1. Advisor: Dr. Abdalla Obeidat Co-Advisor: Prof. Nabil Ayoub
The effect of anisotropy in the Dimer Model on Ferrofluids in One-Dimension.
Dima Al-Safadi
Advisor: Dr. Abdalla Obeidat
Co-Advisor: Prof. Nabil Ayoub

2. Contents Introduction Theory Randomized easy axis Fixed easy axis
Special cases
Discussion and Conclusion

3. Introduction Magnetic anisotropy
In the absence of the applied magnetic field, the magnetic moments of the particles tends to align themselves to the direction which makes the magnetoststic energy minimum, this direction is called the easy direction, briefly, magnetic anisotropy shows how the magnetic properties depend on the direction of measuring the magnetization. There are several types of anisotropy and the most common one is the magnetocrystalline anisotropy.

4. Magnetocrystalline anisotropy (crystal anisotropy)
When single domain fine particles magnetized to saturation, the magnetization has an easy axis along which is prefer to stay. In this case, the total internal energy is minimum. Rotation of the magnetization vector away from an easy axis is possible only by applying an external magnetic field. Thus, the magnetic energy is direction-dependent, and this kind of energy is called magnetic anisotropy energy. It is also called magnetocrystalline anisotropy because it has the same symmetry as the crystal structure of the particle material.

5. one of the simplest expression of the magnetic anisotropy energy that is uniaxial in symmetry is

7. What is the ferrofluids?
Ferrofluid is a stable colloidal suspension consists of single-domain fine magnetic particles (ferromagnetic or ferrimagnetic particles), of diameter (30-150) in a carrier liquid (such as water or kerosene). To prevent the agglomeration and enhance the stability of the suspension, a nonmagnetic surfactant layer (such as polymers) covers particles. Recently, the publications concerned with ferrofluids have increased because these materials have a wide uses in industry, medicine and agriculture

8. Randomize easy axis on ferrofluid dimer model in one-dimension
We will study the effect of the magnetic anisotropy and the effect of the dipole-dipole interaction on the magnetic properties of a dilute ferrofluid consists of a single domain magnetic fine particles.
The easy axis of each particle ( )directed randomly and the magnetic moment ( ) is freely to rotate in three dimensions.

9. Dimer model
To simplify our calculations, we will use a simple theoretical model called (Dimer model) in this model, the uniaxial single domain magnetic fine particle interacts with only one other adjacent particle thus, our assembly consists of ( ) independent non-interacting systems move in one dimension. Since each system composed of two interacting particles, we call it a dimer.

10. For one system we can writex
For one system we can write

11. Theoretical back ground
The partition function for one system is
Where
and

12. Therefore To calculate This integral we assume the following: 1-The interparticle interaction energy is small compared with thermal agitation energy 2-The anisotropy energy is small compared with the thermal energy 3-The applied magnetic field is very small.

13. For system the total partition function is given by
The magnetization is defiand as
And the initial susceptibility is given by

14. Field parallel to the assembly

15. The following expression have been calculated for the partition function
Where

16. And

17. The magnetization M is given by
And the initial susceptibility is calculated to be

18. However, the general form of Curie-Weiss law is:
Therefore our result obeys Curie-Weiss law with
and

19. Magnetic field perpendicular to the assembly

21. Thank you