Elasticity equilibrium

To determine the crosslinking density from the equilibrium elastic modulus, Eq. (3.5) or some of its modifications are used. For example, this analysis has been performed for the PA Am-based hydrogels, both neutral [18] and polyelectrolyte [19,22,42,120,121]. For gels obtained by free-radical copolymerization, the network densities determined experimentally have been correlated with values calculated from the initial concentration of crosslinker. Figure 1 shows that the experimental molecular weight between crosslinks considerably exceeds the expected value in a wide range of monomer and crosslinker concentrations. These results as well as other data [19, 22, 42] point to various imperfections of the PAAm network structure. 

Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

Ronca and Allegra (12) and Flory ( 1, 2) assume explicitly in their new rubber elasticity theory that trapped entanglements make no contribution to the equilibrium elastic modulus. It is proposed that chain entangling merely serves to suppress junction fluctuations at small deformations, thereby making the network deform affinely at small deformations. This means that the limiting value of the front factor is one for complete suppression of junction fluctuations. 

A new stress-relaxation two-network method is used for a more direct measurement of the equilibrium elastic contribution of chain entangling in highly cross-linked 1,2-polybutadiene. The new method shows clearly, without the need of any theory, that the equilibrium contribution is equal to the non-equilibrium stress-relaxation modulus of the uncross-linked polymer immediately prior to cross-linking. The new method also directly confirms six of the eight assumptions required for the original two-network method. 

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

There exist a number of experimental methods for determination of structure sensitive parameters of a system undergoing branching and crosslinking. However, evaluation of some of the results requires application of a theoretical approach to the phenomenon the measurement is concerned with. Then, we may be testing two theories at once. The equilibrium elasticity is one example, since there exist alternative rubber elasticity theories. However, certain conclusions can always be made. 

This is a theoretical study on the structure and modulus of a composite polymeric network formed by two intermeshing co-continuous networks of different chemistry, which interact on a molecular level. The rigidity of this elastomer is assumed to increase with the number density of chemical crosslinks and trapped entanglements in the system. The latter quantity is estimated from the relative concentration of the individual components and their ability to entangle in the unmixed state. The equilibrium elasticity modulus is then calculated for both the cases of a simultaneous and sequential interpenetrating polymer network. 

Further, the equilibrium elasticity of a monolayer film is related to the compressibility of the monolayer (analogous to bulk compressibility) by... 

Among them is the gel point conversion, if multifunctional units are present, as well as accompanying divergence of viscosity, onset of equilibrium elasticity modulus, etc. By comparing the results of modeling with experiment, one can verify to what extent the chemistry is affected by physical interactions which are practically always active in polymerizations. 

Figure 7.12. Elastic and viscous stress-strain curves for normal articular cartilage. Elastic (top) and viscous (bottom) stress-strain curves were obtained by plotting the equilibrium (elastic) and the total-equilibrium (viscous) stresses for visibly normal cartilage.

The oscillating bubble method proves to be very convenient and precise for the evaluation of the non-equilibrium elasticity of surfaces in a wide range of frequencies of external disturbances and surface coverage (adsorption of surfactant) [103-105]. It is based on registration of the sinusoidal variation of bubble volume. The bubble is situated in a capillary containing surfactant solution in which oscillations of different frequencies and amplitudes are created. The treatment of the U = f(ft)) curves (where U is the tension needed to initiate oscillations of constant amplitude) allows the determination of Marangoni elasticities [105]. 

The equilibrium elasticity is observed in the process of extension (or contraction) of the film when there is an equilibrium between the film surface and bulk. It is a consequence of the decrease in equilibrium surfactant concentration when the film is extended. (Nonequilibrium elasticity corresponds to the extension, when there is no equilibrium in the film). 

A detailed analysis of the theoretical concepts of equilibrium elasticity and its role in the stability of various objects is presented by Kitchener [25], Given bellow are the simplest equations of the modulus of the equilibrium elasticity permitting to elucidate the main dependences of the elasticity properties on surface activity and surfactant concentration as well as on film thickness. 

The value of the modulus of equilibrium elasticity of films from typical surfactants (NaDoS, CTAB) of thicknesses from 1 to 4 pm varies within the range of 10 to 60 mN m1. This correlates well with the theory [25]. 

The mechanism of the equilibrium elasticity acts until it is possible to provide a surfactant re-partition between the exterior and interior of the film. In a NBF such a repartition is not possible and this mechanism of elasticity ceases to act. The elasticity properties of bilayer films, in which the hydrodynamic and adsorption processes are characterised with normal time of relaxation, are due to Marangoni effect in the insoluble adsorption layers. That is why stable foams with black films are very sensitive to different local disturbances (heating, vibration, etc.). 

A qualitative evidence of the above are the data reported in [52]. It has been established that there is a correlation between the calculated rate of internal diffusion foam collapse and the experimentally determined rate. To obtain a stable foam from poor surfactants (alcohols, acids, etc.) under these conditions is hardly possible because of either insufficient dynamic elasticity of foam films or the lack of equilibrium elasticity (for films from insoluble surfactants). Furthermore, the n barrier for films from acid or alcohol solutions is low and the typical capillary pressures for a real foam are sufficient to induce disturbance of the film equilibrium and, respectively, foam collapse. 

An elastically active network chain is active in the equilibrium elastic response of the network to deformation. From the topological point of view, an EANC is a chain between two active branch points. An active branch point is a imit from which at least three paths issue to infinity. In the case under consideration, only some of the chemically tetrafunctional diamine units can become active branch points. If the polyepoxide were more than bifunctional, it would also contribute to the number of EANC s. In analogy with Eq. (14), the pgf for the numbo- of bonds with infinite continuation issuing from a diamine unit T,(z) is given by... 

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