Phantom network model relationships

Equating the chemical potential to zero gives a relationship between the equilibrium degree of swelling and the molecular weight Mc. The relation for Mc ph is obtained for a tetrafunctional phantom network model as... 

The relationship of chain dimensions to macroscopic state of deformation has been the focus of a wide body of research. Two classical theories, the affine and the phantom network models, that relate the microscopic deformation to the macroscopic one are described first. More recent theories are then discussed. 

A network is an ensemble of macromolecules linked together, each of them rearranging its configurations by Brownian motion. Classical thermodynamics explains the behavior of elastomers with regard to force, temperatme, pressme, and volume, but does not give the relationship between the molecular structme of the network and elastic quantities such as the moduli. Therefore, statistical mechanics was introduced in the 1940s (16,86,87,107-109), and its theoretical predictions were tested (110-112). Because of the complexity of network structures, two models based on affine and phantom networks were studied. The cross-linking points... 

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

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