The general procedure of the new method is described in this chapter. To process of using the recorded film to data from which the magnetic moment m of the particle can be determined is elaborated. We explain the algorithm used to locate the particles from the images and calculate their exact center.
From their trajectories the distance r(t) between two particles of interest can be calculated. And with a fit of r versus t the magnetic moment is determined.
For a more detailed description of what is mentioned in this chapter, including elaboration of the matlab scripts, see appendix A.2.
3.2.1 Locating the particles
With the camera used in the setup shown in figure 6 a small movie is recorded of particles during repulsive interaction. This film is converted into a sequence of images, preparing to analyse them with matlab. After conversion, the images are cropped to a region of interest including (preferably) two particles, for an example see figure 11(a).
The following procedure is done for every image (frame) of the movie:
1. A convolution is performed with a template image of a particle, see figure 11(b). This template image is a circular image of only one particle which is prepared before processing the image sequence.
2. In this convoluted image is searched for values above a certain threshold value defined by the user, symbolized by the black dots in the figure. This arbitrary threshold factor is defined by the user. A flexible way of selecting points is used, involving the mean and standard deviation of the image:
find (value > mean + std*threshold_factor).
3. All found points are divided in clusters, each cluster represents one single particle.
4. A weighted average is taken of all points in one cluster to determine its center, and therefore the center of the particle. The weightfactor is the actual value of that pixel. So pixels with a greater value, which are closer to the center, have a larger weightfactor. A white plus sign with an index number is plotted for every detected particle.
5. Calculate the distance, center to center, between the two particles in the image by Pythagoras. Or, if there are more particles in the image, calcu-late the distance between the two particles of interest.
The procedure is to repeat step 1 to 5 for every frame in the image sequence.
After this the distance between two particles of interest will be known at every timestep. So actually the data obtained is the time dependend distance r(t), which is exactly what is needed to determine the magnetic moment.
(a) (b)
Figure 11: Both image (a) and (b) are the same particles. Image (a) is the raw data as it is recorded with the camera. Image (b) is the convoluted image of image (a) and here the particles are solid circles, which makes it easier to determine their exact center. In image (b) black dots, points which meet the criterion, and white plus marker, which are the centers of the particles, are plotted aswell.
3.2.2 Fitting r(t)
Two different fitting methods are used to determine the magnetic moment of a particle couple from which r(t) is known.
First, the raw data is fitted with a fifth root curve, see figure 12(a), because of r ∝ √5
t relation, see equation 12 and 13. In the ideal case, that there are sufficient data points to fit the fifth root curve, no problems will be experienced with this fit. However when there happens to be a lack of data points, especially in the steep part, the accuracy of the fit will be lost.
A lack of data in the steep part, which is the start-up phase of the repulsion, occurs often. This might be due to several reasons e.g. when an image with very closely located particles is convoluted the particles are likely to overlap which makes it impossible to part them in clusters and determine their centers.
Another reason is that in the start-up phase of repulsion the particles separate the fastest so it is hard to capture this on film.
To prevent this loss of accuracy a linear fit of r5 versus t, see figure 12(b), is made. Now over the whole range the fit will be linear and because of this, no loss of accuracy will be experienced when there is a small loss of data points.
Now the magnetic moment can be determined with the slope of the fit, see equation 14. So by filling in the slope from the fit a, the dynamic viscosity of the fluid η and the radius of the particle R, the averaged magnetic moment of this particle couple at a certain strength of external field is found.
(a) A non linear fit of r versus t. (b) A linear fit of r5versus t.
Figure 12: Two different fitting methods are used to determine the magentic moment.
4 Results and discussion
As described in the previous section experiments were performed on the breaking of two particle clusters using several magnets to produce magnetic fields which induce attraction or repulsion between the particles. In this section the results from these experiments will be discussed:
• the averaged magnetic moment m of a particle couple was determined,
• magnetization curves are made,
• the magnetic moment variation in the particles is calculated
• and the total number of grains and their size is determined.
Only data from repulsive interaction between the particles is processed because it is more efficient. Particles split up instantly when the external field switches orientation and so a movie of just a few seconds can capture multiple couples splitting up. On the other hand when switching to a horizontal field, which induces attraction, the particles do not start attracting to each other immediatly.
They must be close enough to ‘feel’ each other.
Later on, in section 4.2, we will show that magnetic moments obtained from repulsive interaction are just as accurate as those obtained from attractive in-teraction. So we choose for the most efficient way of experimenting: analysing repulsion only.
4.1 Determining magnetic moment of particle couples
Using the setup shown in figure 6 a small movie is recorded during the magnetic field’s orientation switch from horizontal to vertical, inducing repulsion so the particles split up. Running the scripts locating the particle will supply us with r(t) and as mentioned before two different fit curves are used to obtain the magnetic moment of the particle couple of interest.
The same procedure has been followed but instead of inducing repulsion the external field is switched from a vertical to a horizontal orientation and the particles start attracting each other.
Results from these experiment are given in figure 13 and 14. The two fits, mathematically described in section 3.2.2, of a repulsive interaction are given.
The r ∝ √5
t behavior is clearly visible in the left graph. Of these same particles attraction has been analysed. The fits from this experiment are given in figure 14. Again the fifth root relation is visible.
The magnetic moment determined from these fits are given in table 1.
The magnetic moment determined from the attractive process are slightly higher than those determined from repulsion. However the values from attraction and repulsion, both from linear and non linear fitting still overlap taken in account the errors. Although all four methods are applicable determining the magnetic moment of a pair of particles, we’ll focus on the repulsive part. As stated before
(a) (b)
Figure 13: Fits of the experimental data recorded during repulsion. In figure (a) the distance r between two particles during repulsive interaction is plotted versus t. In figure (b) the relation is made linear, r5 has been plotted versus t.
(a) (b)
Figure 14: Fits of the experimental data recorded during attraction. In figure (a) the distance r between two particles during repulsive interaction is plotted versus t. In figure (b) the relation is made linear, r5 has been plotted versus t.
Linear fitting method (10−14 Am2)
Non linear fitting method (10−14 Am2)
Repulsion 1,90 ± 0,03 1,92 ± 0,03
Attraction 1,96 ± 0,04 1,96 ± 0,03
Table 1: Magnetic moment of one pair of particles determined during attraction and repulsion with a linear and a non linear fitting method.
analysing repulsion is more efficiently because the particles separate simultane-ously.