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