Physics o Compac ion o Fine Cohesi e Pa icles
A. Cas ellanos, J. M. Val e de, and M. A. S. Quin anilla
Depa amen o de Elec onica y Elec omagne ismo, Uni e sidad de Se illa, A enida Reina Me cedes s/n, 41012 Se illa, Spain
(Recei ed 9 No embe 2004; published 22 Feb ua y 2005)
Fluidized ac al clus e s o ine pa icles display c i ical-like dynamics a he jamming ansi ion,
cha ac e ized by a powe law ela ing consolida ion s ess wi h olume ac ion inc emen [ ^c/].
A a c i ical s ess clus e s a e dis up ed and he e is a c osso e o a loga i hmic law (log ^c)
esembling he phenomenology o soils. We measu e @1==@ log ^c/Bo0:2
g, whe e Bogis he
a io o in e pa icle a ac i e o ce (in he luidlike egime) o pa icle weigh . This law sugges s ha
compac ion is uled by he in e nal packing s uc u e o he jammed clus e s a nea ly ze o consolida ion.
DOI: 10.1103/PhysRe Le .94.075501 PACS numbe s: 61.43.G , 45.70.Cc, 61.43.H , 81.20.E
Empi ical s udies on he compac ion o soils da e back o
he beginning o he las cen u y. Walke [1] i ed his da a
by he loga i hmic law 1=logc=c0, whe e
is he pa icle olume ac ion, c he applied consoli-
da ion s ess, and (comp ession index) and c0a e em-
pi ical pa ame e s. This equa ion applies well in loose
samples, whe e compac ion is d i en by ea angemen
o pa icles, and has been adi ionally used in ci il engi-
nee ing [2,3]. An essen ial ing edien in mos g anula sys-
ems is cohesion. Tes s on cohesi e powde s show ha 
dec eases wi h he pa icle olume ac ion o he ini ial
s a e [4,5], indica ing ha in e pa icle a ac i e o ces,
which a o he o ma ion o po ous s uc u es, play a ele-
an ole in he compac ion p ocess. Ye he ini ial s a e in
ypical enginee ing expe imen s in ol es consolida ion
s esses c0>10 kPa [5]. Many indus y applica ions de-
mand esea ch on smalle consolida ions as hese co e-
spond o condi ions o powde low. Fo example, in he
handling o xe og aphic one s, ypical consolida ions ange
om a ew pascals o a ew hund ed pascals. Mo eo e ,
expe imen s a low consolida ions ha e a undamen al in-
e es in o de o cha ac e ize he ansi ion om he luid-
like o he solidlike s a e (jamming) [6] since he s uc u al
p ope ies o he unconsolida ed jammed s a e (c’0),
which is he uly ini ial s a e in any compac ion p ocess,
a e de e minan on he ea angemen o he u he loaded
pa icles. We s udy he compac ion o ine pa icles wi h
con olled a ac i e o ce, ini ially luidized and la e sub-
jec ed o loads om jus a ew pascals up o 10 kPa. Ou
no el expe imen al s udy is aimed o shed ligh on he ole
o he ini ial s a e, i.e., he unconsolida ed jammed s a e, on
compac ion. The powde s es ed a e xe og aphic one s
based on polyme (pa icle densi y p’1g=cm3). They
a e p oduced by an a i ion p ocess, hus ha ing an i egu-
la shape, and size classi ied in a ange o pa icle sizes
(dp) om 19.1 o 7mby ae odynamic classi ica ion,
showing a na ow pa icle size dis ibu ion (see Fig. 1).
Addi ionally, he powde s a e blended wi h umed silica
nanopa icles (ei he 8 o 40 nm nominal diame e s) o coa
uni o mly he polyme pa icle su ace in concen a ions
om 10% o 100% o su ace a ea co e age (SAC).
In he luidized egime he e is an a ac i e o ce F0
be ween he d y and uncha ged pa icles mainly a ising
om he an de Waals in e ac ion F0F dW ’Ada=
24z2
0, whe e z0’4
Ais he dis ance o closes app oach
be ween wo molecules, Ais he Hamake cons an , and da
is he ypical size o he su ace aspe i ies ( ypically A
1019 Jand da0:2m) [7]. An es ima ion o elec o-
s a ic o ces om cha ge spec og aph measu emen s
shows ha hey a e much smalle han an de Waals o ces
as commonly accep ed in he li e a u e o elec oneu al
ine powde s [7,8]. A d y ni ogen a mosphe e minimizes
also capilla y o ces. Because o he s ong in e pa icle
a ac i e o ce as compa ed o pa icle weigh , one pa -
icles a e clus e ed in he luidlike egime [9]. Acco ding o
ou p e ious expe imen al esul s he ypical numbe o
pa icles pe clus e Nand ypical a io o clus e size o
pa icle size depend on he a io o a ac i e o ce o
pa icle weigh F0=mpgBog(g anula Bond num-
be ). In pa icula we ound NBo
g(’0:7) and D
lnN=ln’2:5 o he ac al dimension, in ag eemen
FIG. 1. Typical pic u e om he op ical mic oscope o one
pa icles clus e ed in a nonaqueous liquid suspension ha
eminds one o a di usion-limi ed agg ega e. Fo his one
(12:7mpa icle size and 10% SAC) he ac al dimension
ob ained om se ling expe imen s in gas luidiza ion [9] is
D’2:53.
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0031-9007=05=94(7)=075501(4)$23.00 075501-1 2005 The Ame ican Physical Socie y
wi h he di usion-limi ed-agg ega ion (DLA) model p e-
dic ion. The ypical size o su ace aspe i ies a con ac can
be dec eased down o he size o silica agglome a es co e -
ing he pa icle su ace o la ge enough SAC, hus educ-
ing F dW. (F om scanning elec on misc osopy mic o-
g aphs we obse e silica agglome a es o ypical size da’
50 nm o 8 nm silica nanopa icles and da’200 nm o
50 nm silica nanopa icles.) Since ou ine pa icles a e
a he mal, he limi s o clus e g ow h in he ini ial luidized
s a e esul om he in e play o g a i a ional and low
shea e ec s [10]. In o de o es ima e he limi size o ou
DLA clus e s we will adop a simila app oach o ha used
in Re . [11], whe e he limi s o gela ion in colloidal
agg ega ion we e s udied. In he luidized bed he weigh
o he clus e is balanced by he hyd odynamic d ag om
he su ounding gas. F ac al clus e s sc een ex e nal ields
e y e ec i ely and he luid low inside he clus e is
negligible compa ed o he low ou side, hus he d ag
ac s mainly a he su ace o he clus e whe eas g a i y
is a body o ce ac ing uni o mly h ough he clus e . This
esul s in shea o ces dis ibu ed ac oss he clus e limi ing
i s size. Using a simple sp ing model o he clus e , i has
been shown [10,11] ha he ypical s ain on he clus e is
Nmpg=KcRc, whe e Kcis he clus e sp ing cons an
and Rcdp=2is he clus e adius. Kcis gi en by
k0=, whe e k0is he in e pa icle o ce cons an , and
he elas ici y exponen is 3in he 3D case [12]. Thus
he local shea o ce inside he clus e is Fsk0dp=2
mpgD2. Manley e al. [11] use a c i ical alue, mea-
su ed independen ly, o he maximum s ain sus ainable o
calcula e he maximum size o hei agg ega es. Mo e
gene ally, we may es ima e ha he c i ical shea o ce
mus be o o de o he in e pa icle a ac i e o ce Fmax
s
F0, which leads o BogD2, hus he maximum numbe
o pa icles pe clus e should be NDBoD=D2
g.
Fo DLA clus e s (D2:5) we ob ain NBo0:6
g, in close
ag eemen wi h ou p e ious expe imen al esul s [9]. Now
we can explain why, o a cons an F0(cons an SAC), he
size o ou clus e s measu ed in Re . [9] was weakly
dependen on pa icle size since he c i e ion p edic s
kdp/d0:3
p. Mo eo e , o ou ypical clus e s (<10)
he in e clus e Bond numbe is Bo
gBog=N 2<
100, i.e., in e clus e cohesi eness is small.
In he expe imen al pa o ou wo k we use he luidized
bed es e o measu e as a unc ion o c. A de ailed
epo abou he unc ioning o his appa a us can be ound
in Re . [13]. A d y ni ogen se es o con ol c, being
pumped upwa d o downwa d h ough he powde bed
while he gas p essu e d op pac oss he bed is ead
om a di e en ial p essu e ansduce . is de i ed om
he heigh o he bed, which is measu ed by means o an
ul asonic senso . In o de o subjec he powde o e y
low s esses, like in mic og a i y, he bed is allowed o
se le unde a small upwa ds di ec ed gas low. In his way
cis lowe ed down o cWp, whe e Wis he
powde weigh pe uni a ea and pinc eases as he alue
o he decomp essing gas low used is la ge . In Fig. 2 we
ha e plo ed nea he jamming ansi ion as a unc ion o
c o one Canon CLC700 (100% SAC, in his comme -
cial one he addi i e is TiO2) and o an expe imen al
one wi h simila pa icle size (7:8m) bu only 32%
SAC. The jamming ansi ion is discussed in de ail in a
p e ious wo k [14]. As seen in he da a depic ed in Fig. 2, i
was gene ally obse ed ha in a ange o e y small
s esses he  s c ela ionship ollows a c i ical-like
unc ional o m c/J, p edic ed by simula ions
[15] and eminiscen o an equilib ium c i ical phenomena.
In ou new expe imen al s udy consolida ion s esses
la ge han Wa e applied by a downwa ds di ec ed gas
low. cis hus inc eased up o cWp. We see in
Fig. 2 ha he inc emen o as cis inc eased de ia es
om he c i ical-like powe law, and a a c i ical s ess
c010 Pa c osses o e o a loga i hmic law ’J
log ^c(^cc=c0) ema kably simila o he empi i-
cal equa ion usually employed o desc ibe he compac ion
o g anula ma e ials such as soils [2] in he ea angemen
egime. Likely a ^c’1clus e s ha e eached hei closes
andom packing (
RCP). The c osso e o he loga i hmic
law occu s o olume ac ions o clus e s smalle han
0.64 ( andom close packing o noncohesi e ha d sphe es)
as i migh be expec ed om he exis ence, al hough small,
o in e clus e cohesi eness. Fo example, 
RCP ’0:53 
FIG. 2. Pa icle olume ac ion as a unc ion o he consoli-
da ion s ess. Da a ob ained by consolida ing he powde allow-
ing i o se le unde upwa d di ec ed gas lows (solid sym-
bols) a e join ly plo ed wi h da a ob ained by consolida ing
he powde by means o a downwa d di ec ed gas low (open
symbols). Da a co espond o comme cial one Canon CLC700
(100% SAC) and o an expe imen al one wi h educed su ace
addi i e co e age (32% SAC) and simila pa icle size. The lines
co espond o he powe law c/J ha i s o he da a
in a ange c&c010 Pa and o a loga i hmic i J’
logc=c0 ha i s o he da a in he ange c*c0. Inse :
da a in he loga i hmic egion o one s wi h he same SAC
(32%) bu di e en pa icle size (indica ed).
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0:02 o clus e s o p ima y pa icle size om 7.8 o
19:1mand 32% SAC (see Fig. 4 in Re . [14]), ma ching
he epo ed alue [16] o sphe es o equi alen size o
hese clus e s (kdp’50 m). The inse o Fig. 2 shows
da a o hese one s wi h he same SAC (32%) bu di e -
en pa icle size. I is obse ed ha ’0:04 is almos
independen o pa icle size, sligh ly dec easing when he
SAC is inc eased (see main g aph). Thus he co ela ion
be ween and Bogseems o be a leas a second o de
e ec (no e ha a dec ease o dp om 19.1 o 7:8m
co e s a wide ange o Bog). Howe e , he da a epo ed in
he enginee ing li e a u e e eal clea co ela ions be ween
@1==@ log ^cand he pa icle olume ac ion
o he lowes consolida ion s a e. We plo in he inse o
Fig. 3 he da a o he same one s (32% SAC) o 1= s
^c, which a e also well i ed o a loga i hmic law 1= ’
1=Jlog ^c. The main g aph shows as a unc ion
o Bog(calcula ed assuming F0F dW). A powe law
’0:1Bo0:21
gis clea ly seen. The ex apola ed alue o
Bog1(’0:1) ma ches he ypical alue epo ed
o noncohesi e g anula ma e ials (Bog&1) such as
sand [3]. Wha is he physical o igin o his law? In a
i s o de app oach we may app oxima e 1=’
@1==@ log ^cJlog ^c’1=J2log ^c, whe e
’@=@ log ^cJ. Thus ’1=J2. Le us w i e
J’
RCPc, whe e cN=3D3is he pa -
icle olume ac ion wi hin each clus e . Then ’
1=
RCP2Bo62D=D2
g1=
RCP2Bo0:22
g, whe e
we ha e used he clus e size limi c i e ion (Bog
D2), and D2:5. Clus e s beha e as low cohesi e
e ec i e sphe es and hus 
RCP will be almos independen
on Bog. Using ’0:04 and 
RCP ’0:53 o he one s
wi h 32% SAC, we would p edic ’0:14Bo0:22
g, in good
ag eemen wi h he expe imen al esul . We plo in Fig. 4
he da a o  s Bog o o he one s wi h SAC >30% o
which we admi F0F dW. The new da a also scale wi h
Bogin acco dance wi h he p edic ed law. In summa y, his
law emphasizes he undamen al ole o size and ac al
s uc u e o he jammed clus e s on he dis ibu ion o
oids o be illed in he compac ion p ocess. The case o
highly cohesi e powde s (SAC <30%) needs, howe e ,
addi ional discussion. In he inse o Fig. 4 we include da a
o highly cohesi e one s wi h only 20% SAC using F0
F dW o calcula e Bog. The pa icle size o hese one s is
’7mand he base polyme is polyes e ea ed wi h
di e en amoun s o a c oss-linking agen (gel) ha p o-
duces a sligh inc ease o he polyme ha dness (in any case
small compa ed wi h he e ec o silica). The da a de ia e
clea ly om he scaling law, showing unexpec edly la ge
alues o  ha mus indica e he exis ence o la ge clus-
e s. Mo eo e , in spi e o he simila alues o Bog(same
pa icle weigh and same an de Waals o ce: A’1019 J,
FIG. 3. Comp ession index e sus he g anula Bond numbe
o one s wi h he same su ace addi i e co e age (32% SAC)
and a ying pa icle size (indica ed). The con inuous line is a
powe law i o he da a (’0:1Bo0:21
g). Inse : in e se o
pa icle olume ac ion e sus he consolida ion s ess, whe e
lines a e loga i hmic i s o he da a (c010 Pa).
FIG. 4. Comp ession index e sus g anula Bond numbe o
one s wi h di e en pa icle size and % SAC (indica ed). In he
main g aph Bogis calcula ed assuming ha he in e pa icle
a ac i e o ce in luidiza ion is he an de Waals o ce o
SAC >30%, while con ac memo y (see ex ) is conside ed o
one s wi h SAC <30%. In he inse he an de Waals o ce is
used o all he one s. Fo one s wi h 20% and 100% SAC, he
e ec o inc easing he size o silica nanopa icle addi i es om
8 o 50 nm has been es ed. Addi ionally, ou di e en amoun s
o c oss-linking agen in he pa en polyme ( om 0% o 45%)
ha e been used in hese one s. The con inuous lines a e powe
law i s o he da a, only o one s wi h SAC >30% in he inse
(/Bo0:24
g) and o he whole se o one s in he main g aph
(/Bo0:25
g).
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da’0:2m), he e is a clea di e ence be ween he
comp ession indexes o one s wi h di e en gel con en .
Silica addi i es imp o e powde lowabili y mainly by
inc easing he local ha dness Hon he con ac as i is
p o en by he be e lowabili y o one s blended wi h
200 nm silica agglome a es, o simila size o he ypical
polyme aspe i y size. The s ong a ac i e o ces be ween
loaded ine pa icles cause plas ic de o ma ion o con ac s,
leading o a ele an inc ease o he adhesi e o ce wi h he
applied load [17]. Tone s ei he wi hou o wi h low pe -
cen age o he ha d silica addi i e ha e a e y poo low-
abili y because i is di icul o b eak in e pa icle con ac s
be ween p e iously loaded pa icles, i.e., he in e pa icle
con ac s in highly cohesi e powde will p ese e he mem-
o y o he ini ial loaded s a e. I is likely ha agmen s ha
ea lie exis ed as agg ega es in he loaded powde pe sis in
luidiza ion gi ing ise o la ge clus e s o s ongly adhe ed
pa icles (a simila phenomenon has been ecognized in
agmen a ion o colloidal suspensions o s ongly cohe-
si e pa icles [18]). Thus he a ac i e o ce be ween
clus e ed pa icles in luidiza ion o hese powde s mus
be much la ge han he an de Waals o ce. The inc eased
con ac ha dness by silica addi i e educes he a e o
inc ease o he adhesi e o ce wi h load [17], hus allowing
o an easy b eaking o in e pa icle con ac s by an ex e nal
ene gy sou ce (such as gas luidiza ion), and he e o e
imp o ing lowabili y. Fo hese low cohesi e one s he
an de Waals o ce was indeed a good app oxima ion o
he in e pa icle a ac i e o ce in luidiza ion. Fo one s
wi h only 20% SAC many con ac s a e be ween polyme
su aces. The a e o inc ease o he adhesi e o ce wi h
load inc eases as His inc eased [17], which means ha
con ac s wi h smalle ha dness will gi e ise o la ge
clus e s and hus o la ge alues o . (This explains he
e ec o gel seen on .) The p oblem is, How can we
es ima e Bog o hese highly cohesi e one s wi h ha d-
ness dependen memo ies? One possibili y is o use he
c i e ion o clus e limi size in luidiza ion, BogD2,
whe e D’2:5acco ding o sedimen a ion es s [9], and 
may be ob ained om he in o ma ion on he ini ial
jammed s a e: D=3J=
J. In his equa ion he
mos impo an and a iable pa ame e is J, which is
accu a ely ob ained om he loga i hmic i equa ion J
log ^c, while 
Jcan be expec ed o show a small
a ia ion be ween 0:5(10% SAC) and 0:55 (abo e
60% SAC). Fo low cohesi e one s (SAC *30%), he
es ima ed Bogin his way is simila o he p e iously
calcula ed one assuming F0F dW, bu , o he highly
cohesi e ones, Bogis signi ican ly la ge as we an icipa ed.
The main g aph o Fig. 4 includes he da a o 20% and
10% SAC one s wi h he new es ima ion o Bog, showing
a good i o he p edic ed powe law. We can also dis-
c imina e now be ween he comp ession indices o 20%
SAC one s in he base o he memo y e asing e ec o he
c oss-linking agen added o he pa en polyme .
To conclude, we ha e in es iga ed he compac ion be-
ha io o cohesi e pa icles which a e clus e ed in he
luidlike egime. The numbe o pa icles in ou DLA
clus e s, and hus he clus e packing ac ion (c), is
con olled by he a io o he in e pa icle a ac i e o ce
o pa icle weigh Bog(c’Bo0:1
g). In he close icini y
o jamming and abo e a c i ical s ess c010 Pa, he e
is a c osso e o he loga i hmic law 1= ’1=J
logc=c0. Expe imen al s udies in he enginee ing
li e a u e ha e usually shown his beha io , and he com-
p ession index has been co ela ed o he minimum
pa icle olume ac ion (co esponding o he smalle
s ess applicable, ypically 10 kPa); he la ge i is, he
la ge he . Ou luidiza ion echnique allows o a s udy
o compac ion beha io om jus a ew pascals, hus we
a e able o ela e  o he uly ini ial s a e o ea ange-
men . The ini ial dis ibu ion o oids o be illed, which
ules he compac ion p ocess, is mainly de e mined by he
in e nal packing s uc u e o clus e s ha a e jammed a he
ini ial unconsolida ed s a e. In a i s o de analysis we
es ima e /1=c
J2’Bo0:2
g, in ag eemen wi h ou mea-
su emen s. Ou analysis implies also ha , due o he high
plas ici y o in e pa icle con ac s in highly cohesi e pow-
de s, hese powde s mus e ain memo y in luidiza ion o
p e ious loaded s a es; when jammed, e y la ge clus e s
p oduce e y po ous ini ial s a es and as a consequence
la ge alues o he comp ession index.
We acknowledge he Xe ox Founda ion and he
Spanish Minis e io de Ciencia y Tecnologia (Con ac
No. BMF2003-01739).
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PRL 94, 075501 (2005) PHYSICAL REVIEW LETTERS week ending
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