Simone
Galati‡
*ab,
Marta
Vassallo‡
*ab,
Marta
Vicentini
a,
Marta
Vallino
c,
Federica
Celegato
a,
Gabriele
Barrera
a,
Daniele
Martella
ade,
Elena S.
Olivetti
a,
Alessio
Sacco
a,
Jessica
Petiti
a,
Carla
Divieto
a,
Paola
Tiberto
a,
Alessandra
Manzin
a and
Adriano
Troia
a
aIstituto Nazionale di Ricerca Metrologica (INRiM), Strada delle Cacce 91, Torino 10135, Italy. E-mail: simone.galati@polito.it; marta.vassallo@polito.it
bPolitecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy
cConsiglio Nazionale delle Ricerche (CNR), Strada delle Cacce 73, Torino 10135, Italy
dEuropean Laboratory for Non-Linear Spectroscopy (LENS), Via Nello Carrara 1, Sesto Fiorentino 50019, Italy
eUniversità degli Studi di Firenze, Via della Lastruccia 3, Sesto Fiorentino 50019, Italy
First published on 5th December 2023
Magnetic oxygen-loaded nanodroplets (MOLNDs) are a promising class of nanomaterials dually sensitive to ultrasound and magnetic fields, which can be employed as nanovectors for drug delivery applications, particularly in the field of hypoxic tissue treatment. Previous investigations were primarily focused on the application of these hybrid systems for hyperthermia treatment, exploiting magnetic nanoparticles for heat generation and nanodroplets as carriers and ultrasound contrast agents for treatment progress monitoring. This work places its emphasis on the prospect of obtaining an oxygen delivery system that can be activated by both ultrasound and magnetic fields. To achieve this goal, Fe3O4 nanoparticles were employed to decorate and induce the magnetic vaporization of OLNDs, allowing oxygen release. We present an optimized method for preparing MOLNDs by decorating nanodroplets made of diverse fluorocarbon cores and polymeric coatings. Furthermore, we performed a series of characterizations for better understanding how magnetic decoration can influence the physicochemical properties of OLNDs. Our comprehensive analysis demonstrates the efficacy of magnetic stimulation in promoting oxygen release compared to conventional ultrasound-based methods. We emphasize the critical role of selecting the appropriate fluorocarbon core and polymeric coating to optimize the decoration process and enhance the oxygen release performance of MOLNDs.
In the literature, most of the OLNBs and OLNDs that have been tested are based on perfluoropentane (PFP) core and decafluoropentane (DFP) core, respectively. These fluorocarbons are characterized by different boiling points, comprised in the ranges 28–32 °C for PFP and 51–55 °C for DFP.9 The oxygen release from such systems is based on the acoustic cavitation mechanism, i.e. the growth and oscillation of bubbles in a solution under ultrasound (US) field exposure. Depending on the applied acoustic pressure in the solution, cavitation can be stable or inertial. If the pressure is low, cavitation is stable and the bubbles oscillate around their resonant size, allowing for controlled oxygen release. In contrast, if the pressure is high, inertial cavitation occurs, which leads to violent bubble collapse.14–17
Liquid formulations of OLNDs have shown a series of advantages compared to OLNBs. Among the others, more efficient performance in the oxygen release has been demonstrated, together with the exhibition of a reduced size profile, potentially allowing them to travel through nanoscaled interendothelial gaps, typically found in the fenestrated capillaries associated with tumor environments.2,9,18 OLNDs can undergo acoustic cavitation after their vaporization into microbubbles (diameter in the range of 10–100 μm) due to the interaction with an energy source, which induces an increase in the local temperature up to the fluorocarbon boiling point. Standard methods for inducing vaporization, such as acoustic droplet vaporization or optical droplet vaporization, use US or optical stimuli.19,20 However, these conventional approaches present a limited ability to reach deeper target regions of the human body while maintaining controlled side effects. This limitation arises from the dependence of energy source penetrability on the specific properties of the tissues that need to be crossed.21 A possible solution to overcome this limit is the exploitation of the heat released by the activation of magnetically functionalized OLNDs with radiofrequency magnetic fields, to allow for magnetic droplet vaporization and consequentially to activate the oxygen release.21,22 The advantage of using magnetic fields as stimuli is that their penetration in the body is not affected by the tissue properties, thus greater penetration depths can be achieved. Magnetic functionalization can be obtained by decorating the OLND surface with magnetic nanoparticles (MNPs), which are able to release heat when excited with alternating current (AC) magnetic fields with frequency typically in the range of 100 kHz–1 MHz. In particular, the field induces a cyclic response of the MNP magnetization configuration, which leads to heat generation, mainly due to hysteresis losses.23 The capability of MNPs to release heat is generally expressed with the specific loss power (SLP) parameter, which describes the power dissipated per unit mass of magnetic material and can be derived from thermometric or calorimetric measurements.24
Recently, there have been a few studies investigating the magnetic droplet vaporization mechanism as well as the functionalization and conjugation processes to obtain hybrid multi-responsive nanocarriers with acoustic and magnetic properties.25 While some works have explored the application of magnetic droplet vaporization to other theranostic nanosystems, such as heat mediators for magnetic hyperthermia treatment22 or contrast agents for US imaging,21 none of these studies have considered the oxygen delivery as a therapeutic mean. On the other hand, particular attention has been paid to the evaluation of in vitro biocompatibility, cell internalization, and magnetic hyperthermia efficacy of magnetically decorated OLNDs.26–30
In this framework, we focus on the preparation and characterization of magnetic OLNDs (MOLNDs), obtained by decorating OLNDs with Fe3O4 nanoparticles (NPs). These hybrid nanomaterials were investigated for the application of magnetic droplet vaporization in oxygen delivery. The synthesis of OLNDs was performed by using two types of fluorocarbon cores, based on PFP or DFP, and two polymers as coating, dextran or chitosan.17 The Fe3O4 NPs were prepared via co-precipitation method.31
We conducted a comprehensive physicochemical characterization to investigate the impact of Fe3O4 NP inclusion on the properties of nanodroplets. Furthermore, we indirectly assessed the effectiveness of the magnetic decoration through two distinct experiments. These experiments involved the examination of magnetic features exhibited by the prepared MOLNDs, scrutinizing whether their streaming velocity and trajectory within solution were influenced by the gradient of a magnetic field generated by a permanent magnet. Subsequently, we evaluated the efficacy of oxygen release under both acoustic and magnetic stimuli, analyzing the different performance of MOLNDs with respect to variations in fluorocarbon core and polymer coating types. The results, particularly in terms of oxygen concentration released via the AC magnetic field activation, demonstrated competitive performances compared to the conventional US-mediated method applied to OLNDs.
Representative SEM images of the marks left by DFP-based nanodroplets with chitosan coating on a silicon wafer after drying are shown in Fig. 1B. In the image of OLNDs (left), it is possible to appreciate quasi-circularly shaped halos with dimensions of a few hundred nanometers, made of the solid residues left on the silicon substrate by the liquid droplets after they collapse and evaporation. In the image of MOLNDs (right), clusters of Fe3O4 NPs can be observed in proximity to the droplet halos, as a proof of their localization on the droplet external surface.
Higher resolution images were obtained with TEM, as illustrated in Fig. 1C for DFP-based MOLNDs with chitosan coating. These pictures clearly show the presence of Fe3O4 NP aggregates on the nanodroplet surface, confirming the results obtained with SEM and corroborating the effectiveness of the magnetic decoration.
The histograms of the hydrodynamic size distributions measured with DLS are shown in Fig. 1D and their mean values and standard deviations evaluated over 5 measurements are reported in Table 1. The results indicate that the dimensions of OLNDs with different cores (DFP or PFP) are similar, providing evidence that the fluorocarbon composition does not significantly impact on their size.2 Conversely, the type of the coating affects the droplet size, as confirmed by the larger diameter of chitosan-coated OLNDs compared to that of dextran-coated ones. Moreover, the surface decoration of OLNDs with Fe3O4 NPs is confirmed by an increment in their mean size, regardless of the core and coating employed.
Sample | Diameter ± σ (nm) | ζ Potential ± σ (mV) | |||
---|---|---|---|---|---|
Fe3O4 NPs | 135 ± 35 | −24.8 ± 0.5 | |||
OLNDs | Core | Coating | Decoration | ||
DFP | Chitosan | ✗ | 419 ± 15 | 39.5 ± 2.5 | |
√ | 627 ± 22 | 24.7 ± 1.4 | |||
Dextran | ✗ | 285 ± 16 | −21.7 ± 3.9 | ||
√ | 345 ± 15 | −18.4 ± 0.3 | |||
PFP | Chitosan | ✗ | 407 ± 11 | 54.9 ± 2.8 | |
√ | 638 ± 21 | 40.5 ± 1.0 | |||
Dextran | ✗ | 292 ± 12 | −50.9 ± 1.8 | ||
√ | 563 ± 25 | −25.6 ± 0.6 |
Table 1 also reports the mean ζ potential values and the standard deviations for each sample, evaluated over 5 measurements. The type of coating agent has a strong influence, with dextran resulting in a negative ζ potential, while chitosan in a positive one. Moreover, Fe3O4 NPs exhibits a negative ζ potential, which could result in a higher MNP loading level with the chitosan coating due to attractive electrostatic forces resulting from opposite surface charges. Furthermore, the surface decoration with Fe3O4 NPs affects the ζ potential, with a decrease in its absolute value for MOLNDs in comparison to their corresponding OLNDs, indicating a lower stability of the suspensions.
Additionally, the OLNDs and MOLNDs acoustic response was investigated by collecting the signal emitted during droplet excitation using US fields, with a passive cavitation detector. The results obtained for the acoustic response of the nanodroplets with DFP-core are shown in Fig. 2, while the ones obtained for the PFP-core are shown in Fig. S1 of the ESI.† Three spectra at increasing acoustic pressures from 0.47 MPa to 1.54 MPa are reported for both OLNDs and MOLNDs in Fig. 2A, where it is possible to observe the acoustic spectra emitted by DFP-core samples, characterized by the increase of harmonics bands, as the ultrasonic pressure at 1.1 MHz increases. These bands are associated with stable cavitation activity related to forced bubble oscillations in the solution. Interestingly, for MOLNDs there is an increase in the peak amplitude when considering the lowest acoustic pressures (0.47 MPa and 0.78 MPa), indicating a stronger emitted acoustic signal. The different behavior of OLNDs and MOLNDs can be attributed to the presence of Fe3O4 NPs in the solution, which enhances the acoustic field scattering events, generating a local increase in pressure and, consequently, in cavitation activity. Thus, the magnetic decoration reduces the acoustic pressure threshold for entering the stable cavitation regime towards lower values. As a result, MOLNDs require lower pressures to enable controlled oxygen release while minimizing the potential side effects associated with high acoustic power. However, this difference is less evident at the highest value of acoustic pressure here considered (1.54 MPa), due to the onset of inertial cavitation events for both samples. To quantify inertial cavitation activity, the acoustic spectra were processed using cavitation noise spectrum analysis, which assumes that the shock waves, generated during bubble collapse produce white noise in the spectrum.33 The analysis was carried out through the evaluation of the cavitation noise power (CNP) indicator,33 obtained by first transforming the spectra from linear to logarithmic scale, according to the relationship:
![]() | (1) |
![]() | (2) |
Fig. 2B displays the CNP indicator obtained for the nanodroplets with DFP-core; a large value of CNP means a large acoustic signal generated by inertial cavitation. The results demonstrate that the increase in acoustic pressure causes the CNP to rise after reaching a specific threshold, which varies depending on the type of core and coating. The pressure range before the threshold allows the identification of the optimal working conditions to prevent possible drawbacks from inertial cavitation regime.
Furthermore, the type of coating used affects the cavitation behavior. The chitosan coating provides a more rigid structure compared to dextran, resulting in the triggering of inertial cavitation at higher acoustic pressures for both OLNDs and MOLNDs. Additionally, the magnetic decoration reduces the threshold for the inertial cavitation regime to lower pressures due to the enhancement of acoustic wave scattering events, as already discussed for stable cavitation.
Similar results are obtained for PFP-core samples, as reported in Fig. S1 in the ESI,† but with more intense signals from both stable and inertial cavitation analysis due to the lower boiling temperature of PFP, thus resulting in a higher percentage of cavitated nanodroplets.
The results obtained from the first experiment are shown in Fig. 3. Through the setup reported in Fig. 3A, MOLNDs were excited by US. Upon interacting with the acoustic field, MOLNDs are pushed towards fixed positions along the acoustic field lines, specifically nodes and antinodes, where they tend to agglomerate.34 After the US excitation is turned off, the MOLNDs depart from these fixed points. Their trajectories were recorded both in the absence and in the presence of a permanent magnet and used to determine the streaming velocity variations of the MOLNDs.
In Fig. 3B, three subsequent ecographic frames are presented, selected for a clear visualization of the trajectories of the MOLNDs, recorded for the chitosan-coated DFP-MOLNDs in the presence of the magnet (indicated with the red box). These frames provide evidence of the influence of the magnetic field gradient on the MOLND paths. In this case, the MOLNDs experience an enhanced streaming velocity with a preferred direction towards the magnet (as indicated by green and yellow circles), whereas without the magnet, MOLNDs exhibit random motion in all directions (see .GIF file recorded by ecographic probe present in the ESI†).
The average velocities of the DFP- and PFP-core MOLNDs in the same conditions are reported in Fig. 3C. The influence of the magnetic field leads to a significant increase in streaming velocity, at least twice, confirming the previous preliminary evidence and, thus, the successful magnetic decoration for all the samples. Furthermore, these results also point out a difference due to the coating type. In particular, the chitosan-coated MOLNDs are characterized by a higher streaming velocity than the dextran-coated ones, which could be associated with a higher MNP loading. This confirms the presence of electrostatic attraction forces between chitosan coating and Fe3O4 NPs, in accordance with the ζ potential values reported in Table 1.
The results obtained from the second experiment are shown in Fig. 4. Fig. 4A schematizes the microfluidic setup employed to investigate how the trajectory of MOLNDs is affected by the presence of a permanent magnet, displaced in four different positions with respect to the micro-channel. The deflection angle and the streaming velocity of MOLNDs were extrapolated from the acquired images for each magnet position (Fig. 4B). Fig. 4C and D show the obtained deflection angles and streaming velocities, respectively, for the DFP-MOLNDs. The placement of the magnet laterally to the MOLNDs flow (positions #3 and #4) results in a deflection angle in the investigated frame of approximately 15° from the original trajectory, accompanied by a slight decrease in the average values of the streaming velocity. Conversely, the variations in the deflection angle are not significant in the absence of the magnet or when the magnet is positioned along the flow directions on top of the micro-channel surface (positions #1 and #2). Since the magnet is magnetized out-of-plane, practically no deviations of the MOLND trajectory are indeed expected in the xz-plane, but in the xy-plane, MOLNDs are attracted by the magnet and thus move towards the channel top surface. When the magnet is positioned after the observation point (position #1), this attraction results in a substantial increase in the streaming velocity, which is approximately 25% for chitosan-coated MOLNDs and 8% for dextran-coated ones (Fig. 4D). For position #2, the streaming velocity is reduced in the observation frame by about 22% for chitosan-coated MOLNDs and 5% for dextran-coated ones. The stronger velocity variations observed for the chitosan-coated samples further confirm their greater MNP loading.
As can be appreciated, the standard deviations of measured deflection angles and streaming velocities are significantly wide. This is a consequence of different aspects: the quite broad distribution of the MOLNDs size (as shown from DLS measurements reported in Fig. 1D); the variations in the magnetic moment of the MOLNDs due to different level of MNP loading; the variable position of MOLNDs when injected within the micro-channel; the strong spatial variation in the magnetic field that rapidly decays with the distance from the magnet. In particular, as depicted in Fig. S2 in the ESI,† the magnetic field amplitude decreases from 25 kA m−1 to around 10 kA m−1 along the 1.5 mm width channel, corresponding to a magnetic field gradient of 10 × 103 kA m−2.
As illustrated in Fig. 4E–G, the results of the microfluidic experiments were further corroborated by numerical modeling, which allowed us to have a more extended picture of the trajectories of MOLNDs inside the micro-channel, depending on their initial position at the inlet. Due to the considerable variability in the MNP loading level, there is a strong dispersion in the MOLND magnetic moment at saturation. The reported results provide just an illustrative example, assuming for all chitosan- and dextran-coated DFP-core MOLNDs injected in the micro-channel a saturation magnetic moment of 0.042 pA m2 and 0.0085 pA m2, respectively. The trajectories calculated in the xz-plane when the magnet is located at position #3 are reported in Fig. 4E and F for chitosan and dextran coating, respectively. For the latter, which are characterized by a reduced MNP loading, associated with a lower magnetic moment, the overall trajectories result to be less influenced by the magnet presence. At larger distances from the magnet, the trajectory is slightly deviated, and only the nanodroplets closer to the magnet move towards it. On the contrary, for chitosan-coated MOLNDs, regardless of the initial positions considered, the nanodroplets tend to move towards the channel wall, where the magnet is positioned. In Fig. 4G, we show the simulation results obtained for the chitosan-coated MOLNDs when the magnet is located at position #1. The trajectories in the xy-plane put in evidence an opposed action of the magnet and the gravitational force, which results to be dominant for the MOLNDs near the channel bottom surface.
From simulation results, in all cases the deflection angles strongly vary along the trajectory, reaching values in the order of 90° when the MOLNDs approach the magnet. When comparing experimental and modeling results, strong discrepancies can be observed between the measured and calculated deflection angles, and this can be attributed to various factors. Firstly, the evaluation of MOLND trajectories in the experimental analysis is limited to a small observation window (227 × 167 μm), which is only a fraction of the simulated area inside the channel, falling across the curvature of the simulated trajectories. Secondly, the model involves the assumptions of perfectly inelastic collisions, causing the stream of MOLNDs to stop when they collide with the channel wall. Lastly, the simulations assume a uniform distribution of magnetic decoration across the droplet surface (section 6), whereas SEM and TEM images have indicated the presence of MNP agglomerates with diameters around 150 nm.
Fig. 5C depicts the setup used for evaluating the MOLND oxygen release, induced by the heat generated via hysteresis losses by the Fe3O4 NPs,23 when exposed to AC magnetic fields. The experiment was performed by applying a magnetic field with a peak amplitude of 72 kA m−1 and a frequency of 100 kHz, corresponding to an SLP around 90 W g−1 for the considered Fe3O4 NPs.31 The relative oxygen concentration variations measured at discrete temperatures are shown in Fig. 5D. For each temperature condition, the oxygen concentration was for 5 min before the experiment. Then, the sample was exposed to the AC magnetic field until it reached the desired temperature, and the oxygen concentration was assessed for 5 min. The values shown in the graphs represent the oxygen measurement following exposure to the AC magnetic field, as a variation with respect to the initial concentration. Heating curves for some of these temperatures are reported in Fig. S3 in the ESI.† In this case, the different temperatures at which oxygen is released are associated with the core property: for DFP-based MOLNDs, the oxygen concentration does not increase until 45 °C, while for PFP-based ones, it rises significantly starting from 27 °C. This behavior can be attributed to the different boiling points of the two fluorocarbons, even if the temperatures at which oxygen concentration starts to rise are slightly lower than the two boiling points. The reason is that the temperatures in the solution were recorded with a fiber optic probe, which provides a local measure of the temperature in the solution where the sensing point is placed,35 while on the decorated nanodroplet surfaces (i.e. near to the Fe3O4 NPs), it results in a higher temperature36 and thus closer to the actual fluorocarbon boiling point. In terms of coating, chitosan MOLNDs demonstrate significant oxygen release compared to dextran-coated ones. This behavior confirms the higher level of MNP loading shown by the chitosan-coated MOLNDs, as already supported by DLS measurements.
In order to establish the magnetic droplet vaporization role in the oxygen release, the MOLNDs samples were observed with an optical microscope after being exposed to an AC magnetic field with peak amplitude and frequency values as described above, which enable the target temperatures for oxygen release. Fig. 5E displays relevant images of the treated samples, clearly showing that, due to their increased dimensions (in the order of 10–20 μm), the MOLNDs have undergone the process of vaporization. Additionally, aggregates of Fe3O4 NPs are visible on the surface of the vaporized MOLNDs, confirming the magnetic decoration responsible for the heating release under the AC magnetic field excitation.
From a qualitative analysis, it is possible to derive the relationship between the temperature achieved in the solution through AC magnetic field excitation and the activation of the vaporization process specific for each sample. In particular, vaporization for PFP-MOLNDs starts around 27 °C, with the appearance of a few bubbles visible in the suspension. Their amount progressively increases with temperature, peaking at 29 °C, and then decreases when overcoming the PFP boiling point. For this reason, only a few bubbles can be observed at 37 °C. A similar behaviour is observed for DFP-MOLNDs, but the vaporization process occurs at higher temperature. In particular, the nanodroplets start to vaporize at 45 °C, reach a maximum quantity at 48 °C, and start to diminish at 51 °C. Therefore, the different behavior of the DFP- and PFP-MOLNDs is consistent with the boiling temperature of the two fluorocarbons and with the results obtained in oxygen release analysis.
Furthermore, from our comparative analysis of acoustic and magnetic stimuli for oxygen release, it becomes evident that magnetic droplet vaporization represents a highly promising therapeutic approach, comparable to the established acoustic method. In the reported study, the critical role of selecting the appropriate fluorocarbon core and polymeric coating to optimize the decoration process and enhance the oxygen release performance of MOLNDs is emphasized. In particular, our investigations into different types of cores and coatings have shown that DFP-core MOLNDs are better suitable for biomedical applications at body temperature compared to PFP-based ones due to their higher boiling point. Moreover, the chitosan coating has demonstrated its ability to provide a more controlled and efficient release of oxygen, thanks to its more rigid structure when compared to dextran.
Finally, our in vitro cytotoxicity studies demonstrate that MOLNDs interact non-toxically with A549 cells, particularly when applied at dilutions of 1:
8 or lower.
These findings provide an encouraging result for the future use of MOLNDs in the biomedical field, as nanocarriers that can release drugs under the application of dual stimuli (magnetic and US fields). Furthermore, their hybrid composition could be promising for the integration of additive theranostic features, like US-based imaging and magnetic hyperthermia.
In order to obtain a formulation of MOLNDs, Fe3O4 NPs were dispersed in the same solution used for the OLND preparation, without the fluorocarbon compound (DFP or PFP). The mixture was added to the OLNDs suspension in order to reach a final concentration of Fe3O4 NPs of 1 mg mL−1. Then, the mixture was sonicated for 20 s in an ice-water bath. This procedure was repeated for all the OLND samples, with PFP or DFP-core and chitosan or dextran coating.
SEM images were recorded using the instrument FEI Inspect – F scanning electron microscope through backscattered electron imaging. Prior to the characterization process, a small drop of the suspension was deposited onto a silicon substrate and left to dry completely. This step was crucial in ensuring the sample stability and integrity under the high-vacuum conditions within the SEM chamber.
TEM images were collected with the Philips CM 10 transmission electron microscope, using an electronic beam accelerating voltage of 80 kV. A drop of the suspension was allowed to adsorb for 5 min on a Pelco® carbon and formvar-coated grid, and then rinsed several times with water. The grid was negatively stained with 0.5% w/v uranyl acetate in water and excess fluid was removed with filter paper.
The second setup comprises a microfluidic pump (Flow EZ, Fluigent), used to generate a laminar flow within a single micro-channel chip (Be-Flow, BEOnChip), and an optical microscope (IX73, Olympus) connected to an external camera to record the transit of MOLNDs. The micro-channel has a rectangular cross-section with a width of 1.5 mm and a height of 0.375 mm; its length is 43 mm. A solution of MOLNDs is fluxed inside the micro-channel, with a measured inlet and outlet flow rates of, respectively, 11 and 8.5 μL min−1. The trajectory of MOLNDs was monitored in the absence of a permanent magnet as well as by placing it in four different positions with respect to the observation point positioned at the channel center, where the flow rate is considered as the average between the inlet and outlet values (as displayed in Fig. 4A). Both the MOLNDs deflection angle and streaming velocity were evaluated employing the ImageJ software tracking plug-in.37 Each average value was obtained based on 30 measurements. The used magnet is made of NdFeB and has a disk shape, with a diameter of 5 mm and a thickness of 2 mm, and is magnetized out-of-plane.
![]() | (3) |
Under the assumption of non-interacting MOLNDs and perfectly inelastic collisions with the channel walls, we described the trajectory (i.e. the time evolution of the vector position s and velocity v) of a generic MOLND with mass χ, hydrodynamic radius Rhyd and magnetic moment m by the following equation:
![]() | (4) |
χ = ρcoreVcore + ρMNPsVMNPs | (5) |
Fg = −[χ − ρf(Vcore + VMNPs)]g, | (6) |
Under laminar flow conditions, the profile of the fluid velocity amplitude within the channel rectangular cross-section (yz-plane) was approximated as:
![]() | (7) |
In eqn (4), the MOLND magnetic moment was expressed by means of the Langevin function as:
![]() | (8) |
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3nr04925f |
‡ These authors contributed equally to this work. |
This journal is © The Royal Society of Chemistry 2024 |