Porosity efect on the thermal conductivity of sintered powder materials

Vol.:(0123456789) Applied Physics A (2025) 131:149 h ps://doi.o g/10.1007/s00339-025-08254-y Po osi y e ec on he he mal conduc i i y o sin e ed powde ma e ials J.M.Mon es1 · F.G.Cue as2 · J.Cin as1 · F.Te ne o1 Recei ed: 5 No embe 2024 / Accep ed: 8 Janua y 2025 / Published online: 30 Janua y 2025 © The Au ho (s) 2025 Abs ac In his wo k, he e ec i e he mal conduc i i y o sin e ed powde ma e ials is s udied. The ex ensi e li e a u e ela ed o he p oposed models abou his p ope y in all kind o po ous ma e ials is e iewed, and a new equa ion is p oposed as a unc ion o he ully dense ma e ial conduc i i y, he po osi y o he ma e ial and he ap po osi y o he s a ing powde . This equa ion co e s he po osi y ange o powde agg ega es om he ap po osi y o ze o po osi y, and also applies o sin e ed powde s. The p oposed equa ion has been expe imen ally alida ed by i ing o expe imen al da a o me allic sin- e ed powde ma e ials measu ed a oom empe a u e, esul ing e y good ag eemen s. Also, al e na i e models p oposed by o he au ho s ha e been i ed o he same expe imen al da a o check he ela i e goodness o he p oposed model. The esul s allow o conclude ha a pe cola ion model can desc ibe he beha iou o he e ec i e he mal conduc i i y o sin e ed powde ma e ials wi h low and medium po osi y le els. Keywo ds The mal conduc i i y· Powde ed ma e ials· G anula ma e ials· Sin e ed compac s· Foam ma e ials· Modelling 1 In oduc ion A po ous ma e ial can be de ined as a wo-phase ma e ial consis ing o a i s phase ha p o ides in eg i y o he whole (ma ix) and en elops a second phase, which is he po osi y. The po osi y ange can be e y di e en : om small iso- la ed po es, passing h ough in e connec ed po osi y o ming pa hs inside he ma ix [1], o he ex eme case o oamed ma e ials [2], in which he ma ix can be he mino i y phase o he sys em. The ma ix, based ei he on polyme s [3, 4], ce amics [5, 6] o me als 1, is in mos cases cons i u ed by a ma e ial clea ly con inuous, bu also agg ega es o di e - en weakly bound pa icles, in a s ep p e ious o i s inal consolida ion, can cons i u e he ma ix. Thus, as desc ibed in [7], po ous ma e ials can be classi ied in h ee g oups: packed beds whe e he pa icles con ac s a e poin s, con- solida ed solids wi h small con ac s be ween pa icles, and po ous solids wi h ex ensi e con ac s be ween pa icles. O he classi ica ions o po ous ma e ials can also be ound in he li e a u e, o ins ance based on nume ical c i e ions (low-po osi y ma e ials wi h po osi ies up o 10%, medium- po osi y in he ange 15%–85%, and high-po osi y wi h al- ues highe han 90%) [8], o on hea conduc ion mechanism c i e ions (iso opic po ous ma e ials wi h ‘in e nal po os- i y’, as sponges and oams, and hose o he wi h ‘ex e nal po osi y’, cons i u ed by g ains and pa icula es) [9]. The s udy o he e ec i e p ope ies o po ous ma e i- als, i.e., conside ing he e ec o he po osi y, is o g ea in e es o a wide ange o enginee ing applica ions, mainly including mechanical [1], he mal [10], elec ical [11] and magne ic [12] p ope ies. This wo k ocuses on he he mal p ope ies o po ous ma e ials. Some imes, he s udy o hese e ec i e p ope ies is nec- essa y in he inished p oduc . Thus, he he mal p ope ies o packed beds o ca alys o uel cells elec odes [13], o oams in applica ions in which hea has o be ans e ed o a luid mo ing inside he po ous s uc u e a e o g ea in e es . Fo ins ance, po ous olume ic sola ecei e s made wi h * J. M. Mon es [email p o ec ed] 1 Depa men o Enginee ing andMa e ials Science, Escuela Técnica Supe io de Ingenie ía, Uni e sidad de Se illa, Camino de los Descub imien os, s/n, 41092Se illa, Spain 2 Depa men o Chemical Enginee ing, Physical Chemis y andMa e ials Science, Escuela Técnica Supe io de Ingenie ía, Uni e sidad de Huel a, Campus El Ca men, A da. T es de ma zo s/n, 21071Huel a, Spain J.M.Mon es e al.149 Page 2 o 13 oamed ma e ials a e key componen s o concen a ing sola powe plan s. They ake ad an age o he ene gy pene a - ing deepe inside he ecei e , imp o ing he sola adia ion abso p ion and hea ans e enhancemen o he ci cula ing luid, a he ime ha diminishing he empe a u e o he su ace o he ecei e [14, 15]. Ano he ypical applica ion is in ac ual elec onic componen s, in which big amoun s o hea a e p oduced, needing o be dissipa ed o a oid damage. Mic ochannel hea sinks, using a po ous medium o inc ease he hea ans e capabili y o a coolan in con ac wi h a solid, a e used o ans e he gene a ed hea o he ou side en i onmen [16]. In his con ex whe e luids a e p esen , he he mal conduc i i y o he luid is e y much impo an in packed bed sys ems, in which he con ac he mal esis - ance be ween pa icles is e y big. The he mal conduc i - i y o he solid inc eases in impo ance o be e con ac s among pa icles. Also inished p oduc s in which hea ans e is no ela ed o he passing o a luid a e o in e es . Thus, good he mal p ope ies o cons uc ion ma e ials, wi h s ingen ene gy egula ions and cons i u ed by highly po ous building blocks wi h high esis ance o hea ans e , a e on imes equi ed [17]. O he imes he ma e ials o be s udied, and in pa icula hei he mal p ope ies, a e o in e es in a s age p e ious o hei inal con igu a ion. Thus, sin e ing plan s use i on o e ines and me allu gical was es o p oduce he cha ge ma e- ial o blas u naces in s eel plan s. This cha ge ma e ial is ob ained om a semi-mol en mass ha solidi ies in o po ous pieces wi h he adequa e size and s eng h o eed he blas u nace. The knowledge o he he mal p ope ies o he po ous cha ge is necessa y o achie e a good blas u nace p ocess pe o mance, including he lame on p opaga ion o gas–solid hea ans e [18]. Ano he si ua ion in which he mal p ope ies a e o in e es in an in e media e si u- a ion is me als p ocessing h ough ield assis ed sin e ing echniques (FAST) [19]. These sin e ing echniques use he pass o an elec ical cu en h ough a powde agg ega e o p o oke sin e ing om he hea eleased by he Joule e ec . The knowledge o he he mal p ope ies o he agg ega e is a key ac o in he p ope design o he p ocess [20]. The modelling equi ed is complex o se e al easons. The main di icul y esides in he p ecise knowledge o he mic os uc u e o hese complica ed po ous sys ems [6]. A p esen , Compu ed Axial Tomog aphy equipmen makes i possible o know he h ee-dimensional dis ibu ion o po osi y [21, 22], which leads us o suspec ha he e will be impo an ad ances in his line o wo k. On he o he hand, models de eloped o sys ems wi h open po osi y do no necessa ily ha e o wo k o hose wi h isola ed po os- i y [9, 23], making necessa y speci ics models o di e en si ua ions and ma e ials [6]. When he models ha e a wide applicabili y, in gene al, depend on pa ame e s ha mus be empi ically de e mined o each ma e ial. The complexi y inc eases when he e is a luid lowing h ough he ma ix. I is logical, he e o e, ha a p esen , nume ical simula- ions ha e been ound o be a solu ion o hese si ua ions wi h complica ed h ee-dimensional s uc u e o he po ous ma e ials. Nume ical me hods p ecisely desc ibe he com- plex s uc u e o he po ous ma e ial up o a ce ain esolu- ion, being sol ed by nume ical me hods [23–26]. Also, he applica ion o he ac al geome y, ecognized in se e al aspec s o he mic os uc u e o he ma e ials, is o g ea in e es o po ous ma e ials [6, 27]. This cu en end, based on massi e compu a ion, is absolu ely necessa y, because i allows o know, as in no o he way, he in luence o he mic os uc u al cha ac e is ics o he po osi y. Howe e , he e is no eason why his new end canno coexis wi h simple ma hema ical models, alid especially o he i s app oxima ions, and which p o ide a supe io unde s anding o co ela ions and c i ical pa ame e s. A e iew o hose models can be ound in [13, 28–34]. The p oposi ion o hese simple analy ical exp essions o model he dependence o he he mal (and elec ical) conduc i i y o sin e ed ma e ials on hei po osi y was a esea ch chal- lenge du ing he 1960s–1980s. Du ing hose yea s, many exp essions came o be p oposed. The ealiza ion o he g ea in luence o po e mo phology and po e connec i i y se he b akes on ha wo k, and he subjec was closed in a alse sense, as i i had come o be pe ec ly unde s ood. Table1 lis s some a emp s (some o hem e y old) ha ha e in common o p o ide a simple ma hema ical ela ion- ship o dependence on po osi y. The me hods o deduc ion ha e been e y a ied, and he exp essions we e deduced in he elec ic and he mal con ex , bo h being anspo phenomena in which he in luence o he po osi y could be desc ibed in a simila way. As expec ed, o all hese models, he conduc i i y dec eases as he po osi y inc eases ( he highe he po osi y, he smalle he elec ical/ he mal low ans e c oss-sec- ion and he longe he low pa h i mus ake o bypass he po es). Thus, exp essions in Table1 e i y ha he ela i e conduc i i y ends o 1 when he po osi y ends o 0, and he conduc i i y dec eases o 0 by inc easing he po osi y. The uppe bounda y o Θ is physically es ic ed in powde ed sys ems o a alue lowe han 1, which can be assimila ed wi h he ap po osi y [56]; he po osi y eached by he pow- de a e mode a e ib a ion. (Al hough he so-called he appa en po osi y [57] is sligh ly highe , i s alue is less ep oducible.). Fo his eason, se e al o he exp essions shown in Table1 can only be conside ed alid o powde ed sys ems wi h e y low po osi ies, nea 0. Only he exp essions by Odele skii [39], G oo enhuis e al. [40], Loeb [41], McLa- chlan [45], G uzde e al. [46], Mon es e al. [47, 52, 54, 55], Pabs e al. [50], and Solonin e al. [51] sa is y he uppe Po osi y e ec on he he mal conduc i i y o sin e ed powde ma e ials Page 3 o 13 149 bounda y condi ion and, he e o e, a e o applica ion in he high po osi ies ange. On he o he hand, he exp ession by Odele skii [39], G oo enhuis e al. [40] and Loeb [41] a e expe imen ally e y well alida ed in he low po osi ies ange. The e o e, i would be desi able ha any conside ed exp ession was ans o med o hese equa ions in he low po osi y limi . Some hing o conside ega ding all hese models is he numbe o pa ame e s in ol ed in hem. Acco ding o [23] he conduc i i y models can be di ided in o igid models, hose in ol ing only he mal conduc i i y and po osi y, and lexible models, con aining ex a pa ame e s, in many cases wi hou a clea physical meaning. As can be seen in Table1, mos exp essions in ol e an empi ical pa ame e . This is because he conduc i i y is closely dependen on he mic o- s uc u e (including po e shape and size), and he empi i- cal pa ame e helps o model he mic os uc u al in luence. The e o e, a simple ma hema ical exp ession based only on he po osi y le el, such as Maxwell exp ession wi hou any addi ional empi ical pa ame e , can ha dly desc ibe he e ec i e conduc i i y o high po osi ies. Se e al wo ks on sin e ed ma e ials wi h mo e lexible and mo e complex esul ing equa ions can be ound in he li e a u e [58–65], bu hese equa ions will no be conside ed in his s udy. In his wo k, an almos igid model is p oposed o he he mal conduc i i y o powde agg ega es and po ous Table 1 Simple exp essions o he ( he mal o elec ical) ela i e conduc i i y ( gR ), de ined as he e ec i e conduc i i y ( gE ) no malised by he conduc i i y o he ully dense ma e ial ( g0 ) In hese exp essions, Θ is he po osi y, Θ0 is he ini ial po osi y, ΘM he ap po osi y, Θc a ce ain c i ical alue o po osi y, and he pa ame e s a and n a e cons an s wi h di e en alue and meaning in each case Au ho s Yea Con ex gR = gE/g0gR → 1? gR → 0? Maxwell [35] 1873 Elec ical 2(1−Θ) 2+Θ Θ → 0 Θ → 1 F icke [36] 1924 Elec ical 1−Θ 1−aΘ Θ → 0 Θ → 1 Aus in [37] 1939 The mal 1−Θ 1 − 1 2Θ Θ → 0 Θ → 1 A chie [38] 1942 Elec ical (1−Θ )a Θ → 0 Θ → 1 Odele skii [39] 1951 The mal and elec ical (low po osi y) 1 − 3 2Θ Θ → 0 Θ → 2/3 G oo enhuis e al. [40] 1952 The mal 1−2.1 Θ Θ → 0 Θ → 1/2.1 Loeb [41] 1954 The mal (low po osi y) 1−aΘ Θ → 0 Θ → 1/a Ai azo and Domashne [42] 1968 The mal 1−Θ 1+aΘ2 Θ → 0 Θ → 1 Koh & Fo ini [28] 1971 The mal and elec ical 1−Θ 1+10Θ2 Θ → 0 Θ → 1 Meye [43] 1972 The mal a(1−Θ) a +Θ Θ → 0 Θ → 1 Sko okhod [44] 1974 The mal and Elec ical (1−Θ ) 3 2 Θ → 0 Θ → 1 McLachlan [45] 1985 The mal and elec ical ( 1−Θ / Θ 0) 3 2Θ 0 Θ → 0 Θ → Θ0 G uzde e al. [46] 1989 The mal ( 1−Θ ) 2( 1−Θ / Θ 0)n Θ → 0 Θ → Θ0 Mon es e al. [47] 2003 The mal and elec ical ( 1−Θ ∕Θ M)2 Θ → 0 Θ → ΘM Pabs [48] 2005 The mal 1 − 3 2 Θ+ 1 2 Θ 2 Θ → 0 Θ → 1 Ticha e al. [49] 2005 The mal exp ( − 3 2Θ 1−Θ ) Θ → 0 Θ → 1 Pabs and G ego o á [50] 2006 The mal ( 1−1 2Θ )( 1−Θ / Θc ) Θ → 0 Θ → Θc Solonin and Che nyshe [51] 2006 Elec ical ( 1−Θ ) 3 2 ( 1− ( Θ / Θ0 ) 4 3 ) 1 2 Θ → 0 Θ → Θ0 Mon es e al. [52] 2008 Elec ical ( 1−Θ ∕Θ M) 1+(1−ΘM) 4∕5 Θ → 0 Θ → ΘM Pabs & G ego o á [53] 2012 The mal ( 1−Θ) 2 1+ 1 2Θ Θ → 0 Θ → 1 Mon es e al. [54] 2016 Elec ical ( 1−Θ ∕Θ M) 3 2 Θ → 0 Θ → ΘM Mon es e al. [55] 2018 Elec ical ( 1−Θ ∕Θ M)n Θ → 0 Θ → ΘM J.M.Mon es e al.149 Page 4 o 13 sin e ed compac s. The ap po osi y o he powde (ΘM) is he ex a pa ame e conside ed in he model o powde ed ma e ials. Mo eo e , as in many o he models, he e ec o po e- illing ai conduc i i y will be analysed and conside ed negligible. No much high empe a u es will be conside ed, and he e o e adia ion hea ans e is neglec ed. In addi ion, he Knudsen e ec [66], o accoun o a dec ease in conduc- i i y o nanome ic sized po es will nei he be conside ed. The model now de eloped will be compa ed wi h o he igid o almos igid models a ailable in he li e a u e, among he lis ga he ed in Table1. 2 A new exp ession o e ec i e he mal conduc i i y In p e ious wo ks [54, 55], he e ec i e elec ical conduc- i i y o a po ous me allic sin e ed compac , σE, was expe i- men al and heo e ically s udied. The p oposed equa ion was a unc ion o he elec ical conduc i i y o he ully dense ma e ial, σ0, and he a io be ween he po osi y o he sam- ple, Θ, and o he ap po osi y o he s a ing powde , ΘM, wi h which he compac was manu ac u ed. This ap po os- i y ep esen s he maximum alue ha he powde -mass po osi y can ake in s eady s a es. (Na u ally, his pa ame e depends on he powde pa icle size, shape and dis ibu ion, he e o e ga he ing he mo phog anulome ic in o ma ion.) The p oposed equa ion was: Equa ion(1) sa is ies he expec ed bounda y condi- ions o he conduc i i y, σE → σ0 as Θ → 0, and σE → 0 as Θ → ΘM, when in e pa icle con ac s a e poin s. Equa ion (l) is also applicable o non-sin e ed agg ega es o non-oxidised pa icles, because he me allic phase also exhibi s connec i i y in ha case. On he o he hand, he elec ical beha iou o oxidised me allic powde pa icles unde comp ession was also s udied in [55]. This is he gene al si ua ion when modelling me allic powde s com- pac ion, because pa icles a e usually co e ed wi h a nano- me ic oxide laye (hyd oxides can also be p esen ), which is e i ed (descaling p ocess caused by ic ion) du ing powde comp ession. The equa ion he e p oposed o model his new case was: whe e σ es is he conduc i i y a Θ = 0, wi h a alue some lowe han σ0 because o he mechanical descaling p ocess no being comple ed, and/o because he descaled oxide lay- e s emain in he ma e ial, sligh ly al e ing he conduc i i y alue despi e ep esen ing a e y small olume ac ion. The (1) 𝜎E =𝜎0 ( 1−Θ ∕ΘM ) 3 2 (2) 𝜎E =𝜎 es ( 1−Θ ∕ΘM )n exponen n is a i ing pa ame e desc ibing he descaling a e. Wi h e y insula ing oxide laye s, he conduc i i y will be e y low du ing he i s momen s o compac ion, and he descaling e ec will be e y p onounced. Wi h oxide- ee powde s he exponen n is equal o 3/2, bu wi h he p esence o oxide laye s i akes highe alues. The highe o lowe di e ence be ween he pa ame e s σ0 and σ es, and how a he pa ame e n is om he minimum alue o 3/2, is due o he in luence o he oxide laye s, which in gene al is impo an . Na u ally, Eq.(2) sa is ies he expec ed limi s. Thus, σE → σ es as Θ → 0, and σE → 0 as Θ → ΘM. Res ic ing o si ua ions whe e he empe a u e is no oo high o he ansmission by adia ion o be signi i- can compa ed o he conduc ion mechanism, i is possible o o mula e simila exp essions ha model he he mal conduc i i y o he po ous ma e ial. T ansla ing Eq.(1) and Eq.(2) o he he mal case jus equi es subs i u ing he elec ical pa ame e s o hose o he he mal con ex . Thus, κE and κ0 will be he espec i e alues o he e ec- i e he mal conduc i i y and ully dense ma e ial he mal conduc i i y, and κ es he so-called esidual he mal con- duc i i y. Howe e , he impo an ole played by he oxide laye s in he elec ical case (and which jus i ies he la ge di e ence ha can exis be ween he alues o σ0 and σ es) does no hold in he he mal case. Fo ins ance, in he ex eme case o he pai aluminium- alumina (me al-oxide), wi h a ypical hickness o 4.5nm o he oxide laye [67] and a pa icle adius o 100μm, and conside ing elec ical esis i i ies a a empe a u e o 20ºC o ρ0 = 2.73·10–8 Ωm [68] o he me al and ρX = 1.0·10–12 Ωm [67] o he oxide, a mean elec ical esis i i y o 4.5·107 Ωm is ob ained a e a olume ic a e age, which ep esen s a alue 15 o de s o magni ude highe han ha o he pu e me al. The si ua ion is comple ely di e en o he he mal case. Fo he same ma e ials, he he mal esis i i ies a e 1/κ0 = 4.22·10–3 W−1·m·K [69] and 1/κX = 1 W−1·m·K [70]; a e aging in unc ion o he olume a mean esis i i y o 4.26·10–3 W−1·m·K is ob ained, which is only a 1% highe han ha o he pu e me al. This means ha he in luence o he oxide laye can be neglec ed in he he mal case, and ha he ansla ion o Eq.(1) is he e o e enough o desc ibe he he mal p oblem. Howe e , he simple ansla ion o Eq.(1) could be com- ple ed wi h he con ibu ion o he he mal conduc ion o he ai mass illing he compac po es. This co ec ion was no conside ed in he elec ic model because o he much be e elec ical han he mal insula ing e ec o he ai . Fo exam- ple, o pu e aluminium a 20 ºC, σme al/σai = 7.1·1021 whe eas κme al/κai = 9.1·103 [69], he e o e esul ing an elec ical insu- la ing capabili y a ound 1018 imes highe han he he mal one. Thus, i is necessa y o s udy he e ec o he he mal con- duc ion o he ai illing he po es, mainly o high po osi ies, Po osi y e ec on he he mal conduc i i y o sin e ed powde ma e ials Page 5 o 13 149 when he conduc ion in solid s a e is lowe because o he small a ea o he in e pa icle con ac s. Admi ing ha he ai con ibu ion is p opo ional o he no malised po osi y, an addi ional e m (κai Θ/ΘM) has o be added o he ansla ion o Eq.(1), esul ing in Again, Eq.(3) sa is ies he expec ed bounda y condi ions: κE → κ0 as Θ → 0, and κE → κai as Θ → ΘM. Howe e , aking in o accoun ha κai = 0.026W·m−1·K−1 [69] a 20ºC, o po osi y alues close o ΘM, he second e m will be negli- gible conside ing he much highe expe imen al unce ain y in he de e mina ion o he he mal conduc i i y and he high alues o me allic he mal conduc i i y. A highe em- pe a u es, o ins ance 400ºC, κai inc eases up o a alue o 0.052W·m−1·K−1 [69], esul ing also a small alue in p ac ise. The e o e, he model he e p oposed o desc ibe he he mal conduc i i y o bo h me al powde agg ega es and po ous compac s is he one exp essed by: This exp ession is simila o he equa ions p oposed in he pe cola ion ield. This con ex has been sugges ed many imes o modelling and desc ibing anspo p ope ies, as i co - esponds o hea ans e . Fo low po osi ies (i.e., when Θ → 0), Eq.(4) can be app oxima ed by Taylo expansion a: which desc ibes a linea beha iou , o en obse ed and p o- posed o sin e ed ma e ials wi h low esidual po osi y, and which ag ees wi h he exp essions p oposed and alida ed by G oo enhuis e al. [40], and Loeb [41]. On he o he hand, o sys ems in which he maximum po osi y, ΘM, can ake alues e y close o uni y, as in oamed ma e ials [71], Eq.(4) becomes which o mally i s in o he ca ego y ep esen ed by A chie equa ion[37], and coincides exac ly wi h he exp ession p oposed by Sko okhod [44] and, yea s la e , also de ended by Baue [72]. A he low po osi y limi , Eq.(6) becomes (3) 𝜅E =𝜅0 ( 1−Θ ∕ΘM ) 3 2+𝜅 ai Θ∕Θ M (4) 𝜅E =𝜅0 ( 1−Θ ∕ΘM ) 3 2 (5) 𝜅 E≈𝜅0 ( 1− ( 3 2 ΘM) ⋅Θ ) =𝜅0(1−aΘ ) (6) 𝜅E =𝜅 0 (1−Θ ) 3 2 (7) 𝜅 E=𝜅0 ( 1−3 2Θ ) which coincides wi h he exp ession om Odele skii [39], and using Taylo ’s de elopmen , also wi h he exp ession om Maxwell [35]. 3 Expe imen al alida ion 3.1 Equipmen andexpe imen al p ocedu e The measu emen o he mal conduc i i y in small samples wi h high he mal conduc i i y has been e y di icul un il he ad en o lase measu emen equipmen . This equipmen is based on he ac ha he mal conduc i i y (κ) can be exp essed as he p oduc 𝜅=𝛼 ⋅ cp ⋅ 𝛿 , whe e α is he so- called he mal di usi i y (exp essed in m2/s) o he sample, cp is i s speci ic hea capaci y (exp essed in J/(kg·K)) and 𝛿 is i s appa en densi y (exp essed in kg/m3). Thus, he uni s o κ esul W/(m·K), as can be expec ed. The e o e, o measu e he he mal conduc i i y o samples, h ee measu emen s a e equi ed: he mal di usi i y, speci ic hea capaci y and appa en densi y. A Lase Flash (LFA 1000/1000 HT, om LINSEIS GmbH, Ge many) was used o de e mine he he mal di - usi i y (α) o he compac s. Wi h his echnique, he sample su ace is i adia ed wi h a p og ammed ene gy pulse (lase o xenon lash). This ene gy pulse esul s in a homogeneous empe a u e ise a he sample su ace. The esul ing em- pe a u e ise o he ea su ace o he sample is measu ed by a high-speed IR de ec o and he mal di usi i y alues a e compu ed om he empe a u e ise e sus ime da a (Fig.1). The speci ic hea capaci y was de e mined by Modula ed Di e en ial Scanning Calo ime y, MDSC (Q20-DSC, om TA Ins umen s, USA). This equipmen can measu e he speci ic hea capaci y o a ma e ial in quasi-iso he mal mode, i.e. admi ing only he small empe a u e oscilla ion associa ed wi h modula ion, which ensu es he bes le el o measu emen eliabili y. The measu emen s we e ca ied ou a oom empe a u e, using a modula ion le el o ± 1°C pe 120s. On he o he hand, he appa en densi y was de e mined by dimensional measu emen o he cylind ical compac s and weighing. O all he measu emen s in ol ed, he seem- ingly simple measu emen o appa en densi y may be he main sou ce o e o . Dimensional measu emen s mus be made wi h g ea ca e. In o de o de e mine he po osi y o he compac s, he absolu e densi y, 𝛿0 , o he s a ing powde s mus be known. This was de e mined by pycnome ic echnique (using Accu- pyc II 1340, om Mic ome i ics GmbH, Ge many). The de e mina ion o he ap po osi y is done acco ding o MPIF S anda ds [56]. Essen ially, he me hod consis s o aking 100g o powde which is pou ed in o a g adua ed glass ube J.M.Mon es e al.149 Page 6 o 13 accu a e o 0.2mL. The whole is mechanically apped, a 150 aps/minu e, so ha densi ica ion can ake place wi h- ou any loosening o su ace laye s, un il he heigh o he powde column s ops dec easing. Then he olume and wi h i he ap densi y, 𝛿T , is de e mined. In o de o calcula e he ap po osi y, ΘM , inally we apply Θ M=1−𝛿 T/ 𝛿 0 . 3.2 Powde s Selec ed powde s wi h di e en mo phologies, all in com- me cial g ade, ha e been s udied: NC100.24 spongi o m i on powde om Höganäs, WPL200 i egula -shaped i on powde om QMP, 4SP400 sphe ical nickel powde om No ame , and AS61 i egula -shaped aluminium powde om Ecka -We ke. The e y di e en mo phology o he s udied powde s ob ained by scanning elec on mic oscopy (SEM) is shown in Fig.2. Table2 lis s, o each ype o powde , he absolu e densi y ( 𝛿0 ), he mean pa icle adius ( 0 ), ob ained by lase di ac- ion, and he ap po osi y (ΘM). The absolu e e o made in he de e mina ion o ΘM, con- side ing he p ecision o he ins umen s employed, can be es ima ed in ± 0.01; a ce ainly small alue. Ne e heless, du ing he measu ing p ocess, he way and s eng h o he apping could accoun o a non-con olled inc ease o he Fig. 1 LFA measu ing p inciple: a lase pulse (in blue) hea s he lowe base o he sample, and an in a ed de ec o egis e s he adia- ion (in ed) emi ed by he uppe base o he sample. A compu e cal- cula es he empe a u e signal as a unc ion o ime, om which he he mal di usi i y can be de e mined Fig. 2 Mic og aphs ob ained by Scanning Elec on Mic oscopy (SEM) o selec ed powde s: a NC100.24 i on, b WPL200 i on, c 4SP400 nickel and d AS61 aluminium Po osi y e ec on he he mal conduc i i y o sin e ed powde ma e ials Page 7 o 13 149 expe imen al unce ain y. Expe imen al es s, conce ning he apping e ec , lead o an unce ain y ange o ± 0.05 o ΘM. 3.3 Samples Fo each o he selec ed powde s, cylind ical pa s wi h di - e en po osi ies we e manu ac u ed. The p oduc ion me hod was he con en ional ou e o cold compac ion and u nace sin e ing. Uniaxial cold p essing wi h a 12mm inne diame e die was employed o compac ion. In each case, he wo k- ing p essu e was de e mined based on he comp essibili y cu e [73]. The lowes po osi y was achie ed, in all cases, by applying he maximum a ailable p essu e (1400MPa). The lowes p essu e was chosen as he one ha allowed o ob ain he mos po ous g een compac ha was manipulable. This was achie ed wi h compac s wi h a po osi y alue such ha he Θ/ΘM a io is be ween 0.6 and 0.7, o all powde s. The ange o po osi y s udied o each ma e ial is de ailed in Table3. The p essing p ocess was ollowed by a sin e ing ea men a he empe a u e indica ed in Table3, o 30min and unde a 1.2ba a gon a mosphe e. The empe a u e was chosen jus o inc ease he g een s eng h, a oiding impo - an changes in he inal po osi y and hus achie e a cloud o expe imen al poin s mo e o less equispaced on he po osi y axis (which has no o he a ionale han pu ely aes he ic). Ne e heless, he inal po osi y a e sin e ing has been again de e mined by measu ing and weighing he compac s, and he ob ained alue has been he one conside ed in he la e calcula ions. The high alue chosen o he sin e ing empe a u e o aluminium, e y close o bu lowe han i s mel ing empe a- u e, may come as a su p ise. This is due o he s eng h o he oxide laye s ha su ound he aluminium pa icles, ha e ec i ely passi a e me al, bu also posing an insu moun - able ba ie o sin e ing a low empe a u e (a possible solu- ion would ha e been o use a educing a mosphe e, H2, o example). 4 Resul s anddiscussion Acco ding o he expe imen al p ocedu e desc ibed in Sec .3.1, i s , he appa en densi ies o each compac we e de e mined. Secondly, he speci ic hea capaci y (a oom empe a u e) o a sample o each ype o ma e ial was de e - mined (i does no ma e which sample, since his quan i y does no depend on po osi y and i is p ac ically insensi i e o mic os uc u al de ails, such as g ain size, o example). The alues ob ained we e 470J/(kg·K), 440J/(kg·K) and 906J/(kg·K), o i on, nickel and aluminium, espec i ely. Thi d and inally, he he mal di usi i ies o each compac we e de e mined (a oom empe a u e). Wi h all he in o - ma ion, he alue o he he mal conduc i i ies (a oom empe a u e) was calcula ed. The he mal conduc i i y da a o all compac s o each ma e ial we e i ed by leas squa es o Eq.(4). In his p o- cess, bo h κ0 and ΘM we e conside ed as i ing pa ame e s, al hough bo h we e subjec o cons ain s. On he one hand, ΘM had o be less han 1 and g ea e han he po osi y o he mos po ous compac . On he o he hand, since he selec ed ma e ials a e only comme cially pu e, he alue o κ0 should be less han o equal o he he mal conduc i i y alue o he ully dense pu e ma e ial ound in he li e a u e [68] (78.2 W/(m·K) o i on, 88.5 W/(m·K) o nickel and 238 W/(m·K) o aluminium). Table 2 Absolu e densi y ( 𝛿0 ), mean pa icle adius ( 0) and ap po osi y (ΘM) o s udied powde s Powde 𝛿0 (g/cm3) 0 (μm) ΘM NC100.24 i on 7.86 55.6 0.65 WPL200 i on 7.87 39.2 0.63 4SP400 nickel 8.91 6.6 0.60 AS61 aluminium 2.70 22.2 0.45 Table 3 Po osi y ange, p essu e ange, sin e ing empe a u e and ime o he selec ed powde s Powde Po osi y ange P essu e ange, MPa Sin e ing empe a- u e, ºC Time, min NC100.24 i on 0.03–0.44 1400–110 1150 30 WPL200 i on 0.02–0.43 1400–90 1150 30 4SP400 nickel 0.06–0.37 1400–330 800 30 AS61 aluminium 0.01–0.32 1400–40 650 30 Table 4 Values o he adjus able pa ame e s and he co esponding de e mina ion coe icien , ob ained a e i ing he expe imen al da a o Eq.(4) Ma e ial κ0, W/(m·K) ΘMR2 NC100.24 i on 78.20 0.6058 0.9974 WPL200 i on 78.20 0.6462 0.9920 4SP400 nickel 85.05 0.7614 0.9664 AS61 aluminium 231.71 0.5204 0.9931 J.M.Mon es e al.149 Page 8 o 13 The alues o he adjus able pa ame e s esul ing om he i ings a e shown in Table4, oge he wi h he coe icien o de e mina ion (R2) ha accoun s o he goodness o i in each case. Figu e3 shows he expe imen al he mal conduc i i ies o each ma e ial e sus po osi y. Nex o he expe imen al poin s, he heo e ical cu e, Eq.(4), esul ing om he leas squa es i ing is also shown. In gene al e ms, he p oposed model gi en by Eq.(4) causes a good i ing in all he cases, wi h coe icien s o de e mina ion highe han 0.99 in h ee cases, and nea 0.97 in one case (co esponding o nickel powde ). The alues ob ained o κ0, in acco dance wi h he con- s ain s imposed on he i , a e equal o o sligh ly lowe han he alues epo ed in he li e a u e o he ully dense pu e ma e ial. The la ges di e ence is ound in he case o nickel powde , o which he κ0 alue ob ained is 2.6% lowe han he alue ound in li e a u e. This is, howe e , a pe ec ly accep able alue. Conce ning he ΘM pa ame e , he alues ob ained o he wo i on powde s and he aluminium powde all wi hin he unce ain y in e al (± 0.05) o he expe imen ally de e - mined ΘM alues o hese powde s. Howe e , in he case o he nickel powde , he alue gi en by he i is abo e he Fig. 3 Va ia ion o he expe imen al he mal conduc i i y as a unc ion o po osi y and heo e ical cu e, Eq. (4), ob ained by leas squa es adjus men , o he di e en ma e ials s udied Po osi y e ec on he he mal conduc i i y o sin e ed powde ma e ials Page 9 o 13 149 measu ed alue, conside ing i s uppe unce ain y in e al. I also happens ha his powde is he one ha gi es he wo s i , acco ding o he alue o R2. An analysis o he cha ac e is ics o his powde may lead us o he conclu- sion ha i could be he sphe ical geome y o he ini ial powde ha is esponsible o his g ea e disag eemen . P ecisely, his powde mo phology is no pa icula ly desi - able in Powde Me allu gy, excep o e y speci ic applica- ions, especially aimed a a ou ing he high open po osi y o compac s, such as sel -lub ica ing bushings [74]. On he o he hand, he SEM images in Fig.3 e eal ha he mean pa icle size o he nickel powde is clea ly smalle han he es and exhibi li le size dispe sion. In addi ion, he pa icles a e agg ega ed, o ming clus e s. All hese ac o s, absen in he o he powde s, could be addi ional easons o he la ge disc epancy. Figu e4 shows he expe imen al da a o all he ma e i- als in he same g aph. Fo his pu pose, he ela i e he - mal conduc i i y (calcula ed by no malising by he alue o he ully dense ma e ial conduc i i y esul an o he i ) is plo ed agains he ela i e po osi y (calcula ed by no - malising by he alue o he ap po osi y ob ained in he i ). The collec i e poin cloud has been i ed by leas squa es o Eq.(4) exp essed wi h ela i e a iables, and he coe - icien o de e mina ion o he collec i e i ob ained was R2 = 0.9917. Again, a e y accep able alue. On he o he hand, Table5 shows he esul s o he leas squa es i s using a ious models selec ed om Table1 and he expe imen al da a measu ed in his wo k. To acili a e he compa ison job, he alues ob ained in he i s wi h he model p oposed he e ha e also been included. The o he selec ed models ha e been: he Linea law [41], he A chie model [38], he Pe cola ion law [55], he G uzde e al. model [46], he Pabs and G ego o á model [50], he Solonin and Che nyshe model [51] and he Ai azo and Domash- ne model [42]. Models wi h 2 and 3 adjus able pa ame e s (some o hem subjec o he a o emen ioned cons ain s) ha e he e o e been included. I would be expec ed ha he goodness o i would be g ea e o models wi h mo e deg ees o eedom, ha is, wi h a g ea e numbe o adjus - able pa ame e s. Fo he NC100.24 i on powde , he model ha p o ides he bes i ing among hose selec ed is he Pe cola ion law, wi h 3 adjus able pa ame e s; 2 subjec o cons ain s (κ0 and ΘM) and ano he one (n) comple ely ee. Howe e , he i ing goodness achie ed di e s e y li le om ha achie ed by he model p oposed in his pape , which has only 2 deg ees o eedom and wi h cons ain s (in κ0 and ΘM). The esul ing n alue in he Pe cola ion law is abou 1.4, close o he alue o 1.5 se by he model p oposed he e. On he o he hand, he model o Ai azo and Domashne , wi h 2 adjus able pa ame e s, p o ides an also good ag ee- men , only sligh ly lowe han he commen ed models. Fo ha model, he alue o he adjus able pa ame e a u ns ou o be 11.08, close o he alue o 10 se by Koh and Fo ini [28]. Con a y o expec a ions, i is no he linea law ha p o ides he wo s ag eemen , bu he A chie model; bo h wi h 2 adjus able pa ame e s. Fo he WPL200 i on powde , bo h he Pe cola ion law and he model p oposed he e achie e he same deg ee o ag eemen . Again, he alue o n u ns ou o be abou 1.4, again close o 1.5. An only sligh ly lowe ag eemen is achie ed by he model o Ai azo and Domashne , wi h 2 adjus able pa ame e s, and o which he ee pa ame e a now u ns ou o be wo h abou 9; close again o he alue o 10, se by Koh and Fo ini [26]. Also in his case, he linea law p o ides be e ag eemen han A chie model. Fo he 4SP400 nickel powde , all he selec ed models achie e lowe ag eemen s han he es o he powde s. A chie model, Pe cola ion law and he model o G uzde e al. achie e he highes R2 alues. Howe e , he alues o ΘM and Θ0 esul ing om he i s a e equal o uni y, which is no a alue consis en wi h he meaning a ibu ed o hem by he espec i e models, in he con ex o powde ed ma e ials. The model p oposed he e achie es only sligh ly lowe ag ee- men and a alue o ΘM ha is also oo high, bu less han uni y, hus mo e consis en . The A chie model imp o es i s conco dance compa ed o he o he models. Fo he AS61 aluminium powde he bes ag eemen is p o ided, again, by he iad A chie model, Pe cola- ion law and he model o G uzde e al. As in he case o nickel powde , he esul ing alues o ΘM and Θ0 a e Fig. 4 Expe imen al da a o he mal conduc i i y o all ma e i- als and heo e ical cu e, om no malised Eq.(4), ob ained by leas squa es i ing