer density according to the equation ... 

For semi- and full IPNs made of poly(oxyethylene) and poly(acrylic acid), the effective cross-link densities were determined from the elastic modulus and were compared with values estimated assuming the additivity of crosslink densities of components [113]. Discrepancies between estimated and calculated values are observed. 

Commercially produced elastic materials have a number of additives. Fillers, such as carbon black, increase tensile strength and elasticity by forming weak cross links between chains. This also makes a material stilfer and increases toughness. Plasticizers may be added to soften the material. Determining the effect of additives is generally done experimentally, although mesoscale methods have the potential to simulate this. 

At still longer times a more or less pronounced plateau is encountered. The value of the plateau modulus is on the order of 10 N m", comparable to the effect predicted for cross-linked elastomers in Sec. 3.4. This region is called the rubbery plateau and the sample appears elastic when observed in this time frame. 

Fig. 3. Effect of cross-link density where A represents tear strength, fatigue life, and toughness B, elastic recovery and stiffness C, strength and D,...

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