Dynamic Drop Models

We follow the dynamic evolution of a cluster of Ising spins pointing up surrounded by other spins pointing down, on a lattice. The cluster represents a liquid drop. Under a microscopic point of view, the short range ferromagnetic coupling between these spins plays the role of the van der Waals attraction. Alternatively , under a macroscopic point of view, the same ferromagnetic coupling gives rise to the surface tension along the drop boundary. This naive model is applied to the study of different systems, out of thermodynamic equilibrium. For each such a system, other interaction terms can be included, for instance an external magnetic field with a downwards uniform gradient, representing Earth's gravity. Also, for each system, proper dynamic rules and boundary conditions are adopted. The behaviour of such a drop is monitored as a function of time, through computer simulations. Many quantities of interest, in particular those related to drop fragmentation, were measured and the results were compared with available experimental data. The real systems we have in mind are exemplified by water drops falling from a leaky faucet, nuclear multifragmentation, mercury drops falling on the ground, magnetic hysteresis curves, and interface roughness. The stationary geometry of a water drop pending from a vertical wall is an old mathematical puzzle (see [1] and references therein). A stochastic model [1] was applied as an alternative way to solve this puzzle. The drop is represented by a cluster of Ising spins pointing up (S i = +1) on a square lattice (only the projection of the drop on the vertical plane perpendicular to the wall is of interest). All other lattice spins point down (S i = −1), representing the air around the water drop. These spins are coupled through a first-and-second neighbour ferromagnetic interaction term strong enough to avoid the drop fragmentation. A non-uniform external magnetic field pointing also up is included, with the same intensity along each horizontal lattice row. However, this intensity increases uniformly when one goes from each row to the next one just below, i.e. there is a uniform vertical downward gradient which mimics the Earth's gravity. The Hamiltonian reads H = − S i S j − g r r i S i , (1) where the first sum runs over all pairs < ij > of spins which are first or second neighbours on the square lattice. The index r = 0, 1, …