Ground state configurations of model molecular clusters

Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.

Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories

We consider a uniformly magnetized sphere that moves without friction in a plane in response to the field of a second, identical, fixed sphere, making elastic hard-sphere collisions with this sphere. We seek periodic solutions to the associated nonlinear equations of motion. We find closed-form mathematical solutions for small-amplitude modes and use these to characterize and validate our large-amplitude modes, which we find numerically. Our Runge-Kutta integration approach allows us to find 1243 distinct periodic modes with the free sphere located initially at its stable equilibrium position. Each of these modes bifurcates from the finite-amplitude radial bouncing mode with infinitesimal-amplitude angular motion and supports a family of states with increasing amounts of angular motion. These states offer a rich variety of behaviors and beautiful, symmetric trajectories, including states with up to 157 collisions and 580 angular oscillations per period.

Periodic bouncing modes for two uniformly magnetized spheres. II. Scaling

A uniformly magnetized sphere moves without friction in a plane in response to the field of a second, identical, fixed sphere and makes elastic hard-sphere collisions with this sphere. Numerical simulations of the threshold energies and periods of periodic finite-amplitude nonlinear bouncing modes agree with small-amplitude closed-form mathematical results, which are used to identify scaling parameters that govern the entire amplitude range, including power-law scaling at large amplitudes. Scaling parameters are combinations of the bouncing number, the rocking number, the phase, and numerical factors. Discontinuities in the scaling functions are found when viewing the threshold energy and period as separate functions of the scaling parameters, for which large-amplitude scaling exponents are obtained from fits to the data. These discontinuities disappear when the threshold energy is viewed as a function of the threshold period, for which the large-amplitude scaling exponent is obtained analytically and for which scaling applies to both in-phase and out-of-phase modes.

Periodic nonlinear sliding modes for two uniformly magnetized spheres

A uniformly magnetized sphere slides without friction along the surface of a second, identical sphere that is held fixed in space, subject to the magnetic force and torque of the fixed sphere and the normal force. The free sphere has two stable equilibrium positions and two unstable equilibrium positions. Two small-amplitude oscillatory modes describe the sliding motion of the free sphere near each stable equilibrium, and an unstable oscillatory mode describes the motion near each unstable equilibrium. The three oscillatory modes remain periodic at finite amplitudes, one bifurcating into mixed modes and circumnavigating the free sphere at large energies. For small energies, the free sphere is confined to one of the two discontiguous domains, each surrounding a stable equilibrium position. At large energies, these domains merge and the free sphere may visit both positions. The critical energy at which these domains merge coincides with the cumulation point of an infinite cascade of mixed-mode bifurcations. These findings exploit the equivalence of the force and torque between two uniformly magnetized spheres and the force and torque between two equivalent point dipoles, and offer clues to the rich nonlinear dynamics of this system. Online MagPhyx visualizations illustrate the dynamics.

Stability of vertical magnetic chains

A linear stability analysis is performed for a pair of coaxial vertical chains made from permanently magnetized balls under the influence of gravity. While one chain rises from the ground, the other hangs from above, with the remaining ends separated by a gap of prescribed length. Various boundary conditions are considered, as are situations in which the magnetic dipole moments in the two chains are parallel or antiparallel. The case of a single chain attached to the ground is also discussed. The stability of the system is examined with respect to three quantities: the number of balls in each chain, the length of the gap between the chains, and a single dimensionless parameter which embodies the competition between magnetic and gravitational forces. Asymptotic scaling laws involving these parameters are provided. The Hessian matrix is computed in exact form, allowing the critical parameter values at which the system loses stability and the respective eigenmodes to be determined up to machine precision. A comparison with simple experiments for a single chain attached to the ground shows good agreement.

Vortices and vortex lattices in quantum ferrofluids

The experimental realization of quantum-degenerate Bose gases made of atoms with sizeable magnetic dipole moments has created a new type of fluid, known as a quantum ferrofluid, which combines the extraordinary properties of superfluidity and ferrofluidity. A hallmark of superfluids is that they are constrained to rotate through vortices with quantized circulation. In quantum ferrofluids the long-range dipolar interactions add new ingredients by inducing magnetostriction and instabilities, and also affect the structural properties of vortices and vortex lattices. Here we give a review of the theory of vortices in dipolar Bose-Einstein condensates, exploring the interplay of magnetism with vorticity and contrasting this with the established behaviour in non-dipolar condensates. We cover single vortex solutions, including structure, energy and stability, vortex pairs, including interactions and dynamics, and also vortex lattices. Our discussion is founded on the mean-field theory provided by the dipolar Gross-Pitaevskii equation, ranging from analytic treatments based on the Thomas-Fermi (hydrodynamic) and variational approaches to full numerical simulations. Routes for generating vortices in dipolar condensates are discussed, with particular attention paid to rotating condensates, where surface instabilities drive the nucleation of vortices, and lead to the emergence of rich and varied vortex lattice structures. We also present an outlook, including potential extensions to degenerate Fermi gases, quantum Hall physics, toroidal systems and the Berezinskii-Kosterlitz-Thouless transition.

Active dipole clusters: From helical motion to fission

The structure of a finite particle cluster is typically determined by total energy minimization. Here we consider the case where a cluster of soft-sphere dipoles becomes active, i.e., when the individual particles exhibit an additional self-propulsion along their dipole moments. We numerically solve the overdamped equations of motion for soft-sphere dipoles in a solvent. Starting from an initial metastable dipolar cluster, the self-propulsion generates a complex cluster dynamics. The final cluster state has in general a structure widely different to the initial one, the details depend on the model parameters and on the protocol of how the self-propulsion is turned on. The center of mass of the cluster moves on a helical path, the details of which are governed by the initial cluster magnetization. An instantaneous switch to a high self-propulsion leads to fission of the cluster. However, fission does not occur if the self-propulsion is increased slowly to high strengths. Our predictions can be verified through experiments with self-phoretic colloidal Janus particles and for macroscopic self-propelled dipoles in a highly viscous solvent.

Equilibrium structure of ferrofluid aggregates

We study the equilibrium structure of large but finite aggregates of magnetic dipoles, representing a colloidal suspension of magnetite particles in a ferrofluid. With increasing system size, the structural motif evolves from chains and rings to multi-chain and multi-ring assemblies. Very large systems form single- and multi-wall coils, tubes and scrolls. These structural changes result from a competition between various energy terms, which can be approximated analytically within a continuum model. We also study the effect of external parameters such as magnetic field on the relative stability of these structures. Our results may give insight into experimental data obtained during solidification of ferrofluid aggregates at temperatures where thermal fluctuations become negligible in comparison to inter-particle interactions. These data may also help to experimentally control the aggregation of magnetic particles.

Thermodynamics of a binary monolayer of Ising dipolar particles. II. Effect of relative moment

Thermodynamic behaviors of a binary monolayer of Ising dipolar particles are studied using particle dynamics simulation, varying the relative intensity between the upward and downward dipole moments. The orientational order of the solid phase changes from tetragonal to hexagonal as the moment ratio increases. On the basis of the arguments of the candidates for ground state structures, the energy of the structures are well estimated. The transition point is also determined theoretically, which is consistent with the value obtained from the simulation results. Critical condensation is also studied. While the system whose moment ratio is unity does not exhibit the gas-liquid critical condensation, the transition appears as the moment ratio changes. The local structure of the liquid phase is found to be characterized by the ground state of the tetramer. The above-mentioned results imply that the gas-liquid critical point comes close to the melting transition point as the local structure of the liquid phase becomes closer to the structure of the solid phase, and therefore, the critical condensation is vanished.

Novel Structural Motifs in Clusters of Dipolar Spheres: Knots, Links, and Coils

We present the structures of putative global potential energy minima for clusters bound by the Stockmayer (Lennard-Jones plus point dipole) potential. A rich variety of structures is revealed as the cluster size and dipole strength are varied. Most remarkable are groups of closed-loop structures with the topology of knots and links. Despite the large number of possibilities, energetically optimal structures exhibit only a few such topologies.

Pattern formation in a monolayer of magnetic spheres

Pattern formation is investigated for a vertically vibrated monolayer of magnetic spheres. The spheres of diameter D encase cylindrical magnetic cores of length l. For large D/l, we find that the particles form a hexagonal-close-packed pattern in which the particles' dipole vectors assume a macroscopic circulating vortical pattern. For smaller D/l, the particles form concentric rings. The static configurational magnetic energy (which depends on D/l) appears to be a determining factor in pattern selection even though the experimental system is driven and dissipative.

Structure and scattering in colloidal ferrofluids

The structure of a model colloidal ferrofluid, the dipolar hard-sphere fluid, at low temperature has been investigated using Monte Carlo simulations. Extensive particle association into chainlike and ringlike clusters is observed at low density. The structure factors have been calculated, and are analyzed with the aid of simple scaling arguments. We describe the progression of fluid structures from the low-density associated phase, to the high-density liquid phase. This paper may be of help in obtaining an experimental observation of a fluid-fluid transition in colloidal ferrofluids.

Isotropic fluid phases of dipolar hard spheres

Monte Carlo simulations are used to calculate the equation of state and free energy of dipolar hard sphere fluids at low temperatures and densities. Evidence for the existence of isotropic-fluid-isotropic-fluid phase transitions is presented and discussed. Condensation in the dipolar hard sphere fluid is unusual in that it is not accompanied by large energy or entropy changes. An explanation of this behavior is put forward.