Particle-based Ising model

We characterize equilibrium properties and relaxation dynamics of a two-dimensional lattice containing, at each site, two particles connected by a double-well potential (dumbbell). Dumbbells are oriented in the orthogonal direction with respect to the lattice plane and interact with each other through a Lennard-Jones potential truncated at the nearest neighbor distance. We show that the system's equilibrium properties are accurately described by a two-dimensional Ising model with an appropriate coupling constant. Moreover, we characterize the coarsening kinetics by calculating the cluster size as a function of time and compare the results with Monte Carlo simulations based on Glauber or reactive dynamics rate constants.

Figure 1

(a) Top view of the system (on the $x\text{−}y$ plane), showing the triangular lattice, along with ${\sigma }_{\mathrm{cut}}=1.5\sigma$; $\sigma$ is the particles diameter and ${\sigma }_{\min }=2^{\frac{1}{6}}\sigma$. (b) Side view of the system ($x\text{−}z$ plane). Each dumbbell consists of a bottom particle (type 1, grey) and a top particle (type 2, red) whose $z$ coordinate varies. The former are fixed in position, while the latter interact with potential $V^{22}$. Particles within each dumbbell interact through the potential $V_{\mathrm{mod}}^{12}$. (c) Instantaneous configuration of the dumbbells obtained by using the VMD software [13]. (d) Potential $V^{22}\left(r\right)$, with ${\sigma }_{\mathrm{cut}}, {\sigma }_{\min }$, and the inflection point ${\sigma }_{\inf }={\left(\frac{26}{7}\right)}^{\frac{1}{6}}\sigma$ highlighted. (e) Side view (on the $x\text{−}z$ plane) of two interacting dumbbells in two different minima of $V_{\mathrm{mod}}^{12}$. The minima are located at $z=\sigma$ (spin $s=-1$) and $z=1.54\sigma$ ($s=+1$). The type 2 particles are at a distance of ${\sigma }_{\inf }$. (f) Dumbbell potential $V^{12}$ (purple), and its modified version $V_{\mathrm{mod}}^{12}$ (blue) with the quadratic function used between $r_1=1.0496\sigma$ and $r_2=1.4904\sigma$. The two minima correspond to different $z$ values of type 2 particle.

Figure 2

Equilibrium configurations sampled via molecular dynamics at six different temperatures, $T=0.25,0.285,0.32,0.33,0.335$. The system transitions from a magnetized state to a disordered one around $T=0.335$. Snapshot are taken from a top view in the $x\text{−}y$ axis in which only type 2 particles are shown. Each particle is colored according its $z$ coordinate.

Figure 3

Free energy as a function of the average magnetization $M$ for 6 temperatures (see key), computed using metadynamics. Two distinct minima are visible with an energy barrier between them larger than $2k_{B}T$ up until $T=0.33$.

Figure 4

Magnetization and heat capacity as a function of the temperature, obtained from unbiased simulations. The red bar identifies the region where the critical temperature is located.

Figure 5

Finite size scaling analysis for system sizes $L=10$, 20, 30, 40, $100\sigma$ (in the key). (a) Binder cumulant, along with dashed line highlighting the crossing point between lines. Scaling of (b) absolute magnetization, (c) susceptibility, (d) heat capacity, (e) heat capacity from pairwise interactions (p) and (f) from bonded ones (b).

Figure 6

(a) Illustration of the setup used to study single spin-flip dynamics. The central spin is allowed to flip, while the six neighbor spins have fixed spins, see text for detail. (b) Time evolution of $z$ for the central spin, with setup as in (a). (c) Zoom-in of (b) over a smaller time interval. (d) Free energy profile, as a function of $z$, with 0 to 3 neighbor spins with negative magnetization. (e) Rate of central spin-flip with 0 to 6 neighbor spins with opposite magnetization, along with Glauber and reactive dynamics MC rates adjusted to fit the MD, see text. In green, Kramers prediction based on Eq. (10). (f) Rate of central spin-flip with same setup as (a), as function of $\gamma$. The temperature is fixed to $T=0.185$.

Figure 7

(a) Illustration of the setup used to study cluster-flip dynamics. A single cluster is placed in the center of a system with opposite magnetization. (b) Time required to invert the cluster's magnetization, as a function of the cluster size, in term of spins number, for MD and reactive and Glauber MC.

Figure 8

Cluster growth, starting from an initial random configuration at $T=0.185<T_c$, as a function of time. (a) Three snapshots are shown at $t={10}^3,{10}^4,{10}^5{\tau }_{LJ}$ for MD simulations, and the corresponding snapshots for Glauber and reactive dynamics MC. (b) Average cluster size $L\left(t\right)$, (c) growth exponent $\alpha \left(t\right)$, (d) Ising model energy, and (e) absolute value of magnetization $\left|M\right|$ as a function of time for the three different dynamics considered. In cyan, reactive dynamics MC shifted by a factor of 2.75, see main text for details. In (b) the dashed line shows the power law fit with exponent 0.5. In (d) it shows a power law with exponent $-0.5$.