Influence of the cluster formation in a magnetic particle suspension on heat production effect in an alternating magnetic field

Abstract

In the present study, we concentrate on the results of the unexpected characteristic that the stable particle cluster formation induces a decrease in the degree of heating effect in an alternating magnetic field, by means of Brownian dynamics simulations. If chain-like clusters are stably formed in the system, whether or not a large heating effect is obtained is dependent on the magnitude relationship between the magnetic particle-particle and the magnetic particle-field interaction strengths. As the magnetic particle-particle interaction strength increases and becomes more dominant than the influence of the magnetic field, the magnetic moments of stable chain-like clusters do not have a sufficient tendency to inline in the field direction. This leads to a smaller area of the hysteresis loop, and therefore the stable cluster formation induces a significant decrease in the heating effect.

Appendix

Figure 6 shows results of the dependence of potential curves on the magnetic particle-particle interaction strength for the two cases of λ = 10 and 16, where the common steric interaction strength of λ_V = 150 is used. In the figure, U^m* and U^V* are the non-dimensional magnetic particle-particle and steric repulsive interaction energy based on the thermal energy k_BT, respectively, and U^* is the total interaction energy as U^* = U^m* + U^V*. A representative snapshot of particle aggregates is also shown in the figure for each case of the magnetic interaction strength for reference. It is noted that the magnetic particle-particle interaction energy is evaluated under the assumption that the magnetic moments of the two magnetic particles of interest incline in the same direction along the line connected between the centers of the particles, and the abovementioned energies are evaluated as a function of the non-dimensional distance between the two particles, r_ij^* (= r_ij/d). The cluster formation is mainly governed by the depth of a potential curve, and if the depth is much deeper than thermal energy k_BT, stable clusters more strongly tend to be formed in the system. The depth of the potential curve is U^* = −9.12 and − 14.7 at r_ij^* = 1.295 and 1.29 for λ = 10 and 16, respectively. Since these depths are much deeper than unity (i.e., thermal energy), the snapshots exhibit significantly similar network (chain-like) clusters between these cases, but slightly clearer network structures are recognized in the case of λ = 16. Moreover, it is noted that the minimum energy in each net potential energy arises in close vicinity to the contact surface of the steric layer (i.e., r_ij^* = 1.3)

Fig. 6

Dependence of potential curves on the magnetic particle-particle interaction strength for the two cases of λ = 10 and 16, where the common steric interaction strength of λ_V = 150 is used. The distance giving rise to a minimum energy in each net potential energy is in close vicinity to the contact surface of the steric layer. Snapshots are not significantly different between these two cases, but slightly clearer network structures are observed for λ = 16 than for λ = 10

Figure 7 shows results of the time change in the system energy to a steady state situation for the case of R_B = 5 in no applied magnetic field ξ = 0, where results are shown for the magnetic interaction strengths λ = 10 and 16. Two representative snapshots of particle aggregates are also shown in the figure at the cycle number of N_cycl = 5 and 25 for each case of the magnetic interaction strength in order to assess the rate of the convergence. From the results regarding the change in the system energy, it is clearly evident that the convergence to a steady-state situation is approximately accomplished in an early stage of the simulation steps, at N_cycl ≃ 10 and 5 for λ = 10 and 16, respectively. Moreover, it is seen that there is no essential difference between the snapshots at N_cycl = 5 and 25 for both the cases. The snapshots in the figures are essentially quite similar to those for thermodynamic equilibrium in no applied magnetic field in Fig. 6. However, it is recognized that slightly looser network structures are formed in the case of an alternating magnetic field (Fig. 7) since in this case, the viscous forces and torques function to decrease the stability of network clusters. These characteristics may clearly validate the accuracy of the present results in respect to the snapshots, the hysteresis loops, and so forth.

Fig. 7

Time change in the system energy to a steady-state situation for the case of R_B = 5 in no applied magnetic field. The convergence to a steady-state situation is accomplished in an early stage of the simulation steps, at N_cycl ≃ 10 and 5 for λ = 10 and 16, respectively. There is no essential difference between the snapshots at N_cycl = 5 and 25 for the both cases

Figure 8 shows the dependence of potential curves on the steric repulsive interaction strength for the common value of the magnetic interaction strength, where two representative cases of λ_V = 100 and λ_V = 200 are addressed: Figs. 8 a and b are results for λ = 10 and λ = 16, respectively. A representative snapshot of particle aggregates in thermodynamic equilibrium is also shown in the figure for each case of the steric repulsive interaction strength for reference. The method of evaluating the magnetic particle-particle interaction energy has already been described in Fig. 6. Although in the Brownian relaxation mode of the magnetic moments the surface roughness may be expected to have an influence on the heating performance, it seems to be reasonable to consider that the structure of network clusters is essentially the dominant factor for determining the motion of the network clusters in the situation of an alternating applied magnetic field. This is because these clusters are stably formed in the system due to magnetic interactions under the influence of the disturbance of an alternating field in the present cases of λ = 10 and λ = 16, not due to the surface roughness of magnetic particles. Hence, we here discuss the influence of the steric repulsive interaction strength λ_V on the potential curves and particle aggregates; a larger value of λ_V implies a more numerous number of surfactant molecules, i.e., a larger value of the number density n_s. The depth of the potential curve is U^* = − 9.15 and − 9.08 at r_ij^* = 1.285 and 1.295 for λ_V = 100 and 200, respectively, in the case of λ = 10, and similarly, U^* = − 14.76 and − 14.61 at r_ij^* = 1.285 and 1.295 for λ_V = 100 and 200, respectively, in the case of λ = 16. It is seen that the energy depth is not strongly dependent on the value of the steric repulsive interaction strength λ_V for both the cases of λ = 10 and 16. This characteristic clearly exemplifies that the snapshots of particle aggregates are not significantly different among those for the three cases of λ_V = 100, 150, and 200

Fig. 8

Dependence of potential curves on the steric repulsive interaction strength for the common value of the magnetic interaction strength, where two representative cases of λ_V = 100 and λ_V = 200 are addressed: Figs. 8 a and b are results for λ = 10 and λ = 16, respectively. A representative snapshot of particle aggregates in thermodynamic equilibrium is also shown in the figure for each case of the steric repulsive interaction strength for reference. The energy depth is not strongly dependent on the value of the steric repulsive interaction strength λ_V for both the cases of λ = 10 and 16, which clearly exemplifies that the snapshots of particle aggregates are not significantly different among those for the three cases of λ_V = 100, 150, and 200