Comparisons of characteristic timescales and approximate models for Brownian magnetic nanoparticle rotations

Abstract

Magnetic nanoparticles are promising tools for a host of therapeutic and diagnostic medical applications. The dynamics of rotating magnetic nanoparticles in applied magnetic fields depend strongly on the type and strength of the field applied. There are two possible rotation mechanisms and the decision for the dominant mechanism is often made by comparing the equilibrium relaxation times. This is a problem when particles are driven with high-amplitude fields because they are not necessarily at equilibrium at all. Instead, it is more appropriate to consider the "characteristic timescales" that arise in various applied fields. Approximate forms for the characteristic time of Brownian particle rotations do exist and we show agreement between several analytical and phenomenological-fit models to simulated data from a stochastic Langevin equation approach. We also compare several approximate models with solutions of the Fokker-Planck equation to determine their range of validity for general fields and relaxation times. The effective field model is an excellent approximation, while the linear response solution is only useful for very low fields and frequencies for realistic Brownian particle rotations. V C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4922858] I. DESCRIBING DRIVEN NANOPARTICLE ROTATIONSIn many magnetic nanoparticle (MNP) applications like biosensing, 1-7 hyperthermia, 8,9 and magnetic particle imaging, 10-13 nanoparticles are driven to rotate by oscillating magnetic fields.14 Understanding the resulting magnetic particle dynamics is important to advance these applications. A typical way to discuss the dynamics is through the timescales of the nanoparticle rotations. [15][16][17] In particular, we often consider the relaxation time: the timescale for a sample of particles to return to equilibrium after some perturbation (e.g., alignment with a field). Conventional magnetic particles are understood to have two rotational mechanisms. The entire particle can rotate as a rigid body by Brownian rotations, 18 and the particle's magnetic moment can also rotate internally due to restructuring of electronic states in N eel rotation. 19,20 The equilibrium relaxation time is different for each mechanism and depends on many parameters. 21,22 However, because most applications involve magnetically excited particles, it is more important to examine non-equilibrium timescales determining the speed of movements in varying driving fields-these timescales can be very different from the relaxation time. One only needs to imagine that in a stronger field, the particles will align faster to see why this is true. We will hence refer to those non-equilibrium timescales as the "characteristic times" of the rotations.In reality, the possibility for N eel rotations complicates the matter and it is important to understand which mechanism is dominant for chosen nanoparticles. 23,24 This is an open problem because these processes will in general not be decoupled. If the processes did truly happen independently (in parallel)...

Mixed Brownian alignment and Néel rotations in superparamagnetic iron oxide nanoparticle suspensions driven by an ac field

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“…Measurement of the relaxation time for magnetic nanoparticle biosensing can be achieved using one-dimensional ac susceptibility [ 4 ]. For a small oscillating applied field as typical in ac susceptibility measurements, the magnetization parallel to the applied field (as above we specify ) can be written with the Debye equation [ 35 , 47 , 48 ], so that when Ω > 1, the Debye magnetization reduces to and the dynamics are again completely described by the master variable.…”

“…Writing the white noise torque in Eq 1 in terms of the white noise process, and multiplying both sides by k B T , the Langevin equation can be rewritten in terms of two commonly used variables [ 38 , 39 ]: the unitless field ξ t = μ H t / k B T and the zero-field Brownian relaxation time τ B = 3 ηV / k B T , so that we have Note that we see from Eq 2 that the white noise process has dimensions of so the term containing the square root of the relaxation time in Eq 3 is dimensionally correct.…”

Magnetic control of heterogeneous ice nucleation with nanophase magnetite: Biophysical and agricultural implications

“…Simulation of the Brownian motion of magnetic nanoparticles in an applied magnetic field is an important topic for numerous experiments using magnetic resonance imaging (e.g., refs. 17 and 18 ), and so there is an extensive literature on it. We need to know how fast these particles rotate in the surrounding water, which is the time constant for Brownian motion, τ B .…”