Robert J. Deissler

Dependence of Brownian and Néel relaxation times on magnetic field strength

PURPOSE In magnetic particle imaging (MPI) and magnetic particle spectroscopy (MPS) the relaxation time of the magnetization in response to externally applied magnetic fields is determined by the Brownian and Néel relaxation mechanisms. Here the authors investigate the dependence of the relaxation times on the magnetic field strength and the implications for MPI and MPS. METHODS The Fokker-Planck equation with Brownian relaxation and the Fokker-Planck equation with Néel relaxation are solved numerically for a time-varying externally applied magnetic field, including a step-function, a sinusoidally varying, and a linearly ramped magnetic field. For magnetic fields that are applied as a step function, an eigenvalue approach is used to directly calculate both the Brownian and Néel relaxation times for a range of magnetic field strengths. For Néel relaxation, the eigenvalue calculations are compared to Browns high-barrier approximation formula. RESULTS The relaxation times due to the Brownian or Néel mechanisms depend on the magnitude of the applied magnetic field. In particular, the Néel relaxation time is sensitive to the magnetic field strength, and varies by many orders of magnitude for nanoparticle properties and magnetic field strengths relevant for MPI and MPS. Therefore, the well-known zero-field relaxation times underestimate the actual relaxation times and, in particular, can underestimate the Néel relaxation time by many orders of magnitude. When only Néel relaxation is present--if the particles are embedded in a solid for instance--the authors found that there can be a strong magnetization response to a sinusoidal driving field, even if the period is much less than the zero-field relaxation time. For a ferrofluid in which both Brownian and Néel relaxation are present, only one relaxation mechanism may dominate depending on the magnetic field strength, the driving frequency (or ramp time), and the phase of the magnetization relative to the applied magnetic field. CONCLUSIONS A simple treatment of Néel relaxation using the common zero-field relaxation time overestimates the relaxation time of the magnetization in situations relevant for MPI and MPS. For sinusoidally driven (or ramped) systems, whether or not a particular relaxation mechanism dominates or is even relevant depends on the magnetic field strength, the frequency (or ramp time), and the phase of the magnetization relative to the applied magnetic field.

Modeling the Brownian relaxation of nanoparticle ferrofluids: Comparison with experiment

We obtain good agreement between the calculated and measured ratio of harmonics only when the model includes nanoparticles which have a distribution in the hydrodynamic diameter - that is polydisperse. We are unable to find good agreement if the diameter of the nanoparticles is constrained to only one value - that is monodisperse. In Fig. 1 we plot the measured [3] ratios of 5th/3rd harmonics of the magnetization for samples using “100 nm” iron oxide particles (top plot) and “40 nm” iron oxide particles (bottom plot). The measurements were collected with the nanoparticles in ferrofluid solutions with a range of water/glycerol ratios corresponding to different viscosities (and therefore different Brownian relaxation times.) As described in [3] the data are plotted as a function of ωτB. Also shown in Fig. 1 are the calculations of the ratios of 5th/3rd harmonics assuming both monodisperse and polydisperse ferrofluids. Details of the calculations and parameters used in the models can be found in [5].

Numerical study on the quench propagation in a 1.5 T MgB2 MRI magnet design with varied wire compositions

To reduce the usage of liquid helium in MRI magnets, magnesium diboride (MgB2), a high temperature superconductor, has been considered for use in a design of conduction cooled MRI magnets. Compared to NbTi wires the normal zone propagation velocity (NZPV) in MgB2 is much slower leading to a higher temperature rise and the necessity of active quench protection. The temperature rise, resistive voltage, and NZPV during a quench in a 1.5 T main magnet design with MgB2 superconducting wire was calculated for a variety of wire compositions. The quench development was modeled using the Douglas–Gunn method to solve the 3D heat equation. It was determined that wires with higher bulk thermal conductivity and lower electrical resistivity reduced the hot-spot temperature rise near the beginning of a quench. These improvements can be accomplished by increasing the copper fraction inside the wire, using a sheath material (such as Glidcop) with a higher thermal conductivity and lower electrical resistivity, and by increasing the thermal conductivity of the wires insulation. The focus of this paper is on the initial stages of quench development, and does not consider the later stages of the quench or magnet protection.

A multiscale and multiphysics model of strain development in a 1.5 T MRI magnet designed with 36 filament composite MgB2 superconducting wire

High temperature superconductors such as MgB2 focus on conduction cooling of electromagnets that eliminates the use of liquid helium. With the recent advances in the strain sustainability of MgB2, a full body 1.5 T conduction cooled magnetic resonance imaging (MRI) magnet shows promise. In this article, a 36 filament MgB2 superconducting wire is considered for a 1.5 T fullbody MRI system and is analyzed in terms of strain development. In order to facilitate analysis, this composite wire is homogenized and the orthotropic wire material properties are employed to solve for strain development using a 2D-axisymmetric finite element analysis (FEA) model of the entire set of MRI magnet. The entire multiscale multiphysics analysis is considered from the wire to the magnet bundles addressing winding, cooling and electromagnetic excitation. The FEA solution is verified with proven analytical equations and acceptable agreement is reported. The results show a maximum mechanical strain development of 0.06% that is within the failure criteria of −0.6% to 0.4% (−0.3% to 0.2% for design) for the 36 filament MgB2 wire. Therefore, the study indicates the safe operation of the conduction cooled MgB2 based MRI magnet as far as strain development is concerned.

Mechanical Analysis of MgB2 Based Full Body MRI Coils Under Different Winding Conditions

The winding of composite superconducting wire around a mandrel is one of the first stages of manufacturing processes of a superconducting magnet. Depending on the method of mechanical support conditions during winding, the strain development at the final stage in a superconducting magnet may vary significantly. Therefore, proper selection of the winding process is important to increase the feasibility for a conduction cooled full body MRI magnet based on magnesium diboride (MgB2), a strain sensitive high-temperature superconductor. A multiscale multiphysics finite element analysis) model of an 18 filament MgB2 wire is developed for strain estimation. The computationally homogenized representative volume element of the composite wire is used in the coil bundle in place of the actual MgB2 wire. The simulation considers winding, thermal cool-down and electromagnetic charging to estimate total strain developed at the final step—electromagnetic charging. Four different types of support conditions are studied and strain development is reported. Results suggest that a combination of radial and axial support at the inner radial surface and outermost axial surfaces of the mandrel, respectively, is the most favorable winding condition with a minimum strain development of 0.021%, which is half in comparison to no mandrel support.

Dependence of the Magnetization Response on the Driving Field Amplitude for Magnetic Particle Imaging and Spectroscopy

To predict the response of magnetic nanoparticles to changes in the external magnetic field in magnetic particle imaging (MPI) and magnetic particle spectroscopy, it is important to understand the relaxation mechanisms and relaxation times. Often, the zero-field formulas for Brownian and Néel relaxation are employed when theoretically estimating the relaxation times. However, as reported previously, the relaxation times depend on the magnetic field strength. The Néel relaxation time can change by many orders of magnitude even for magnetic field strengths typically used in MPI. Here, we report on numerical simulations of the Fokker-Planck equations governing Brownian and Néel relaxation for an externally driven system. We find that when only Néel relaxation is present-as occurs if the particles are embedded in a solid-a strong magnetization response can occur even if the zero-field equation predicts a weak response. For a system of particles suspended in a fluid, the dominate relaxation mechanism, either Brownian or Néel, depends on the magnetic field strength, the driving frequency, and the phase of the magnetization relative to the driving field. In addition, some analytical expressions for the relaxation times are evaluated.

Brownian and Néel relaxation times in magnetic particle dynamics

The equations for the well-known Brownian and Néel relaxation times T_B and T_N (given in the Introduction) are not sufficient to determine whether or not the Néel relaxation mechanism may be neglected. In fact, for a sufficiently large magnetic field strength, the Néel relaxation time can change by many orders of magnitude, whereas the Brownian relaxation time is comparatively insensitive to the magnetic field strength. Therefore, even if τ_N >> τ_B, the Néel relaxation mechanism can still be important and even dominate.

Magnetic particle spectroscopy of magnetite-polyethylene nanocomposite films: A novel sample for MPI tracer design

Several harmonic spectra are plotted in Figure 2. An important result is the comparison of free particles embedded in agar and films (Figure 2c). The particles embedded in the films are “frozen” in place, while those in agar, unless bound within the matrix, should still have some freedom to rotate. Particles in both should have identical Neel relaxation times, as that is dependent only on the structure of the iron core and not local environment, provided magnetic interactions can be excluded. A calculation of the Neel relaxation time yields values in the range of 0.12-171 microseconds (for crystal anisotropies in the range of 11-21kJ/m3). The Brownian relaxation time for the films is assumed to be infinite, such that relaxation is only possible through the Neel mechanism. However, the Brownian relaxation times for particles in agar are long enough that relaxation is also assumed to be dominated by the Neel mechanism. While Brownian relaxation may play a role in explaining the signal difference, it is likely due to magnetic interactions between neighboring particles that steepen the magnetization curve. Figure 2c demonstrates that a low interparticle distance generally results in a higher signal. In samples with high local concentrations (such as the films presented here, or samples that exhibit significant aggregation), the average distance between two particles is small enough that dipole-dipole interactions must be considered. Such interactions are one possible explanation for the performance of Feridex IV and Resovist.

A Computational Study to Find an Optimal RRR Value for a 1.5-T Persistent-Mode Conduction-Cooled MgB 2 MRI Magnet From a Quench Protection Point of View

The effect of magnetoresistance, magnetothermal conductivity (MTC), and RRR value on the active quench protection of a persistent-mode conduction-cooled 1.5 T MgB₂ superconducting magnet is studied using numerical simulation. It is found that inclusion of magnetoresistance in the simulation can significantly decrease the calculated maximum temperature reached in this magnet during a quench, e.g., by 9 K for <italic>RRR</italic> = 100. The inclusion of both magnetoresistance and MTC similarly decreases the peak temperature, but by a smaller amount, e.g., by 7 K for <italic>RRR</italic> = 100. Using copper with a lower RRR value can also significantly decrease the peak temperature reached in this magnet, e.g., by 16 K for <italic>RRR</italic> = 50 as compared to <italic>RRR</italic> = 200. However, decreasing the RRR value to much less than 50 results in a significantly higher voltage, e.g., decreasing the RRR value from 50 to 10 results in a 1000 V increase. A RRR value between 50 and 100 may provide a good balance.

Corrections to “Quench Protection Using CLIQ of a MgB2 0.5 T Persistent Mode Magnet” [Jun 17 Art. no. 4700605]

Manuscript received February 15, 2017. Date of current version March 8, 2017. This work was supported by the National Science Foundation under Grant PFI:BIC 1318206 and the Ohio Third Frontier. (Corresponding author: C. Poole.) The authors are with the Department of Physics, Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: crp68@case.edu; tnb@case.edu; rjd42@case.edu; mam18@case.edu). Digital Object Identifier 10.1109/TASC.2017.2673258 Corrigendum IV): Eq. 11 should be: