RULE OF MIXTURE FOR COMPOSITE THERMOELECTRICS

Transcription

1 TH 19 TH INTRNTIONL CONFRNC ON COMPOIT MTRIL RUL OF MIXTUR FOR COMPOIT THRMOLCTRIC Y. Lu 1 *, K. agara 1, L. Hao 1, H. Yoshida 2, Z.W. Ji 3, F.. Pan 4 1 Graduate hool & Faulty of ngineering, Chiba University, 1-33, Yayoi-ho, Inage-ku, Chiba, , Japan, 2 Chiba Industrial Tehnology Researh Institute, 889, Kasori-ho, Wakaba-ku, Chiba, , Japan, holl of Physis, handong University, Jinan, 251, China, 4 College of Materials iene and ngineering, Chongqing University, Chongqing, 444, China * Corresponding author (luyun@faulty.hiba-u.jp) Keywords: omposite thermoeletris, series model, parallel model, random model, rule of mixture, effetive medium theory 1 Introdution Generally, thermoeletri performane is evaluated by the figure of merit Z ( = 2 ρ -1 κ -1 ) or the dimensionless figure of merit, ZT, where is the eebek oeffiient, ρ is the eletrial resistivity, κ is the thermal ondutivity and T is the absolute temperature. For onveniene, the power fator P ( = 2 ρ -1 ) is also used as a riterion to evaluate thermoeletri performane. Thus, to improve thermoeletri performane, the eebek oeffiient should be inreased, meanwhile, the eletrial resistivity ρ and the thermal ondutivity κ should be dereased. In reent years, the omposite method has been frequently used to enhane the performane of thermoeletris and thermoeletri devies. The improvement of thermoeletri performane by adding metal powder, suh as u and g to oxide thermoeletris, has been tried [1-5]. In these works, the eletrial resistivity was dereased by adding metal powder. Indeed, the eebek oeffiient showed an inrease trend in some works [1,2,4]. It therefore lad to enhane the power fator. In our previous work, it was revealed that the power fator had the biggest value when the amount of adding Cu into Cu/TiO 2-x omposite was 12%, at whih the transition from semiondutor to metal appeared [6]. On the other hand, the rule of mixture (ROMs) and effetive medium theory (GT) have been ommonly used to evaluate and disuss the eletrial ondutivity/resistivity and the thermal ondutivity of laminated/sandwih-strutured omposites and partile omposites [7-1]. lso, the effetiveness and usefulness of these theories have been onfirmed by a tremendous amount of investigation. Therefore, the ROMs have been applied to the eletrial ondutivity/resistivity and the thermal ondutivity of sandwih-strutured omposite thermoeletris [11-14]. We have used the GM to evaluate the eletrial resistivity and the thermal ondutivity of Cu/TiO 2-x partile omposites [6]. Further, the ROMs for the eebek oeffiient of the sandwih-strutured metal/semiondutor/metal omposites were proposed [12-15]. In those studies, it was revealed that the power fator an be enhaned by sandwih strutures, a series struture of metal/semiondutor/metal. The ROMs are simple and pratiable to explain thermoeletri properties of the series omposite thermoeletris. However, the ROMs were only given as a funtion of the thikness of the sandwih-strutured omposites. esides, no general ROMs for the eebek oeffiient of the laminated/sandwih and the partile omposite thermoeletris have been given. In the present work, the ROMs and GT were applied to the eletrial resistivity and thermal ondutivity of omposite thermoeletris with the series/parallel and powder-distributed strutures respetively. The general ROMs for the eebek oeffiient of the series and parallel omposite models were dedued. We also tried to apply the GT to disuss the eebek oeffiient of the partile omposite thermoeletris with random distribution. 2 ROMs for laminated omposites 2.1 ROMs for the eletrial resistivity and thermal ondutivity The series and parallel models for laminated omposites are shown in Fig.1. It was assumed that the sattering of the arriers and phonons, and reations never our at the boundaries between

2 material and. In Fig.1, σ ( = 1/ρ ) [Ω -1 m -1 ], κ [Wm -1 K -1 ] and [VK -1 ] denote the eletrial ondutivity, the thermal ondutivity and the eebek oeffiient, respetively. ubsript,, orrespond to the material, and the omposite, respetively. If eletrial urrent J ours along axis x in the models, the ROMs for the eletrial ondutivity of the series and parallel models an be presented as follows [7-1]. model: model: 1 (1 ) (1) ( 1 ) (2) where ϕ denotes volume fration of material. If heat flow Q ours along axis x in the models, the ROMs of the thermal ondutivity for the series and parallel models an be given by replaing κ with σ in qs. 1 and 2, respetively. 2.2 ROMs for thermoeletri fore hemati diagrams of thermoeletri fore and their equivalent iruits of the series and parallel models are shown in Fig.2. H, L, T and V express width, length, temperature differene and thermoeletri fore, respetively. For onveniene, the thiknesses of the models was set to 1. In the series model (Fig.2(a) and (b)), the total thermoeletri fore of the omposite ( V = V + V, i.e. T = T + T ) an be given as follows. T T T T T T T T T T (3) assumed that the heat flow Q along axis x is stati, the temperature differene is estimated as T = Q H/κ. T /( T + T ) and T /( T + T ) in q.3 an be expressed as follows. y Heat flow Q letri urrent J Heat flow Q letri urrent J ρ ρ (a) model ρ ρ (b) model x Fig.1 Models for laminated omposite thermoeletris. ρ ρ T T T T T T y QH QH QH (4) QH QH QH H H (5) H (a) model L T V T V I a () model x L ΔH H H V V V V (b) quivalent iruit of series model V I a V (d) quivalent iruit of parallel model Fig.2 hemati diagrams of generation of thermoeletri fore in laminated omposite thermoeletris.

3 RUL OF MIXTUR FOR COMPOIT THRMOLCTRIC aording to H / H = (1-ϕ )/ϕ, the ROM equation for the total thermoeletri fore of the series model by substituting qs.4, 5 into q.3 is expressed as 1 ( ) ( ) (6) 1 The ROM of q.6 is more general omparing the equations in referenes [11-14], also the results are onsist with referenes [11-14] as mentioned later. On the other hand, in the parallel model (Fig.2() and (d)), assumed that T = T = T and V > V (i.e. > ), a irulation urrent I a will form between material and due to the differene of thermoeletri fore, and I a = ( V - V )/( R + R ), where R and R are the eletrial resistane of the two materials along x axis respetively. The total thermoeletri fore of the parallel model an be given as follows. a V V I R V R (7) then R R V V R R R T T ( T T ) (8) aording to T = T = T, q.8 an be rewritten as then R ( ) (9) R R R R R L H L L H H H H ( 1 1 ) (1) by substituting q.1 into 9, the ROM for the total thermoeletri fore of the parallel model is given as ffetive medium theory has been used to alulate the eletrial and thermal ondutivities for partile omposites with random distribution as shown in Fig.3 and onsists with the experimental results [7-1]. speially, the general effetive medium equation (GM) is used to apply to the eletrial and thermal ondutivities of partile omposites as q.12. In our previous work, the GM was used to disuss the eletrial resistivity and the thermal ondutivity of Cu/TiO 2-x omposite thermoeletris, and the results from GM are onsisted with those from the experiment and the analysis by FM [6, 11]. The GM of the thermal ondutivity an be obtained by replaing κ with ρ -1 in q (1 ){( ) ( ) } 1 1 ( ) {(1 ) / } ( ) 1 1 {( ) ( ) } 1 1 ( ) {(1 ) / }( ) (12) where ϕ C is the ritial volume fration, whih takes aount an effetive perolation threshold. The value of t determining the effetive perolation slope, is usually loated between 1 and 3. We know that the thermoeletri effet originates in transport events aused by thermal and eletrial flows under a temperature potential. That is, the thermoeletri events are similar physial ones with the thermal or eletrial transport. Therefore, we propose to apply the GM to disuss thermoeletri y Heat flow Q letri urrent J x ρ ρ ρ 1 ( 1 ( 1) 1) ( 1 1) (11) Fig. 3 Random model for partile omposite thermoeletris. 3 ffetive medium equation of random model for partile omposites fore by replaing ρ -1 in q.12 with the eebek oeffiient as follows.

4 letrial resistivity, ρ / μωm letrial resistivity, ρ / μωm (1 ){( ) ( ) } ( ) {(1 ) / } ( ) ( {( ) ( ) } ) {(1 ) / }( ) (13) with the volume fration of the addition metal. In the ase of the series, the resultant eletrial resistivity by the ROM is onsistent with those of the alulation and the experiment as well as referene [12]. In the ases of the series and parallel Certainly, ϕ C and t should be deferent values for the thermal ondutivity, eletrial resistivity and the eebek oeffiient. Taking the aount of the influene extent of the added phase in a wide range, the ritial volume fration ϕ C was set.3,.5 or.7 in the following alulation. lso, the value of t was given 1 or 2. 4 Thermoeletri properties of the omposites by ROMs and GMs GM ϕc t In this work, thermoeletri properties, inluding eletrial resistivity, the thermal ondutivity and the eebek oeffiient of Cu/i and Ni/i series omposites, parallel omposites and partile omposites, were alulated by the ROMs and GMs as mentioned above. In the ase of the series omposite, the thermoeletri properties were also ompared with those from referene [12]. Thermoeletri properties of the soure materials for alulation are listed in Table 1. In the ase of partile omposites, substituting the thermoeletri parameters of the soure materials into qs. 12 and 13, and the thermoeletri properties of the omposites were alulated by Newton-Raphson method. Table 1 Thermoeletri properties of the soure materials at 298 K [11]. Property i Cu Ni ρ (μωm) (μvk -1 ) (Wm -1 K -1 ) P (mwm -1 K -2 ) ZT Calulations of the power fator of the omposites The eletrial resistivity of Cu/i and Ni/i omposite thermoeletris by the ROMs and GM are shown in Fig.4 and Fig.5, respetively. On the whole, the resultant eletrial resistivities derease 1-2 Cu volume fration, ϕ Fig.4 letrial resistivity of Cu/i omposite thermoeletris by the ROMs and GM Ni volume fration, ϕ GM ϕc t Fig.5 letrial resistivity of Ni/i omposite thermoeletris by the ROMs and GM. omposites, the eletrial resistivities beame the upper limit and the lower limit, respetively. The resultant eletrial resistivities of the partile omposites by the GM loated between the upper limit and the lower limit. They are similar with Cu/TiO 2-x omposites [6]. esides, when t = 1, the trend with the volume fration takes steeper ompared with that when t = 2.

5 RUL OF MIXTUR FOR COMPOIT THRMOLCTRIC The resultant eebek oeffiient of Cu/i and Ni/i omposite thermoeletris by the ROMs and GM are shown in Fig.6 and Fig.7, respetively. The resultant eebek oeffiients derease with the volume fration of the addition metal, and the trends are similar to these of the resultant eletrial resistivity above. In the ase of the series, on the whole the resultant eebek oeffiient from the eebek oeffiient, - / μvk Cu volume fration, Fig.6 eebek oeffiient of Cu/i omposite thermoeletris by the ROMs and GM. GM ϕc t of the eebek oeffiient are 1%~2% higher than these from the ROM. It may be aused by the welding in the interfaes of Cu/i and Ni/i omposite thermoeletris [11]. In both the omposites of the series and parallel, the resultants eebek oeffiients beome the upper limit and the lower limit, respetively. The resultant eebek oeffiients of the partile omposites loated between the upper limit and the lower limit. Interestingly, when t = 2, the eebek oeffiients have a sudden fall and then get lose to that of the addition metal phase around ϕ C (Fig.6). These events were also observed in Cu/TiO 2-x omposites [6]. s a onlusion, the ROMs and GM for the eebek oeffiient of the omposites seem reasonable. The resultant power fators of Cu/i and Ni/i omposite thermoeletris by the ROMs and GM are shown in Fig.8 and Fig.9, respetively. In both of the ROMs and the experiment, the power fators of the series omposites have a similar trend, and show a peak in the range over ϕ =.8, and is higher than those of the soure materials. esides, in the ases of ϕ C =.3, t = 1 and ϕ C =.5, t = 2, the power fators of the partile omposites by the GM have a peak, and are higher than those of the soure materials in a ertain range. It means that the power fator an be enhaned by the series-strutured omposite or eebek oeffiient, - / μvk Ni volume fration, Fig.7 eebek oeffiient of Ni/i omposite thermoeletris by the ROMs and GM. GM ϕc t ROMs has the same trend with the experiment, however, in the ϕ region from.2 to.8, the values Power fator, P / mwk -2 m GM ϕc t Cu volume fration, ϕ Fig.8 Power fator of Cu/i omposite thermoeletris by the ROMs and GM.

6 Power fator, P / mwk -2 m GM ϕc t Ni volume fration, ϕ Fig.9 Power fator by the ROMs and GM for Ni/i omposite thermoeletris. partile omposite. Indeed, it has been verified by experiments in the series omposites [11-14] and partile omposites [1-5]. However, the power fator annot be inreased by the parallel-strutured omposite. 4.2 Calulations of dimensionless figure-of-merit of the omposites The thermal ondutivities of Cu/i and Ni/i omposite thermoeletris by the ROMs and GM are shown in Fig.1 and Fig.11, respetively. In the ases of the series and parallel omposites, the resultant thermal ondutivities are inverse to those of the resultant eletrial resistivities and eebek oeffiients, beame the lower limit and the upper limit, respetively. There is a steeper trend against the volume fration in the ase of a small t value (t = 1). s attention point, it is a negative effet for thermoeletris that the thermal ondutivity is inreased by metal addition in all omposite strutures. The dimensionless figure-of-merits of Cu/i and Ni/i omposite thermoeletris by the ROMs and GM are shown in Fig.12 and Fig.13, respetively. In the ases of the series and parallel omposites, the resultants ZT beame the upper limit and the lower limit respetively. lthough the power fator an be inreased by metal addition as disussed, there is no improvement in thermoeletri performane by metal addition due to the inrease of the thermal ondutivity. In this work, the dedutions of the ROMs and GM were based on the assumptions without the sattering of arriers and phonons, and no reations at the boundaries between the soure Thermal ondutivity, κ / WK -1 m GM ϕc t Cu volume fration, ϕ Fig.1 Thermal ondutivity by the ROMs and GM for Cu/i omposite thermoeletris. Thermal ondutivity, κ / WK -1 m GM ϕc t Ni volume fration, ϕ Fig.11 Thermal ondutivity by the ROMs and GM for Ni/i omposite thermoeletris.

7 Dimensionless figure-of-merit, ZT Dimensionless figure-of-merit, ZT RUL OF MIXTUR FOR COMPOIT THRMOLCTRIC.2.1 Cu volume fration, ϕ GM ϕc t Fig.12 Dimensionless figure-of-merit by the ROMs and GM for Cu/i omposite thermoeletris (at 298 K)..2.1 Ni volume fration, ϕ GM ϕc t Fig.13 Dimensionless figure-of-merit by the ROMs and GM for Ni/i omposite thermoeletris (at 298 K). materials. That is, although the power fator an be enhaned for simple omposites by omposite effet, the improvement of the performane is diffiult. 5 Conlusions In this work, the ROMs of the eebek oeffiient for the series and parallel omposite models were dedued. The GM was applied to the eebek oeffiient of partile omposite model. The ROMs of the series and parallel models take values of the upper limit and lower limit of the eletrial resistivity and the eebek oeffiient, respetively. On the other hand, the GM takes values between those from the ROMs. The upper limit and lower limit of the thermal ondutivity is inverse to those of the other properties. The ROMs and GM for the eebek oeffiient of the omposites seem reasonable. lthough the power fator an be enhaned by omposite effet for simple omposites, however, the thermal ondutivity is inreased. Thus, the performane improvement is diffiult by only simple omposite method. The thermoeletri figure of merit of M/T/M (M = Cu or Ni and T = i.88 b.12 ) was enhaned beause of the interfae effet aording to the reation at the interfae. In the other work, high-performane bulk thermoeletris was abtained through all-sale proess inluding nano- and meso-sale [16]. Therefore, to improve the thermoeletri performane by omposite method, the nanoomposite and reations at the interfaes should be introdued. Conferenes [1] M. Ito and D. Furumoto "ffet of mehanial milling and g adding on thermoeletri properties of Na x Co 2 O 4 ". ripta Materialia, Vol.55, pp , 26. [2] M. Ito and D. Furumoto "ffet of noble metal adding on mirostruture and thermoeletri properties of Na x Co 2 O 4 ". Journal of lloys and Compounds, Vol.45, pp , 28. [3] P. H. Xiang, Y. Kinemuhi, H. Kaga and K. Watari "Fabriation and thermoeletri properties of Ca 3 Co 4 O 9 /g omposites". Journal of lloys and Compounds, Vol.454, pp , 28. [4] Y. ong, Q. un, L. R. Zhao, F. P. Wang and Z. H. Jiang "ynthesis and thermoeletri power fator of (Ca.95 i.5 ) 3 Co 4 O 9 /g omposites",. Materials Chemistry and Physis, Vol.113, pp , 29. [5].F. Wang, Z.L. ai, H.F. Wang, Q. Lv, J.L. Wang and G.. Fu "High temperature thermoeletri properties of i 2 r 2 Co 2 O y /g omposites". Journal of lloys and Compounds, Vol.554, pp , 213. [6] Y. Lu, K. agara, Y. Matsuda, L. Hao, Y. R. Jin and H. Yoshida "ffet of Cu powder addition on thermoeletri properties of Cu/TiO 2 x omposites". Ceramis International, Vol.39, pp , 213. [7] D.. MLahlan, M. laszkiewiz and R. Newnham "letrial resistivity of omposites". Journal of the merian Cerami oiety, Vol.73, No.8, pp , 199. [8] D.G. Han, G.M. Choi "Computer simulation of the eletrial ondutivity of omposites: the effet

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