Monte Carlo study of the droplet formation-dissolution transition on different two-dimensional lattices

Abstract

In 2003 Biskup [Commun. Math. Phys. 242, 137 (2003)] gave a rigorous proof for the behavior of equilibrium droplets in the two-dimensional (2D) spin-1/2 Ising model (or, equivalently, a lattice gas of particles) on a finite square lattice of volume V with a given excess delta M identical with M-M 0 of magnetization compared to the spontaneous magnetization M 0=m0V . By identifying a dimensionless parameter Delta(delta M) and a universal constant Delta c , they showed in the limit of large system sizes that for Delta<Delta c the excess is absorbed in the background ("evaporated" system), while for Delta>Delta c a droplet of the minority phase occurs ("condensed" system). By minimizing the free energy of the system, they derived an explicit formula for the fraction lambda(Delta) of excess magnetization forming the droplet. To check the applicability of the asymptotic analytical results to much smaller, practically accessible, system sizes, we performed several Monte Carlo simulations of the 2D Ising model with nearest-neighbor couplings on a square lattice at fixed magnetization M . Thereby, we measured the largest minority droplet, corresponding to the condensed phase in the lattice-gas interpretation, at various system sizes (L=40,80,...,640) . With analytical values for the spontaneous magnetization density m 0 , the susceptibility chi , and the Wulff interfacial free energy density tau W for the infinite system, we were able to determine lambda numerically in very good agreement with the theoretical prediction. Furthermore, we did simulations for the spin-1/2 Ising model on a triangular lattice and with next-nearest-neighbor couplings on a square lattice. Again finding a very good agreement with the analytic formula, we demonstrate the universal aspects of the theory with respect to the underlying lattice and type of interaction. For the case of the next-nearest-neighbor model, where tau W is unknown analytically, we present different methods to obtain it numerically by fitting to the distribution of the magnetization density P(m) .

Planar ordering in the plaquette-only gonihedric Ising model

“…The possibility of using an order parameter akin to the m sq in Eq. (14) had also been suggested by Lipowski [9], who confirmed that it appeared to possess the correct behaviour in a small simulation.…”

“…1 we show the full time series for the magnetisation m and the fuki-nuke parameter m x abs of the multicanonical measurements for an intermediate (L = 20) and the largest (L = 27) lattice size in the simulations along with the system energy. The time series of the energy per system volume e = E/L 3 of the larger lattice can be seen to be reflected numerous times at e −0.9 (coming from the disordered phase) and e −1.2 (coming from the ordered phase) which shows qualitatively that additional, athermal and non-trivial barriers may be apparent in the system [14]. It is clear that the standard magnetisation is not a suitable order parameter, since it continues to fluctuate around zero even though the system transits many times between …”

Shape effects on the cluster spreading process of spin-crossover compounds analyzed within an elastic model with Eden and Kawasaki dynamics

“…49 In Fig. 10, we represent the total elastic energy of the spin crossover systems as a function of the relative size of the LS domain and for different ratios of the major and minor axes of the ellipse.…”

Free-Energy Barrier at Droplet Condensation

“…16) When all quantities are normalized to the total number of spins, the Δ parameter requires a geometric correction factor α = 2/ √ 3 ≈ 1.075, which is just the inverse square root of the hexagonal unit cell volume √ 3/2 for links of unit length. Hence, here we have to plot λ against…”

“…And (ii ) does the Ising model with an extended range of interaction or on different two-dimensional lattice types behave similarly? To answer these questions we have performed Monte Carlo computer simulations for increasing system sizes of (i ) square lattices with NN interactions 12), 16) and (ii ) next-nearest-neighbour (NNN) interactions as well as of triangular lattices with NN interactions. 16) In addition, we also analyzed in case (i ) the finite-size scaling behaviour of the associated free-energy barrier.…”