Dumbbell model

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 ${\mbox{\boldmath$R$}} = {\mbox{\boldmath$R$}_1} - {\mbox{\boldmath$R$}_2}$. The evolution of ${\mbox{\boldmath$R$}}$ 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 ${\mbox{\boldmath$R$}}$ reads:

$$\zeta \dot{\mbox{\boldmath$R$}} = - {\partial E \over \parti... ...\mbox{\boldmath$R$}}} + \sqrt{2 k_B T}{\mbox{\boldmath$\xi$}}$$ (3.5)

where $\zeta$ is the friction coefficient, $E$ is the potential energy of the spring, and the thermal noise has been modeled by means of the zero average Brownian process ${\mbox{\boldmath$\xi$}}$ with correlation $\langle \xi_i(t) \xi_j (t') \rangle = \delta_{ij} \delta(t-t')$.

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:

$$\dot{\mbox{\boldmath$R$}} = {\mbox{\boldmath$v$}}(\mbox{\boldmath$R$}_1,t) - {\mbox{\boldmath$v$}}(\mbox{\boldmath$R$}_2,t)$$ (3.6)

Since the typical size of polymers (order $0.1-0.2 \mu m$), is usually smaller than the viscous scale of turbulence $\eta \equiv (\nu ^3 / \epsilon)^{1/4}$, the velocity field is smooth on the scale ${\mbox{\boldmath$R$}}$ and the velocity difference can be estimate by the velocity gradient $v_i(\mbox{\boldmath$R$}_1,t) - v_i(\mbox{\boldmath$R$}_2,t) = \nabla_j v_i R_j + O({R^2})$.

With the addiction of the stretching term the equation (3.5) became:

$$\zeta \left[ \dot{\mbox{\boldmath$R$}} - ({\mbox{\boldmath$R... ...{\mbox{\boldmath$R$}}} + \sqrt{2 k_B T}{\mbox{\boldmath$\xi$}}$$ (3.7)

As long as the elongation is smaller than the maximum length $R \ll R_{max}$, the polymer can be modeled by an elastic spring of Hook modulus $K_0$ so that $E= K_0 R^2/2$. The value of the elastic modulus is fixed by requiring that the zero-shear equilibrium length, estimated from the balance between elastic end thermal energy $E \sim k_b T$, must be equal to the radius of gyration of the molecule $R_0$:

$$R_0 = \sqrt{ k_B T / K_0}$$ (3.8)

The explicit expression for the Hook modulus $k_0 = K_B T / R_0^2$ points out the entropic origin of the ``elastic'' behavior of polymers.

Substituting the quadratic shape for potential energy into Eq. (3.7) we get the equation for the elongation ${\mbox{\boldmath$R$}}$ [56]:

$$\dot{\mbox{\boldmath$R$}} = ({\mbox{\boldmath$R$}} \cdot \na... ...math$R$}} + \sqrt{2 R_0^2 \over \tau} {\mbox{\boldmath$\xi$}}$$ (3.9)

where has been introduced the polymer relaxation time $\tau$

$$\tau = {\zeta \over K_0} = {\zeta R_0^2 \over k_B T}\;.$$ (3.10)

The relaxation time $\tau$ in general is dependent on the elongation $R$, because the friction coefficient $\zeta$ changes with the size of the molecule, and when the elongation grows to values close to $R_{max}$ the elastic potential is no longer quadratic, so the Hook modulus $K_0$ changes with $R$. To take in account these effects the Finite Extendible Nonlinear Elastic model (FENE model) [57] assumes $\tau \propto (R^2_{max} - R^2) / (R^2_{max} - R^2_0)$. Nevertheless, the linear model is supported by the experimental evidence of a constant relaxation time in the regime $R \ll R_{max}$.

In the Zimm model, considering the entraining of fluid within the coiled polymer, the friction coefficient is estimated as $\zeta= 4 \pi R_0 \mu$, 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.