Anisotropic dynamics of magnetic colloidal cubes studied by x-ray photon correlation spectroscopy

Herein we present our results on the investigation of the influence of external magnetic fields on the anisotropic collective dynamics of core/shell colloidal cubes having a hematite core and silica shell. Owing to the hematite cores, these micrometer-sized particles possess permanent dipole moments, which are at an angle with respect to the long diagonal of the cubes. As a result, they self-assemble into chains, which subsequently sediment to form higher-order structures. Using multispeckle ultrasmall-angle x-ray photon correlation spectroscopy, the anisotropic dynamics within these structures at the nearest-neighbor length scale was probed. The relaxation of the intermediate scattering function follows a compressed exponential behavior along all the different directions with respect to the external field—parallel, perpendicular, and at an angle $\approx {45}^{\circ }$—indicating hyperdiffusive behavior. We believe that the inhomogeneous distribution of stress points originating from the interplay of external field-induced (both gravitational and magnetic) alignment of the chains are responsible for the anomalous dynamics. The effective diffusion coefficients along and at $\approx {45}^{\circ }$ angle exhibit mild de Gennes narrowing, which is not very common for hyperdiffusive dynamics. We rationalize our observations by considering a superposition of diffusive and stress-induced ballistic processes and argue that depending on the azimuthal direction the relative contribution from these two processes changes.

Figure 1

(a) A representative TEM image of core-shell hematite-silica cubic colloids. (b) A schematic showing the direction of the magnetic moment (red), which is oriented at an angle $\alpha$ with respect to the long diagonal ([111] direction) of the cube. The orange triangle indicates the [111] plane. (c) Experimental scheme for the multispeckle ultrasmall-angle x-ray photon correlation spectroscopy experiment. In this case, the x-ray beam, external magnetic field (B) direction, and the direction of gravity (g) are perpendicular to each other.

Figure 2

Dynamics at an angle $0^{\circ }$ (parallel) with respect to the applied field. (a) Variation of scattered intensity $I$ as a function of scattering vector $q$ at different magnetic fields. The vertical dashed lines represent the $q$ values at which the dynamics was measured. The inset shows a representative 2D diffraction pattern where the magnetic field (B) is along the horizontal direction at a field value of 50 mT. An azimuthal sector average has been taken in the shaded region to study the dynamics along the direction of the field. (b) Intensity autocorrelation functions for different $q$ values indicated by the dashed lines in (a). The symbols represent the experimental data while the lines correspond to the fit with the KWW model. (c) Observed variation of the relaxation rate $1/〈t_{c\parallel }〉$ as a function of $q$ (symbols), which is fitted with the expression $1/〈t_{c\parallel }〉\propto q^{n_{\parallel }}$ (continuous lines). (d) The variation of the exponent $n_{\parallel }$ as a function of B. (e) Variation of the effective diffusion coefficients as a function of $q$ for different field values. The lines are a guide to the eye. (f) Variation of the compressed exponent, $\gamma$, as a function of $q$. Identical color scheme has been used for (a), (c), and (d). Error bars are smaller than the symbol size for (c) and (e).

Figure 3

Dynamics at an angle ${90}^{\circ }$ (perpendicular) with respect to the applied field. (a) Variation of scattered intensity $I$ as a function of scattering vector $q$ at different magnetic fields. The vertical dashed lines represent the $q$ values at which the dynamics was measured. The inset shows a representative 2D diffraction pattern where the magnetic field (B) is along the horizontal direction at a field value of 50 mT. An azimuthal sector average has been taken in the shaded region to study the dynamics along the direction of the field. (b) Variation of the compressed exponent, $\gamma$, as a function of $q$. (c) Observed variation of the relaxation rate $1/〈t_{c\perp }〉$ as a function of $q$ (symbols), which is fitted with the expression $1/〈t_{c\perp }〉\propto q^{n_{\perp }}$ (continuous lines). (d) The variation of the exponent $n_{\perp }$ as a function of B. The continuous line is a guide to the eye. Identical color scheme has been used for (a), (b), and (c). Error bars are smaller than the symbol size for (c) and (e).

Figure 4

Dynamics at an angle $\approx {45}^{\circ }$ (diagonal) with respect to the applied field. (a) Variation of scattered intensity $I$ as a function of scattering vector $q$ at different magnetic fields. The vertical dashed lines represent the $q$ values at which the dynamics was measured. The inset shows a representative 2D diffraction pattern where the magnetic field (B) is along the horizontal direction at a field value of 50 mT. An azimuthal sector average has been taken in the shaded region to study the dynamics at an angle $\approx {45}^{\circ }$ with respect to the field. (b) Intensity autocorrelation functions for different $q$ values indicated by the dashed lines in (a). The symbols represent the experimental data while the lines correspond to the fit with KWW model. (c) Observed variation of the relaxation rate $1/〈t_{c_D}〉$ as a function of $q$ (symbols), which is fitted with the expression $1/〈t_{c_D}〉\propto q^{n_D}$ (continuous lines). (d) The variation of the exponent $n_D$ as a function of B. (e) Variation of the effective diffusion coefficients as a function of $q$ for different field values. The lines are a guide to the eye. (f) Variation of the compressed exponent, $\gamma$, as a function of $q$. Identical color scheme has been used for (a), (c), and (d). Error bars are smaller than the symbol size for (c) and (e).

Figure 5

Interplay between the external magnetic force, which is trying to align the chains at an angle, and the gravitational force, which is trying to minimize the gravitational energy by aligning them parallel to the external magnetic field.