Unzipping an adsorbed polymer in a dirty or random environment

The phase diagram of unzipping of an adsorbed directed polymer in two dimensions in a random medium has been determined. Both the hard-wall and the soft-wall cases are considered. Exact solutions for the pure problem with different affinities on the two sides are given. The results obtained by the numerical procedure adopted here are shown to agree with the exact results for the pure case. The characteristic exponents for unzipping for the random problem are different from the pure case. The distribution functions for the unzipped length, first bubble, and the spacer are determined.

Figure 1

(Color online) Schematic diagram of a direct polymer in a random medium with an attractive wall. By construction, one direction along the polymer is special. The wandering is in the transverse $d$ direction and therefore it is a $d+1$ dimensional model. Here $d=1$. There is a line at $x=0$ (a wall or an interface) that attracts the polymer. (a) Hard wall: polymer is not allowed in the region $x<0$. (b) Soft wall: polymer is allowed in the whole region. There may be an extra potential $V(>0)$ on one side, say $x<0$. (c) Homopolymer in random medium. There is a unique (sample independent) ground state for this case. (d) Hetropolymer in pure medium. The ground state is sample dependent. In (c) and (d), randomness is in the shaded regions.

Figure 2

(Color online) Schematic diagram showing the first bubble of length $m_b$, spacer of length $m_s$, and the unzipped segment of length $m$ when a pulling force $g$ is applied at one end of the polymer in the random medium.

Figure 3

(Color online) Dimensionless force $\left(g\right)$ vs temperature $\left(T\right)$ phase diagram for an adsorbed polymer. Each lattice site on one side $(x<0)$ of the wall has a repulsive potential of strength $V>0$. Positive $g$ corresponds to the case of pulling the polymer away from the wall on the favorable side while the negative $g$ (note the different scale) is for pulling on the wrong side. The two limiting cases are $V\to 0$ (soft-wall) and $V\to \infty$ (hard-wall). The points are from numerics and the lines are the exact analytical results. Note the reentrance on the positive side for $V=0.67$ (shown in inset).

Figure 4

(Color online) Critical force vs disorder strength for both the hard-wall and the soft-wall cases at $\beta ϵ=15$, i.e. $(T=0.067)$. The thin slice of reentrance is shown in the inset.

Figure 5

(Color online) Phase surface for (a) hard-wall and (b) soft-wall in the presence of disorder.

Figure 6

(Color online) Force distance isotherms in (a) fixed force ensemble and (b) fixed distance ensemble, for a polymer of length $N=2000$ at $\beta ϵ=1.5(T=0.067)$. Each lattice site on one side of the wall contains a repulsive potential of strength “V” and polymer is pulled away from the wall on the other side. The inset in (a) shows the knees which developed in the fixed force ensemble for low $x$ but are absent in the fixed distance ensemble.

Figure 7

(Color online) Data collapse of extensibility at $\beta ϵ=15(T=0.067)$ for the pure case (a) soft-wall and (b) hard-wall. The lengths of the chain used are $N=1000$, 1500, and 2000. For both cases, the exponents are $d_{\chi }=2$ and $\varphi =1$.

Figure 8

(Color online) Specific heat up to a multiplicative constant for $g=1.0$. (a) There is a peak developing as $N$ increases, over and above a discontinuity at $T_c\left(g\right)=\mathrm{0.872\; 04}$. (b), (c) Data collapse at $T_c\left(g\right)$ for chains of length $N=1500$, 2000, and 2500, with ${\varphi }_t=1$ and $\alpha {\varphi }_t=0.93$ in (b) but $\alpha {\varphi }_t=1$ in (c).

Figure 9

(Color online) Data collapse for two different correlation functions for the random system. (a), (b) are for hard-wall and (c), (d) are for soft-wall. The chain length $N=1000$, 1500, and 2000. In (a) and (c) the disorder averaging is done at the response function level of an individual sample (by definition it is the extensibility up to a factor of $T^{-1}$). The exponents are $d_{\chi }=1.5,\varphi =0.5$. In (b) and (d) averaging is done at moment level. The exponents are $d_{\mathcal{C}}=2.0,\varphi =0.5$. For all these correlation functions, the disorder strength $\Delta =2∕3$ and averaging is done over ${10}^5$ realizations at $\beta ϵ=15(T=0.067)$.

Figure 10

(Color online) Specific heat for the random system at (a) $g=0$ and (b) $g=1$. There is a finite size dependence which, however, does not scale to yield a good data collapse. The averaging is done over ${10}^4$ samples.

Figure 11

(Color online) (a) Rescaled force $g{N}^{1∕3}$ vs rescaled average distance $⟨x⟩∕N^{2∕3}$, from the wall for four different samples of size $N=1024$ and $N=2048$ at $\beta ϵ=15(T=0.067)$ for the disorder strength $\Delta =2$ when there is binding to the wall. (b) Extensibility for the same samples.

Figure 12

(Color online) (a) Rescaled force $g{N}^{1∕3}$ vs rescaled average distance $⟨x⟩∕N^{2∕3}$ from the wall for four different samples of size $N=1024$ and $N=2048$ at $\beta ϵ=15(T=0.067)$ for the disorder strength $\Delta =2$ and there is no binding to the wall $(ϵ=0)$. (b) Extensibility for the same samples.

Figure 13

(Color online) Log-log plot of the probability distribution $P\left(⟨\Delta x⟩\right)$ of the rescaled average step height $⟨\Delta x⟩$. This is for a polymer of length $N=1024$ in a fixed force ensemble at $\beta ϵ=15(T=0.067)$, disorder strength $\Delta =2$, and there is no binding to the wall $(ϵ=0)$. ${10}^5$ samples are used to generate the statistics. The slope of the fitted line is 1.

Figure 14

(Color online) Rescaled force $g{N}^{1∕3}$ vs rescaled average distance $⟨x⟩∕N^{2∕3}$, from the wall for a polymer of length $N=2048$ at $\beta ϵ=15(T=0.067)$. The disorder is kept the same (at a disorder strength $\Delta =2$) for both binding and no binding ($ϵ=0$) to the wall. The steps are for a single sample and smooth curves are after averaging over ${10}^4$ samples.

Figure 15

(Color online) Data collapse for the probability distribution of pseudocritical force $P\left(g_0\right)∕N^{d_g}$ vs $(g_c-g_0){\widehat{N}}^{{\varphi }_g}$ for the hard-wall case for $N=512$ and 1024. $d_g=0.41$ and ${\varphi }_g=0.31$. The averaging is done over ${10}^5$ samples.

Figure 16

(Color online) Average separation from the wall, $⟨x_i⟩$, of $i\mathrm{th}$ monomer of chain of length $N=300$ at $\beta ϵ=3(T=1∕3)$ and disorder strength $\Delta =5∕3$ when a stretching force $g=0.5$ is applied at one end. Four different chain configurations are shown by the dotted and dashed lines. The solid line is when averaging is done over 90 000 samples.

Figure 17

(Color online) Probability distribution of the unzipped length $m$. (a) Plot of $P\left(m\right)$ vs $m^p$, at $\beta ϵ=3(T=0.333)$ and disorder strength $\Delta =5∕3$, when a stretching force $g=0.5$ is applied at the end. The points are for $N=200$ and 300 and the solid line is the fit using the functional form $f\left(m\right)=A{e}^{-\alpha m^p}$. Here $p=0.51\pm 0.01$. (b) Plot of $P\left(m\right)$ vs $m$ at $\beta ϵ=3(T=1∕3)$, a force $g=g_c=0.82$ is applied. This is the force on the phase boundary. The solid curve is the fit using the functional form $f\left(m\right)=c_0∕m^{b_0}+c_1∕{(N-m)}^{b_1}$. To obtain the fit, we have excluded $m=0$ and $m=N$. For this plot $c_0=0.026\pm 0.001,b_0=0.37\pm 0.01,c_1=0.028\pm 0.001$, and $b_1=0.71\pm 0.02$ are the fitting parameters. $N=300$ for both the figures.

Figure 18

(Color online) Probability distribution of length of the spacer, and the length of the first bubble, at critical force $g_c=0.82$ and $\beta ϵ=3(T=1∕3)$ for a disorder strength $\Delta =5∕3$. A fitting function of the form $f\left(m\right)=A{e}^{-\alpha m^c}$ is used to fit the data. The best fit gives the value $c=0.72\pm 0.01$ for zipped segment and $c=0.31\pm 0.01$ for the first bubble.