Network of chains

Next let us consider the differences in molecular architecture between polymers which exclusively display viscous flow and those which display a purely elastic response. To attribute the entire effect to molecular structure we assume the polymers are compared at the same temperature. Crosslinking between different chains is the structural feature responsible for elastic response in polymer samples. If the crosslinking is totally effective, we can regard the entire sample as one giant molecule, since the entire volume is permeated by a continuous network of chains. This result was anticipated in the discussion of the Bueche theory for chain entanglements in the last chapter, when we observed that viscosity would be infinite with entanglements if there were no slippage between chains. 

A thermoset polymer does not flow when it is heated and subjected to pressure. Thermoset polymers consist of an interconnected network of chains that are permanently chemically connected to their neighbors, either directly or via short bridging chains, as shown in Fig. 1.4. We refer to such networks as being crosslinked. Thermoset polymers do not dissolve in solvents, but they can soften and swell. 

The data presented in Figure 4.14a are consistent with the following mechanism. The dispersion that emerges from the blender is fundamentally unstable with respect to coagulation and coagulates rapidly to form a volume-filling network throughout the continuous phase. Except for the size and structure of the chains, the situation is comparable to a cross-linked polymer swollen by solvent. In both, the liquid is essentially immobilized by the network of chains, and the system behaves as an elastic solid under low stress. The term gel is used to describe such systems whether the dispersed particles are lyophilic or lyophobic. 

The structure of an elastomer comprises a network of chains, meaning that there are gaps between adjacent chains. Indeed the elasticity of rubber relies on substantial thermal motion of the chains, which would not be possible if the chains were closely packed. The free volume available in the rubber means that some liquids can enter the rubber and cause swelling - sometimes very large amounts of swelling. For example the ability of oil to swell natural rubber is well known. 

In the ideal case, when one considers the network of chains of equal lengths, the stresses under the given deformation can be obtained in a very simple way. In virtue of the speculations of the previous section, free energy of the whole network can be represented as the sum of free energy of all the chains, while each of the equal chains of the network can be characterised by the same equilibrium distribution function W(s), where s is the separation between adjacent junctions. In the state without deformation, the function has the form (1.5), while in a deformed state, it depends on the displacement gradients (1.39). The free energy of the whole network can be written down simply as... 

The resulting physical picture of a rubber-like system as a close-packed collection of mers is radically different from the two-phase image introduced by James and Guth [10]. The latter represents mbber as a network of chains, which act as entropic springs in tension, embedded in a bath of simple liquid. The bath gives rise to an isotropic pressure, whereas the network is responsible for the deviatoric stress. More recent physical pictures consider as well the distribution of network junctions in the liquid and the action of these junctions as constraints on the free motion of a generic chain of the network. The current description is on the mer or atomic level and treats the full stress tensor, both the mean and deviatoric portions, in terms of atomic interactions. 

With some natural starches or with glycogen it is possible that the action of the /5-amylase is stopped before all end chains are removed because the substrate molecule has so complicated a network of chains that the enzyme, which presumably has a large molecule, can not penetrate into the inner parts of the substrate molecule. This point has been stressed in the case of glycogen by Meyer and Jeanloz. These authors degraded the substrate by a short treatment with hydrochloric acid to the extent that all end chains were accessible for the enzyme and a true saccharification limit was reached. Experiments with different starches and... 

Fig. 6.11. Typical packing structures in concentrated suspensions (a) simple network of chains (b) denser floes, cormected by chains (c) dense packing of spheres surroimded by a thick immobile layer of solvent or absorbed species (redrawn from Sormtag and Strenge [6]).

Precipitated silicas produced the highest melt elasticity that we have observed from simple Cr/silica catalysts [521]. Silica gels are usually "set (gelled) under acidic or neutral conditions, as the pH from mineral acid is adjusted upward by the addition of sodium silicate. Primary silica particles form a network of chains that occupies the entire volume of the solution as illustrated in Scheme 23. In contrast, precipitated silicas are usually made under basic conditions as the pH of sodium silicate solution is adjusted downward by the addition of acid. Primary silica particles coagulate into strong secondary aggregates that precipitate out of solution as a fine flocculent, but they do not gel. The precipitated silica does not occupy the entire reaction volume as a gel does. The process is illustrated in Scheme 29. After precipitation, the secondary aggregate structure can sometimes be "reinforced" by deposition of a further silica layer. 

Increasing the concentration of metal particles in an insulating adhesive matrix changes the electrical properties of the composite in a discontinuous way. Assuming a random dispersion of the metal filler, as the concentration increases no significant change occurs until a critical concentration, pc, is reached. This point, where the electrical resistivity decreases dramatically, called the percolation threshold, has been attributed to the formation of a network of chains of conductive particles than span the composite. A two-dimensional cartoon of a conductive adhesive below p and just above pc is shown in Fig. 3. A typical plot showing the relationship between particle concentration and electrical resistivity is shown in Fig. 4. 

The existence of a united polymer network of chains and filler particles with higher density. 

Individual polymer chains in an ensemble can also be covalently joined to other chains around it at discrete points along it. This yields a 3D network of chains (or open-tree structures of chains or a mix of both). Cross-linking is desirable where insolubility and high mechanical strength are demanded of aplastic. Ideally, each and every chain will be linked to each other so that the entire ensemble of chains is a single giant molecule (this actually does occur in natural rubber when vulcanized or cross-linked.) An automobile or aircraft tire is an example of a fully cross-linked polymer. On heating, cross-linked polymers do not convert into a viscous liquid melt as the molecules are chemically linked to one another and cannot flow independently. 

An expression for the force as a function of strain can be established by statistical thermodynamic analysis of the chain, and then of a network of chains. 

The theory of rubber elasticity is largely based on thermodynamic considerations. It will be briefly discussed as an example of how thermodynamics can be applied in polymer science. Eor more detailed information the reader is referred to the various textbooks [10-13]. It is assumed that there is a three-dimensional network of chains, that the chain units are flexible and that individual chain segments rotate freely, that no volume change occurs upon deformation, and that the process is reversible (i.e., true elastic behavior). Another usual assumption is that the internal energy U of the system does not change with deformation. Eor this system the first law of thermodynamics can be written as ... 

Several models have been proposed to describe the motion of a single polymer chain in an entangled network of chains. In the reptation model, proposed by De Gennes (51),... 

R. P. Fellmann, W. H. Stass, and D. R. Kory, Impact-Resistant Polymer Mass, Ger. Offen. 2,748,751 (1978). Interpenetrating networks with poly(Me methacrylate). Co-continuous interpenetrating networks of chains. Ethylene vinyl acetate copolymer/poly(Me methacrylate) blends. 

The basic theory of gel formation from colloidal particles has been formulated by Thomas and McCorkle (228), who show that the Verwey-Overbeek theory for the interaction of two spherical double layers around adjacent spherical colloidal particles leads to isotropic flocculation. New particles can be attached more readily to the ends of a chainlike floe where the repulsion energy barrier is at a minimum. It is this type of aggregation that converts a sol to a gel at a certain point by forming an infinite network of chains of particles throughout the sol volume. (See also Chapter 3.)... 

Quantitative evaluation of the stress-strain characteristics of the rubber network then involves calculation of the configurational entropy of the whole assembly of chains as a function of the state of strain. This calculation is considered in two stages calculation of the entropy of a single chain and calculation of the change in entropy of a network of chains as a function of strain. 

We wish to calculate the strain-energy function for a molecular network, assuming that this is given by the change in entropy of a network of chains as a function of strain. 

A rubber consists of a cross-linked network of chains each of RMM = 2 x lO" the density of the specimen is 900 kg m . Calculate the shear modulus at 0 C. Assume a Gaussian network and... 

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