In section 4.2 the magnetization curves were compared to reference measure-ments to check their reliability. The Langevin curve (see equation 17) fitting allows us to determine the number of grains n, their magnetic moment µ aswell as the radius of a grain. The grains consist of magnetite, Fe3O4, and because they are smaller than the magnetic domain wall, which is 80 nm [10], they may be considered single domain areas. For this reason the volume of a grain is given as the ratio between the magnetic moment and the saturation magnetization:
Vgrain= µ Msat
. (20)
For magnetite the saturation magnetization Msat= 6,2 · 105J m−3 T−1 [11].
The summed weighted Langevin curve, see equation 19 , has been used to fit the magnetization curve of the PEG and MyOne particles. Advantages of this summed Langevin fitting is that it provides several properties of the grains inside the particles. With the fitting, the grain moment distribution is determined and with this the mean volume and radius of a grain can be calculated using equation 20.
The second useful variable which can be determine by fitting with the summed Langevin equation is the total number of grains which lay inside one particle n.
The volume of a grain together with the total amount of grains will provide the total volume occupied by the grains and with that the filling factor of magnetic content in a particle. These properties are given in table 2.
A typical grain size is about 6 to 12 nm, so this corresponds with calculated values for grains in the MyOne and PEG particles.
MyOne PEG radius r 6,3 nm 9,6 nm µ (10−18 Am2) 0,66 2,2
number n 6, 7 · 104 1, 1 · 104 Filling factor 27% 0,4%
Table 2: Grain properties of MyOne and PEG particles
5 Conclusion
Here, we have studied the analysis of couples of superparamagnetic particles.
We have been able to determine the magnetization curve of these. Two types of particles were investigated:
• Micromer PEG particles — 2,6 µm diameter,
• Dynabeads MyOne particles — 1,0 µm diameter.
By exploiting the magnetic dipole–dipole interaction of the superparamagnetic particles, especially repulsion, it is possible to determine the magnetic moment of one pair of particles. This magnetic moment depends on the applied external magnetic field and a magnetization curve, the curve of the magnetic moment versus the applied field, has been made for both types of particles.
The magnetization curves have been compared to those made with a VSM measurement. They corresponded almost perfectly and we conclude our new method is succesful for determination of the magnetic moment of two particles.
Compared to the VSM measurement the method described in this report has a great advantage. The averaged magnetic moment of a pair of particles has been determined. With a VSM measurement this is not possible. The magnetic moment is determined of a large number of particles.
Therefore, with a VSM measurement it is also impossible to obtain an estimation for the deviation or variation in the magnetic moment of these particles. And with our new method it is. By applying statistics on a sample of particle couples it is possible to calculate the standard deviation of the magnetic moment, and therefore the magnetic content, of a superparamagnetic particle.
Several properties of the grains, which are embedded in the particle, like grain size and magnetic moment, are determined with weighted Langevin curve fitting.
With these properties information like the total volume of the grains and the number of grains in a particle could be calculated.
In conclusion, the method we developed in this project has been proved succesful in producing a magnetization curve for magnetized superparamagnetic particles, which was the primairy goal. And previously impossible determinable properties like the number and size of grains inside one particle can now be calculated.
References
[1] E. Pollert, K. Kn´ıˇzek, M. Maryˇsco, P. Kaˇspar, S. Vasseur, E. Duguet, 2007, J. Magn. Magn. Mater. 316 (2), 122–125.
[2] J.A. Abels, F. Moreno–Herrero, T. van der Heijden, C. Dekker, N.T.
Dekker, 2005, Biophys. J. 88.
[3] G. Fonnum, C. Johansson, A. Molteberg, S. Mørup, 1997, J. Magn. Magn.
Mater. 5–14.
[4] M.W.J. Prins, M. Megens, 2007, Chapter in Encyclopedio of Materials:
Science and Technology. Elsevier, 1–6.
[5] P.A. Besse, G. Boero, M. Demierre, V. Pol, R. Popvic, 2002, Appl. Phys.
Lett. 4199.
[6] S.P. Mulvaney, R.L. Cole, M.D. Kniller, M. Malito, C.R. Tamanaha, J.C.
Rife, M.W. Stanton, L.J. Whitman, 2007, Biosens. Bioelectron, 191–200 [7] G. Fonnum, C. Johansson, A. Molteberg, S. Mørup, E. Aknes, 2005, J.
Magn. Magn. Mater. 41.
[8] H. Swagten, 2008, Course syllabus Magnetism & Magnetic Materials [9] J. Leach, H. Mushfique, S. Keen, R. Di Leonardo, G. Ruocco, J.M. Cooper,
M.J. Padgett, 2009, Phys. Rev. E 79, Comparison of Fax´en’s correction for a microsphere translating or rotating near a surface.
[10] R.F. Butler & S.K. Banerjee, 1975, Theoretical single-domain size range in magnetite.
[11] C.E. Housecroft & A.G. Sharpe, 2005, Inorganic chemistry 2nd Ed. Pear-son.
[12] G. Mihajlovic, K. Aledealat, P. Xiong, S von Moln´ar, M. Field, 2007, Appl.
Phys. Lett 91, Magnetic characterization of a single superparamagnetic bead by phasesensitive micro-Hall magnetometry.
A matlab scripts
A.1 The protocol
The protocol for tracking particles in an image is as following:
1. Crop the image to a region of interest only including the particles. This can be done by using
• cutregion.m, which simply cuts out the ROI of a sequence of images or
• cutregion_backwards.m, which does exactly the same but names the image in reverse order. This is useful when repulsive interaction is analysed,
2. Load the first image (read_image.m) and define a template particle (template_particle.m) used in the convolution,
3. Perform the convolution (convolution.m),
4. Search for values in the convoluted image im_conv above a certain thresh-old. This threshold depends on the standard deviation of the image and an user define factor, threshold_factor,
5. Part all found locations in clusters, each cluster is a particle, and calculate the center of each particle by taking the weighted mean of the correspond-ing cluster.
The centers then are stored and step 3 to 5 is repeated for every image in the sequence.
The output of the script (particle_tracking.m), which includes looping through step 2 to 5, will be the locations of the center (in pixels) of each particle in time and the number of particles found.
An option to run the script data_analysis.m at the end of
particle_tracking.m is provided. It will use the locations of the particles to determine the desired magnetic moment m. See subsection A.2 for more information about this script.
A.2 Data analysis
The script data_analysis can be run at the end of the particle tracking script to analyse the output of this. It takes all the positions of the particles (which is the output of particle_tracking) and calculates the distance r (in meters) between two particles taking in account the resolution.
A weighted linear fit of r5 versus the time t is made with lscov. Weightfactors are taken 1/σ2i, with σi the error in data points (the distance r, or r5depending on if the non linear or linear fitting method is used) i. The slope of the fit is used to determine the magnetic moment m. As explained in section 2.3.1 this actually is the average of the magnetic moments of the two particles.
To check the value of m a fifth root fit of r versus t is made aswell. The command nlinfit is used to fit the data on a custom equation.
The calculated data, like particle position, distances and calculated magnetic moments, is printed in a file which is saved in the on the disk, optionally together with the convoluted images.
B Data from non linear fitting
In this appendix the magnetization curves, Langevin fits and reference VSM fit from MyOne and PEG particles obtained from the non linear fitting method are given. For the magnetization curve of the PEG particles, see figure 19, and the magnetization curve of the MyOne particles is given in figure 20.
The graphs corresponding to the non linear fitting method are very similar to those corresponding to the linear fitting method (figures 19 and 20).
Figure 19: The magnetization curve of PEG particles. The Langevin fit of the data (black) aswell as the reference VSM measurement (purple and green) is plotted.
Figure 20: The magnetization curve of MyOne particles. The Langevin fit of the data (black) aswell as the reference VSM measurement (purple) is plotted.