Nonequilibrium interactions between ideal polymers and a repulsive surface

We use Newtonian and overdamped Langevin dynamics to study long flexible polymers dragged by an external force at a constant velocity $v$. The work $W$ performed by that force depends on the initial state of the polymer and the details of the process. The Jarzynski equality can be used to relate the nonequilibrium work distribution $P\left(W\right)$ obtained from repeated experiments to the equilibrium free energy difference $\mathrm{\Delta }F$ between the initial and final states. We use the power law dependence of the geometrical and dynamical characteristics of the polymer on the number of monomers $N$ to suggest the existence of a critical velocity $v_c\left(N\right)$, such that for $v<v_c$ the reconstruction of $\mathrm{\Delta }F$ is an easy task, while for $v$ significantly exceeding $v_c$ it becomes practically impossible. We demonstrate the existence of such $v_c$ analytically for an ideal polymer in free space and numerically for a polymer which is being dragged away from a repulsive wall. Our results suggest that the distribution of the dissipated work $W_{\mathrm{d}}=W-\mathrm{\Delta }F$ in properly scaled variables approaches a limiting shape for large $N$.

Figure 1

Illustration of the distribution of work $P\left(W\right)$ (solid line) and the shifted function $G=e^{-\beta W}P\left(W\right)$ (dashed line). $P$ is measured by repeating the experiment, and $G$ needs to be reconstructed from the approximate knowledge of $P$.

Figure 2

A beads-and-springs model of a Gaussian (Rouse) chain is being dragged in free space by pulling its end point $x_0$ with a constant velocity $v$, such that $x_0=vt$. Note, that we consider only a one-dimensional dynamical problem, and the second dimension in this figure is for illustration purpose only.

Figure 3

Normalized mean work ${\mu }_{\mathrm{ND}}$ for $N=100$ as a function of time $t$ (see text). For large $N,{\mu }_{\mathrm{ND}}$ resembles a triangular wave with a period $T=4N/\omega$. (For large $t$ the apparent periodicity disappears.)

Figure 4

Normalized mean work ${\mu }_{\mathrm{OLD}}$ as function of time $t$ for $N=100$ (see text). At short times, ${\mu }_{\mathrm{OLD}}$ is parabolic, and for large times it grows linearly with $t$.

Figure 5

At $t=0$ a polymer is at equilibrium with one of its ends attached to the wall. At $t>0$ it is dragged away from the wall with a constant velocity $v$ until it reaches a distance $L=5R$, where it is equilibrated. We consider a one-dimensional problem, and the transverse dimension is only for illustration purposes.

Figure 6

Results for the dissipated work distribution in the ND case extracted from a sample of $\mathcal{N}={10}^3$ repeated calculations for a short polymer ($N=10$) dragged away from a repulsive wall. The solid line histograms depict the numerically calculated probability density $\tilde{P}\left(y\right)$ measured as a function of $y=\beta W_{\mathrm{d}}$, where $W_{\mathrm{d}}$ is the dissipated work. Dotted lines represent the shifted function $\tilde{G}\left(y\right)=e^{-y}\tilde{P}\left(y\right)$. The polymer was dragged at a constant velocity (a) $v=0.1v_c$ and (b) $v=2v_c$.

Figure 7

Results for the dissipated work distribution in the OLD case. All the notations are the same as in Fig. 6.

Figure 8

Results for the dissipated work distribution in the ND case extracted from a sample of $\mathcal{N}={10}^3$ repeated calculations for polymers of three different $N\mathrm{s}$ as indicated in the legend. Each calculation was performed with the same relative velocity $u=1$, or $v=v_c$.