Adsorption of interacting self-avoiding trails in two dimensions

We investigate the surface adsorption transition of interacting self-avoiding square lattice trails onto a straight boundary line. The character of this adsorption transition depends on the strength of the bulk interaction, which induces a collapse transition of the trails from a swollen to a collapsed phase, separated by a critical state. If the trail is in the critical state, the universality class of the adsorption transition changes; this is known as the special adsorption point. Using flatPERM, a stochastic growth Monte Carlo algorithm, we simulate the adsorption of self-avoiding interacting trails on the square lattice using three different boundary scenarios which differ with respect to the orientation of the boundary and the type of surface interaction. We confirm the expected phase diagram, showing swollen, collapsed, and adsorbed phases in all three scenarios, and confirm universality of the normal adsorption transition at low values of the bulk interaction strength. Intriguingly, we cannot confirm universality of the special adsorption transition. We find different values for the exponents; the most likely explanation is that this is due to the presence of strong corrections to scaling at this point.

Figure 1

Examples of trails for the three cases of surface interaction studied. The trail starts at the solid circle in the surface. Bulk and surface interactions are indicated by dashed circles and are denoted by $\omega$ and $\kappa$, respectively. Panel (a) shows monomer-surface interactions in a horizontal surface, panel (b) shows bond-surface interactions in a horizontal surface, and panel (c) shows monomer-surface interactions in a diagonal surface. Note that the starting point does not contribute for the surface energy.

Figure 2

Finite-size fluctuation map for 128-step trails for the three boundary cases MS, BS, and DS. Darker colors represent regions of small fluctuations, while brighter colors (yellow and orange) represent regions with strong fluctuations.

Figure 3

(a) Graphs of ${\mathrm{\Gamma }}_n$ as a function of $\kappa$ for $\omega =2$. The black circles denote the maximum points $({\kappa }_n,max{\mathrm{\Gamma }}_n)$ of the ${\mathrm{\Gamma }}_n$ curves. (b) Graphs of the length scale exponents ${\nu }_{\perp ,n}$ (solid line) and ${\nu }_{\parallel ,n}$ (dashed line) as a function of $\kappa$ for $w=3$. The inset shows the crossing points (black circles) of these exponents. (c) The average number of contacts $〈m_s〉$ as a function of $n$ for $\omega =1$ for five different values of $\kappa$. (d) The bulk specific heat per monomer for $\kappa =1$. The inset shows the location of the maximum values of the bulk specific heat as a function of the length $n$.

Figure 4

The phase diagram for the MS case. The red square is the location of a multicritical point where the special surface transition occurs. The solid lines are critical transitions, i.e., the coil-collapsed transition and the normal surface transition. The dashed line is the coexistence line of the collapsed-adsorbed transition.

Figure 5

Surface exponents as a function of the bulk interaction parameter $\omega$ for trails with up to 10 240 steps. The red circles are the values for the ${\varphi }^{\left(a\right)}$ and the black circles are the values for $1/\delta$. The dashed line represents the expected value in two dimensions. (a) The MS case. (b) The BS case. (c) The DS boundary scenario.

Figure 6

Surface exponents $1/\delta$ and ${\varphi }^{\left(a\right)}$ as a function of $\omega$ for the BS case for trails with 2048 and 10 240 steps. The black and red circles are the values of $1/\delta$ and ${\varphi }^{\left(a\right)}$, respectively, for trails with 2048 steps (lower symbols), and the blue and orange circles are the values of $1/\delta$ and ${\varphi }^{\left(a\right)}$, respectively, for trails with 10 240 steps (upper symbols).

Figure 7

(a) The surface exponent ${\varphi }^{\left(s\right)}$ for different sizes as a function of $n^{-0.5}$ on the special surface point. Black circles are the values for the MS case, red squares are those for the BS case, and blue triangles are those for the DS case. The dashed black line is the expected value of 0.44. (b) Exponent $1/{\delta }^{\left(s\right)}$ as a function of $n^{-0.5}$ for the boundary scenarios.

Figure 8

Bulk specific heat per monomer normalized with $\varphi \approx 0.78$ as a function of $\kappa$ for $\omega =3$ for the MS case. Two different sizes are shown, 1024 steps (black curve) and 2048 steps (lighter red curve).