ARTICLE IN PRESS Physica B 403 (2008) 4053–4058
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Modeling of microwave heating of metallic powders V.D. Buchelnikov a,b, D.V. Louzguine-Luzgin c,, A.P. Anzulevich a, I.V. Bychkov a, N. Yoshikawa d, M. Sato e, A. Inoue c a
Chelyabinsk State University, Chelyabinsk 454021, Russian Federation Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-Ku, Sendai 980-8577, Japan d Graduate School of Environmental Studies, Tohoku University, Sendai 980-8579, Japan e National Institute for Fusion Science, 322-6 Oroshi, Toki, Gifu 509-5292, Japan b c
a r t i c l e in fo
Article history: Received 13 June 2008 Received in revised form 28 July 2008 Accepted 9 August 2008
As it is known from the experiment that bulk metallic samples reﬂect microwaves while powdered samples can absorb such a radiation and be heated efﬁciently. In the present paper we investigate theoretically the mechanisms of penetration of a layer of metallic powder by microwave radiation and microwave heating of such a system. & 2008 Elsevier B.V. All rights reserved.
Keywords: Microwave radiation Metal Heating
1. Introduction In materials science, microwave (MW) radiation has been traditionally applied to ceramics. It has been shown that MW energy could be used for processing full-scale ceramic products . As MW radiation causes internal heating of the material then lower temperatures and shorter times can be used compared to those applied at conventional heating. In some cases MW processing can reduce sintering time by a factor of 10 and minimize grain growth. Owing to so-called skin-effect bulk metals reﬂect MWs and can hardly be heated at room temperature. They can undergo only surface heating due to limited penetration of the MW radiation. However, recently MW heating has been successfully applied to powdered metals and fully sintered samples were obtained in 1999 in a multimode cavity [2,3]. Later MW heating of metallic powders in separated electric (E-) ﬁeld and magnetic (H-) ﬁeld was performed in a single-mode applicator [4,5]. The MW sintering of various metallic powders, steels and non-ferrous metallic alloys helped to produce sintered samples within tens of minutes at sintering temperature ranges between 1370 and 1570 K . Moreover, nanomaterials and some composite materials can also be produced by such a technique [7,8].
Corresponding author. Tel.: +81 22 215 2592; fax: +81 22 215 2381.
E-mail address: [email protected]
(D.V. Louzguine-Luzgin). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.08.004
The heating mechanism of metallic powders has not been clariﬁed fully yet. Here, for explanation of MW heating of metallic powders we propose the following theoretical model. We consider the metallic powder as some composite medium. This composite medium consists of the mixture of spherical metallic particles covered by thin dielectric oxide shell and gas (or vacuum). The effective permittivity, permeability and electric conductivity of the mixture correspond to the permittivity, permeability and electric conductivity of the effective medium. So in the present work, we theoretically studied using a model of conductive composite the possible heating mechanisms of metallic powders situated in a single mode 2.45 GHz MW applicator and provide some theoretical explanation of the MW heating behavior for iron powder .
2. Theoretical model For explanation of experimental results , let us consider a problem of electromagnetic wave transmission through a plate of composite conductive material and reﬂection from an ideal (fully reﬂective) massive conductive medium (Fig. 1). As was mentioned above, the composite consists from the mixture of spherical metallic particles covered by thin oxide dielectric shell and gas (or vacuum) which compose the effective medium. If the composite plate is absent then there is a standing wave only. In this case, we can ﬁnd the positions of electric (E-) and magnetic
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region 2 ε = εeff , μ = μeff
region 1 ε = 1, μ = 1
region 3 ε = 1, μ = 1
Here e0, h0 are the amplitudes of alternative electric and magnetic ﬁelds in the initial incident wave. In the present case, when the electric ﬁeld polarizes along y-axis and the z1 z2 2 magnetic ﬁeld polarizes along x-axis e0 ¼ h0. Tezp10z ;pk , Thp0 ;pk z1 z2 z1 z2 and Rep0 ;pk , Rhp0 ;pk are the transition and reﬂectance matrices for layer z1z2
region 4 reflector
z1 z2 1 z2 1 z2 1 z2 1 2 Tezp10z þ pk Mz12 Þ þ ðMz21 þ pk M z22 Þ , ;pk ¼ 2p0 ½p0 ðM 11 z z
1 z2 1 z2 1 z2 1 z2 þ pk M z12 Þ þ ðM z21 þ pk Mz22 Þ1 , Thp10 ;pk2 ¼ 2pk ½p0 ðM z11 2 Rezp10z ;pk ¼
Fig. 1. The scheme for the theoretical model. The regions 1 and 3 correspond to the gas (vacuum). Region 2 is the plate from composite conductive material and the region 4 is the ideal massive conductive medium (reﬂector). The electromagnetic wave falls on these regions from the left side.
(H-) ﬁelds maximums in the standing wave. Knowing these positions will allow us to place the composite plate either in maximum of E- or H-ﬁeld. When the composite plate exists, we divide our problem onto four parts. First region lies on the left side of the plate. Second region corresponds to the plate and third region is a distance between right side of the plate and fully reﬂecting massive medium (designated as a reﬂector, region 4). As it can be seen from the analysis of the problem in the region 1, the electromagnetic ﬁeld is a sum of four waves: the incident from the left wave, the two waves reﬂected from the left and the right side of the plate and the wave reﬂected from the reﬂector. In the region 2, we also have four waves: the transmitted wave from the region 1, the wave reﬂected from the right surface of the plate, the wave which is reﬂected from the reﬂector and transmitted into the plate from the right side, the wave reﬂected from the reﬂector, transmitted into the plate and reﬂected from the left side of the plate. And ﬁnally in the region 3, we have also four waves: the wave transient from the ﬁrst region and transmitted from the plate, the two waves reﬂected from the left and right sides of the plate and the wave reﬂected from the reﬂector. For the calculation of the electromagnetic ﬁelds in all regions we used the transition matrix method . According to this method, the transition matrix between two points z1 and z2 of any medium can be expressed as ! 1 z2 1 z2 M z12 M z11 z1 z2 ¼ M , (1) 1 z2 1 z2 M z21 M z22 where 1 z2 M z11 1 z2 M z12
1 z2 M z21
and for layer 0L, we have taken the transition matrix M0L ¼ M0h MhðhþbÞ MðhþbÞL . For region 2, the amplitudes of electric and magnetic ﬁelds are expressed as 0z zðhþbÞ E2 ðzÞ ¼ e0 ½Te0z p0 ;pk þ Tep0 ;pk Rep0 ;pk Lz zh 0L þ ðTeLz p0 ;pk þ Tep0 ;pk Rep0 ;pk ÞTep0 ;pk expðipÞ, 0z
H2 ðzÞ ¼ h0 ½Thp0 ;pk þ Thp0 ;pk Rhp0 ;pk Lz
þ ðThp0 ;pk þ Thp0 ;pk Rhp0 ;pk ÞThp0 ;pk . ðTe0z p0 ;pk Þ
ðTeLz p0 ;pk Þ
and third waves coefﬁcients Here for the ﬁrst pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eff =meff ; eeff, meff are the effective permittivity
are p0 ¼ 1, pk ¼
Þ and and permeability of the composite. For the second ðRepzðhþbÞ 0 ;pk pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eff =meff , pk ¼ 1. For ﬁfth fourth ðRezh p0 ;pk Þ waves we have p0 ¼ ðTe0L p0 ;pk Þ wave these coefﬁcients are p0 ¼ pk ¼ 1. Analogously the same coefﬁcients are used for H-ﬁeld in region 2. For layers 0z, z(h+b), Lz, zh (Fig. 1) in region 2 we used the following transition matrices: M0z ¼ M0h Mhz ; MzðhþbÞ ; MLz ¼ MLðhþbÞ MðhþbÞz
1 z2 M z22
þ Thp0 ;pk Thp0 ;pk .
H3 ðzÞ ¼ h0 ½Thp0 ;pk þ ðThp0 ;pk þ Thp0 ;pk Rhp0 ;pk ÞThp0 ;pk .
1 z2 M z11 ,
0z zðhþbÞ 0L E1 ðzÞ ¼ e0 ½Te0z þ TeLz p0 ;pk þ Tep0 ;pk Rep0 ;pk p0 ;pk Tep0 ;pk expðipÞ, 0z
¼ MLðhþbÞ MðhþbÞh Mhz ¼ MðhþbÞL MhðhþbÞ Mzh
Lz Lz zh 0L E3 ðzÞ ¼ e0 ½Te0z p0 ;pk þ ðTep0 ;pk þ Tep0 ;pk Rep0 ;pk ÞTep0 ;pk expðipÞ,
Using the transition matrix method we can ﬁnd the expressions for the amplitudes of electric and magnetic ﬁelds in each regions. The amplitudes of the electric and magnetic ﬁelds in region 1 are
M0z ; MzðhþbÞ ¼ Mzh MhðhþbÞ ; MLz
And ﬁnally for region 3, we obtain the following amplitudes of the electromagnetic wave:
o is the frequency of waves, c is the speed of light, e is the permittivity and m is the permeability of medium.
H1 ðzÞ ¼ h0 ½Thp0 ;pk þ Thp0 ;pk Rhp0 ;pk
1 z2 1 z2 1 z2 1 z2 p0 ðM z11 þ pk M z12 Þ ðMz21 þ pk M z22 Þ (3) z1 z2 z1 z2 z1 z2 z1 z2 , p0 ðM 11 þ pk M 12 Þ þ ðM21 þ pk M 22 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p0 ¼ 0 =m0 ; pk ¼ k =mk , e0, m0 are the permittivity and the permeability of the region in which the wave incident on the layer exists and ek, mk are the permittivity and the permeability of the region in which the transmitted wave exist (after the layer). In the region 1 for all layers we have p0 ¼ pk ¼ 1. For layers 0z, z(h+b), Lz (Fig. 1) we used the following transition matrices:
Rhp10 ;pk2 ¼
¼ MðhþbÞL MzðhþbÞ ; Mzh ¼ Mhz .
ho pﬃﬃﬃﬃﬃﬃ i ¼ cos mðz2 z1 Þ , c rﬃﬃﬃﬃ i m ho pﬃﬃﬃﬃﬃﬃ ¼ i mðz2 z1 Þ , cos c rﬃﬃﬃﬃ ho pﬃﬃﬃﬃﬃﬃ i ¼ i mðz2 z1 Þ ; cos m c
1 z2 1 z2 1 z2 1 z2 p0 ðMz11 þ pk M z12 Þ ðM z21 þ pk M z22 Þ z1 z2 z1 z2 z1 z2 z1 z2 , p0 ðM11 þ pk M 12 Þ þ ðM 21 þ pk M 22 Þ
In the third region, the coefﬁcients p0 and pk are equal to 1 for all waves and for layers 0z, Lz, zh (Fig. 1) we used the following transition matrices: M0z ¼ M0h MhðhþbÞ MðhþbÞz ; MLz ¼ MzL ; Mzh ¼ MzðhþbÞ MðhþbÞh ¼ MðhþbÞz MhðhþbÞ . It is known  that the volume heat density Qeh absorbed in the plate from the external electromagnetic wave in 1 s is a sum of electric Qe and magnetic Qh parts of heat and it is expressed as Q eh ¼ Q e þ Q h ¼
ð00 jEj2 þ m00 jHj2 Þ.
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Here, the squares of modulus of electric and magnetic ﬁeld intensities in the plate are determined by Eq. (4). From Eqs. (4) and (6) it follows that the absorbed heat and the modulus of amplitudes of electric and magnetic ﬁelds in the composite plate depends on the real and imaginary parts of dielectric permittivity and magnetic permeability. So, in case when the plate consists of the mixture of metallic powdered particles with dielectric shells (core–shell particles) and gas (vacuum) are situated in the effective medium it is necessary to know the effective values of dielectric permittivity, magnetic permeability and electric conductivity of such a composite medium. For calculations of the effective dielectric permittivity, magnetic permeability and electric conductivity we will use the effective medium approximation (EMA) model [11–15].
3. Effective medium approximation for core–shell composite Let us consider the core–shell particles which are randomly distributed in gas (for example, in air) or vacuum. According to EMA model, an average value of electric displacement of effective medium connect with an average value of electric ﬁeld strength as hDi ¼ eff hEi ¼ eff E0 ,
where eeff is the effective permittivity of the composite, E0 is the R external electric ﬁeld, /DS ¼ (1/V) VD dV, V is the volume of whole composite. It follows from Eq. (7) that for the calculation of the effective permittivity of the composite we need to know the expressions for electric ﬁeld strength in metallic core, dielectric shell and gas (vacuum). Firstly, we consider one type of spherical core–shell inclusion. If the radius of metallic core is R1, the external radius of dielectric shell is R2, then the width of the shell is R2R1. The core–shell particle is in the external electric ﬁeld E0. According to the electrostatic theory , expressions for electric potential inside and outside the core–shell particle can be presented as
j1 ¼ C 1 r cos y; roR1 , j2 ¼ ðC 2 r þ C 3 =r2 Þ cos y; R1 oroR2 , j3 ¼ ðE0 r þ C 4 =r2 Þ cos y; r4R2 ,
where the constants Ci are calculated from the standard boundary conditions
j1 jr¼R1 ¼ j2 jr¼R1 ; j2 jr¼R2 ¼ j3 jr¼R2 , qj qj2 qj2 qj3 1 1 ¼ 2 ; 2 ¼ 3 qr
Here, e1 is the effective dielectric permittivity of the metallic core, e2 is the dielectric permittivity of shell, e3 is the dielectric permittivity of gas (vacuum). After placing the boundary conditions (9) in Eq. (8) we ﬁnd the constants Ci. Final expressions for electric potentials are 92 3 z j1 ¼ E0 r; roR1 , 2a3 þ b2 " # 3z3 R31 j2 ¼ ð1 þ 22 ÞE0 r ð1 2 Þ 3 E0 r , 2a3 þ b2 r R1 oroR2 ,
j3 ¼ E0 r
a3 b2 R32 E0 r; 2a3 þ b2 r 3
z ¼ ðR2 =R1 Þ3 ¼ ð1 þ lÞ3 ; l ¼ ðR2 R1 Þ=R1 , a ¼ ðz 1Þ1 þ ð2z þ 1Þ2 ; b ¼ ð2 þ zÞ1 þ 2ðz 1Þ2 .
The electric ﬁeld strength and the electric potential connect by help of equation E ¼ rj. Substitution of expressions (10) in this equation gives us the following results: 92 3 z E0 ; roR1 , 2a3 þ b2 " ! 3z3 R3 E2 ¼ 1 þ 22 ð1 2 Þ 31 E0 2a3 þ b2 r # 3 R þ3ð1 2 Þ 51 rðE0 rÞ ; R1 oroR2 , r E1 ¼
E3 ¼ E0 þ
a3 b2 R32 a3 b2 R32 E0 3 rðE0 rÞ; 3 2a3 þ b2 r 2a3 þ b2 r 5
In EMA, we deal with a mixture of two types of spherical particles which are randomly distributed in the effective medium. As the second type of particles we will consider spherical inclusions of gas (vacuum). It is considered that the permittivity of such composite is equal to the permittivity of the effective medium. The electric ﬁeld inside the second type of particles is determined as  Eg ¼
3g E . g þ 2eff 0
Here, eg is the dielectric permittivity of gas or vacuum. The electric ﬁelds inside the ﬁrst type particles are expressed by the initial two formulas (12) in which e3 ¼ eeff (because of the bough kinds of particles are distributed in the effective medium). After substitution of electric ﬁelds (12) and (13) in Eq. (7) and their integration we ﬁnd the ﬁnal equation for calculating the effective dielectric permittivity pz
2 ½31 þ ðz 1Þð1 þ 22 Þ eff ½32 þ ðz 1Þð1 þ 22 Þ 2aeff þ b2 g eff þ ð1 pzÞ ¼ 0, g þ 2eff
where p is the volume fraction of a metal in the effective medium. Note that the problem of calculation of the core–shell composite effective dielectric permittivity was studied in a number of papers. We think that the ﬁrst classical paper, in which the spherical core–shell system was considered, is the paper of Hashin and Strikman . In this paper, the effective magnetic permeability of the composite medium consisting of core–shell spherical particles was calculated at condition that the particles do not disturb the external magnetic ﬁeld. The classical Hashin’s and Shtrikman’s result  for effective dielectric permittivity follows from the conditions E3 ¼ E0 and eeff ¼ e3 into Eq. (12). It is possible when ae3 ¼ be2. In this case, we obtain the following expression for effective permittivity:
HS eff ¼ 2
ðz þ 2Þ1 þ 2ðz 1Þ2 . ðz 1Þ1 þ ð2z þ 1Þ2
In Hashin and Shtrikman’s model, the spherical core–shell particles have different size so they occupy whole volume without air/gas or vacuum regions. Recently Peng et al.  studied the effective permittivity of a composite matter with core–shell inclusions by self-consistent method. The spherical dielectric core–shell particles were randomly distributed in a dielectric matrix. The formula for the effective permittivity for such composite was obtained. But in this work , the metallic core–dielectric shell system was not considered. The authors made conclusion that the permittivity of shell has very little effect on the effective permittivity. In the same year Rybakov and Semenov  and 1 year later Rybakov et al.  using EMA calculated the effective permittivity of the powdered sample in which the spherical particles have a metallic
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core and dielectric shell. In last year, Li et al.  also considered core–shell system but their results for effective permeability do not agree with [17–19] and ours. In our opinion, the reason for disagreement is the separate consideration of core and shell in Ref. , whereas both core and shell compose one system. Note that according to EMA we can use the effective magnetic permeability and effective electric conductivity in the same formula (14) for the considered composite. In this case instead of e1, e2, and e3 we need to use the permeability or conductivity of the metallic core m1, s1, the dielectric shell m2, s2, and the gas m3, s3.
4. Heat conduction equation For calculation of temperature distribution in the composite plate we used the heat conduction equation which can be written as qðC rTÞ q qT ¼ þ Q, (16) k qt qz qz where C is the heat capacity, r is the density, k is the heat conductivity and Q ¼ Q eh Q c Q r
is the whole heat density absorbed into plate and emitted from the plate. First heat density term is determined by formula (6). Second and third heat density terms describe the power loss per unite volume due to convection and radiation. They are expressed as (see, for example, Ref. ) Q c ¼ aðT T 0 Þ;
Q r ¼ b1 sSB ðT 4 T 40 Þ,
where a is the heat-transfer coefﬁcient, b1 is the effective emissivity of powder, sSB is the Stefan–Boltzman constant, T0 is the surrounding temperature.
5. Numerical results and discussion Here we ﬁrst consider the dependencies of the effective permittivity of powder and the electric and magnetic ﬁelds inside the powder from different parameters which are contained in Eqs. (5), (6) and (14). We perform such simulations for iron powder. In this case, the core of particles is pure iron, the shell is the iron oxide Fe2O3. The permittivity of oxide can be approximately written as e2 ¼ e0 2+ie00 2C10+i0.1 . As we do not know the values of magnetic permeability of iron and iron oxide for MW frequencies, the effective magnetic permeability was taken from paper  m ¼ m0 +im00 ¼ 2+i0.6. This value can also be well estimated in our model from Eq. (14). For example, if we take the following values of the permeability for iron, iron oxide and air m1 ¼ 5+i1, m2 ¼ 2+i0.5 and m3 ¼ 1 then with relative thickness of the shell equals l ¼ 0.002 and the volume fraction of iron in the powder equals p ¼ 0.375, the value of the effective permeability of the sample meff ¼ 2+i0.2, i.e. being of the same order of magnitude as in the work . Let us take the frequency of the external alternative electromagnetic ﬁeld to be o/2p ¼ 2.45 GHz and the permittivity of gas (vacuum) eg to be equal 1. The permittivity of metallic core can be written as 
1 ¼ 1
4ps 4ps þi , gðo2 =g2 þ 1Þ oðo2 =g2 þ 1Þ
where s is the static conductivity, g is the collision frequency of electrons. For well conducting metals the collision frequency of
electrons is 10131014 s1 (for Fe g ¼ 4.17 1014 s1 at room temperature ). In this case, we can neglect the term (o/g)2 in Eq. (20) at frequency o/2p ¼ 2.45 GHz. The conductivity of the iron core depends on the temperature and can be expressed as 
s ¼ s0 =½1 þ 6:51 103 ðT T 0 Þ,
where s0 is the conductivity of the iron core at the room temperature T0 ¼ 295 K; s0 ¼ 8.7 1016 s1. Because the ﬁrst two terms in Eq. (20) is smaller then third one we can also neglect them in our calculations. In the experiment , the temperature changes in a wide range and we need take into account the temperature dependences of main physical values which are contained in the heat conduction Eq. (17). These dependences were taken from Eq.  for pure iron and in the considered temperature interval they are
r ¼ r0 ½1 þ 4:35 105 ðT T 0 Þ, C ¼ 4:25 106 1:61 103 T þ 7:1 T 2 ðerg g1 K1 Þ,
k ¼ 1:175 107 1:36 104 T þ 5:12 T 2 ðerg s1 cm1 K1 Þ, (22) 3
where r0 is the iron powder density ; r0 ¼ 2.926 g cm . At such powder density the volume fraction of iron particles together with oxide shells is approximately p2 ¼ pz ¼ 0.375. We used the following values for the remainder parameters in the conduction Eqs. (16)–(18): b1 ¼ 0.5, a ¼ 3 104 erg s1 cm2 K1 [21,25]. The characteristic sizes in the theoretical model (Fig. 1) were taken as: L ¼ 12.245 cm, b ¼ 1 cm, h ¼ 5.6225 cm. The thickness of plate b is approximately the same as the size of iron powder sample in the experiment . Since the input power in the experiment was 200 W, the value of amplitude of magnetic ﬁeld was h0E0.25 Oe . But the best agreement between the theoretical and experimental results at selected parameters have been received for h0 ¼ 0.5 Oe. Perhaps this difference can be explained by neglecting the heat which is transmitted to iron powder from the crucible. The average size of particles in 200 mesh iron powder is 74 mm . The thickness of oxide shell for iron powder particles is 10–100 nm. So, we will take the relative thickness of the oxide shell approximately at l ¼ 2 103. Fig. 2 presents the dependence of the effective permittivity from the volume fraction of core–shell particles p2. It is seen from Fig. 2 that the effective permittivity at small volume fraction of the core–shell particles is close to the effective permittivity of gas (vacuum). After a percolation threshold at p ¼ 13 (this value is estimated when eg ¼ 0) fast increase of the effective permittivity is observed. In this region, the dependence of the effective permittivity from the volume fraction of core–shell particles is almost linear. It can be shown by the numerical simulations that the value of the effective permittivity also depends strongly on the thickness of dielectric shell. Fig. 3 presents the distribution of square of modulus of electric and magnetic ﬁelds inside the plate of the core–shell composite. The plate is placed to the maximum of magnetic ﬁeld and the minimum of electric ﬁeld in the standing wave according to Fig. 1 (curves 1 in Fig. 3 present the amplitudes of electric and magnetic ﬁelds without the composite plate). It is clearly seen that the electric and magnetic ﬁelds can penetrate almost fully into the plate at large enough relative thicknesses of the shell of spherical particles (curves 2 and 3). At small shell thicknesses the magnetic and electric ﬁelds do not penetrate into the plate (curves 4). It is seen that the values of the electric ﬁeld in the plate are smaller than those of magnetic ﬁeld. Fig. 4 presents the numerical modeling of time dependence of iron powder temperature with using Eq. (16). It is seen that the experimental and theoretical results are in a qualitatively good
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Fig. 2. The dependence of real (a) and imaginary (b) parts of effective permittivity of powdered sample from the volume fraction of core–shell particles.
6.2 z, cm
6.2 z, cm
Fig. 3. The distribution of square modulus of magnetic (a) and electric (b) ﬁelds inside the plate from core–shell powder for different thickness of shell l: (1) the ﬁelds without plate; (2) l ¼ 0.02; (3) l ¼ 0.002; (4) l ¼ 0.0002. The thickness of plate is 1 cm, it sides shown by arrows.
agreement. Note that the iron powder in the maximum of magnetic ﬁeld is heated mainly by magnetic losses. In our simple theoretical model, we cannot simulate the heating of composite in the maximum of electric ﬁeld. It follows from the fact that in case when we place the composite plate into the maximum of electric ﬁeld the redistribution of electric and magnetic ﬁelds take place. This redistribution occurs because the plate in our problem has inﬁnite sizes in two dimensions. In real experiment , the powder sample has the ﬁnite size and the redistribution of electric and magnetic ﬁeld is smaller. Upon the experiment in the maximum of electric ﬁeld the iron powder was almost not heated. This fact can be explained in our model by the fact that the amplitude of electric ﬁeld in the composite plate is smaller than the amplitude of magnetic ﬁeld (Fig. 2). Note that the difference between the theoretical and experimental results at short time (Fig. 3 at to200 s) is caused by the fact that in the experiment the temperature was measured by the
optical pyrometer  and the temperature can be measured only approximately from 650 K. As well as other kinds of electromagnetic waves MWs are reﬂected from the surface of the bulk sample because the alternating magnetic ﬁled of the incident wave produces eddy currents in the skin layer of the bulk sample which in their turn generate alternating magnetic ﬁled around the surface which B vector is opposite to that of the incident wave and owing to the interference process an incident wave propagating inside the sample attenuates, while the opposite (‘‘reﬂected’’) wave created by the alternating magnetic ﬁled produced by eddy currents becomes almost as strong as incident one with the deduction of ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the eddy current losses in the skin layer d ¼ c= 2pmso of the bulk sample. Thus, an electromagnetic wave is ‘‘reﬂected’’ from the surface of bulk sample. In case of powder samples, eddy currents can be generated only inside the conductive cores of the metallic particles which
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This work was supported by Grant-in-Aid ‘‘Priority Area on Science and Technology of Microwave-Induced, thermally Non-Equilibrium Reaction Field’’ of Ministry of Education, Sports, Culture, Science and Technology, Japan.
   
400 t, s
     
Fig. 4. The dependence of temperature from the time for iron powder. The solid line is the modeling results; the dark square is the experimental ones .
  
volume fraction (and thus surface fraction as well) can be as low as 0.4–0.5. At the same time, the maximum radius vector and the area of the eddy current inside the particles, which is limited by a particle size, is also smaller than the area of the eddy currents in bulk samples which will deteriorate the ability of the surface to reﬂect MWs.
     
6. Conclusion The experimental results obtained and the theoretical evaluation both indicate that powdered metallic samples can be penetrated by MW radiation. The results of computer simulation indicate that the theory provides good ﬁtting of the experimental curves.
  
  
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