Coil-Stretch transition

The dumbbell model is able to reproduce the coil-stretch transition, which occurs when the stretching exerted by the flow overcomes the relaxation of polymer molecules.

The evolution of the polymer elongation can be obtained following the trajectories of polymer molecules, and Eq. (3.9) can be solved by characteristics as:

$${\mbox{\boldmath$R$}}(t) = \int_0^t ds W(t,s) \sqrt{ R_0^2 ... ... e^{-(t-s)/\tau} + W(t,0) {\mbox{\boldmath$R$}}(0) e^{-t/\tau}$$ (3.11)

where $W(t_2,t_1)$ is the evolution matrix of the linearized flow along the fluid trajectory ${\mbox{\boldmath$X$}}(t)$ from time $t_1$ to time $t_2$.

We are interested in the tail of the probability distribution function of elongation $R$ for $R \ll R_0$. Following Balkovsky et al. [52], at time large enough, when the initial condition is forgotten, the events contributing to the right tail of the pdf are those which have experienced large stretching. The gradients of velocity field tends to orientate the vector ${\mbox{\boldmath$R$}}$ in the direction of the leading Lyapunov vector, and its length is determined by the local value of the finite-time Lyapunov exponent $\gamma(s)$ as

$$R(t) \sim R_0 \int_0^{\infty} e^{\gamma s - s / \tau } \;ds$$ (3.12)

The values dominating the tail are those where $\gamma(s) s - s / \tau$ takes a sharp maximum at some $s_*$ before relaxing to its typical negative values. The probability of those events is given by the asymptotic form of the finite-time Lyapunov exponents

$$P(\gamma(s_*)) \sim \exp^{S(\gamma(s_*))s_*}$$ (3.13)

where $S(\gamma )$ is the Cramér function (see Section 2.4). With logarithmic accuracy one can then replace $\gamma(s_*) s_*= \ln (R/R_0) + s_* /\tau$. The maximum value $X_* \equiv (s_*)^{-1} \ln (R/R_0)$ is fixed by the saddle-point condition that $S-X_*S'$ should vanish at $X_* + \tau^{-1} + \lambda$. The final answer for the pdf is:

$$P(R) \propto R_0^{q} R^{-1 -q}$$ (3.14)

with

$$q = S'(X_* + \tau^{-1} + \lambda)$$ (3.15)

The convexity of the entropy function ensures that $q$ is positive if $\lambda < 1/ \tau$. In accordance with Eq. (3.15), the exponent $q$ decreases when $\lambda$ increases, and it tends to zero as $\lambda \to 1/\tau$. This means that when the value of the Weissenberg number $Wi = \lambda \tau$ exceeds unity the the integral of the pdf diverges at large $R$, and most of the polymer molecule are stretched. Below unity the pdf is stationary and only high moments of polymer elongation diverge. This is called ``coil-stretch'' transition.

The exponent $q$ can be expressed via the equation $L_{q}=q/\tau$, where $L_{q}=\max_{\gamma}[q\gamma-S(\gamma)]$ is the generalized Lyapunov exponent of order $q$. Its value indicates the highest converging moment of polymer elongation.

In the coiled state, moments of polymer elongation $\langle R^n \rangle$ with $n < q$ reach stationary values, while high moments with $n >q$ diverge exponentially in time as:

$$< R^n > \sim e ^ {(L_{n} - {n \over \tau})t}$$ (3.16)

In the Gaussian approximation of the Cramér function $S(\gamma )$, which holds near its minimum

$$S(\gamma) = {(\gamma - \lambda)^2 \over 2 \Delta}$$ (3.17)

the generalized Lyapunov exponents read

$$L_q = q \lambda + {q^2 \over 2} \Delta$$ (3.18)

and the exponent $q$ is $q = 2(1- \lambda \tau) / (\tau \Delta)$.