Immittance Model of Anodic Dissolution of Ferromagnetic Materials

Transition elements of d-group (Fe, Co, Ni) and f-group (Gd, Tb, Dy, Ho, Er, Tm), which are ferromagnetic elements, show a spontaneous magnetization. Under their anodic dissolution the magnetization of domains changes. The immitance of the simple model of the anodic dissolution of polycrystalline ferromagnetic materials in a constant magnetic field is presented. Moreover, the expressions describing the impedance and admittance of this model are developed. 

When alternating current flows through a ferromagnetic material with a magnetization different from zero, there must be appearance of Emf of electromagnetic self-induction E [1]:

E = - dΨ/dt     (1)

where Ψ is magnetic flux-linkage and t is time.

The magnetic flux-linkage can be determined from the following equation [2]:

 Ψ = -μ_o∫(H·M)dv/I     (2)

where I is the current in ferromagnetic;
H is intensity of the external magnetic field;
M is magnetization of the ferromagnetic;
μ_ois magnetic constant.

The integration is carried out over the whole volume of the ferromagnetic material v.

From equation (2) it can be seen that in constant external magnetic field, H = const, the magnetic flux-linkage is a function of current and magnetization only:

 Ψ = Ψ(I,M)     (3)

Then in the linear approximation, for the complete differential  Ψ, we obtain:

dΨ = (∂Ψ/∂I)_MdI+(∂Ψ/∂M)_IdM     (4)

After substituting expression (4) to equation (1) we get:

E = - dΨ/dt=(∂Ψ/∂I)_MdI/dt+(∂Ψ/∂M)_IdM/dt     (5)

By definition,(∂Ψ/∂I)_M is nothing else but inductance of ferromagnetic L[1]:

L = dΨ/dI     (6)

The ferromagnetic magnetization is the sum of magnetic moments of the atoms. Therefore, change in the magnetization of a ferromagnetic, associated with the transition of metal ions into electrolyte solution under anodic polarization equals to:

dM = kM_BN_Adn     (7) 

where M_B = eh/(4πm_e) is Bohr magneton [1];
kM_B is magnetic moment of the atom, which consists of k number of Bohr magnetons;
N_A is Avogadro number
n is mole number.

Material balance of metal ions in the vicinity of the electrode layer is:

dn/dt = -I/zF     (8)

where F is a Faraday constant.
z is a number of electrons during the ionization of ferromagnetic atom.
Taking into account the expressions (6), (7) and (8), the equation (5) takes the form:

E = -LdI/dt+kM_B(∂Ψ/∂M)_II/ze     (9)

When the system is perturbed by an harmonic current of small amplitude, one can use the Laplace transform, and then equation (9) in phasor variables [3] can be written as:

δE=-jωLδI+aδI     (10)   

Thus, for the immittance of the considered model of anodic dissolution of ferromagnetic materials the following expressions are obtained:
For the impedance:

Z = δE/δI=-jωL+a     (11)

For the admittance:

Y=δI/δE=(-jωL+a)^-1     (12)

An electrical equivalent to the expressions (11) and (12) is a series connection of an inductance, L and an active resistance, R = a (Figure1).

In the present work the immittance of the simple model of the anodic dissolution of polycrystalline ferromagnetic materials in a constant magnetic field is developed. The ferromagnetic materials that include transition elements of d (Fe, Co, Ni) and f(Gd, Tb, Dy, Ho, Er, Tm) groups show spontaneous magnetization. The expressions for the impedance and admittance of this model are obtained. The electrical equivalent of the impedance model is a series connection of inductance and active resistance.

 References

1. J. D. Jackson, Classical Electrodynamics 3rd ed., p. 791, Wiley, New York, (1998)
2. Encyclopaedic Dictionary of Physics, B.A. Vvedensky, Editor, p.187, Moscow, Soviet Encyclopaedia (1965).
3. D.C. Giancoli. Physics for Scientists and Engineers. 2nd Ed., p. 921, Prentice-Hall, Englewood Cliffs, N.J. (1989).