Ferrofluids under oscillatory magnetic fields

Abstract

Ferrofluids exhibit two canonical interfacial instabilities, a static Rosensweig (normal-field) instability that produces a lattice of peaks and a dynamical Faraday instability that produces parametrically excited standing waves. Here we present a systematic phase diagram of ferrofluid surface states driven by a purely AC vertical magnetic field with zero mean. Scanning a broad range of frequencies and field amplitudes, we resolve two robust branches: a Faraday-wave regime that includes a stable square lattice and a Rosensweig-like peak--valley regime indistinguishable in morphology from Rosensweig peaks. The Faraday-onset boundary is well described by a power law close to $\sqrt{f}$, while the Rosensweig-like peak onset becomes essentially frequency independent at low viscosity. The wave vector of the square lattice grows linearly with frequency over our accessible band. We present a surface-wave theory that captures the full phenomenology, including the emergence of Rosensweig peaks under zero-mean AC driving, the near-$\sqrt{f}$ scaling of the phase boundaries, the linear growth of the selected wave vector with frequency, and the preference for square over hexagonal lattices.

Figures (3)

• Figure 1: Representative surface states under vertical sinusoidal magnetic driving $B(t)=B_0\sin(2\pi f t)$ with $f = 50 \, \mathrm{Hz}$. (a) Faraday-wave state showing boundary-influenced wavefronts ($B_0=14.0\,\mathrm{mT}$, $\eta=3.6\,\mathrm{cP}$). (b) Square-lattice Faraday state observed within the Faraday-wave regime ($B_0=14.0\,\mathrm{mT}$, $\eta=2.2\,\mathrm{cP}$). (c) Rosensweig-like peak state with localized peaks at larger drive ($B_0=16.0\,\mathrm{mT}$, $\eta=2.2\,\mathrm{cP}$). The container diameter is $60\,\mathrm{mm}$ and the liquid depth is $5\,\mathrm{mm}$.
• Figure 2: Phase diagrams in the $(f,B_0)$ plane for two viscosities. Blue symbols indicate the Faraday instability onset and orange symbols indicate the Rosensweig-like peak onset as defined in the text. (a) Lower viscosity ($\eta=2.2\,\mathrm{cP}$, $\rho=0.87\,\mathrm{g/mL}$). (b) Higher viscosity ($\eta=3.6\,\mathrm{cP}$, $\rho=0.91\,\mathrm{g/mL}$). A stable square lattice appears within the Faraday-wave regime in an intermediate window between the two onsets. Dotted lines show fits (see main text for explanation).
• Figure 3: Wave number of the square-lattice state. (a) Real-space definition of the lattice spacing $a$ and the wave number $k\equiv 2\pi/a$. (b) Measured $k$ versus drive frequency $f$ for the low-viscosity sample ($\rho=0.87\,\mathrm{g/mL}$, $\eta=2.2\,\mathrm{cP}$) within the square-lattice window of the Faraday-wave regime. The data are well fit by a linear relation $k(f)=k_0+\omega/c_{\mathrm{eff}}$. Orange symbols show the square-branch data of Pi et al.PiEtAlPRL2000 replotted after converting their wave-number convention by a factor of $2\pi$ to match $k=2\pi/a$.