^{1,2}

^{2}

^{2}

^{1}

^{1}

^{2}

Magnetic separation has gained much attention due to its implications in different fields, becoming feasible as an alternative to existent technologies at the industrial and lab scale. Substantial efforts are focused to improve the magnetic particles used in these applications. Here we show how a relatively simple and low-cost simulation strategy (tracer simulations) can be employed to predict the effect of various key factors in magnetic separation processes, namely, particle properties and magnetic separator designs. For concreteness, we consider here specific problems in magnetic separation. The first one is the effect of different profiles of the magnetic field in the separation of magnetic nanoparticles, and the second one is the magnetophoresis of colloidal particles in a dispersion of magnetic nanoparticles.

The manipulation of magnetic particles by the use of inhomogeneous magnetic fields has emerged as a topic of great interest in a wide range of research and technological areas [

The idea behind magnetic separation is to take advantage of the distinctive magnetic response of the particles in solution to remove them from complex mixtures by the use of applied inhomogeneous magnetic fields [

The basic ingredients in any magnetic separation application are two: the selection of appropriate magnetic particles and the design of the magnetic separator. Typically, the particles employed in these applications are superparamagnetic particles. One has to bear in mind that superparamagnetism emerges as a quantum effect in some ferromagnetic and ferrimagnetic materials, below the single domain size. This implies that this phenomenon is limited to nanocrystals of size below a certain critical size which depends on the material [

A second issue about the separation process is the application of a specific magnetic field over the target sample, inducing a magnetic moment in the carrier particles. This magnetic field has not only to induce a magnetic moment but also to generate a magnetic gradient (which produces a magnetic force on the particle) in order to drive carrier particles apart from solution. Then, the conditions to be fulfilled by the magnetic field source are two: it has to induce large magnetizations but also a gradient in the intensity of the magnetic field. The simplest option to induce magnetophoresis in a lab tube or vial is by the application of a simple bar magnet, but this option is highly inefficient, since typically only those particles near the magnet really experience enough magnetic force to move. However, it is possible to obtain efficient magnetic separators by combining permanent magnets in convenient arrangements in order to generate magnetic fields suitable for magnetic separation. Among the possible arrangements of the magnets, we will discuss here the advantages and drawbacks of two possible cylindrical tube geometries, which have been called open and closed arrangements. Essentially, the closed structure consists of an arrangement of magnets around the tube containing the suspension, which generates a uniform magnetic gradient pointing towards the wall of the tube (see, e.g., [

Here, we propose a simple simulation methodology which allows to model the magnetophoretic separation of nanoparticles inside different designs of magnetic separators. As a first application, the methodology proposed here will be employed to compare the performance of open and closed separator designs. In a second one, we will study the magnetophoretic separation process of a mixture containing particles with different sizes and magnetic responses.

Let us start by describing the equations of motion of superparamagnetic nanoparticles (NPs) in a liquid dispersion under the effects of an external magnetic field. In this situation, NPs will move in the direction of the magnetic gradient (magnetophoresis). As shown experimentally and theoretically in previous works [

In this work, we will consider only the noncooperative case characterized by _{2}O_{3} NPs (diameter 12 nm) employed in [

Under these conditions, we thus need to consider only the individual motion of NPs in the magnetic gradient to obtain the magnetophoretic behavior, ignoring the interaction between NPs. The magnetophoretic velocity of a NP immersed in a fluid with viscosity

The magnetophoretic velocity of a particle in the steady state is obtained by balancing the magnetic force

In the case of a very simple geometry for the magnetic field, it has been possible [

The simulation technique is as follows. First of all, we need to know the geometry of the magnetic separator and the magnetic field

In this subsection, we will consider a magnetophoretic separation problem for which we obtained both experimental results and an analytical solution. Comparison of the results of our simulations with previously known results is a necessary step in order to ensure the validity of our simulation approach. After this validation step, we will employ our simulation method in the following subsection to explore other situations in which previous theoretical results are not available.

Here we consider the closed geometry for the magnetic separator, similar to the actual separators employed in recent experimental works [_{2}O_{3} NPs (diameter 12 nm) was placed inside the separator. The magnetization curve for these NPs was given also in [^{−1}. The employed solvent was water (with viscosity

Profile of the magnitude (modulus) of the quadrupolar magnetic field in Tesla (a) and its gradient in Tesla/m (b) for the closed type magnetic separator (top view) employed in the simulations. Note that the magnitude of gradient of the field intensity corresponds to

Magnetophoretic separation of superparamagnetic _{2}O_{3} NPs of diameter 12 nm inside a 30 T/m magnetic separator. Comparison between the simulations performed here, the analytical solution and the experimental results reported in [

Now, we employ our simulation methodology to compare the performance of two different designs of magnetic separators. The first design we consider here is the one considered in the previous subsection, which we will call “closed type’’ separator from now on. As we said previously, the main advantage of this geometry for the magnetic separator is the fact that the magnetic gradient is approximately uniform inside the system. The second design we will consider here is an “open type’’ separator. In this case, the geometry of the separator is the same as the closed type considered in the previous subsection, but now part of the magnets were removed. As we have mentioned in the introduction, this partial removal of magnets is made in commercial separators in order to facilitate visual contact with the dispersion during the separation process so that the separation can be monitored easily by eye inspection [

Profile of the magnitude of the magnetic field

In order to compare the different performance between the open and the closed type separators, we consider the same suspension of _{2}O_{3} NPs of diameter 12 nm described in the previous subsection. Now we perform simulations for this suspension in the case of open-type geometry of the magnetic separator. The technical details (number of tracer particles, time step, etc.) were the same as employed in the simulation of the previous subsection. Here, the simulations were performed until a simulation time of

Series of snapshots extracted from simulations comparing the time evolution of the separation process in the closed-geometry (upper row) and open-geometry (bottom row) schemes. The snapshots are taken from a top view of the cylindrical separator with radial geometry. The different snapshots correspond to different times during separation (

Comparison of the fraction of particles in solution as a function of time as obtained in tracer simulations for open and closed type magnetic separators with magnetic profiles shown in Figures

The problem we would like to consider in this section is the motion of colloidal particles (with sizes of the order of hundreds of nm or larger) in a dispersion containing superparamagnetic NPs. This particularly asymmetric mixture has a fascinating behavior which has received significant attention in recent years. For example, it is possible to induce the assembly and transport of nonmagnetic colloids immersed in a dispersion of superparamagnetic NPs by applying external magnetic fields [

Here, our interest will be the study of the behavior of a mixture of colloidal particles and NPs in a magnetic separator. Experimentally, this system has been studied in [

The simulation methodology employed here is based on particle tracers simulations as developed in the previous section. Now, we will have two different types of tracer particles, one corresponding to NPs and another one corresponding to colloidal particles. For simplicity, we will consider the simulation of this mixture only in the case of closed type magnetic separators. The profile of the magnetic field is shown in Figure

Due to the symmetry of the problem, we consider only the radial motion of the particles so the simulations can be performed in 2 dimensions (the vertical coordinate

In this case, we have performed a single simulation for a particular case of interest which is now being realized experimentally [_{2}O_{3} NPs of diameter 12 nm) and colloidal particles similar to commercial latex micro spheres (1 g/L dispersion of colloids with diameter 900 nm). In these conditions, the initial value of the quantity

We have performed different simulations with a total number of 10^{6} tracer NPs together with

Experimentally [

Snapshots from simulations of the magnetophoresis of an aqueous dispersion of _{2}O_{3} superparamagnetic nanoparticles (grey) and latex polystyrene particles (orange) under a magnetic gradient of 30 T/m at different times (1 h, 10 h, 20 h, and 24 h resp.).

The observed profiles of latex particles can be understood from the analysis of the trajectories of individual tracers. Typical trajectories for the radial distance

Examples of 15 trajectories obtained in simulations corresponding to tracer latex particles immersed in a dispersion of NPs starting from different distances to the center of the system (see details in the text.)

Although our simulations described in the previous section are not particularly costly from the computational point of view, they require the use of relatively large amounts of disk space to store the tracer trajectories and later analysis to obtain relevant quantities such as concentration profiles and number of particles remaining inside the magnetic separator. For this reason, it could be convenient to develop a simplified approach amenable of solution without the need of performing computer simulations.

The motion of the latex particles observed in the simulations can be described with reasonable accuracy with a simple equation. The basic idea is to disregard the radial dependence in the magnetophoretic velocity of the NPs in (_{2}O_{3} NPs under 30 T/m considered in our previous subsection, we have

Comparison of trajectories of two tracer latex particles as obtained in the simulations and by numerical solution of (

In this work, we have presented a low-cost simulation strategy based on the concept of particle tracers aimed to tackle the magnetophoresis process in the noncooperative magnetophoretic regime. We have successfully validated this simulation approach by comparing the results obtained against existing experimental and also analytical results obtained for the separation process of a colloidal dispersion of _{2}O_{3} superparamagnetic nanoparticles in an aqueous solution. Thanks to this methodology, we have been able to evaluate different key factors involved in the magnetophoretic separation process. Regarding the separator design, we have shown that the homogeneous magnetophoretic conditions created by a closed type separator (high magnetic field over almost the whole sample and constant magnetic gradient) enhance the separation process, providing more control over the process and reducing the expected separation time when compared to the open-type version of the separator. We have also extended that methodology to solutions of colloidal particles in aqueous solutions of superparamagnetic nanoparticles in the closed type geometry. The simulation performed in this case is able to account for the ring-like structure expected in some experimental situations and agrees with the simplified numerical model proposed.

This work was supported by the Spanish Government Grants no. FIS2009-13370-C02-02, PET2008-02-81-01/02, and CONSOLIDER-NANOSELECT-CSD2007-00041. J. Faraudo and J. Camacho are also supported by the Catalan Government Grant no. 2009SGR164. The authors thank LL. M. Martínez from SEPMAG Technologies for help regarding the calculation of the profiles of the magnetic fields and for extensive discussions.

_{3}O

_{4}nanocrystals

_{2}O

_{3})/shell(SiO

_{2}) nanoparticles as high relaxivity T

_{2}-contrast agents for magnetic resonance imaging