Abstract
Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.
The Fourier transforms of the Kohlrausch function are calculated to obtainη′ andη″. A three-parameter model forη′ andη″ is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions.
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Abbreviations
- D :
-
rate of deformation tensor
- G(t) :
-
mechanical relaxation modulus
- H :
-
relaxation spectrum
- I(t):
-
flux of defects
- P n (s) :
-
probability of finding a walker ats aftern-steps
- P :
-
generating function ofP n (s)
- s(t) :
-
fraction of surviving defects
- Γ, (γ) :
-
gamma function (incomplete)
- η 0 :
-
zero shear viscosity
- η * (ω) :
-
complex viscosity
- ω :
-
frequency
- 〈t〉 n :
-
n-th moment
- F[]:
-
Fourier transform
- f * (u) :
-
Laplace transform off(t)
- η′, η″ :
-
components ofη *
- G f, η *f :
-
fractional model
- G 3, η *3 :
-
three parameter model
- \(\bar z\) :
-
complex conjugate ofz
- \(\mathop D\limits^ \circ \) :
-
material time derivative ofD
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Stastna, J., De Kee, D., Powley, M. et al. Fractal-time stochastic processes and dynamic functions of polymeric systems. Rheol Acta 29, 137–144 (1990). https://doi.org/10.1007/BF01332380
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DOI: https://doi.org/10.1007/BF01332380