Unified mathematical model of the kinetics of nanoparticle phase condensation in magnetic fields

Abstract

In this paper, we aim to present a unified mathematical modeling and description of the kinetics of magnetic nanoparticles phase condensation (conducting to the appearance of bulk elongated aggregates) under homogeneous permanent or alternating magnetic field. For such case, the aggregate growth rate usually takes the form dV/dt = G(V)∆(t), with V and t being the aggregate's volume and time, respectively, ∆(t)—the supersaturation of the nanoparticle suspension, and with the function G(V) depending on the precise configuration of the applied field. The Liouville equation for the aggregate size distribution function is solved by the method of characteristics. The solution is obtained in parametric form for an arbitrary function G(V), providing a general framework for any type of the applied magnetic field. In the particular case of low‐frequency rotating magnetic field (G(V)~V2/3), an explicit expression of the distribution function is obtained, while the dimensionless average aggregate volume 〈V〉 is found by the method of moments allowing a complete decoupling of the system of equations for the statistical moments 〈Vn〉 of the distribution function. Numerical examples are provided for the cases of permanent and low‐ or medium‐frequency rotating fields. It is shown that in all cases, the average volume 〈V〉 only slightly depends on the relative width of the initial size distribution. Nevertheless, at any times, t > 0, the size distribution shows a significant spreading around the average value 〈V〉, which increases progressively with time and achieves a final plateau at long times. This model can be helpful for several biomedical or environmental applications of magnetic nanoparticles in which the nanoparticle suspension undergoes a field‐induced phase condensation.

Adsorption of Organic Dyes on Magnetic Iron Oxide Nanoparticles. Part II: Field-Induced Nanoparticle Agglomeration and Magnetic Separation

“…It should be noted that our experimental setup does not allow detecting the aggregates of the size lower than ∼1 μm. However, the recent theoretical model 39 predicts a considerable increase of the minimal agglomerate size with time such that nanosized aggregates likely disappear at elapsed times t approximately larger than the agglomeration time scale τ a (determined below). We believe therefore that the chosen optical microscopy method gives a relevant picture of the aggregate size distribution at least at times t ≥ τ a .…”

Kinetics of field-induced phase separation of a magnetic colloid under rotating magnetic fields

“…7b. Such a shift is typically reproduced by the model considering only the aggregate growth mechanism ignoring the coalescence [Kuzhir et al (2020)]. This allows us to suppose that the relative contribution of coalescence in the present model is overestimated especially at short times, possibly due to approximations ̇∝ (1 − 〈 〉) [Eq.…”