6.1.3 Hysteresis

In general, relaxation processes in non-equilibrium systems are characterized by the intrinsic relaxation time ($ \tau_0$) of the system and the external characteristic time at which the system is driven ( $ \tau_\mathrm{ext}$). In the case of the DNA unzipping at CF, there is an intrinsic relaxation time that indicates the average time the molecule needs to reach the thermal equilibrium and an external time related with the loading rate of the pulling experiment. The hysteresis is a phenomenon that appears when the driven rate is faster than the relaxation time of the system ( $ \tau_\mathrm{ext} \ll \tau_0$). In this situation, the system gets stuck in metastable states because it is not capable of reacting fast enough to the changes induced by the external control parameter. This phenomenon is specially observed in systems with rough free energy landscapes (e.g., disordered or glassy systems) that have energetic barriers much higher than the thermal noise. Under this circumstances, the barriers prevent the system to explore the other regions of the free energy landscape.

Here we are precisely facing this effect. In the CF unzipping experiments, the rate of change of the force, although very slow to our lab time scale, is much faster than the relaxation time of the DNA molecule. The slowest feasible experimental unzipping/rezipping cycles do not reproduce the equilibrium FDC. And even if we further reduce the pulling rate, there is no guarantee that we will be able to obtain it. In fact, the relaxation time of a system depends exponentially on the height of the typical energy barriers involved, which is about $ 20$-$ 30$ $ k_BT$ at the coexistence force (see Fig. 3.20). So there are firm reasons to believe that the equilibrium FDC$ _f$ will be obtained after performing pulling experiments that last orders of magnitude much longer than the ones described here (see Eq. 6.1 and the corresponding discussion for an estimation of the time required).

So far, we have focused on the experiments performed at low pulling rates. Considering that we will obtain similar results by decreasing the loading rate, we can start looking at the other side. By increasing the loading rate, the irreversibility of the process increases. The cycles of hysteresis become wider and the total amount of dissipated work increases. Figure 6.3a shows several unzipping/rezipping cycles at different loading rates and the corresponding area of the cycles of hysteresis. The unzipping of DNA at CF has already been studied for short hairpins [153,154] and a linear relation between the dissipated work and the loading rate has been found in the limit of slow loading rates. Moreover, the dissipated work tends to zero when the time spent in the pulling tends to infinity (zero pulling rate). However, in our case this does not hold, at least for the range of loading rates explored (0.05-5 pN$ \cdot $s$ ^{-1}$). In fact, the dissipated work tends to saturate at a value of $ \simeq 1000$ $ k_BT$.

Figure 6.3: Hysteresis. (a) Pulling cycles at different loading rates. (Inset) Dissipated work (area enclosed by the cycle) vs. loading rate. (b) Hysteresis in logarithmic scale. Same graph as the inset in panel a. The continuous curve is just a visual guide.
\includegraphics[width=\textwidth]{figs/chapter6/hysteresis.eps}

Figure 6.3b shows a logarithmic representation of the loading rate, which is indicative of the trend of the dissipated work at very low loading rates. According to this, the pulling rate must be decreased orders of magnitude to let the system equilibrate and reduce the dissipated work down to zero. A detailed study of the free energy landscape of the system would give valuable information about the energetic barriers that cannot be surpassed during an unzipping process at CF.

JM Huguet 2014-02-12