Collapse transition in polymer models with multiple monomers per site and multiple bonds per edge

We present results from extensive Monte Carlo simulations of polymer models where each lattice site can be visited by up to $K$ monomers and no restriction is imposed on the number of bonds on each lattice edge. These multiple monomer per site (MMS) models are investigated on the square and cubic lattices, for $K=2$ and 3, by associating Boltzmann weights ${\omega }_0=1$, ${\omega }_1=e^{{\beta }_1}$, and ${\omega }_2=e^{{\beta }_2}$ to sites visited by 1, 2, and 3 monomers, respectively. Two versions of the MMS models are considered for which immediate reversals of the walks are allowed (RA) or forbidden (RF). In contrast to previous simulations of these models, we find the same thermodynamic behavior for both RA and RF versions. In three dimensions, the phase diagrams, in space ${\beta }_2\times {\beta }_1$, are featured by coil and globule phases separated by a line of $\mathrm{\Theta }$ points, as thoroughly demonstrated by the metric ${\nu }_t$, crossover ${\varphi }_t$, and entropic ${\gamma }_t$ exponents. The existence of the $\mathrm{\Theta }$ lines is also confirmed by the second virial coefficient. This shows that no discontinuous collapse transition exists in these models, in contrast to previous claims based on a weak bimodality observed in some distributions, which indeed exists in a narrow region very close to the $\mathrm{\Theta }$ line when ${\beta }_1<0$. Interestingly, in two dimensions, only a crossover is found between the coil and globule phases.

Figure 1

Rescaled average squared end-to-end distance $〈R^2〉/N$ against ${\beta }_1$ for (a) RA and (b) RF models in three dimensions with $K=2$. In both panels, the insertions show the extrapolation of ${\beta }_1^{*}$ to $N\to \infty$, where the solid lines are linear fits of the data for the largest chain lengths, and the dashed lines show the central values of ${\beta }_{1,\mathrm{\Theta }}$ estimated from the intersections in the main plots.

Figure 2

(a) Exponent $\nu$ against ${\beta }_2$ for fixed ${\beta }_1=-0.2$ in the RF model in three dimensions. The vertical (dotted blue) line indicates the value of ${\beta }_2$ where the probability distribution of sites with three monomers seems to have two peaks (see Fig. 5). (b) Metric ${\nu }_t$ (main plot) and crossover ${\varphi }_t$ (inset) exponents versus the ratio $R={\beta }_{2,\mathrm{\Theta }}/{\beta }_{1,\mathrm{\Theta }}$, for RA (blue triangles) and RF (black circles) models in three dimensions, calculated along the $\mathrm{\Theta }$ lines (see Fig. 4). The horizontal (dashed, red) lines indicate the $\mathrm{\Theta }$ class exponents, and open (full) symbols are data for ${\beta }_{1,\mathrm{\Theta }}>0$ (${\beta }_{1,\mathrm{\Theta }}<0$).

Figure 3

(a) Exponent $\gamma$ versus the connectivity coefficient $\mu$ calculated at the $\mathrm{\Theta }$ point $({\beta }_{1,\mathrm{\Theta }},{\beta }_{2,\mathrm{\Theta }})=(-0.2,0.55)$ for the RF model in three dimensions. (b) Tricritical entropic exponent ${\gamma }_t$ against the ratio $R={\beta }_{2,\mathrm{\Theta }}/{\beta }_{1,\mathrm{\Theta }}$ for RA (blue triangles) and RF (black circles) models in three dimensions. In both panels, the horizontal (dashed red) line indicates the $\mathrm{\Theta }$ exponent ${\gamma }_t=1$. In panel (b), open (full) symbols are data for ${\beta }_{1,\mathrm{\Theta }}>0$ (${\beta }_{1,\mathrm{\Theta }}<0$), and the insertions display the connectivity constants ${\mu }_t$ as functions of ${\beta }_{1,\mathrm{\Theta }}$ (left) and ${\beta }_{2,\mathrm{\Theta }}$ (right) for both RA and RF models in three dimensions.

Figure 4

Phase diagrams for (a) RA and (b) RF models in three dimensions with $K=3$. In both panels, circles (blue) and triangles (red) are the tricritical points obtained, respectively, from the intersections of $〈R_N^2〉/N$ curves and from the second virial coefficient. The dashed vertical lines (black) indicate the values of ${\beta }_{1,\mathrm{\Theta }}$ for the case $K=2$. A comparison between the $\mathrm{\Theta }$ lines for RA and RF models is shown in the insert in (b).

Figure 5

Probability distributions of the number of sites visited by three monomers, for the RF model in three dimensions with fixed ${\beta }_1=-0.2$ and several values of ${\beta }_2$.

Figure 6

Metric exponents $\nu$ against ${\beta }_2$ for (a) RA and (b) RF models in two dimensions and several chain lengths. The parameter ${\beta }_1$ is fixed at ${\beta }_1=0$ in (a) and ${\beta }_1=-0.20$ in (b). The dashed and dotted horizontal lines indicate the exponents expected for the Duplantier-Saleur (DS) and Blöte-Nienhuis (BN) universality classes.