Intrinsic permeability of composites filled with nickel powders

The microwave permeability of nickel (Ni) is reconstructed from constitutive parameter measurements of paraffin-binded composites filled with flakes or spheres of carbonyl nickel. The metal permeability is assumed to be equal to the intrinsic permeability of inclusions. The mean diameter of spheres is about 12μm, the mean diameter of flakes obtained from spheres by ball milling is about 50μm, the thickness is about 2μm. The shape and size are determined by microscopy and by laser particle sizer Analysette 22. The measurements of composite permittivity ε
_mix
and permeability μ
_mix
are performed with MS2028 VNA applying the reflection-transmission coaxial-line technique within 0.05-20GHz frequency band. The calculations are performed for isotropic mixtures filled with spheres and for plane-isotropic mixtures filled with flakes. The assumption of plane isotropy for flakefilled samples is reasonable as for the pressed washer sample the thickness (~0.8mm) is much smaller than the diameter (7mm). The effects of filling factor, particle shape and size on permittivity and permeability are analyzed. The reconstruction of inclusion permeability μ
_incl
is based on treatment of the measured dependence of μ
_mix
and μ
_mix
on frequency f and volume fraction of inclusions p.The procedure is based on generalized Sihvola mixing model [1] simplified for a plane-isotropic composite. The model is valid for wide range of susceptibility contrast and accounts for inclusion shape and geometric spectral linewidth [2] described by parameter a in equation (Eq.1). The dependence of composite susceptibility X
₃₂
on frequency is defined by the corresponding dependence of inclusion susceptibility X
₁₂
. The susceptibilities X
₁₂
and X
₃₂
are normalized by the corresponding constitutive parameter of a binder: permittivity ε
₂
or permeability ε
₂
of polymer. Equation (Eq.1) relates susceptibility of a composite X
₃₂
to the filling factor p, inclusion susceptibility X
₁₂
, inclusion depolarization factor N and mixture structure characterized by parameter a. X
_m32
/X
_m12
= p(1+aX
_m32
)/[1+aX
_m32
+ N
_Xm12
(1-p)] (1) Indexes 1, 2 and 3 relate to inclusion, binder and mixture. The inclusions are assumed to be identical spheres or identical oblate ellipsoids of rotation; their shape defines the depolarization factor N [3]. The similar equation (Eq.2) describes composite electric susceptibility Xe
₃₂
. As metal conductivity is about 10
⁸
Ohm×m,thus X
_e12
> >{e32} at practicable fillings, and Eq.1 may be simplified as shown below. X
_e32
= p
_e12
(1+X
_e32
) / [1+ X
_e32
+ N
_chie12
(1-p)][p/N (1-p)-ap] (2) The difference in electric and magnetic susceptibility of metals ε
_chie
12/c
_him12
>> 1 makes it possible to estimate intrinsic permeability of metals at microwaves, where the measurements with bulk metals are impossible because of skinning. Moreover the mixing model (Eq.1) makes it possible to estimate from the measured data the spectral linewidth a as well. It is easy to show that for metal inclusions (X
_ε12
→α∝) the inclusion depolarization factor N and percolation threshold p
_c
are related to linewidth parameter a as Eq.3 a = N(1-pc/p
_c
(3) The parameters N and p
_c
are readily determined experimentally and are used in the generalized Maxell Garnett (Odelevskiy) equation to relate a composite susceptibility to that one of inclusion [4]. The measured at 50MHz quasistatic permittivity and permeability ε
_mix
(p) and μ
_mix
(p) data are shown in Fig.1 by circles and squares correspondingly. As the imaginary permeability μ
_mix
at 50MHz is comparable to real part μ'
_mix
, the value μ"
_mix
is presented by the absolute value of μ
_mix
. In the vicinity of percolation threshold p
_c
the samples become electrically conductive and the measured permittivity has poor reprodusability and jumps from high positive to negative values. The effect of negative permittivity is known [5] but we relate it to inductive resistance of long clusters similar to that of wire sections [6]. The dashed lines in Fig.1 present the permittivity curves calculated from Eq.2 with N and p
_c
as fitted parameters. The fitted values of depolarization factor N
_sph
=0.149 and N
_flk
=0.04 agree with microscopic measurements. The percolation threshold values are correspondingly p
_c-sphere
=0.56, and p
_c-flake
=0.19; thus Eq.3 gives a
_sphere
=0.111 and a
_flake
=0.167. The measured frequency dependence of microwave permeability for composites filled with flakes and spheres (the data in Fig.2 are selected for composites with approximately the same ε
_mix
- values) shows that the absorption peak shape and frequency strongly depend on inclusion shape and on thickness because of skinning. The absorption frequency is much lower than that of composites with carbonyl iron [1] and the permeability values measured at 50MHz are far from static ones. The reconstructed inclusion permeability at this frequency is also far from the static one as the loss tangent for spheres and flakes is close to unity, while the permeability real value is about 5 for both inclusion types; the value is about 2 times lower than the reported data [7]; the difference may be attributed to skinning and structural defects introduced in flakes while ball milling. The measured microwave constitutive parameters of composites show that Ni flakes and spheres both are promising fillers for interference suppressors and UHF-band absorbers similar to sendust powders [4].