Self-Assembled Magnetic Surface Structures
M. Belkin^, A. Glatz*, A. Snezhko*, and I. S. Aranson* (^Department of Chemical Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208; *Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439)

Abstract: We propose a first-principles model for self-assembled magnetic surface structures at the water-air interface reported in earlier experiments [2-4]. The model is based on the Navier-Stokes equation for liquids in shallow water approximation coupled to Newton equations for interacting magnetic particles suspended at the water-air interface. The model reproduces most of the observed phenomenology, including spontaneous formation of magnetic snake-like structures, generation of large-scale vortex flows, complex ferromagnetic/antiferromagnetic ordering of the snake, and self-propulsion of bead-snake hybrids. The model provides valuable insights into self-organization phenomena in a broad range of non-equilibrium magnetic and electrostatic systems with competing interactions.

Details on the computations: (see also [1])
The computational algorithm is implemented for graphics processing units (GPUs). The dynamic equations in [1] were solved in a periodic x,y domain by a quasispectral method. We used a domain area of 160x160 in dimensionless units (the length is normalized by the layer height h0), on a grid with 1024x1024 points, and up to 225 magnetic particles. The algorithm was implemented for massive parallel GPUs and run on a NVIDIA GTX285 GPU with a peak performance of 1 TFlop. Typically, a speed up more than 100 times was achieved compared to a fast Intel i7 CPU. This GPU setup allowed us to harness a supercomputer-like power on a single desktop computer. The equations were solved in discrete time steps by a semi-implicit algorithm using a quasi-spectral method. The algorithms were implemented using the (NVIDIA) CUDA programming model which allowed us to parallelized the code by unwrapping all mesh point and particle loops into so-called kernel routines supplying a thread index which replaced the loop variables. The linear part of equation (2) was solved in Fourier space using the CUDA-fast Fourier toolbox routines. An important point in the implementation of the CUDA kernel routines is to avoid simultaneous write access to the used data structures and data transfer between the host CPU system and the graphics card.

Simulation parameters: (see also [1])
  • mp=1
  • β=1
  • σ=0.6
  • Ip=1/80
  • μr=3/80
  • ε=5
  • ω=1
  • a=0.8
  • κ=3
  • η=0.01
  • μ0=0.26
  • dt=0.01
  • system length=160 (in units of h0=1)
  • grid size: 10242

supplementary materials:

  • Multimedia:
    Simulation and experimental movies
    1. Movie 1: Self-assembly of a snake from initially 225 randomly distributed particles. [13MB, xvid]
    2. Movie 2: Formation and motion of a snake-head hybrid (64 particles). [13MB, xvid]
    3. Movie 3: Experimental self-assembly of a snake-head hybrid from 90μm Nickel particles. In the first part of the movie the frequency of the magnetic field was 60Hz which was increased later to 100Hz to speed up the motion of the snake. [9MB, xvid]

  • Experimental movies and information

[1], tba
[2] A. Snezhko, I.S. Aranson, and W.-K. Kwok, Phys. Rev. Lett., 96, 078701 (2006).
[3] M. Belkin, A. Snezhko, I. S. Aranson, and W.-K. Kwok, Driven Magnetic Particles on a Fluid Surface: Pattern Assisted Surface Flows, Phys. Rev. Lett., 99, 158301 (2007). [PDF]
[4] A. Snezhko, M. Belkin, I.S. Aranson, and W.-K. Kwok, Phys. Rev. Lett., 102, 118103 (2009).