Magnetic force imaging of a chain of biogenic magnetite and Monte Carlo analysis of tip–particle interaction

Magnetotactic bacteria form chains of magnetite nanoparticles that serve the organism as navigation tools. The magnetic anisotropy of the superstructure makes the chain an ideal model to study the magnetic properties of such an organization. Magnetic force microscopy (MFM) is currently the technique of choice for the visualization of magnetic nanostructures, however it does not enable the quantitative measurement of magnetic properties, since the interactions between the MFM probe and the magnetic sample are complex and not yet fully understood. Here we present an MFM study of such a chain of biological magnetite nanoparticles. We combined experimental and theoretical (Monte Carlo simulation) analyses of the sample, and investigated the size and orientation of the magnetic moments of the single magnetic particles in the chain. MonteCarlo simulations were used to calculate the influence of the magnetic tip on the configuration of the sample. The advantage of this procedure is that analysis does not require any a priori knowledge of the properties of the sample. The magnetic properties of the tip and of the magnetosomes are indeed varied in the calculations until the phase profiles of the simulated MFM images achieve a best match with the experimental ones. We hope our results will open the doors towards a better quantification of MFM images, and possibly a better understanding of the biological process in situ.


Introduction
Magnetosomes are magnetite crystals, embedded in an organic membrane, that are synthesized by magnetotactic bacteria [1]. The bacteria are ubiquitous in sediments of freshwater and marine aqueous environments [2]. The bacteria form chains of magnetosomes by a process regulated by a combination of physical and biological forces, while particles, when isolated from the cells, often form closed ring structures [3][4][5][6][7][8]. The magnetosome properties (dimension, morphology, magnetic properties), as well as the magnetosome chain pattern (single chain, multiple chains and their cellular distribution), are biologically controlled so the same strain of bacteria always forms magnetosomes with the same mature size and morphology, and have the same magnetosome arrangements [2]. Therefore, the magnetosome chains have advanced as a model system for magnetic nanostructures, because a variety of magnetic properties arise from the varying particle dimension and organization observed in different strains, and particularly in terms of magnetic anisotropy [9][10][11].
Atomic force microscopy (AFM) [12] is widely used in surface science to characterize the topography and the physical properties of a broad spectrum of inorganic and organic materials at the nanoscale level. Its variant, magnetic force microscopy (MFM) [12], in turn enables the visualization of magnetic structures even in complex materials [13]. Besides MFM, there are several other techniques which allow the simultaneous imaging and quantification of magnetic structures. Bose-Einstein condensates, as magnetic sensors, take advantage of the high sensitivity of cold atoms to changes in the magnetic field, and allow the measurement of magnetic fields down to 300 pT, but with a spatial resolution of only 3 µm [14]. Electron Holography allows imaging of samples at a high spatial resolution of 1 nm, but with the drawback of having to work with extremely small and thin samples under a high vacuum [15]. Optical methods have very recently achieved a spatial resolution of 400 nm and a magnetic sensitivity of 100 µT [16]. Thus, MFM appears to be the technique of choice since it has the potential to achieve both high spatial (20 nm) and high field sensitivity (1 µT) resolution. In addition, these excellent resolutions are coupled with the advantage of working under environmental conditions, which is particularly important for the study of biologically relevant processes. However, MFM functionality is reduced because the interaction between the MFM probe and the magnetic sample are complex and not yet fully understood.
The magnetosomes have therefore been tested for numerous bio-and nano-technological applications [25,26]. The first attempts to quantify the magnetic properties of nanoparticles in general, and of chains of nanoparticles in particular, were performed on simple model systems using MFM, e.g. on magnetosomes [27][28][29]. Proksch et al initiated the study of biogenic magnetite chains by MFM [27]. The magnetotactic bacterium studied was freeze-dried in order to ensure that the magnetite chain was close enough to the MFM tip for detection of any magnetic signal. Proksch et al assumed that the magnetic moments of all magnetosomes were oriented along the chain axis, and did not consider the effect of re-orientation of magnetic moments due to the particle-particle and particle-tip interactions [27]. More recently, Wei et al focused on the magnetization reversal of a two-particle assembly. The particles were studied both in the absence of, and while applying, an external magnetic field [28]. In contrast to the work of Proksch, they assumed independent magnetic moments of the individual magnetosomes, and showed that the dipolar magnetic interaction between neighboring particles is strong enough to at least partly keep the magnetization along the chain axis.
Here, we present an AFM/MFM study of isolated magnetosomes assembled in a chain. By a combination of experimental and theoretical (Monte Carlo simulation) analyses, we investigate the size and orientation of the magnetic moments of the single magnetic particles in the chain. The Monte Carlo simulations were used to quantify the influence of the magnetic tip on the magnetic configuration of the sample during measurement. The analysis does not require any a priori knowledge of the properties of the sample. In the simulations, the magnetic properties of the tip and of the magnetosomes are varied until the phase profiles of the simulated MFM images best match the experimental ones. The geometric parameters needed are obtained by the topographic analysis done by the AFM simultaneously with the magnetic measurements obtained by MFM.

Magnetosome sample preparation
Magnetosomes were isolated from the Magnetospirillum gryphiswaldense strain MSR-1 as described in [30]. The isolated magnetosomes were washed with MiliQ water. The magnetosomes were concentrated magnetically by placing a strong neodymium magnet, and physically by centrifuging at 6000g for 5 min, the supernatant was removed and the magnetosomes were washed with MiliQ water. Washed magnetosomes were pipetted onto freshly cleaved mica which was placed between two magnets in order to get aligned magnetosome chains.

Atomic force microscopy
Atomic and MFM images were recorded using the commercially available nanowizzard III (JPK instruments, Berlin, Germany). For MFM experiments, silicon cantilevers with a hard magnetic, medium momentum cobalt alloy coating on the tip (MagneticMulti75-G, NanoAndMore, Wetzlar, Germany) were used. The thickness of the coating of the tip is ca. 50 nm and the tip radius is less than 60 nm. Prior to the measurements, the magnetization of the tip was saturated using a strong magnet. The topographical images were obtained in alternating current (ac) mode. MFM images were obtained in the hover mode, i.e. on the first scan, the cantilever directly scans the surface of the sample, and on the second scan (hover mode retrace), the cantilever is raised to a user-defined height, and follows the topographical pattern from the previous trace. On the first trace, short range interactions (i.e. Van der Waals forces) have the most significant effect and the topography is imaged. On the hover mode retrace, long range interactions such as magnetic forces are most prevalent, and the MFM image will therefore reflect the magnetic properties of the sample. These lift heights were in the range of 30-100 nm.
Evaluation of the cantilevers' spring constant k, and the quality factor Q, was performed with the internal software of the JPK nanowizzard, applying the thermal method [31]. The obtained values were Q = 230 (±10%) N m −1 and k = 6.7 (±5%) N m −1 , respectively.
The AFM and MFM images were analysed with the open source program Gwyddion [32]. The images were typically flattened and leveled by the three-point plane method to obtain the zero height and phase in the plane level. 3D images were generated using the same program. Profiles were exported from Gwyddion and analysed with Origin and Python.

Monte Carlo simulations
Monte Carlo simulations were performed to understand the phase contrast images obtained by the MFM measurements. All magnetic objects were treated as point dipoles, having a dipole moment m. The magnitude of the dipole moments was kept constant during the simulations, while the direction of the moments of the magnetosomes was variable. For the calculation of the energy, only magnetic interactions were considered. The potential energy of a dipole moment m in a magnetic field B is defined as: The magnetic field produced by a magnetic dipole at position r is given by: where µ 0 is the vacuum permeability. Consequently, the energy of N magnetosomes is given by: where m i and m j are the magnetic moments and r ij is the connecting vector between the two magnetosomes i and j . The energy between the magnetosomes and the tip is: where r i is the connecting vector between magnetosome i and the tip, and m tip is the magnetic dipole moment of the tip. The total energy of the system is the sum of these two contributions. During the simulation, the positions of the magnetosomes and the trace of the tip were defined. One simulation step consisted of choosing one magnetosome randomly and changing the direction of its dipole moment from m old i to m new i . Then the energy difference E = E new − E old was calculated according to the former equations. Following the Metropolis algorithm [33] the new dipole moment was accepted with a probability: where the temperature T was set to room temperature and k B is the Boltzmann constant.

Calculation of the phase shift.
The oscillating MFM cantilever is sensitive to force gradients, since these lead to a change of the effective spring constant: k eff = k − F z [34].
If the cantilever is driven at its resonant frequency, small force gradients F z k lead to phase shifts of where k is the spring constant and Q is the quality factor of the MFM cantilever, which were obtained experimentally [35].
Prior to the measurements, the tip was magnetically saturated by a strong magnet with its dipole moment pointing into the negative z-direction. Because the magnetic moment of the tip is usually considerably larger than the magnetic moments of the magnetosomes, we assume a constant value of m tip = −m tip e z .
As F = −∇E = ∇(m · B), the z component of the force is given by: and results in a phase shift of: is the z component of the magnetic field at the position of the tip, produced by the dipole moment of magnetosome i, connected to the tip position via vector r i .
The dipole moment of one of the magnetosomes (the index i is neglected for clarity) at the position (P x , P y , P z ) is described by the vector components (m x , m y , m z ). The z component of the magnetic field B(r) = µ 0 /4π((3(m · r)r/r 5 )−(m/r 3 )) at the position of the tip (x, y, z) is given by: The second partial derivative of this expression with respect to z was solved analytically with the help of Mathematica (Wolfram, 2010). For every position of the tip, three runs with 10 4 iterations over the rotation of the dipole moment of individual spins were performed. Whether the system had attained its equilibrium state was verified by monitoring the energy and checking its convergence to a constant mean value. At one tip position, the phase was calculated for each of the three runs, and the mean value was taken as the phase signal. The tip was then moved to the next position. The final dipole configuration of the last run was used as the starting configuration for the new position.

Results and discussion
Magnetosomes isolated from Magnetospirillum gryphiswaldense MSR-1 were studied [36]. Isolation of the magnetosomes was performed following a published protocol [37] (see section 2). Figure 1 shows a topographic image of the sample obtained by a standard AFM scan, where nine magnetosomes can be seen, arranged in a rather straight line. The measured magnetosome height is 39.9 (±3.0) nm (n = 9). The edge-to-edge distance between two consecutive particles is found to be 48.9 (±4.8) nm. For isotropic particles like the magnetosomes, it is expected that height and distance have the same value. The observed difference arises most probably from the magnetosome membrane, since the magnetosomes are arranged on the substrate with intact membranes with a thickness of ≈4 nm [38]. The reduced height can be related to the shrinking of the magnetosome membrane during the drying process. The diameter for magnetosome magnetite particles from MSR-1 is typically 40 nm [1]. Thus, considering the thickness of the membrane, the observed average sizes are in good agreement with literature values.
The chain is then measured with a magnetically coated tip (from right to left in our case) (figure 2). In MFM measurements, which are recorded in the two-step hover mode (see section 2.2), each topography scan line is re-scanned with a user-defined distance between the tip and the sample (lift height). In this case, magnetic forces are dominant and can be quantified by the phase shift of the cantilever oscillation. Phase images of the scans obtained at different lift heights are shown in figure 2(a)-(c). The chain axis is also indicated in the figure, together with the corresponding one-dimensional phase profile along the axis and the previously obtained height profile (figure 2(d)-(f )). The detected phase shifts change dramatically with increasing lift height. The measured phase shift is negative, indicating an attractive force between tip and sample. Furthermore, the absolute value of the detected phase shift decreases with increasing lift height (from about −0.6 • at 30 nm to about −0.2 • at 100 nm). This is expected, since the magnetic field of the magnetosomes decays with the distance to the power of 3.
More interestingly, the shape of the profiles also change with different lift heights.
These different shapes are determined by the interaction between the tip and sample. Depending on the interaction strength, i.e. the lift height, the magnetic interaction may change the orientation of the magnetic moment of the magnetosome. Qualitatively, at a 30 nm lift height when the tip is approaching the magnetosomes, the phase shift first decreases, indicating an attractive interaction, and then oscillates ( figure 2(d)). When the lift height is increased to 50 nm, this oscillation becomes extremely small, eventually vanishing in the noise level (left end of chain in figure 2(e)). However, the signal is not constant along the chain axis. Small bumps can be seen towards both ends of the chain (around positions of 350 nm and 600 nm, respectively). The shape of the signal obtained at the 100 nm lift height is very different than that obtained at lower lift heights (figure 2(f )). The phase profile takes a minimum close to the starting point of the scan at a position of ∼600 nm and increases towards the other end of the chain, even taking slightly positive values, which indicates repulsion between sample and tip.
Monte Carlo simulations have been performed to gain a better understanding of the interaction between the MFM tip and the individual magnetic moments of the magnetosomes in the chain. This interaction indeed determines the different shapes of the phase profiles we measured experimentally. Monte Carlo simulations allow detection of the equilibrium state of a system at a given temperature (here ambient temperature) by minimizing its free energy [33] (see section 2). The field of the tip is described as a magnetic dipole, with its magnetic moment pointing in the negative z-direction (the (positive) z-direction is assumed to be perpendicular to the plane of the magnetosomes directed to the tip). All the magnetosomes are assumed to have the same (geometric) size, and the size of their dipole moment, m = M s V , is typically calculated using either the saturation magnetization of magnetite (M s = 480 kA m −1 ) or the oxidized version of the iron oxide, namely maghemite, (M s = 380 kA m −1 ) [39] to test for the nature of the particles. The volume of the magnetosomes is given by a sphere of radius r Ms = 20 nm (see section 2). The distance between two magnetosome dipoles is set to d = 50 nm, and this is consistent with the topographical information. The value of the magnetic moment of the tip is given by the manufacturer as m tip = 10 −16 A m 2 .
In the simulations, he tip position is virtually moved at a constant lift height above the magnetosomes dipoles. This is a slight difference from the experiment, in which the tip position is adjusted to take the topography into account and keeps a constant height above the magnetosomes, with slight drops between the particles. Thus, in the simulations the distance between tip and sample is a bit larger than in the experiment, which reduces the effect of the non-uniform size of the magnetosomes. For every tip position, the average phase signal of the final configuration is determined. We varied the moment of the tip as well as the moment of the magnetosomes in the simulations in order to find out the parameters that best match our experimental results (see online Supplementary table S1 (stacks.iop.org/ JPhysD/47/235403/mmedia)). The main characteristic of a good match between experiment and simulation is (next to the phase shift values) the shape of the phase profiles at different heights. Typically, the shape of these phase profiles at all three heights can only be reproduced for a given set of parameters. The best fit was obtained for a value of 10 −16 A m 2 for the magnetic moment of the tip, corresponding to the value provided by the probe manufacturer, and a value of 1.6 × 10 −17 A m 2 for the magnetic moment of the single magnetosomes corresponding to magnetite, in agreement with values from literature [1]. This is indeed the only set of parameters that enables the qualitative match of the profiles at all three lift heights. Figure 3 shows a comparison of simulated and measured phase profiles along the chain axis for the set of parameters explained above.
Despite the simplifications used in the theoretical analysis, e.g. the assumption of uniform magnetosome size, and the point dipole approximation, the phase profiles obtained from the simulations qualitatively match the experimentally obtained ones. The characteristic shapes of the profiles are reproduced: first, oscillations at low lift heights (figures 3(a) and (d)), second, jumps at medium lift heights (figures 3(b) and (e)) and, third, the constant decrease of the signal at large lift heights (figures 3(c) and (f )). A quantitative satisfactory match is only obtained for intermediate lift heights (figures 3(b) and (e)), while for small (figures 3(a) and (d)) and large (figures 3(c) and (f )) lift heights, the absolute values of simulated and experimental phase shifts do not coincide. The experimentally obtained maximum of the phase shifts is twice that of the simulated one at 30 nm, and only a tenth at 100 nm lift height. These differences might arise from an effective screening of the magnetic fields due to the finite size of the magnetosomes and of the tip, which would be distancedependent. Alternatively, it could be the consequence of contamination of the tip for some measurements, which in fact enlarges the probe to magnetosome distance. However, this disagreement between simulation and experiment is most probably due to the point dipole approximation that was used in describing the magnetic tip. In Häberle et al [40], it was shown that close to the tip the magnetic field decays with 1/r instead of with the 1/r 3 dependence of a classical dipole. Thus, in the simulations the magnetic field effectively decays too fast, explaining why the simulations underestimate the tip-magnetosome interaction with increasing distance. Most probably the agreement between simulation and experiment can be significantly improved by introducing a lift height dependent calibration factor, as proposed by Sievers et al in order to realize quantitative MFM measurements [41].
Interestingly, the simulations also reveal the influence of the magnetic moment of the tip on the orientation of the moments of the individual magnetosomes. At low lift heights, the influence of the tip is large, as expected (figure 4). In the absence of the magnetic field of the tip, the energetic ground state of the system is given by all magnetic moments of the magnetosomes, aligned along the chain. When the tip is directly above one magnetosome, its magnetic field is strong enough to re-orient the magnetic moment of the magnetosome. The magnetic moment then aligns with the field in the negative z-direction, leading to a strong field gradient at the tip and therefore to a big phase shift signal ( figure 4(b)). When the tip is above the gap between two magnetosomes, the field is still large enough for the moments of the magnetosomes to align with the-now tilted-field ( figure 4(b)). Because the magnetic moment is now not fully oriented in the z-direction, the phase shift signal is smaller than in the case when the tip is directly above the magnetosome. This effect leads to the observed oscillations. When the tip is moved across the magnetosome, its magnetic moment re-aligns with the field of the tip, so it now essentially points in the x-direction, meaning an effective flip of the moment compared to the starting configuration. Therefore, a symmetric configuration would typically be expected when the tip scans from the other direction, even if the chain is oriented the same way, because the same types of interactions are obtained.
The jumps of the signal at medium lift heights (50 nm) are qualitatively explained by the competing influences of the magnetic moment of the tip and the moments of the other magnetosomes on any single particle ( figure 4(c)). In contrast to the lower lift heights, the magnetosomes at the end do not completely align in the z-direction when the tip is directly above. The field of the other particles is strong enough to partly hold it back, and the moment only gets tilted. In particular, the field of the tip is not strong enough to directly flip the moment of the magnetosome when the tip passes the particle. It is only after the tip also passed the second magnetosome that the combined field is strong enough to flip the moments, resulting in a jump in the phase signal. For each following magnetosome, the adjacent dipole moments point in opposite directions, therefore there is no competition with the tip, and the dipole moments align with the tip.
At large lift heights (100 nm), the influence of the magnetic field of the tip on the orientation of the magnetosome dipoles is small ( figure 4(d)). The magnetic moments of the other magnetosomes in the chain stabilize the dipole direction and prevent its alignment with the tip dipole. Thus, after the tip has passed all magnetosomes, the magnetic moments of these are anti-parallel to the field of the tip, resulting in the observed positive phase shift, indicating repulsive forces.
The combination of MFM measurements and Monte Carlo simulations permits a much deeper understanding of the magnetosome-MFM tip interaction in a chain of particles than has been previously reported. Proksch et al quantified the magnetic moment of the chain without any description of the interaction between the magnetosomes and the tip [27]. Furthermore, their approach cannot easily be used for systems for which no a priori magnetic information is available. In fact, in the present approach no a priori information is needed.
Wei et al in turn proposed a more comprehensive description of the tip-magnetosome interaction, showing that isolated particles would easily be polarized by the MFM tip, whereas the orientation of the magnetic moments of magnetosomes in a chain would be stabilized by dipolar interactions among the particles [28]. However, the authors did not present any description of the phase profiles, certainly not at different lift heights.

Conclusions
We have experimentally studied a single chain of magnetic particles by MFM, and theoretically simulated the associated phase shift profiles by the use of Monte Carlo simulations. Despite the inevitable approximations made to make the problem theoretically tractable (e.g. describing the magnetic field of tip and magnetosomes as point dipoles), the simulated data fits well with the phase profiles that were observed experimentally. The simulations are rapid and enable the most comprehensive description of the interaction between an MFM tip with the magnetosomes in a chain.
In particular, our model involves inter-particle interactions that were previously ignored, and thereby gives insights into the magnetic configuration of the single magnetic moments in the chain. The model clearly shows a more pronounced interaction of the tip with particles at the end of the chain as expected, and as experimentally observed. This is explained by the lower number of stabilizing neighbors of the particles close to the chain ends. The best correlation between experiments and simulations was obtained for a magnetic moment of the magnetosomes in the chain of about 1.6 × 10 −17 A m 2 , corresponding to magnetite.