Efficiency of High Gradient Magnetic Separation Applied to Micrometric Magnetic Particles

This article presents two prototypes of laboratory magnetic separators that generate high gradient magnetic fields. Such a field is created in a separation cell via steel wool. The efficiency of separators was tested on a water suspension containing weakly magnetic Fe2O3 nano/micro-particles, prepared in three size fractions in a size range of 60 nm – 10 μm. The separation process was evaluated via optical transmittance of the suspension before and after sequential separation processes. Repeated separations on the same sample exhibit an asymptotic trend that results in the conclusion that it is not possible to trap all solid content. According to the decrease of solid particles concentrations during cyclic separation we set the efficiency of the process. It is maximally 46% for fine fraction, 65% for medium fraction, and 40% for coarse fraction after infinity separation cycles.


INTRODUCTION
Magnetic separation is a well-established technology (1). In laboratory and biomedical experiments, however, this technique provides a limited tool for separation of nano/micro-particles (2,3), magnetic cells (4), or blood cells from liquids (5). This limitation is caused by forces that have an opposite direction than magnetic forces.
Generally, the separation condition is that magnetic attraction forces must be larger than the other forces. The result of the process is the separation of input mixture to magnetic and non-magnetic components. The magnetic attraction force depends on the magnetic field gradient. A low gradient magnetic separation can be applied in the most simply way by a permanent magnet located in the side of the test-tube. As such, low-gradient separation principle is used in commercial laboratory low-gradient magnetic separation technologies. This method is suitable for strongly magnetic suspensions. However, in order to separate weakly magnetic materials or paramagnetic materials, a high gradient magnetic separation (HGMS), using a strongly non-homogeneous magnetic field must be created in the vicinity of the magnetic particle (6). Such HGMSeparators are composed of a source of magnetic field and steel wool that is placed into a separation cell. This wool creates a complicated high gradient magnetic field (see Fig. 1) inside of the separation cell (7).
In the works focused on HGMSeparation the physical process is well described (7,8). On the other hand, very little attention has been paid to other parameters of HGMSeparators such as separation capacity, separation efficiency, size of particles that can be effectively separated out, etc. Theoretical description of these parameters is generally missing in the literature. With the rapid evolution of nanotechnology, these parameters must play a key role in laboratory magnetic separation of nanoparticles. Therefore, the aim of this work is to test magnetic separation efficiency of two self-manufactured HGMSeparator prototypes on weakly magnetic particles, working in dynamic regimes, and to determine some parameters that are important for effective separation of nano/micro-particles.

Theoretical Background (SI Unit System)
In the first approximation, two forces act on the particles in the separator: magnetic F m and fluidic resistance F st . Gravitational force is neglected. We suppose a spherical magnetic particle (paramagnetic or weakly ferromagnetic) having the radius R in the external field with the magnetic induction B. The magnetic force F m is given by where V = 4/3πR 3 is the volume of particle, grad = i•∂/∂x + j•∂/∂y + k•∂/∂z is the gradient operator and M is the magnetization, especially below saturation we can write where χ is the effective magnetic susceptibility of particle, µ 0 is the magnetic permeability of free space. Next the magnetic force is When the particle is saturated, M = M s , Eq. (2) must be rewritten as Generally, field gradient is very difficult to determine in cases of HGMS based processes that use the steel wool located in a separation cell. It is difficult to simply express the magnetic force for superparamagnetic particles since they exhibit paramagnetic behavior in a multiparticle amount. Interparticle magnetic interactions are usually taken to be negligible and nonsignificant for the result of the separation process (9,10). For a single spherical particle, the Stokes' resistance force is valid where η is the dynamic viscosity of the liquid, R is the particle radius and v is the particle speed related to steel wool or magnets. The magnetic forces pull the suspension particles towards steel wool and the Stokes' resistance restricts this motion. Steady state, when the magnetic force and the Stokes' resistance force are equal, sets in almost immediately. The quantity of the separated particles depends on the dimensions of the separation cell and the proportion between the velocity of the particle and fluid, respectively (see Fig. 2).

Particle Flow Ratio (SI Unit System)
We have made several assumptions to make a rough estimate of the quality of fraction separation: 1. The velocity of fluid v y is expressed as the medial value of the drift velocity (velocity profile in the separation cell is neglected), 2. Only cross-coordinate of F mx and v x is non-zero because it is connected with the separation process, 3. Magnetization M is a linear function of B (weak field), which means that Eq. (2) is valid.
A solid particles flow ratio κ defined for a selected point in separation cell is See Fig. 2, where I x and I y are cross-flow (transversal separation flow) and direct flow of particles and C is the concentration of particles. Now we can express the coordinate v x using the Eqs. (3) and (5) Considering simple field condition B x •∂B x /∂x ≫ B y •∂B x /∂y and Finally, the flow ratio κ is expressed via substitute Eq. (8) into Eq. (6) by equation A simple gradient magnetic separation (without steel wool), working in dynamic regime, which was put into action in cylindrical tube. The trajectory of magnetic particle that is attracted to the tube upper wall subject to magnetic force generated via linear gradient of magnetic field. The capturing of the particle in the separation cell depends on the initial position of the particle, axial flow rate v y , grad(B) that is parallel to drift velocity v x , length L and diameter D of the tube, and other parameters presented in Eq. 9.

HGMS APPLIED TO MICROMETRIC MAGNETIC PARTICLES
This means that the quality of separation linearly depends on χ, R 2 , B x , and ∂B x /∂x, and hyperbolically decreases according to η and v y .

Experimental Arrangement
The experimental apparatus for magnetic separation includes glass reservoir vessels (1 liter), one of which is placed in Ultrasonic Cleaner VGT 1200, 1.3 liter volume, 60 W, 40 kHz, peristaltic pump of PCD 32 type, max flow-rate 50 mL/min, magnetic separator with a separation cell filled by steel wool that is compressed into the plastic separation cell (Fig. 3). The separation cell volume is approximately 50 mL. The ferromagnetic steel wool has packing density 0.08 g/ mL and is composed of wire-strips sized 0.5 × 3 mm. The individual parts are connected by silicone hoses having an inner diameter of 4 mm. A detailed description of the separation process is given in the section titled "Description of the Separation Process."

Electromagnetic Separator
The coil separator ( Fig. 4) consists of a wooden stand Pertinax (synthetic resin bonded paper), cylinder with wire winding and a separation cell containing steel wool inside. The coil was wound with a copper wire with the diameter of 0.8 mm. The total number of turns is 1485 and the maximum coil resistance is 4.5 Ω. The separation cell (tube) has 20 mm in diameter. The magnetic field in the central point of the coil was measured by the Hall probe and reached the value of 50 mT at the 3 A electric current (measured without presence of steel wool).

Description of the Separation Process
Separation processes using both separators function as follows: Prepared suspension, approximately 1 liter, was brought into the separator using the peristaltic pump after its homogenization by ultrasound in Ultrasonic Cleaner, while the same flow rate of 40 mL/min was maintained for the whole period of separation. The mixture further continued through the separation cell filled by steel wool to the collection vessel, where the separation process ended. The separation process for the same sample was taking place in five identical separation cycles. The steel wool was changed after these five separation steps, while its mass was maintained. The degree of separation was evaluated via optical density of suspension immediately after separations.

Material
The Fe 2 O 3 powder was used for the measurement. The material was provided by Sigma Aldrich Company, product number 310050. The composition of this powder determined via X-ray diffraction is 96% of hematite and 4% of maghemite. Such material was selected for its significantly weak magnetic fraction, since hematite is weakly ferromagnetic and maghemite is ferrimagnetic material under room temperature. The magnetization of powder was performed using the Vibrating Sample Magnetometer (VSM, EV9 Microsense), Fig. 6. The saturation magnetization reached 2 A.m 2 /kg and initial susceptibility was approximately 10 −5 m 3 /kg. In the case of ferrimagnetic magnetite, these parameters are presented to be in the order 100 times stronger (11).
The powder suspension was prepared using distilled water. In the first step the suspension was undergone to sedimentation. This method is based on the speed of particle sedimentation according to size. The next step was the ultrasound cavitation disintegration that lies in the periodic compression and release of the fluid, while cavitation bubbles are created, primarily on the particles. Collapse of the bubbles on the particles then leads to creation of local temperature and pressure extremes that result in destruction of the particles (12,13). Three Fe 2 O 3 fractions were prepared-coarse, medium, and fine. Their size distributions are presented in Fig. 7 and were determined by employing the dynamic light scattering method by a Malvern Zetasizer NanoZS (type ZEN3600). The distribution of coarse fraction was out of measurement limits of this optic device and has mode of approximately 5 μm, based on the knowledge of initial particles size distribution.

Evaluation of Separation Processes
The evaporation and photometry techniques were selected for the evaluation of the separation processes.
The evaporation method provides information about mass concentration C of particles in suspension and is based on the measurement of dry powder weight after fluid evaporation, from 50 mL, C = (mass of solid powder)/(volume of suspension) in mg/mL. While the method does not depend on coagulation and sedimentation of particles, only the densest samples were determined using this method in the range of calibration measurement. Other samples were excluded from such HGMS APPLIED TO MICROMETRIC MAGNETIC PARTICLES evaluation process since it is highly time-consuming and unreliable for low concentrated samples. The accuracy of its method is limited by resolution of weight scales, 10 −5 g.
Standard photometry was selected as a reliable method to evaluate separation process since the turbidimetry, i.e., measurement of intensity of upright dispersed light, is improper for Fe 2 O 3 suspension, as we tested (14). The photometry is based on transmittance of light through a water suspension containing magnetic particles. The transmittance T(%) is defined in the literature as T(%) = 100•Φ /Φ 0 where Φ is the intensity of light coming out of the sample and Φ 0 is the intensity in front of the sample. The measurement was done using Spekol Carlzeiss Jena 376581 tuned to 500 nm that appeared as the most appropriate light for the selected material. The optic layer of samples in the cuvette was 10 mm. In this method the value of 0% was calibrated to completely opaque suspension (using opaque stop), and the value of 100% corresponds to distilled water without powder particles. Resolution of the device is 1%. The chosen optical measurement is fast, but it has to be done immediately after the process of separation due to very fast sedimentation and coagulation of particles in the suspension sample, which significantly influences measurement results.

RESULTS AND DISCUSSION Results
The results of magnetic separation on Fe 2 O 3 powder are presented in Fig. 8. It is apparent that devices separated the magnetic component from the suspension after each separation cycle, which shows an asymptotic trend in ideal cases. In several cases the asymptotic trend was not observed, which was caused by unknown process errors. Those non-asymptotic processes are designated as w.t. in Fig. 8 and are taken to be invalid in subsequent theoretical computations. In spite of the fact that measurement repetitions A, B, C, D, E were done using the same initial suspension sources (coarse, medium and fine), we found differences in T 0 values and subsequently in the individual separation processes. For this reason these repetitions could not be averaged for the subsequent data processing.
Since the separation process depends on many parameters, an appropriate physical function describing the repetitive process is hardly possible to derive. To understand a separation process more precisely we decided to describe it via a mathematical equation. Thus, we created appropriate asymptotic function expressing transmittance T (%) of light via suspension in form where T 0 is the initial transmittance, b (%) and k (-) are the constants and N (-) is number of separation cycles applied. The separation limit T ∞ = T 0 + b expresses transmittance after an infinite number of separation cycles. The function represented by Eq. (10) provides a good theoretical fit to experimental points since in several cases the sum of square deviations r 2 → 1, using Matlab software. The parameters T ∞ , evaluated base of measurements, are displayed in the figure legends. While the parameter T 0 was optically determined to be 0-30% in cases of coarse and medium fraction, in the case of fine fraction T 0 > 60%. The reason for these results is in low initial concentration of particles in suspensions with fine fraction. Additionally, we expected that coarse fraction should have a higher value of T ∞ since the separation quality coefficient is according to Eq. (9) proportional to particles size R 2 . However, the general trend is opposite (see Table 1). The fine fractions have (ostensibly) higher separation limits. The separation trends displayed in Fig. 8 for fine fraction do not have an asymptotic, but rather chaotic trend. In our opinion, such results could be explained via limited wool surface area of the separators, i.e., during the separation of coarse and medium fraction the wool surface is quickly covered by strongly concentrated suspensions. Additionally, the fine fraction probably was not really separated, since the fraction was prepared above the particle-size separation limit of the separator, which must be limited by Brownian force (see below).
In Table 2, the separation ranges after five cycles are presented. It is apparent that separation ranges T r are significantly higher in cases of coarse and medium fractions than in cases of fine fraction. It is not reliably possible to compare the efficiency between both types of separators.

Solid Concentrations in Suspensions
Real concentration of solid matter in suspension can be estimated using the well-known Lambert-Beer law that defines transmittance as T(%) = 100•10 −α•C where α (mL•mg −1 ) is the   Note: The abbreviation w.t. denotes "without asymptotic trend", PM denotes separator with permanent magnets and C denotes the separator with coil. HGMS APPLIED TO MICROMETRIC MAGNETIC PARTICLES coefficient depending on optical properties of dispersed material, length of absorption layer and wavelength of light, and C (mg/mL) is the concentration of matter in suspension. Using this law, the concentration after N-th separation cycles is and depends on transmittance T N of suspension after N-th separation cycle. Equation (11) provides the simplest approximation of C since the parameter α depends on particle size distribution.

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C 0 do not correspond to samples having transmittance T 0 in main separation measurements in Fig. 8. Fitting the relation between C and log(T/100) in equation C = − log(T/100)/α + β, one can find the parameters α and β and subsequently estimate the real concentrations of suspensions from main separation measurements. The parameter β calibrates the photometry device for the real concentration of suspension when T = 100%. Finally, the separation efficiency per single or several separation steps η n (%), between cycles N and N + n, where n = 0, 1, 2, ..∞ is the count of other separation steps, is derived to be valid in interval 0−100%, where change of the concentration is Using these last equations, for example, medium fraction sample B separated in five cycles using an electromagnetic separator having measured transmittances T 0 = 17%, T 5 = 71%, and T ∞ = 82%, estimated concentrations and efficiency are C 0 = (0.0888 ± 0.0083) mg/mL, C 5 = (0.0406 ± 0.0065) mg/mL, ΔC = (− 0.0481 ± 0.0106) mg/mL, and η 5 = (54 ± 13) %. Its concentration after infinite separation cycles is C ∞ = (0.0358 ± 0.0064) mg/mL and η ∞ = (60 ± 13) %, taking into account calibration parameters α and β valid for medium fraction and uncertainty of transmittance 0.577%. The efficiency after infinite separation cycles is shown in Table 3. In spite of the fact that we measured and processed the data carefully, standard confidence intervals are large.

Importance of Parameters Equation 9
Separation ratio κ, defined via Eq. (9), is proportional to B x , ∂B x /x, χ, and R 2 , and inversely proportional to η and v y . The parameter v y is flow rate q divided to hose cross-section area. The effect of B x depends on magnetic field generated by magnets/coil. The effect of ∂B x /x depends on packing density of steel wool and its magnetic properties. The effect of material susceptibility can be tested using different materials and the effect of viscosity using different liquids. The effect of particle size R is hard to verify since it is difficult to prepare a monodispersion, but our results are partially discussed in the result section. Finally, we tested the effect of a flow rate of liquid (see Fig. 10). In accordance with expectations, slower flow of suspension results in more effective separation.

Equation 10
The magnetic force acting on a single particle must overcome Stokes' viscous force, and, in a limited case, the Brownian force as well. This Brownian force may be estimated by the term k B T/R (2). From equilibrium condition between magnetic and Brownian force the size of the particles that cannot be separated may be estimated. The physical meaning of parameters presented in Eq. (10) could be considered in this way: In case of hypothetical separator having infinite capacity and wool surface area (see below), the parameter T ∞ = T 0 + b probably depends only on Boltzmann factor exp[−wR 3 /(k B T)], where the term wR 3 represents potential magnetic energy of a particle in the vicinity of steel wool, where w (J/m 3 ) depends on the position of the particle in the separator, its magnetic behavior, external magnetic field strength, and gradient of this field. The parameter k probably depends namely on flow-rate of suspension in cases of separators working in dynamic regimes. Separator capacity K(mL) = Vm/ρ, which must be proportional to separation cell volume V (mL), steel wool packing density ρ (g/mL), and mass of wool in the cell m (g/mL). The parameter K really has to influence both constants from Eq. (10). If the K is sufficient volume, concentration of solid matter small and solid particles are only paramagnetic or magnetically weak; next, more important must be the active surface area of the wool. In case of our separators, 4 g of wool equals 24 cm 2 of the surface area.

Effect of Filtration
To estimate the importance of the filtration effect during separation, we performed a special measurement. The suspension was flowing through the separation cell containing the steel wool without the presence of permanent magnets. This measurement was supposed to provide us with information whether the capture of particles is caused by magnetic field. We found a certain level of a filtration effect (see Fig 11). We attribute its existence to internal magnetic field of steel wires, their geomagnetic magnetization, adhesion between steel wool and the particles, and to adhesion of particles on inner walls of the separation cell and hoses. The separation range T r of such filtration effect for medium fraction was determined to be 20% for five separation cycles. A true filtration based on mechanical capturing of particles similar to filtration of suspension via filtration paper does not exist in our separation systems.

CONCLUSIONS
Two self-manufactured prototypes of HGMSeparators were tested on water suspensions containing magnetically weak Fe 2 O 3 particles, in dynamic regimes. Using a new theoretical approach to evaluation of separation efficiency via optical transmittance T(%) of suspension we discovered that separation process exhibits an asymptotic trend. The asymptotic equation, T = T 0 + b(1exp(− k•N)], constructed in this work, provides good information about the parameters of the separation process. This asymptoticity is caused mainly by the wool surface area capacity of separators and size separation limits of particles connected with forces acting on them. The separation without magnetic field exhibits a certain level of filtration effect whose origin is unclear, but it is a result of inner magnetic and adhesion forces. The separation efficiency depends on the flow-rate of the suspension, slower flow of suspension results in a better separation effect. The transmittance value can be recomputed to real concentration of solid powder matter in suspension using Lambert-Beer law. In spite of the fact that this law is used mainly for solutions, for our Fe 2 O 3 nano/micro particles dispersed in water it appears to be the simplest but useful relation between concentration and transmittance. Based on this theory we estimate that maximal efficiency after infinity separation cycles on fine fraction reaches η ∞ = (46 ± 18) %, on medium fraction reaches η ∞ = (65 ± 13) % and on the coarse fraction η ∞ = (40 ± 13) %. The efficiency in case of fine fraction is influenced by the negative fact that samples were prepared on/above the level of size separation limits of Fe 2 O 3 solid particles.