Network elastic force

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

In Eq. (III-9) the deformation ratios are defined with respect to a reference state in which the chain dimensions are such that they do not exert any elastic forces on the crosslinks (state of normal coiling). In general, the chains in a network may not actually be in this state at the beginning of a deformation experiment, because the ciosslinking process may quite well exert a, largely unknown, influence on the chain dimensions. 

This section seeks to make a quantitative evaluation of the relation between the elastic force and elongation. The calculation requires determining the total entropy of the elastomer network as a function of strain. The procedure is divided into two stages first, the calculation of the entropy of a single chain, and second, the change in entropy of a network as a function of strain. 

The aim of this section is to find the relation between the elastic force and the deformation for a polymer network. For that purpose the change in entropy associated with deformation of the chains in the network must be evaluated. Figure 3.7 shows the distribution of the chain end-to-end vectors in the deformed (stretched) and undeformed (unstrained) states. The distribution has spherical symmetry in the undeformed state, and when the... 

Description of the mechanics of elastin requires the understanding of two interlinked but distinct physical processes the development of entropic elastic force and the occurrence of hydrophobic association. Elementary statistical-mechanical analysis of AFM single-chain force-extension data of elastin model molecules identifies damping of internal chain dynamics on extension as a fundamental source of entropic elastic force and eliminates the requirement of random chain networks. For elastin and its models, this simple analysis is substantiated experimentally by the observation of mechanical resonances in the dielectric relaxation and acoustic absorption spectra, and theoretically by the dependence of entropy on frequency of torsion-angle oscillations, and by classical molecular-mechanics and dynamics calculations of relaxed and extended states of the P-spiral description of the elastin repeat, (GVGVP) . The role of hydrophobic hydration in the mechanics of elastin becomes apparent under conditions of isometric contraction. 

The equation for the elastic force is similar to Eq. (5.59) for the affine network with v replaced by... 

Formally speaking, the reaction of polymerization seems most effective at P -> oo, and on the T vs V2 state diagram (Figure 3.59), the asymptote T —> oo corresponds to v n responding to P ,c. At lower uj < V2,n, the curve A of the solution-gel transition heis a positive first derivative. The specific shape of the curve A depends on the model s details. J his curve ends on the binodal curve of the two-phase gel state due to the elasticity forces of the network chains and the interaction between polymer and LMWL (see above). The numerical values of g have been determined for different types of lattice. It has also been established that the inequality f J/j < V2,c holds true (de Gennes, 1979). 

Schematic stress-strain isotherms in elongation for a unimodal elastomer in the Mooney-Rivlin representation of modulus against reciprocal elongation. The isotherms are represented as the dependence of the reduced stress ([f ] = f /(a - on reciprocal elongation. (f = f/A, f = elastic force, A = undeformed area, a = elongation). The top three are for a crystallizable network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent. Isotherm D is for an unswollen unimodal network that is inherently noncrystallizable.

Ion-exchange resins are highly crosslinked polyelectrolytes. When placed in water they take up water and swell until the osmotic forces are balanced by the elastic forces of the polymer network. In this swollen form they are best viewed as solid solutions. 

The distance CyC, measured in units of the mesh size, is S. The displacement Co C reduces the entropy of the side group by an amount S In z, where z is the number of " gates surrounding one unit cell in the network. As soon as p exceeds a few units, this entropy defect is large, and there is a strong elastic force that tends to bring C back to Co-... 

Hydrogels with hydrophilic functional groups swell in water exclusively as a result of polymer—water interaction forces. Overall, three forces expand the hydrogel network polymer—water interactions, electrostatic interactions, and osmosis. In other words, infinite solubility of hydrogels is prevented by elastic forces, which originate... 

Rubber elasticity of a polymer network is one of the most distinctive features of long polymer chains. The elastic force of such a network is mainly due to, the change of conformational entropy of network strands which are connected to other strands by chemical linkages or topological constraints. The theoretical models to clarify the relationship between... 

Equation (7.6) predicts that the elastic force is directly proportional to the temperature and the total number of chains in the network. Both of these have been experimentally observed. 

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