Excluded volume effects and fractional viscoelasticity in polymers
The excluded volume effect is added to a fractional viscoelastic model for modeling fractal polymers. This reveals a physical connection between the fractional time derivative, fractal geometry, and excluded volume effect. This derivation is a general theoretical framework based on the Scott-Blair fractional model of viscoelasticity when the excluded volume and the hydrodynamic interaction are explicitly taken into account to derive the microscopic stress within the molecular theory of Rouse and Zimm. The methodology extends the generalized molecular theory of Zimm by adding the effect of excluded volume where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. The modified distribution of the end-to-end vector of a monomer contained within a polymer network is used for pre-averaging approximations of the mobility matrix in the Zimm model. The pre-averaging approximation is important since the mobility matrix is a nonlinear function and it is difficult to explicitly calculate. Through application of thermodynamic laws, we derive the linear fractional model of viscoelasticity based on its spectral dimension, fractal dimension, and the excluded volume parameter for fractal media. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is strongly correlated with fractal structure and excluded volume effects.
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