Tunable dynamic response of magnetic gels: Impact of structural properties and magnetic fields

Ferrogels and magnetic elastomers feature mechanical properties that can be reversibly tuned from outside through magnetic fields. Here we concentrate on the question of how their dynamic response can be adjusted. The influence of three factors on the dynamic behavior is demonstrated using appropriate minimal models: first, the orientational memory imprinted into one class of the materials during their synthesis; second, the structural arrangement of the magnetic particles in the materials; and third, the strength of an external magnetic field. To illustrate the latter point, structural data are extracted from a real experimental sample and analyzed. Understanding how internal structural properties and external influences impact the dominant dynamical properties helps to design materials that optimize the requested behavior.

Figure 1

Dynamic relaxatory behavior for three different linear elastic chains of $N=10$ magnetic particles of $m=1.68$. The chains differ by orientational memory $(D,\tau )$ leading to qualitatively different energetic ground states: ferromagnetic “F” ($D=0.1$, $\tau =0.04$), anti-ferromagnetic “AF” ($D=0.6$, $\tau =0.0004$), and spiral-like “Sp” ($D=0.6$, $\tau =0.04$). (a) Dynamic relaxation spectra, where $n$ labels the modes. (b) Example of a characteristic eigenmode ($n=8$) that appears very differently in the three cases due to the varying orientational memory. $i$ labels the particles, $\delta r$ denotes displacements along the chain axis, $\delta m_{\theta }$ and $\delta m_{\varphi }$ mark the angular deviations of the magnetic moments in spherical coordinates. (c) Illustration of the three different energetic ground states (light gray) and the resulting different modes $n=8$ as characterized in (b). In all cases the lengths of the unstrained linking springs between the particles are $r_{ij}^{\left(0\right)}=2$.

Figure 2

Dynamic relaxatory behavior of (from left to right) a small quadratic, rectangular (aspect ratio 2:3), and hexagonal lattice of $N=9$ particles. Magnetic moments are oriented perpendicular to the plane and of magnitude $m=1$. (a) Changes in the relaxation spectra for the three different particle distributions. (b) Different appearance of an example mode ($n=5$) for the three lattices (undeformed energetic ground states indicated in light gray). In all cases the lengths of the unstrained linking springs between the particles are $r_{ij}^{\left(0\right)}=2$, except for the long edges of the rectangular lattice, where they are $r_{ij}^{\left(0\right)}=3$.

Figure 3

Chain-like structures observed by x-ray microtomography in the experimental sample referred to in the main text [36]. On the left-hand side, three cross-sectional images at different heights $H$ from the base of the sample are depicted. Bright spots label the positions of magnetic particles. On the right-hand side, a three-dimensional reconstruction of the chain-like structures formed by the magnetic particles in the sample is shown. For details of the data acquisition see Ref. [36]. Taken from Ref. [36], Fig. 5. ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

Figure 4

Tunability of the dynamic behavior by an external magnetic field oriented perpendicular to the plane and affecting the magnetic moments. (a) Positions of the magnetic moments are extracted from the x-ray microtomographic cross-sectional image of an anisotropic real experimental sample [36] displayed for $H=3$ mm on the left-hand side of Fig. 3. Only a fraction of the image is shown for illustration. Gray areas correspond to the microtomographic spots. (b) Tunability of the spectrum by changing the magnetization. (c) The density of dynamic modes gets shifted in the frequency direction by adjusting the magnetization. Dynamic modes for $\omega \approx 1.3$ and $\omega \approx 2.5$ ($m=1$) are illustrated in Figs. 5 and 5, respectively. [The tomography data in panel (a) are taken from Ref. [36], Fig. 5 ($H=3$ mm), ©IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.]

Figure 5

Illustration of dynamic relaxational modes for the 969-particle planar irregular lattice extracted from the experimental sample. Colored illustrations of the deformed lattices for each mode are superposed to the black undistorted lattice corresponding to the energetic ground state. (a) Examples of lower modes show the expected global collective deformations, here of elliptic (${\lambda }_4$), triangular (${\lambda }_{10}$), quadratic (${\lambda }_{13}$), pentagonal (${\lambda }_{15}$), hexagonal (${\lambda }_{22}$), and heptagonal (${\lambda }_{24}$) shape. Selected eigenmodes (b) around the end of the “Debye plateau” ($\omega \approx 1.3)$ and (c) around the small hump that might be connected to a “boson peak” ($\omega \approx 2.5$), cf. Fig. 4, show a much more localized character. The initial spring lengths $r_{ij}^{\left(0\right)}$ were set according to the values extracted from the experimental sample, while the magnetic moment was chosen as $m=1$.