Phase transition and surface sublimation of a mobile Potts model

We study in this paper the phase transition in a mobile Potts model by the use of Monte Carlo simulation. The mobile Potts model is related to a diluted Potts model, which is also studied here by a mean-field approximation. We consider a lattice where each site is either vacant or occupied by a $q$-state Potts spin. The Potts spin can move from one site to a nearby vacant site. In order to study the surface sublimation, we consider a system of Potts spins contained in a recipient with a concentration $c$ defined as the ratio of the number of Potts spins $N_s$ to the total number of lattice sites $N_L=N_x\times N_y\times N_z$. Taking into account the attractive interaction between the nearest-neighboring Potts spins, we study the phase transitions as functions of various physical parameters such as the temperature, the shape of the recipient, and the spin concentration. We show that as the temperature increases, surface spins are detached from the solid phase to form a gas in the empty space. Surface order parameters indicate different behaviors depending on the distance to the surface. At high temperatures, if the concentration is high enough, the interior spins undergo a first-order phase transition to an orientationally disordered phase. The mean-field results are shown as functions of temperature, pressure, and chemical potential, which confirm in particular the first-order character of the transition.

Figure 1

Ground state of the system with $N_x=N_y<N_z$.

Figure 2

Average number of particles per site $n$ (curve 1, red, left scale) and order parameter $Q$ (curve 2, blue, right scale) versus temperature $T$ with (a) $\mu =-0.3$, (b) $\mu =-0.4$, and (c) $\mu =-0.51$. See text for comments.

Figure 3

Phase diagram in the plane ($T,\mu$). The solid line is a first-order transition line.

Figure 4

Phase diagram in the plane ($T,p$).

Figure 5

Isotherms ($p,n$) are shown by thick curves for $T=0.25$ (curve 1, violet), 0.23 (curve 2, green), 0.2 (curve 3, blue), and 0.18 (curve 4, red). Thin broken lines indicate discontinuities of $n$.

Figure 6

Phase diagram in the plane ($T,n$). See text for comments.

Figure 7

Entropy (left) and energy (right) as functions of temperature and chemical potential.

Figure 8

Surface in the space “temperature, pressure, and chemical potential.”

Figure 9

Magnetization $M$ versus temperature $T$ (in unit of $\text{J}/k_B$) for a lattice of $15\times 15\times 30$ sites with a spin concentration $c=50\%$. The system is completely ordered at $T=0$, and then surface spins are evaporated little by little with increasing $T$. The phase transition of spin orientations of the solid core occurs at $T_c\simeq 1.234$.

Figure 10

Energy histogram $P\left(U\right)$ recorded at $T_c=1.234$ for a $15\times 15\times 30$ lattice with $c=50\%$. The presence of the two peaks indicates that the transition is of first order.

Figure 11

Comparison of the evolution of (a) the magnetization and (b) energy versus the temperature of a half-filled lattice $N_x\times N_y\times N_z$ for several sizes $N_x=N_y=N_z/2=20$ (red void circles), 30 (blue filled circles), 40 (green void diamonds), 50 (black stars), 60 (magenta crosses), and 70 (sky-blue void squares).

Figure 12

Transition temperature versus lattice size $N_x\times N_y\times N_z$ at $c=50\%$ where $N_x=N_y=N_z/2$, with $N_z=30$, 40, 50, 60, 70, and 80.

Figure 13

Simulation for a half-filled lattice size $35\times 35\times 70$ ($c=50\%$): (a) snapshot at $T=1.3128$ close to the transition and (b) $R$ vs $Z, R$ being the percentage of lattice sites having $Z$ nearest neighbors, at $T=1.3128$. Note that the solid phase is well indicated by the number of sites with six neighbors.

Figure 14

Effect of concentration. (a) Magnetization versus temperature for a lattice $15\times 15\times N_z$ where $N_z=20$ (red crosses), 30 (black stars), 40 (green diamonds), 50 (blue filled circles), and 60 (red void circles). For each case, only the 15 first layers are filled, corresponding to concentrations $15/N_z$. (b) Transition temperature versus the recipient height $N_z$.

Figure 15

(a) Total magnetization $M$ and (b) energy per spin $U$ versus $T$. Red void circles indicate results of localized spins and blue filled circles indicate those of the completely mobile model. Between these two limits, green void diamonds, black stars, and magenta crosses correspond respectively to the cases where one, two, and four surface layers are allowed to be mobile.

Figure 16

Diffusion coefficient $D$ versus $T$. Red void circles (lowest curve) indicate results of localized spins and blue filled circles (topmost curve) indicate those of the completely mobile model. Between these two limits, from below green void diamonds, black stars, and magenta crosses correspond, respectively, to the cases where one, two, and four surface layers are allowed to be mobile.

Figure 17

(a) Magnetic susceptibility $\chi$ and (b) heat capacity $C_V$ versus $T$. Red void circles indicate results of localized spins, and blue filled circles indicate those of the completely mobile model. Between these two limits, green void diamonds, black stars, and magenta crosses correspond, respectively, to the cases where one, two, and four surface layers are allowed to be mobile.

Figure 18

Layer magnetizations for the first four layers. Red void circles, blue filled circles, green diamonds, and black stars are the magnetizations of the first, second, third, and fourth layers. See text for comments.

Figure 19

(a) Magnetization, (b) energy, and (c) diffusion coefficient for two system shapes, $20\times 20\times 40$ (red void circles) and $40\times 20\times 20$ (blue filled circles), at $c=50\%$.