Attraction-driven aggregation of dipolar particles in an external magnetic field

We present a detailed experimental study of the aggregation of particles of permanent magnetic moment in an external magnetic field. The experiments are performed with millimeter-sized particles floating on the surface of water. Due to the large size of the particles, thermal noise does not have any relevance in the system; the particles undergo deterministic motion. Experiments are carried out varying the concentration which also controls the relative importance of the dipole-external field and the dipole-dipole interactions. We determined the exponents characterizing the aggregation process and found that the attraction driven aggregation of dipolar particles obeys the Vicsek-Family dynamic scaling. The exponents are found to have a concentration dependence which can be attributed to the change of mobility of clusters and their interaction at higher concentrations.

Figure 1

Composite particles used in the experiments. Metal cylinders magnetized along their axis are attached to cork disks which can then float on the surface of water. The top of the disk is colored blue on which the white and yellow spots indicate the orientation of the vector of the dipole moment.

Figure 2

Time evolution of the particle system at two different concentrations $\varphi =0.0265$ (a),(b),(c) and $\varphi =0.1060$ (d),(e),(f). Cluster-cluster aggregation can be observed where the structure of aggregates strongly depends on the concentration of the system.

Figure 3

Average cluster size $\langle S\rangle$ as a function of time $t$ for all the concentrations considered (the value of $\varphi$ increases from bottom to top at small values of $t$). At the time of saturation the aggregation process slows down and the configuration gets practically frozen. For intermediate times the functional form of $\langle S\rangle$ can be well approximated by power laws which are indicated by the straight lines.

Figure 4

Average number of clusters $\langle n\rangle$ as a function of time $t$ for all the concentrations considered (the value of $\varphi$ increases from bottom to top at small values of $t$). At intermediate times good quality power law fits can be obtained.

Figure 5

Dynamic exponents characterizing the attraction driven aggregation of dipolar particles as a function of concentration $\varphi$. For clarity, error bars are only shown for the exponent $w$. For comparison, we also show the product of the exponents $Z\Delta$ which should be compared to the value of $w$. It can be observed that the exponents fulfill the scaling relation Eq. (6).

Figure 6

Time evolution of the number of clusters of a fixed size $n_S(t)$ for four different concentrations: (a) $\varphi =0.0796$, (b) $\varphi =0.1060$, (c) $\varphi =0.1327$, (d) $\varphi =0.1592$. Power laws can be fitted with a reasonable accuracy.

Figure 7

Data collapse analysis of the cluster size distributions obtained at different time intervals. A reasonable collapse of the data is obtained. The value of the crossover exponent $\Delta$ was determined by fitting straight lines.