INCLUSION OF ELECTROSTATIC FORCES TO ASSESSMENT OF RATE OF MAGNETIC FORCES IMPACT TO IRON NANOPARTICLE AGGREGATION Dana ROSICKA a, Jan SEMBERA a a Technical University of Liberec, Studentska 2, 461 17 Liberec, Czech Republic, dana.rosicka@tul.cz Abstract Iron nanoparticles used for remediation of soil and groundwater aggregate. Reasons of the aggregation are adhesive forces which apply when the particles get close to each other. In the case of iron particles, the magnetic attractive forces play the main role in aggregation; however the magnetic forces have long range. We assessed the range of main influence of magnetic forces on aggregation and called it limit distance. This value was extended by the influence of electrostatic forces caused by surface charge around the particles. The electrostatic forces cause repulsion of particles and decrease the limit distance. Keywords: iron nanoparticles, magnetic force, electrostatic force, limit distance 1. INTRODUCTION Iron nanoparticles which composed from zero-valent iron and its oxides (nzvi) are produced by the company TODA Kogyo Corp. and are used for decontamination of groundwater and soil, especially for decontamination of organic pollutants such as halogenated hydrocarbons [1.]. During a remedial intervention, transport of the iron nanoparticles is slowed down due to rapid aggregation of them. Since the particles are iron, the aggregation is caused mainly by the long range attractive magnetic forces [2.-6.]. In our previous work [7.], we established the range of the forces by the use of value limit distance which represents the range around the aggregating particle in which the magnetic force overweighs other forces and causes the aggregation. For simplicity, the derivation of the value has been done as comparison of gravitational and magnetic force. However, a surface charge is established on the surface of the particles which causes repulsive electrostatic forces. The surface charge is dependent on ph value of water [8.]. Zerovalent iron provides alkaline reaction in water and that induces mainly negative value of the ζ potential of nzvi [9.]. As a consequence, all the particles have the same polarity of the charge and repulse each other. That is why the repulsive electrostatic forces have been added into the model of limit distance and the influence of the extension of the model has been determined. 2. METHODS AND MODELS 2.1 Magnetic properties of iron nanoparticles Every nanoparticle has a nonzero vector of magnetization. For TODA iron nanoparticles, the saturation magnetization and average diameter was measured [5.]: saturation magnetization, average diameter. For simplicity we suppose the same magnitude of vector of polarization for all nanoparticles. Our model of magnetic field around the iron nanoparticle is based on the model of magnetic field around a magnet. [10.] The electromagnetic potential in the point near a permanent magnet of volume is equal to ( ) (1) where is the vector of magnetic polarization at the point, the vector is the difference between the source of magnetic field and the point, is the length of. Intensity of the magnetic field can be subsequently computed as
( ) ( ( )) (2) Finally, the magnetic force between the source of the intensity of magnetic field and a permanent magnet of volume with the vector of polarization at the ( ) ( ) ( ) point is equal to and magnetization vector in the direction : (3) In prior work [9.], we derived scalar potential of the magnetic field around one homogeneous spherical iron nanoparticle with radius located at the point for spherical nanoparticle with radius ( ) ( ( )) ( ) ( ) (4) where is the radius of the nanoparticle and are the coordinates of the point. From equations (2) and (3), the analytical computation of magnetic force between two iron nanoparticles can be obtained. Since the nanoparticles aggregate, the magnetic force between aggregates was derived. One aggregate can be composed of millions of nanoparticles. It is time-consuming and difficult to analytically compute these forces. As a consequence, the forces were computed numerically, either as a sum of magnetic forces between every nanoparticle in one aggregate with every nanoparticle in second aggregate or as one magnetic force between two averaged aggregates. Averaged aggregate is a big homogeneous particle with direction of polarization computed as a vector sum of vectors of polarization of all nanoparticles in the aggregate divided by number of nanoparticles in the aggregate : (5) 2.2 Electrostatic properties of iron nanoparticles Electrostatic forces among iron particles are given by the surface charge. The surface charge was determined by ζ potential measured by the equipment Malvern ZetaSizer. ζ potential depends on ph of water so the dependence of ζ potential on ph of the solution for TODA nanoiron particles was measured (Fig. 1). A point of zero charge for the TODA particles is between ph 6 and 7. Fig. 1 Dependence of ζ potential of nzvi on ph measured with Malvern ZetaSizer. The surface charge density of particle is related to the potential as follows [11.]
( ) (6) When the potential is low, the last equation is approximately (7) where (8) is molar gas constant, is absolute temperature, is dielectric constant of water, is molar concentration of electrolyte, is charge number, is Faraday's constant, and is ionic strength. We calculated the limit distance for the surface charge which corresponds to value of ζ potential and for the surface charge which corresponds to value of ζ potential. These are two extremes. For the value surface charge of particles is close to the iso-electric point, electrostatic forces have small influence, and the particles aggregate. For the value, surface charge causes stabilization of aggregating particles. 2.3 Configuration of nanoparticles in aggregates On the basis of previous research [7.] we choose the unstructured model of aggregate. Unstructured model of aggregate ( ) composed of nanoparticles with its centre at position is a set of nanoparticles so that all of the nanoparticles are balls of constant radius and constant saturation magnetization. Their centres ( ) are uniformly randomly distributed inside the ball with centre in and radius, and the directions of saturation magnetization vectors are uniformly randomly distributed in the unit sphere. The radius of nanoparticles is set to magnetization. and all nanoparticles in our model have the same size of 2.4 Limit distance The influence of magnetic forces was assessed by one number - limit distance. This dimension expresses the range of the magnetic forces among particles. The definition of this quantity follows: up to this distance from the centre of an aggregate, the attractive magnetic forces cause the aggregation of the aggregate and a particle placed in the range. So in range larger than the limit distance, other forces outweigh the magnetic forces. In our previous work, the limit distance was defined the following way: the limit distance was the distance of the point in which the gravitation and magnetic forces effecting on the aggregate are equal (Fig. 2). The magnetic force decreases with biquadrate so the limit distance is estimated by the eq. (9.) where is magnetic force, their centers aggregates. is the gravitation force, and distance of is equal to sum of radii of both Fig. 2 Illustration of comparison of attractive magnetic force and counteracting gravitation force. In the limit distance, the two forces are balanced. (9)
3. RESULTS AND DISCUSSION 3.1 Model of limit distance extended by electrostatic forces The extension was done by addition of Coulomb s law (10) into the balance of (10) forces. The electrostatic force is computed for two interacting particles with diameters and with surface charges. represents distance between the two interacting particles as they would be two point charges. (11) Since the magnetic force decreases with biquadrate and the Coulomb s force with square, the balance of forces has this form (11). Hence we obtain the extended formula for limit distance (12). (12) The magnetic force was obtained from the formulas (3), (2), and (4). For the size of the equation, we do not show the formula here. is sum of radii of both aggregates. 3.2 Comparison of limit distances with and without influence of electrostatic forces The limit distance was computed for the values of surface charge of all particles and. The computation program was set for unstructured model of aggregates with the size of magnetization. For larger accuracy, statistical computation using Monte Carlo method was done. The computed limit distances are shown in Fig. 3 and Fig. 4 where they are compared with the old model of limit distance (10). In the graphs, the limit distance is not expressed in total values but as ratio of limit distance and distance of centers of interacting particles attached to each other. This represent how many times the distance between the centers of particles had to be increased not to aggregate due to the magnetic forces. The comparison was done for one nanoparticle interacting with aggregate of different sizes compound of nanoparticles. Fig. 3 Comparison of limit distances with and without consideration of influence of electrostatic forces with surface charge of particles.
It is visible that in the case of higher ζ potential, the limit distance of magnetic forces is decreased. Hence it is important to include the electrostatic forces into the determination of the rate of influence of magnetic forces on the rate of aggregation of particles. Fig. 4 Comparison of limit distances with and without consideration of influence of electrostatic forces with surface charge of particles. 4. CONCLUSION Range of attractive magnetic forces which influence rate of aggregation of iron particles is studied. The range is estimated by value limit distance in which attractive magnetic forces are equal to repulsive forces and the magnetic forces do not cause the aggregation. In this paper, the limit distance was extended by the repulsive electrostatic forces which have also long range. The importance of this extension was assessed for different surface charge of particles. The extension cannot be generally neglected since the repulsive electrostatic forces limit the attractive magnetic forces in some cases. ACKNOWLEDGEMENTS This work was supported by Grant Agency of the Czech Republic, project: Non-standard application of physical fields analogy, modelling, verification and simulation no. 102/08/H081 and by the Ministry of Education of the Czech Republic within the project no. 7822 of the Technical University in Liberec. LITERATURE [1.] [2.] [3.] Zhang W-x: Nanoscale Iron Particles for Environmental Remediation: an Overview, J. Nanopart. Res. 5, 3-4 (2003) Li L., Fan M., Brown R. C., Van Leeuwen J. (H.), Wang J., Wang W., Song Y., Zhang P.hang: Synthesis, Properties, and Environmental Applications of Nanoscale Iron-Based Materials: A Review, Crit. Rev. Env. Sci. Technol. 36, 405-431 (2006) Horak D, Petrovsky E, Kapicka A, Frederichs T: Synthesis and Characterization of Magnetic Poly(Glycidyl Methacrylate) Microspheres, Journal of Magnetism and Magnetic Materials 500-506 (2007)
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