The simplest model to describe the behavior
of a molecule of polymer is the so called
Dumbbell model.
It consists in a couple of massless beads connected by a spring,
which corresponds to the end-to-end vector of the polymer
.
The evolution of
is determined by the sum
of three forces: the hydrodynamic drag force acting on the molecule,
the thermal noise, and the elastic force of the spring.
In absence of external flow the equation for
reads:
If the polymer is surrounded by a non homogeneous flow, we
must add to Eq. (3.5) the stretching force determined by
the difference of velocities of the external flow between the two beads:
With the addiction of the stretching
term the equation (3.5) became:
Substituting the quadratic shape for potential
energy into Eq. (3.7) we get the equation
for the elongation
[56]:
The relaxation time in general is dependent on the elongation , because the friction coefficient changes with the size of the molecule, and when the elongation grows to values close to the elastic potential is no longer quadratic, so the Hook modulus changes with . To take in account these effects the Finite Extendible Nonlinear Elastic model (FENE model) [57] assumes . Nevertheless, the linear model is supported by the experimental evidence of a constant relaxation time in the regime .
In the Zimm model, considering the entraining of fluid within the coiled polymer, the friction coefficient is estimated as , which substituted in (Eq (3.10) gives the Zimm relaxation time (Eq (3.3)).
The dumbbell model retains only the fundamental elastic mode of the polymer chain, with the slowest relaxation time. Although higher oscillatory modes, with faster relaxation times, have been observed experimentally [47] in DNA chains, they can be only weekly excited by the gradients of velocity in a turbulent flow, thus for a simplified rheological model it is sufficient to retain only the principal mode.