Pulling self-interacting polymers in two dimensions

We investigate a two-dimensional problem of an isolated self-interacting end-grafted polymer, pulled by one end. In the thermodynamic limit, we find that the model has only two different phases, namely a collapsed phase and a stretched phase. We show that the phase diagram obtained by Kumar et al. [Phys. Rev. Lett. 98, 128101 (2007)] for small systems, where differences between various statistical ensembles play an important role, differs from the phase diagram obtained here in the thermodynamic limit.

Figure 1

(Color online) Schematic phase diagram proposed by Kumar et al. [6].

Figure 2

(Color online) The model of a polymer on the two-dimensional square lattice pulled by the last monomer. The arrow indicates the direction of the pulling force. The dark (blue) filled circles on lattice sites denote monomers interacting via nearest-neighbor interactions.

Figure 3

The fluctuations in the number of contacts as a function of temperature for fixed force $F=0.0$ (upper left panel) and $F=0.5$ (lower left panel). Each panel contains curves for ISAWs of length (from bottom to top) $N=20,25,\dots ,55$. In the upper right panel we show a log-log plot of the growth in the peak value of the fluctuation curve with chain length $N$. The lower right panel shows the peak position (critical temperature value) vs $1∕N$.

Figure 4

(Color online) The finite-size phase diagram for flexible chains as obtained from the position of the peak in the contact fluctuation curves for $N=55$. The solid black curve and the dashed curve are obtained by fixing the force and varying the temperature.

Figure 5

Same as in Fig. 3 but for fixed force $F=1.1$ (upper left panel) and $F=1.2$ (lower left panel).

Figure 6

The fluctuations in the number of contacts as a function of force for fixed temperature $T=1.0$ (upper left panel) and $T=0.5$ (lower left panel). Each panel contains curves for ISAWs of length (from bottom to top) $N=20,25,\dots ,55$. In the upper right panel we show a log-log plot of the growth in the peak value of the fluctuation curve with chain length $N$. The lower right panel shows the peak position (critical force value) vs $1∕N$.

Figure 7

The average extension per monomer $⟨h⟩∕N$ as a function of temperature $T$ for different values of the force. Each panel contain four curves for, from top to bottom, $N=25$, 35, 45, and 55.

Figure 8

The average number of contacts per monomer $⟨m⟩∕N$ as a function of temperature $T$ for different values of the force. Each panel contain four curves for, from bottom to top, $N=25$, 35, 45, and 55.

Figure 9

(Color online) Schematic phase diagram conjectured by Kumar et al. [6, 9]. The conjectured phase diagram has three different phases: “collapsed,” “extended,” and “stretched.” We have performed simulations of the whole phase space up to length $N_{\max }=128$. The dashed lines at fixed temperatures $T=0.7$ and $T=1.71$ display lines along which simulations were performed for walk lengths up to $N_{\max }=1024$. The six points displayed are those at which we have focused our attention.

Figure 10

End-to-end distance divided by $N^{2\nu }$ against $1∕N$ for the point $(T,F)=(2.0,0.0)$ with, from top to bottom, 2$\nu =1.48$, 1.50, and 1.52. We see that $2\nu =1.5\left(2\right)$. This point is thus in the extended phase.

Figure 11

End-to-end distance divided by $N^{2\nu }$ against $N$ for two points, $(T,F)=(0.7,0.0)$ and $(T,F)=(0.7,0.6)$, for lengths up to $N=1024$. A clear linear dependence is seen implying $2\nu \approx 1$. These points are in the collapsed phase and this exponent result is consistent with this assumption.

Figure 12

End-to-end distance divided by $N^{2\nu }$ against $1∕N$ for three points, $(T,F)=(0.7,1.5)$, (1.71, 1.5), and (1.71, 0.6) with, from top to bottom, 2$\nu =1.99$, 2.00, and 2.01 for chain lengths up to $N=1024$. We conclude that all these points belong to a stretched phase of the same type where $\nu =1.0$.

Figure 13

(Color online) Here is a second possible phase diagram assuming there are two phases for nonzero forces, both with $\nu =1$. Instead of extended and stretched we have fully stretched and stretched. In such a hypothesized fully stretched phase the polymer is effectively in a rodlike conformation where the average height is approximately equal to the length of the polymer.

Figure 14

The fluctuations in the number of contacts $m$ for $F=1.4$ fixed plotted against temperature $T$. There is no sign of any growing singularities which would indicate a phase transition in the thermodynamic limit.

Figure 15

Plots of the fluctuations against force $F$ for the high temperature $T=1.71$. The chain lengths are, from top to bottom, $N=1024$, 512, 256, and 128.

Figure 16

Plots of the fluctuations against force $F$ for the low temperature $T=0.7$. At the peak the chain lengths are, from top to bottom, $N=512$, 256, 128, and 64.

Figure 17

The maximum of the fluctuations in the number of contacts $m$ for lengths from $N=32$ up to $N=128$ for four different forces (from top to bottom $F=1.10$, 1.12, 1.15, and 1.20) in the region where the original series data analysis detected a phase transition.

Figure 18

(Color online) The height of the last monomer at six different points depicted in Fig. 9. We see that in all cases the height of the last monomers does not converge to 1, which we would expect for a fully stretched phase for figures at $F=1.5$, $T=0.7$, and $T=1.71$ and for $F=0.6$, $T=1.71$.

Figure 19

The behavior of the average number of bulk contacts $⟨m⟩$ for length $N=32$ up to $N=128$ for the point $T=0.1$, $F=1.2$ as a function of $1∕N$. We observe an increase in the average number of contacts $⟨m⟩$ with $N$ even for very low temperatures.

Figure 20

(Color online) The finite-size phase diagram of a self-interacting self-avoiding walk under tension in two dimensions for length $N=128$. We have estimated the position of the maximum force $F_p$, and the force $F_c$ for $T=0.0$. The $T_{\theta }^N\approx 1.47$. One would expect that $F_c=1.0$ in the thermodynamic limit.

Figure 21

The logarithm of the fluctuation in the number of contacts per monomer $m∕N$ against $\ln \left(N\right)$ at $T=0.074$ for the force $F$ at which the fluctuations are maximal. From this curve we can estimate the exponents $\alpha =0.62\left(10\right)$ and $\varphi =0.72\left(6\right)$, which we note are not those of the two-dimensional $\theta$ transition.

Figure 22

(Color online) The final conjectured phase diagram for a two-dimensional self-interacting polymer under a stretching force arising from our analysis in this work.