Magnetic Particle-particle Interactions Research Articles

Elucidation of the trapping characteristics of magnetic disk-like particles in a Hagen-Poiseuille flow via Brownian dynamics simulations (usage of multi-magnetic poles)

In the present study, we elucidate the behaviour of a suspension composed of magnetic disk-like particles that flow in a Hagen-Poiseuille flow in a gradient magnetic field and the trapping characteristics of the magnetic particles. To this end, we utilise multiple pairs of magnetic poles located outside the cylindrical system. Brownian dynamics simulations were performed in order to clarify the dependence of these characteristics on a variety of factors. It is seen that an increase in the magnetic field strength improves the trapping characteristics of disk-like particles if the influence of an applied magnetic field is sufficiently more dominant than that of a flow field and that of the magnetic particle-particle interaction. In the case of a strong magnetic particle-particle interaction, thin chain-like clusters are formed from an anchored particle trapped around a magnetic pole. If the two poles are sufficiently close to each other, an arch-type cluster is formed between them. We expect that this arch-type cluster may give rise to good trapping performance even in the presence of a strong flow field if the formation arises at a sufficiently large separation distance between the pair of magnetic poles. Highlights We have performed Brownian dynamics simulations for a magnetic disk-like particle suspension in a Hagen-Poiseuille flow. If the magnetic particle-particle interaction is sufficiently more dominant than the flow field, thin chain-like clusters are formed from an anchored particle. As the pole separation distance decreases, an arch-type configuration tends to be formed along the surface of the wall. This arch-type cluster that is formed between the pair of magnetic poles may be expected to improve the trapping characteristics depending on the pole separation distance.

Regime change in the aggregate structures of spherical magnetic Janus particles (quasi-2D Monte Carlo simulations)

We have investigated a suspension composed of spherical magnetic Janus particles in thermodynamic equilibrium. The magnetic moment of Janus particles treated here is located at a position shifted from the particle centre, which is on the line perpendicular to the magnetic moment direction. We conducted quasi-2D Monte Carlo simulations to investigate the change in the regime of aggregate structures and the dependence of the regime on various factors, such as the strength of the magnetic particle-particle interaction, magnetic particle-field interaction, and the distance of the magnetic moment from the particle centre. In the absence of an applied magnetic field, the Janus particles aggregate to form stable clusters composed of a few particles with increased values of the magnetic particle-particle interaction strength. As the strength of the magnetic field is increased, these clusters collapse, and chain-like clusters are formed aligned in the direction of the magnetic field. The appearance of this regime change requires a stronger magnetic field as the position of the magnetic moment approaches the particle surface. This can be attributed to the fact that the magnetic interaction between neighbouring particles becomes more dominant as the magnetic moments of the constituent particles in the cluster units approach each other. Highlights We have addressed the aggregation phenomena in a suspension composed of spherical magnetic Janus particles. We have investigated the regime change in the aggregate structures of the Janus particles under various conditions. In a strong magnetic field, thick chain-like clusters are formed as the magnetic moment is shifted away from the particle centre. The present results are useful for elucidating the internal structure of complex aggregates in 3D systems.

球状磁性ヤヌス粒子からなる分散系を対象にした擬2次元モンテカルロ・シミュレーション（磁気モーメントが粒子中心から横方向にずれた粒子モデル）

I have addressed a magnetic Janus particle suspension in the thermodynamic equilibrium situation. The magnetic moment of Janus particles treated in the present study is shifted from the particle center, which is perpendicular to the magnetic moment direction. From the viewpoint of elucidating the internal structure of particle aggregates in a three-dimensional (3D) system, I have treated a dispersion composed of magnetic Janus particles with a full 3D rotational ability in a quasi-two-dimensional (2D) system. The quasi-2D Monte Carlo simulations have been performed in order to investigate the dependence of a regime change in the aggregate structures on various factors such as the magnetic particle-particle interaction strength, magnetic particle-field interaction strength and the distance of the magnetic moment from the particle center. In no applied magnetic field, the Janus particles aggregate to form a stable cluster unit composed of 2-4 particles if the magnetic particle-particle interaction strength is increased. As the magnetic field is increased, the cluster units collapse and chain-like clusters are formed aligned in the magnetic field direction. In the situation of a strong magnetic field, the chain-like clusters composed of Janus particles tend to be thicker as the magnetic moment is shifted away from the particle center. This is because the magnetic interaction between neighboring particles increases due to the magnetic moments of constituent particles in chain-like clusters being closer to each other. The present results are useful for elucidating the internal structure of complex aggregates in a 3D system.

Aggregation phenomena and regime change in a magnetic cubic particle suspension in an alternating magnetic field via quasi-two-dimensional Brownian dynamics

We have addressed the aggregation phenomena of a cubic haematite particle suspension in an alternating magnetic field in order to investigate the regime change in the aggregate structure. Brownian dynamics simulations have been performed for a quasi-two-dimensional system with an alternating magnetic field applied in the plane of the system. In a weak alternating field, the cubic particles aggregate to form closely-packed structures tending to an aligned face-to-face configuration with increasing magnetic particle-particle interaction strength. In a closely-packed cluster, the orientation of the magnetic moment of a constituent particle tends to be more strongly influenced by the magnetic interaction between neighbouring particles and less influenced by the direction of the alternating magnetic field. As the strength of the applied field is increased, the particle moments become more restricted to the direction of the alternating field and the closely-packed structures transform into elongated aggregates with an offset face-to-face configuration. As the direction of the alternating magnetic field switches from the positive to the negative x-direction, the elongated aggregates tend to collapse temporarily and when the orientation of the moments is again strongly restricted to the switched field direction, elongated aggregate structures with offset face-to-face contact are re-formed in the system. Highlights of the present paper We have investigated a regime change in the internal structure of cubic particle aggregates in an alternating magnetic field. As the magnetic field strength is increased, closely-packed aggregates with an aligned face-to-face configuration transform into elongated aggregates with an offset face-to-face configuration. We have clarified that the elongated aggregates collapse temporarily and then re-form when the direction of the alternating magnetic field switches. We have suggested that the frequency of the magnetic field may be used as a means for controlling the local internal structure of particle aggregates.

Improvement of trapping performance of magnetic particles by magnetic multi-poles via Brownian dynamics simulations of magnetic rod-like particles in a Hagen-Poiseuille flow

In the present study we have implemented a Brownian dynamics simulation of a Hagen-Poiseuille flow in a cylindrical vessel and have investigated the feasibility of multi-pairs of magnetic poles for exhibiting a good particle trapping performance. We have focused on two representative types of configuration consisting of two pairs of magnetic poles. The first is a vertical configuration of two pairs of poles arranged perpendicular to the direction of the flow field. The second is a parallel configuration, where each pair of magnetic poles is arranged along the surface of the vessel wall. In the case of the vertical pole arrangement, although densely-packed aggregates were formed between the poles, their location was unstable in the flow field and they tended to be carried away downstream. In contrast, a densely-packed cluster formed in the parallel arrangement tended locate in a more stable manner along the surface of the wall where the flow field is weaker. We conclude that under appropriate conditions of the magnetic particle-particle interaction strength, the applied magnetic field strength and the pole separation distance, the parallel arrangement of magnetic poles is the configuration of choice for giving rise to a superior trapping performance.

Quasi-two-dimensional Brownian dynamics simulations of the regime change in the aggregate structures of cubic haematite particles in a rotating magnetic field

In the present study, by means of quasi-two-dimensional Brownian dynamics simulations, we have addressed the physical phenomena of a suspension composed of cubic haematite particles in a rotating magnetic field in order to elucidate the relationship between aggregate structures and their response to a rotating magnetic field. In the case of a relatively weak magnetic particle-particle interaction, single particles remain without aggregating to form specific clusters, and the magnetic moment of single particles tends to follow a change in the direction of the rotating magnetic field. In the case of a strong magnetic particle-particle interaction, closely-packed clusters with a face-to-face configuration tend to form in the system, although the individual magnetic moments of the constituent particles do not follow the change in the magnetic field direction. As the magnetic field strength is increased, closely-packed structures are transformed into aggregate structures with an offset face-to-face configuration. In this situation, the magnetic moments of constituent particles are strongly restricted to the magnetic field direction, and the aggregates themselves may significantly rotate to follow the change in the orientation of the rotating magnetic field.

The behaviour of magnetic spherical particles and the heating effect in a rotating magnetic field via Brownian dynamics simulations

The present study addresses the physical phenomena of a suspension composed of magnetic spherical particles in a rotating magnetic field in order to elucidate the influence of particle aggregation phenomena on the heat production from Brownian relaxation by means of Brownian dynamics simulations. In the case of a weak magnetic particle-particle interaction, particles do not aggregate to form specific cluster configurations. In this situation, the magnetic moment of each particle quickly inclines toward the field direction, and single particles do not give rise to a sufficiently large heating effect. With an increasing magnetic interaction between particles, chain-like clusters tend to form in the system and function to offer a larger resistance to the orientation of the magnetic moments, which leads to an increase in the heating effect. In the case of a significantly strong magnetic particle-particle interaction, the particles tend to aggregate to form stable ring-like clusters where the magnetic moments of the constituent particles are not able to be so responsive to the rotating magnetic field and this leads to a decrease in the heating effect. Highlights of the present paper The characteristics of particle aggregation have been clarified. The relationship between the strength of the magnetic particle-particle interaction and the frequency of the rotating magnetic field on the aggregation phenomena has been clarified. The response of the particle aggregates to an applied rotating magnetic field has been clarified. The relationship between the performance of the degree of heat production and the internal structure of magnetic particle configurations has been clarified.

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

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.

Monte Carlo simulations of magnetic particle suspensions with a simple assessment method for the particle overlap between magnetic spheroids

We have developed a simple assessment method for the overlap between spheroidal particles, which neither requires the complex manipulation of vectors and matrices that is indispensable in the ordinary methods, nor is based on a model potential. Moreover, we have developed an evaluation method for the interaction energy arising from the overlap of the steric layer coating spheroidal particles. This is based on a sphere-connected particle model, but some modifications are introduced in order to express an appropriate repulsive interaction energy at the deepest overlapping position. We have investigated the phase change in a magnetic spheroidal particle suspension for a two-dimensional system by means of Monte Carlo simulations. In the case of no external magnetic field, if the magnetic particle-particle interaction is sufficiently strong to favour cluster formation, long raft-like clusters tend to be formed in a dilute situation. With decreasing values of area fraction, a chain-like structure in a dense situation transforms into a raft-like structure within a narrow range of the particle area fraction. Similarly, the raft-like clusters are preferred in a weak applied magnetic field, but an increase in the field strength induces a phase change from a raft-like into a chain-like structure.Highlights of the present paper:A simple assessment method has been proposed for the overlap between two spheroidal particles.The particle overlap assessment is free from a complex mathematical manipulation regarding vectors and matrices.A modified sphere-connected model has been proposed in order to more accurately evaluate a repulsive interaction due to the overlap of the steric layers coating spheroidal particles.2D Monte Carlo simulations have been performed to elucidate the phenomenon of a phase change by magnetic spheroidal particles on a material plane surface.A phase change between a raft-like and a chain-like aggregate structure is able to be controlled by the area fraction of particles and an external magnetic field.

Separation of two attractive ferromagnetic ellipsoidal particles by hydrodynamic interactions under alternating magnetic field.

In applications where magnetic particles are used to detect and dose targeted molecules, it is of major importance to prevent particle clustering and aggregation during the capture stage in order to maximize the capture rate. Elongated ferromagnetic particles can be more interesting than spherical ones due to their large magnetic moment, which facilitates their separation by magnets or the detection by optical measurement of their orientation relaxation time. Under alternating magnetic field, the rotational dynamics of elongated ferromagnetic particles results from the balance between magnetic torque that tends to align the particle axis with the field direction and viscous torque. As for their translational motion, it results from a competition between direct magnetic particle-particle interactions and solvent-flow-mediated hydrodynamic interactions. Due to particle anisotropy, this may lead to intricate translation-rotation couplings. Using numerical simulations and theoretical modeling of the system, we show that two ellipsoidal magnetic particles, initially in a head-to-tail attractive configuration resulting from their remnant magnetization, can repel each other due to hydrodynamic interactions when alternating field is operated. The separation takes place in a range of low frequencies f_{c1}<f<f_{c2}. The upper frequency limit f_{c2}τ_{r}≈0.04 (where τ_{r} is the rotation time scale) depends weakly on the ratio of magnetic field to particle magnetization strength, whereas f_{c1} tends to zero when this ratio increases.

Regime of aggregate structures and magneto-rheological characteristics of a magnetic rod-like particle suspension: Monte Carlo and Brownian dynamics simulations

In the present study, we address a suspension composed ferromagnetic rod-like particles to elucidate a regime change in the aggregate structures and the magneto-rheological characteristics. Monte Carlo simulations have been employed for investigating the aggregate structures in thermodynamic equilibrium, and Brownian dynamics simulations for magneto-rheological features in a simple shear flow. The main results obtained here are summarized as follows. For the case of thermodynamic equilibrium, the rod-like particles aggregate to form thick chain-like clusters and the neighboring clusters incline in opposite directions. If the external magnetic field is increased, the thick chain-like clusters in the magnetic field direction grow thicker by adsorbing the neighboring clusters that incline in the opposite direction. Hence, a significant phase change in the particle aggregates is not induced by an increase in the magnetic field strength. For the case of a simple shear flow, even a weak shear flow induces a significant regime change from the thick chain-like clusters of thermodynamic equilibrium into wall-like aggregates composed of short raft-like clusters. A strong external magnetic field drastically changes these aggregates into wall-like aggregates composed of thick chain-like clusters rather than the short raft-like clusters. The internal structure of these aggregates is not strongly influenced by a shear flow, and the formation of the short raft-like clusters is maintained inside the aggregates. The main contribution to the net viscosity is the viscosity component due to magnetic particle-particle interaction forces in relation to the present volumetric fraction. Hence, a larger magnetic interaction strength and also a stronger external magnetic field give rise to a larger magneto-rheological effect. However, the dependence of the viscosity on these factors is governed in a complex manner by whether or not the wall-like aggregates are composed mainly of short raft-like clusters. An increase in the shear rate functions to simply decrease the effect of the magnetic particle-particle and the particle-field interactions.

Mesoscopic simulation for the structures of magnetic fluids

The microstructures of magnetic fluids are simulated using a dissipative particle dynamics (DPD)-based method and are fundamentally important for controlling the macroscopic properties of magnetic fluids and understanding the corresponding rheological behaviors in diverse engineering applications. The cubic polynomial spline function often used as smoothing function in smoothed particle hydrodynamics (SPH) is employed as the conservative force potential function, which can provide a stronger conservative force weight function than the conventional weight function by choosing properly the cutoff radius between the dissipative particles. By employing the above method, the desired results are obtained for both stronger and weaker magnetic particle-particle interaction under the condition of varying the mass of the dissipative particles. In addition, the influences of the magnetic particle-particle interaction and of the magnetic particle area fraction on the microstructure of magnetic fluids are also investigated, respectively, and the obtained results agree qualitatively well with those in the literature obtained by other numerical approaches and experiments. The numerical solutions of the mean equilibrium velocities of the magnetic and dissipative particles are also calculated and approximate the corresponding theoretical values very well. Therefore the employed DPD-based method is highly effective in the simulation of the microstructure of magnetic fluids.

3次元モンテカルロ・シミュレーションによるキューブ状ヘマタイト粒子分散系の内部構造の検討

We have investigated a regime change in the aggregate structures of a suspension composed of magnetic cubic particles in thermodynamic equilibrium, by means of Monte Carlo simulations. In concrete, we have addressed the dependence of the regime change on a variety of factors such as the magnetic field strength and the magnetic particle-particle interaction strength. The orientational distribution function and order parameters have been focused on for quantitatively discussing these characteristics. The main results obtained here are summarized as follows. If the magnetic interaction strength is sufficiently large for cluster formation, closely-packed clusters are formed under the combination and expansion of a cluster unit composed of 8 particles. A regime change in the internal structure of aggregates appears in a narrow range of the magnetic interaction strength, which is clearly exhibited by the order parameter employed here. A closely-packed configuration can be clearly characterized by the orientational distribution function; 8 high peaks appear in the orientational space in the case of the closely-packed structure. As the magnetic field is increased, the closely-packed clusters are collapsed and transformed into wall-like clusters along the magnetic field direction. This is because the magnetic moment of each particle has a strong tendency to incline in the magnetic field direction in the situation of a strong magnetic field.

G0500302 強磁性棒状粒子からなるサスペンションにおける凝集構造の相転移に関するモンテカルロ・シミュレーション

Magneto-rheological properties of a magnetic particle suspension are strongly dependent on the formation of aggregates of magnetic particles. A suspension of ferromagnetic rod-like particles can be expected to exhibit strong magneto-rheological characteristics, which is a significantly important factor for application of magnetic particle suspensions to mechanical dampers and actuators. Hence, in the present study, we address a suspension composed of ferromagnetic rod-like particles in thermodynamic equilibrium. Monte Carlo simulations have been carried out in order to investigate the dependence of the phase change hi aggregate structures on the magnetic field strength for various cases of the magnetic particle-particle interaction strength. The cluster formation of magnetic particles significantly appears for a strong magnetic interaction larger than a certain value. As the applied magnetic field strength is increased, the clusters come to dissociate into small clusters, and finally at a certain strong applied magnetic field, all clusters completely dissociate into single-moving particles. This clearly exhibits that an applied magnetic field induces the phase change in a magnetic rod-like particle suspension.

0109 ディスク状ヘマタイト粒子分散系の磁気粘性効果に関するブラウン動力学シミュレーション

We have investigated the orientational distribution and magneto-rheological characteristics of a disk-like hematite particle dispersion by means of Brownian dynamics simulations. In the situation of a strong applied magnetic field, the particles can freely rotate about the magnetic moment direction together with their magnetic moment aligning along the magnetic field direction, which leads to a linear peak-type orientational distribution, not a single peak-type distribution that is usual for a ferromagnetic particle dispersion. Since the magnetic moment is more strongly restricted to the magnetic field direction with increasing field strength, the viscosity due to the magnetic properties of the particles increases more significantly. An increase in the magnetic particle-particle interaction strength leads to a decrease in the viscosity, and in the range of a magnetic interaction strength where column-like clusters are formed, the viscosity does not show a significant change with increasing magnetic interaction strength.

Brownian Dynamics Simulations for Investigating Aggregate Structures in a Suspension Composed of Rod-Like Hematite Particles

We have investigated aggregation phenomena in a suspension composed of rod-like hematite particles, which have a magnetic moment normal to the particle axis direction, by means of Brownian dynamics simulations. The present Brownian dynamics method takes into account the spin rotational Brownian motion around the particle axis in addition to the ordinary translational and rotational Brownian motion. First, the validity of the present simulation method has been clarified by comparing with results obtained by the fully-established Monte Carlo method. Then, from performing the simulations, we have investigated the influences of magnetic particle-field and particle-particle interactions, shear rate and volumetric fraction of particles on particle aggregation phenomena. Snapshots of aggregate structures are used for qualitative discussion and cluster size distribution, radial distribution function and orientational correlation functions of the direction of particle axis and magnetic moment are focused on for quantitative discussion. The main results obtained here are summarized as follows. The raft-like clusters are significantly formed at a magnetic particle-particle interaction much larger than that for a magnetic spherical particle suspension. This is because the rotational Brownian motion has a significant influence on the formation of clusters for a suspension composed of rod-like particles with large aspect ratio. An applied magnetic field enhances the formation of raft-like clusters. A shear flow does not have a significant influence on the internal structures of clusters, but influences the cluster size distribution of raft-like clusters. The volumetric fraction of particles has a significant effect on the growth of raft-like clusters and clear cluster formation appears from a volumetric fraction between 0.01 and 0.05 in the present simulation circumstances.

G0804 ブラウン動力学シミュレーションによる扁平ヘマタイト粒子分散系の凝集現象の解明(GS8 粒子分散・超流動)

We have investigated aggregation phenomena in a suspension composed of disk-like hematite particles by means of Brownian dynamics simulations. We have investigated the influence of the magnetic particle-field and the particleparticle interactions, shear rate and the volumetric fraction of particles on particle aggregation phenomena. The significant column-like clusters are formed at a magnetic particle-particle interaction much larger than in the case of a magnetic spherical particle suspension. This is because the rotational Brownian motion has a significant influence on the formation of clusters for a suspension composed of disk-like particles with large aspect ratio. An applied magnetic field is found to decrease the formation of column-like clusters. A shear flow is found not to have a significant influence on the internal structures of clusters, but it does influence on the cluster size distribution of the column-like clusters.

On the Negative Magneto-Rheological Effect of a Suspension Composed of Rod-Like Hematite Particles by Means of Brownian Dynamics Simulations with Spin Brownian Motion

We have investigated the behaviors of rod-like hematite particles in a simple shear flow as well as in an applied magnetic field, in order to clarify the dependence of the negative magneto-rheological effect on particle aggregation and orientational distribution of particles. The present Brownian dynamics method has a significant advantage that it takes into account the spin rotational Brownian motion around the particle axis in addition to the ordinary translational and rotational Brownian motion. The net viscosity is decomposed into three components and discussed at a deeper level and in more detail: these three viscosity components arise from (a) the torque due to the magnetic particle-field interaction and (b) the torque and (c) the force due to the interaction between particles. In contrast to the ferromagnetic rod-like particle that exhibits a single-peak-type orientational distribution, the present hematite particle has a linear-peak-type distribution with a much more gentle profile. Hence, the orientational distribution does not change significantly. However, this slightly reformed distribution makes a significant influence on the negative magneto-rheological effect. In a dilute suspension, the effect of the magnetic field strength on this negative magneto-rheological effect is qualitatively and quantitatively in good agreement with that obtained by the previous theory and Brownian dynamics simulations without translational Brownian motion. In a relatively dense suspension, the viscosity components arising from an applied magnetic field and the interaction between particles come to change rapidly since certain strength of magnetic particle-particle interaction, which is due to the formation of raft-like clusters.

Lattice Boltzmann Simulation of a Two-Dimensional Flow of a Magnetic Suspension between the Two Parallel Walls under a Non-Uniform Magnetic Field

First we have developed the Lattice Boltzmann method based on the fluctuation hydrodynamics which is applicable to a flow problem of a particle suspension. In this method, we have introduced the viscosity-modifying method, instead of the velocity-scaling method, in which the modified viscosity is used for generating the random forces in the lattice Boltzmann simulation. This viscosity-modifying method has been seen to be available for a magnetic particle suspension. Then we have applied this method to a two-dimensional Poiseuille flow of a magnetic suspension between the two parallel walls in order to investigate the behavior of magnetic particles in a non-uniform applied magnetic field situation. From the results of the snapshots, the pair correlation function between the magnetic pole and the magnetic particles and the averaged local particle velocity and magnetization distributions, it is seen that the behavior of magnetic particles drastically changes due to which factor dominates the phenomenon much more significantly among the magnetic particle-particle interaction, the non-uniform applied magnetic field strength and the translational and rotational Brownian motion.

扁平粒子からなる強磁性コロイド分散系の凝集構造に関する3次元モンテカルロ・シミュレーション(流体工学,流体機械)

We have investigated aggregate structures of a colloidal dispersion composed of ferromagnetic plate-like particles with a magnetic moment normal to the particle axis at the particle center, by means of three-dimensional Monte Carlo simulations. Such plate-like particles have been modeled as disk-like particles with the side section shape of a spherocylinder. In concrete, we have attempted to clarify the influences of the magnetic field strength, magnetic interactions between particles and volumetric fraction of particles. In order to discuss quantitatively the aggregate structures of clusters, we have focused on the radial distribution and orientational pair correlation function, etc. For no applied magnetic field cases, long column-like clusters are formed as magnetic particle-particle interactions increase. Characteristics of these clusters are that particles incline in a certain direction and their magnetic moments alternate in direction between neighboring particles. For applied magnetic field cases, the particles incline in the magnetic field direction so that the column-like clusters are not formed easily. When the magnetic field has the same order of influence as the magnetic particle-particle interactions, the particles do not form the clusters. However, as the influence of the magnetic field and magnetic particle-particle interactions become significantly strong, thick chain-like clusters are formed along the magnetic field direction.