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S a is ics o disloca ion a alanches
in FCC and BCC me als: disloca ion
mechanisms and mean swep
dis ances ac oss mic osample sizes
and empe a u es
Jo ge Alcalá1*, Jan Očenášek2, Ja ie Va illas1,2,6, Jaa a A. El‑Awady3,
Je ey M. Wheele 4,5 & Johann Michle 5
Plas ic de o ma ion in c ys alline ma e ials consis s o an ensemble o collec i e disloca ion glide
p ocesses, which lead o s ain bu s emissions in mic o‑scale samples. To un a el he combined ole o
c ys alline s uc u e, sample size and empe a u e on hese p ocesses, we pe o med a comp ehensi e
se o s ic displacemen ‑con olled mic opilla comp ession expe imen s in conjunc ion wi h la ge‑
scale molecula dynamics and physics‑based disc e e disloca ion dynamics simula ions. The esul s
indica e ha plas ic s ain bu s s consis o nume ous indi idual disloca ion glide e en s, which span
o e minuscule ime in e als. The size dis ibu ions o hese e en s exhibi a g adual ansi ion om
an incipien powe ‑law slip egime (spanning
≈
2.5 decades o slip sizes) o a la ge a alanche domain
(spanning
≈
4 decades o emission p obabili y) a a cu ‑o slip magni ude
sc
. This cu ‑o slip p o ides
a s a is ical measu e o he cha ac e is ic mean disloca ion swep dis ance, which allows o he
scaling o he a alanche dis ibu ions is-à- is he a che ypal disloca ion mechanisms in ace‑cen e ed
cubic (FCC) and body‑cen e ed cubic (BCC) me als. Ou s a is ical indings p o ide a new pa hway o
cha ac e izing me al plas ici y and owa ds comp ehension o he sample size e ec s ha limi he
mechanical eliabili y in small‑scale s uc u es.
C ys al plas ici y is cha ac e ized by he onse o in e mi en disloca ion glide e en s o a alanches, which lead
o p ominen s ain bu s emissions when he sample size is educed o a ew mic ome e s. The emission p ob-
abili y,
P
, o a disloca ion bu s o size
s
is asc ibed o he powe -law ela ion
P(s)∝s−
, whe e he exponen ,

, is a uni e sal pa ame e wi h scale-in a ian alue
≈
1.51–11. This a guably led o he in e p e a ion in ha he
size dis ibu ion o he disloca ion bu s s ul ils a mean- ield depinning phase ansi ion pa adigm cha ac e ized
by a c i ical

exponen 3,7–10,12–14. T unca ion o in e mi en esponses is a key ea u e in he scaling o physical
phenomena anging om ea hquakes o neu al ac i i y15. In c ys al plas ici y,
P(s)
is exponen ially unca ed
when he s ain bu s s app oach a cu -o alue. The no ion o s ess- uned c i icali y (STC) hen a ises when
his cu -o slip is go e ned by he applied s ess, while olume- uned c i icali y (VTC) implies ha he cu -
o depends on sample size. In bo h models, uned c i icali y may a ise a c i ical alues o s ess and sample
olume3,7,8,12,13. Al e na i ely, in he ad en o sel -o ganiza ion p ocesses o he disloca ion s uc u e du ing
s aining, disloca ion glide may occu a a ixed c i icali y le el. The slip dis ibu ion hen becomes independen
o he applied s ess, leading o he hypo hesis o sel -o ganized c i ically (SOC)12,15–18. Along hese lines, i may
be a gued ha he sel -simila scaling o he disloca ion cells a ising a la ge shea s ains du ing single c ys al
de o ma ion may en a i ely esul in SOC.
OPEN
1Depa men o Ma e ials Science and Me allu gical Enginee ing, InSup, ETSEIB, Uni e si a Poli ècnica de
Ca alunya, 08028 Ba celona, Spain. 2New Technologies Resea ch Cen e, Uni e si y o Wes Bohemia in Pilsen,
30614 Plzeň, Czech Republic. 3Depa men o Mechanical Enginee ing, Whi ing School o Enginee ing, The Johns
Hopkins Uni e si y, Bal imo e, MD 21218, USA. 4Labo a o y o Nanome allu gy, Depa men o Ma e ials, Swiss
Fede al Ins i u e o Technology in Zü ich, Vladimi -P elog-Weg 5, CH-8093 Zü ich, Swi ze land. 5Labo a o y
o Mechanics o Ma e ials and Nanos uc u es, Empa, Swiss Fede al Labo a o ies o Ma e ials Science and
Technology, 3602 Thun, Swi ze land. 6P esen add ess: Ins i u e o The momechanics, Czech Academy o
Sciences, 182 00 P ague 8, Czech Republic. *email: [email p o ec ed]
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The ull- ledged e olu iona y cha ac e o he disloca ion ne wo k becomes mani es du ing plas ic s aining
o mac oscopic ma e ial samples, whe e he mobile disloca ions a e in e mi en ly pinned by immobile o es
coun e pa s h ough junc ion o ming p ocesses18. While he applied s ess and disloca ion densi y bo h luc u-
a e du ing subsequen a alanche emissions, a ne inc ease in disloca ion densi y p e ails due o he s o age o
he mobile disloca ions wi hin he s eady ne wo k, yielding a ne posi i e slope
θ
in he s ess–s ain cu e ha
quan i ies onse o s ain ha dening19. This concep ion is ma kedly a ec ed by he sample size. In he smalle
mic ome e and submic ome e -sized ee-s anding c ys als, plas ici y is go e ned by he sca ci y o disloca-
ion sou ces and by he onse o a single-ended disloca ion sou ce a angemen pinned a he sample bulk
ha in e mi en ly p oduces disloca ion a alanches. The e m con ined plas ici y is hen coined o emb ace he
no ion in ha he o ma ion o he disloca ion ne wo k is d as ically a ec ed by he limi ed sample size. Unde
hese condi ions, su ace annihila ion o mobile disloca ions becomes a undamen al mechanism ha p e en s
disloca ion ne wo k de elopmen 20–27.
A ansi ion o bulk-like plas ici y may occu in mic ome e -sized samples whe e he incipien disloca ion
s uc u e e ol es in o a hea ily en angled (dense) ne wo k con aining a my iad o disloca ion sou ces ha ac i-
a e a la ge applied s esses. Disloca ion mul iplica ion hus ou weighs su ace annihila ion. Bulk-like plas ic-
i y also de elops in la ge mic oc ys als wi h mildly en angled ne wo ks, whe e he in e ac ion o he mobile
disloca ions wi h he o es a angemen al eady hinde s signi ican su ace annihila ion. Since he disloca ion
densi y may app oach simila alues as in mac oscopic scales, plas ic s aining occu s a simila le els o applied
s esses2,18,19,28–30.
Compu a ional simula ions ha e shown ha he in e ac ion be ween mobile and o es disloca ions a ec s
he a alanche dis ibu ions. The seminal mean- ield unde s a ing o his esul is ha wi h he inc easing dislo-
ca ion en anglemen ha occu s upon he onse o subsequen s ain bu s s, he dis ance o he c i ical d i ing
s ess emains cons an which esul s in he a ainmen o an in e mi en , s eady-s a e esponse in he con ex
o SOC. The cu -o slip size hen becomes independen o he applied s ess4,7,12,16–18. Ten a i ely, STC would
hen p e ail unde anishing ha dening (
θ
→0
), whe e g ea e cu -o slip magni udes esul wi h inc easing
applied s esses. Recen in es iga ions ha e howe e challenged he abo e mean- ield depinning scena io13,14,
showing ha he a alanche dis ibu ion c osses o e be ween wo powe -law a alanche egimes a a c i ical
cu o slip size14. Expe imen al e idence is s ill needed in suppo o hese concep ions, including he in luence
o he cha ac e is ic ace-cen e ed cubic (FCC) and body-cen e ed cubic (BCC) disloca ion glide mechanisms
in mic o-scale c ys als de o ming unde con ined and bulk-like plas ici y.
He e, we in es iga e he disloca ion mechanisms and s a is ical ea u es o he indi idual disloca ion glide
e en s which we e concealed in he p e ious in es iga ions o s ain bu s s in mic o-scale sample sizes. A com-
p ehensi e se o s ic displacemen -con olled mic oc ys al comp ession expe imen s and physics-based simu-
la ions was pe o med o gain access o hese indi idual in e mi en e en s, spanning o e ew nanoseconds,
which a e ega ded as he undamen al cons i uen elemen s o disloca ion-media ed plas ici y. We show ha
he size dis ibu ion o he indi idual disloca ion glide e en s is cha ac e ized by a ansi ion om an incipi-
en powe -law slip egime o a la ge a alanche domain. This ansi ion is a inge p in o he disloca ion glide
phenomenology de eloping in FCC and BCC mic oc ys als, which go e ns he onse o con ined and bulk-like
plas ici y in mic ome e -sized samples along wi h he compe i ion be ween STC and SOC.
Me hods
Mic opilla comp ession expe imen s. Ou mic oc ys al comp ession expe imen s, pe o med unde
s ic displacemen con ol, enabled he iden i ica ion o indi idual disloca ion glide e en s. These a e he con-
s i uen plas ic in e mi encies which occu wi hin he s ain bu s s obse ed in expe imen s d i en wi h con-
en ional dynamically- eac i e es ing sys ems (see Supplemen a y Ma e ial, Sec ion1, and he cu en Resul s
and discussion sec ion o a desc ip ion on he in luence o he s aining mode and es ing sys em on he plas ic
in e mi encies). The cylind ical pilla -like mic oc ys als used in ou comp ession expe imen s we e ocused
ion beam (FIB) milled om [111]-o ien ed Cu and Ta single c ys als, a [100]-o ien ed Al single c ys al, and a
[100]-o ien ed W single c ys al. Pilla -like ec angula mic oc ys als wi h squa e c oss-sec ions we e also FIB-
milled om a [100]-o ien ed Cu single c ys al and a [123]-o ien ed Al su ace. Beam cu en in he Ga sou ce
was educed o less han 200 pA in all inal machining s eps o educe damage. The cylind ical Al mic oc ys als
had a diame e
D≈
8µm whe eas he Cu, Ta and W coun e pa s had
D
anging om 0.5 o 20µm. The e ec-
i e diame e s o he ec angula Cu mic oc ys als anged om
D≈
0.5 o 4µm. In all cases, he sample heigh ,
l
, was
≈
3
D
. The ec angula samples we e hea ea ed a 600°C o 1h o emo e any FIB-induced disloca ion
loops31.
The comp ession expe imen s we e pe o med using a modi ied Alemnis nanoinden e (Alemnis GmbH,
Thun, Swi ze land) i ed inside he high acuum chambe o a Zeiss DMS 962 scanning elec on mic oscope
(SEM). This sys em is capable o ope a ing unde s ic , in insic displacemen con ol, hus main aining inden e
ip posi ion du ing sudden a alanche emissions (Sec ion1 in Supplemen a y Ma e ial). The nanoinden e was
cus om modi ied o enable es ing a high empe a u es wi h he sample and inden e empe a u e ma ched o
minimize any in luence o he mal expansion d i on applied displacemen s32. The applied s ain a es,
˙ε
, anged
om
10−4
o
10−3
, which is su icien ly small so as o neglec addi ional s aining p oduced du ing a alanche
p opaga ion (Sec ion2 in Supplemen a y Ma e ial). A de ailed accoun o some o he s ess–s ain cu es is
published elsewhe e18,31,33.
MD and DDD simula ions. Molecula dynamics (MD) simula ions we e conduc ed by uniaxially com-
p essing cuboidal [110]-o ien ed Al and cuboidal [100]-o ien ed Ta simula ion cells wi h pe iodic and non-pe i-
odic bounda y condi ions; whe e he disloca ion densi y,
ρ
, anged om
≈1015
o
1016
m−2
o mimic bulk-like
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plas ici y a ex eme disloca ion densi ies. The simula ion cell size was 70
×
70
×
40nm comp ising
≈
11 million
a oms o Al and
≈
10 million a oms o Ta, modelled h ough he embedded-a om me hod (EAM) po en ials
in Re s.34,35. All simula ions we e ca ied ou wi h he LAMMPS code36 unde an NPT ensemble, whe e he
uniaxial s ess
σ
is ob ained om he p essu e enso in he di ec ion o he applied displacemen (wi h a e-
quency o 200 s) while he ba os a ac s in he o he main di ec ions o he simula ion cell. The Al cells we e
de o med a RT and he Ta coun e pa s a RT and 630°C wi h
˙ε
o 1.5
×106s−1
. The ne wo ks we e in oduced
in o he compu a ional cells by b inging one o he ee su aces in o con ac agains a sphe ical inden e un il
a dense de ec ne wo k a ained (see37 and Sec s.6 and 7 in he Supplemen a y Ma e ial). Upon applica ion o
pe iodic bounda y condi ions o he simula ion cells, he s ess–s ain cu es we e aken o ep esen bulk ma e-
ial esponses. The ees anding MD simula ions we e hen exclusi ely employed o assess slip ace o ma ion
a he su ace. De ails o he s eng hening p o ided by he incep ed ne wo ks (see Re .38) a e gi en in Sec .6 o
he Supplemen a y Ma e ial.
The 3D DDD simula ions we e conduc ed wi h an in-house e sion o he open sou ce code Pa aDis30,39 o
cap u e con ined plas ici y esponses whe e disloca ion annihila ion occu s a he ee su aces. All disloca ion
eac ions a e plana in his in-house e sion o he open sou ce code, u he inco po a ing a se o a omis ically-
in o med, physics-based c oss-slip mechanisms, he de ails o which a e desc ibed in30. The assumed elas ic
ma e ial p ope ies ma ched hose o FCC Ni (Young’s modulus E
=
210 GPa, and Poisson’s a io ν = 0.31). The
s ess–s ain cu es we e ob ained by uniaxially comp essing ec angula compu a ional cells wi h a squa e
c oss-sec ion unde a nominal s ain a e con ol o
˙ε=
200s−1 (whe e he s ess
σ
along he s aining di ec ion
is e alua ed wi h a equency o 5ns). F ee-s anding [100]-o ien ed mic oc ys als we e comp essed wi h
D=
2
and 5μm wi h aspec a io
l/D=3
. The ini ial disloca ion mic os uc u e was in oduced as F ank-Reed (FR)
sou ces ha ing andom line di ec ions andomly assigned o one o he wel e FCC slip sys ems. The disloca ion
densi y,
ρ
, was a ied in he ange o
1014
m−2 o ep oduce he a ainmen o bulk-like plas ici y and he slip
dis ibu ions om he expe imen s wi h Cu mic oc ys als, while he applied
˙ε
emained ixed a 2
×102
s−1. DDD
simula ions unde se e e disloca ion s a a ion we e also pe o medin smalle mic oc ys als wi h
D=
1μm
(
ρ=
1012
m−2). Since he long- ange image ield esul ing om he ac ion- ee bounda y condi ions gi e ise
o ela i ely small co ec ions in he cu en simula ion cells40,41, his image ield is igno ed he e.
Measu emen o he slip dis ibu ions. Rep esen a i e s ess–s ain cu es unde ue displacemen
con ol in conjunc ion wi h he associa ed su ace slip aces a e shown in Fig.1. In ou expe imen s and simula-
Figu e1. S ess–s ain cu es and slip phenomenology in FCC and BCC mic oc ys als. (a) Con inuum(bulk-
like) plas ici y cha ac e ized by cons an shea s ain γ along he sample heigh and by a homogeneous slip ace
dis ibu ion. No e he onse o ma ked s ain-ha dening o Cu (D ≈ 20μm, la ge θ) and mild s ain-ha dening
o Al (θ → 0). (b) Con ined plas ici y cha ac e ized by ma kedly se a ed, he e ogeneous slip aces along he
{111} <0
1
1> slip sys em amily31. No ice he de elopmen o s ess d ops in he s ess–s ain cu es, a ea u e ha
is associa ed wi h ecu en single-ended disloca ion sou ce ope a ion and he occu ence o sudden disloca ion
ne wo k des abiliza ion p ocesses, which lead o la ge alues o he Pea son co ela ion coe icien , (see Fig.5).
(c) BCC plas ici y ac oss sample sizes33. Inc easingly luc ua ing s ess–s ain cu es, cha ac e ized by he la ges
alues, a e obse ed wi h dec easing mic opilla diame e and inc easing empe a u e. The inse illus a es
he emission o an indi idual slip e en unde p og essi e s aining (wi h he cu en , in insic displacemen
con olled es ing sys em) cha ac e ized by Young’s modulus, E, s ess d op, Δσ, and plas ic s ain inc emen ,
ε. The schema ic ep esen a ion o a cylind ical mic opilla illus a es he al e na ing sc ew/edge disloca ion
cha ac e s o an ac i e singleended sou ce e ol ing a he mic opilla cen e . This leads o he onse o wa y
slip as shown o D = 0.5μm a 400°C. He e ogeneous slip ace dis ibu ions along he <111> {11
2
} slip sys em
amily is also illus a ed o D = 5μm a 400°C.
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ions, an indi idual slip e en in ol es sudden ins abili y o he disloca ion a angemen p oducing s ess d op,
�σ
, a ixed displacemen (see inse o Fig.1c). The onse o a slip magni ude,
s
, hus educes he sample heigh ,
l
, so as o induce s ess d op,
�σ
, as he elas ic ene gy is eleased37,42. This p ocess is desc ibed h ough
whe e
SF
is he Schmid ac o and
E
is he e ec i e elas ic modulus o he mic oc ys al (see Sec .2 in Supple-
men a y Ma e ial). Equa ion(1) indica es ha he s ess d op magni ude
�σ
inc eases when he same slip e en
size occu s in samples o a smalle heigh . This suppo s he no ion in ha he plas ic in e mi encies may only
become no iceable in small (mic oc ys alline) sample sizes. Acco ding o coa se-g ained c ys al plas ici y43, he
ollowing co ela ion be ween he slip magni udes and he shea s ain
γ
can hen be exp essed
whe e
b
is he magni ude o he Bu ge s ec o ,
N
is he o al numbe o mobilized disloca ions in a gi en a a-
lanche,
L
is he mean dis ance swep by each disloca ion,
As=LD
is he swep a ea, and
V
is he sample olume.
Equa ion(2) p esc ibes he possible onse o ac ional slips (
s<1b
) when a small numbe o mobile disloca ions
glide ac oss a dis ance
L≪D
. In all coa se-g ained analysis based upon Eq.(2), he pa ame e
L
becomes a linea
ep esen a ion o he a ea swep by he expansion o he unzipped h ee-dimensional disloca ion segmen s along
wi h he e ol ing o he su ace unca ed single-ended disloca ion sou ces in he ac i e slip plane.
The a alanche size dis ibu ion was in es iga ed by ecou se o he complemen a y cumula i e dis ibu-
ion unc ion,
C(s)
, which gi es he emission p obabili y o a slip e en
s≥smin
(whe e
smin
is he minimum
measu able slip). Hence,
C(s)→1
when
s→
smin
. Fo a powe -law p obabili y densi y unc ion
P(s)∝s
−

,
whe e
C
(s)≡
∞
s
P(s)
ds
, i ollows ha
C(s)∝s−κ
whe e
κ=−1
. Fu he analyses in he ime domain we e
pe o med h ough he cu en ly de ined a alanche emission equency dis ibu ion unc ion
ν(s)=n(s)/ T
,
whe e
n(s)
is he numbe o a alanches ha ca ies slip g ea e han
s
and
T
is he o al s aining ime. A de ailed
desc ip ion o hese unc ions and hei p ope ies a e gi en in he Supplemen a y Ma e ial.
Resul s and discussion
Indi idual a alanche emissions e sus dynamically‑d i en bu s s. An impo an inding om he
cu en MD and DDD simula ions is ha mic oscale plas ici y is essen ially cha ac e ized by he onse o indi-
idual a alanche e en s associa ed wi h ecu en s ess d ops
�σ
. Ou simula ions show ha hese indi idual
a alanches a e emi ed ega dless o he ex e nally applied de o ma ion a e, which may become se e al decades
g ea e han hose in con en ional dynamically-d i en expe imen s. In hese expe imen s, he sample is apidly
s ained a e he onse a plas ic ins abili y, he magni ude o which is hen con olled by he ac i a ion he
eedback loop and mechanical dynamics o he es ing sys em (see he discussion gi en in Sec .1 o he Sup-
plemen a y Ma e ial).
Since he MD simula ions indica e ha he p opaga ion ime ame is
≈
20ns o a medium-sized a alanche
e en sweeping ac oss a mic oc ys al wi h
D=
2µm, i is a gued ha a la ge , dynamically-d i en, s ain bu s
(
�ε ≈
0.1) occu ing o e a ime ame se e al decades g ea e would be comp ised o mo e han a hund ed o
indi idual a alanches. Wi hin he ange o ime ames unde conside a ion, each cons i uen indi idual a a-
lanche would essen ially p opaga e a ixed displacemen , as ep oduced wi h he cu en expe imen s, which
esul s in he
C(s)
dis ibu ions being much na owe han hose o he dynamically-d i en s ain bu s s. A co -
espondence is he e o e an icipa ed be ween he
C(s)
dis ibu ions om expe imen s and simula ions pe o med
unde s ic displacemen con ol and hose o he cons i uen indi idual a alanche e en s concealed in he
dynamically-d i en expe imen s. Much smalle
ν(s)
a e howe e eached in he cu en displacemen con olled
expe imen s, whe e low s ain- a es o
˙ε=10−3
o
10−4s−1
a e ex e nally applied, han in he compu a ional
simula ions o in he dynamically-d i en s ain bu s s whe e
˙ε
is se e al decades g ea e .
Finally, he MD simula ions illus a e he ene ge ics o he cu en indi idual slip e en s. I is ound ha wi hin
he la ge a alanche domain, he elas ic ene gy elease a e is 20 imes g ea e han he elas ic ene gy s o age a e
p io o he onse o he in e mi ency. The a io be ween hese elas ic ene gy a es is educed o 4 wi hin he
small slip domain, while i app oaches uni y o
s<0.1b
whe e plas ic de o ma ion p oceeds h ough smoo h
disloca ion bowing p ocesses (
�σ →0
).
Uni e sal scaling o slip dis ibu ions. Mas e slip dis ibu ions as a unc ion o mic oc ys al o ien a-
ion, size and c ys alline s uc u e a e shown in Figs.2, 3 and 5. These igu es inco po a e he indi idual slip
e en s om ou s ic displacemen con olled expe imen s, MD and DDD simula ions. O e all, he slip dis i-
bu ions comply wi h
whe e he uni e sal powe -law exponen
κ≈
0.4–0.5
±0.03
(
≈
1.4–1.5) desc ibes he incipien slip egime
a aining o
s<sc
,
n
app oaches 1.0, and
A
is a no maliza ion cons an which ensu es
∞

0
P(s)ds ≡
1
, so ha by
i ue o he scaling p ope ies o unc ion
C(s)
in Sec .3 o he Supplemen a y Ma e ial,
A
=1/s
−κ
min
. The cu -o
slip size
sc
hen se s he exponen ial unca ion o
s>sc
, which esul s in he onse o a la ge a alanche egime
wi h educed emission p obabili y. I is no ed ha while a simila o mula ion as in Eq.(3) was p oposed in Re .4,
he dis ibu ion o he indi idual disloca ion glide e en s is in be e acco d wi h he cu en ly employed ela ion
(1)
s=[l�σ/(E×SF)],
(2)
s
=γl=(NbAs/V)l=
NbL
D,
(3)
C
(s)=As−κexp

−
s
scn,
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Figu e2. (a) Mas e dis ibu ions o he indi idual slip e en s in FCC mic oc ys als om expe imen s and
simula ions unde bulk-like and con ined plas ici y. In acco dance wi h he uni e sal C(s) unc ion included in
he main igu e (Eq.(4) in he main ex ), no ice he onse o an incipien slip domain, cha ac e ized by powe
law exponen κ ≈ 0.5, and an exponen ially- unca ed la ge a alanche domain which se s-in beyond he cu -o
slip sizes> sc (whe e he band wi h powe -law exponen κ = 0.5 is d awn o isual guidance). The ag eemen
be ween he C(s) dis ibu ions om expe imen s, MD and DDD simula ions o mic oc ys als de o ming unde
bulk-like plas ici y is u he shown, whe e he pa ame e s Lc and Nc in he Table p o ide he scaling o he C(s)
dis ibu ions as unc ion o sample diame e D. The inse o (a) p o ides a alanche emission equency unc ion
ν(s) om key expe imen s wi h he same colo code as in he main igu e. Sca e ed slip e en s a e hen gi en
in (b) and (c), whe e he possible co ela ion be ween he slip sizes and he applied shea s ess, τ, sugges s
he onse o s ess- uned c i icali y while he lack o co ela ion suppo s sel -o ganized c i icali y (SOC), as
discussed in he main ex .
Figu e3. Ex ension o he slip dis ibu ions in Fig.2 using a di e en se o [100]-o ien ed Cu mic oc ys als
es ed as a unc ion o empe a u e and size. These mic oc ys als we e hea ea ed o emo e he FIB-induced
disloca ions, and exhibi ed con ined plas ici y when subjec ed o uniaxial comp ession (see Fig.1b). As
compa ed o in he Cu mic opilla s in Fig.2, he cu en incipien slip egime (κ ≈ 0.4) is clea ly p olonged
o la ge cu -o slip sizes in he ange o sc ≈ 10b–20b. This ea u e becomes mo e e iden as he es ing
empe a u e is inc eased (D = 3.5μm a T = 400°C). No ice ha when he mic oc ys al diame e is educed
owa ds sub-mic ome e sizes, he cu -o slip sc is d as ically educed. This is suppo ed by he DDD simula ions
pe o med wi h a educed numbe o disloca ion segmen s o cap u e such sou ce s a a ion esponses (g ey
ci cles, whe e D = 1μm, ρ = 1 × 1012 m−2and N = 1 along wi h he mechanisms illus a ed in Fig.4e, ). The inse
p o ides a alanche emission equency unc ion C(s) om he expe imen s in he main igu e. The sca e ing o
he slip e en s as a unc ion o he applied shea s ess τ in he ac i e slip sys ems in shown in (b), in suppo o
s ess- uned c i icali y (STC). La ge slip e en s hus occu when τ > τc.

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(see Sec .5 in he Supplemen a y Ma e ial). Along hese lines, he s a is ical sca e o he powe -law exponen
κ
in he small slip domain is compu ed he e h ough he s anda d boo s apping esampling me hod, whe e he
hypo hesis o he powe -law scaling sa is ies he Kolmogo o –Smi no es . This in ol es compa ison o he
expe imen ally measu ed slip dis ibu ions agains a syn he ic se o 5000 powe -laws wi h 5% signi icance le el44,
see Sec .3 in he Supplemen a y Ma e ial.
The backg ound o he scaling o he slip dis ibu ions s ems om Eq.(2), whe e i ollows ha
sc=b(NL)c/D
.
The subs i u ion o his ela ion in o Eq.(3) yields he explici dependency o he slip dis ibu ions upon he
mic osample diame e ,
D
, and he o al dis ance a elled by he mobile disloca ions in an in e mi ency wi h
s=sc
, as gi en by he p oduc
(NL)c
. Along he seminal disloca ion s o age- eco e y model45, de ailed in es-
iga ions in FCC c ys als ha e e ealed he in ica e connec ion be ween a mean, de e minis ic alue o he
disloca ion swep dis ance
L
and he s ain ha dening esponse p oduced by he expansion o pinned disloca-
ion segmen s ac oss an a angemen o o es disloca ions, leading o he o ma ion o o ien a ion-dependen
junc ions19,46,47. In he ligh o hese in es iga ions, we ind
L=
0.7μm and 2.3μm o he Cu and Al mic osa-
mples wi h
D=
2μm and 8μm, espec i ely. This is consonan wi h he es ima ed
ρ≈
7
×
1014m−2 and
ρ≈
8
×
1013m−2 which de elop unde he applied shea s esses o
τ◦≈
65MPa and 27MPa in he mic oc ys al
comp ession expe imen s o Cu and Al, espec i ely (see Sec .6 in he Supplemen a y Ma e ial). I is no ed ha
he s ain ha dening pa ame e
θ
is hen educed as compa ed o ha a aining a smalle disloca ion densi ies18.
Mechanisms and s a is ics o bulk‑like plas ici y in FCC mic oc ys als. Figu e2 shows he
C(s)
dis ibu ions o he indi idual slip e en s om expe imen s and simula ions o bulk-like mic oscale plas ic-
i y, whe e a a he uni o m a angemen o non-localized, ain ing slip aces de elops along he mic oc ys al
heigh in conjunc ion wi h plas ic ba eling (Fig.1a). A mas e slip dis ibu ion is cons uc ed o mic oc ys als
ha exhibi in ense in e ac ions be ween he mobile disloca ions and he o es disloca ions. This dis ibu ion
is i ed wi h
sc≈
1
b
, and inco po a es he expe imen al esul s om Cu mic oc ys als (
D≈
2μm) comp essed
along he [111] o ien a ion, he DDD simula ions o ees anding Ni mic oc ys als con aining a la ge disloca-
ion densi y, and he MD simula ions o an ex emely en angled Al bulk modeled h ough pe iodic bounda y
condi ions.
The disloca ion mechanisms unde lying he abo e mas e slip dis ibu ion a e shown in Fig.4. The MD
simula ions illus a e ha he incipien slip domain is dis inguished by co ela ed disloca ion glide e en s in ol -
ing he expansion o pinned loops o e
L<30
nm. These p ocesses occu h oughou he en i e cell olume
(Fig.4a–d). In he MD simula ions, s ess d ops
�σ
a e no longe de ec ed below he minimum slip size
smin
≈0.07b
whe e he s ess–s ain cu es acqui e a con inuous appea ance and de o ma ion is accumula ed h ough
e e sible bowing o pinned disloca ion segmen s along wi h he onse o g adual, smoo h disloca ion glide
e en s o e
L≈5
nm. As he sudden elease o elas ic ene gy anishes in hese p ocesses, hey a e no ega ded
as ue a alanches.
While he abo e obse a ions conce n bulk-like esponses, he DDD simula ions inco po a e he ole o he
ee su aces in a alanche p opaga ion a disloca ion densi y le els ende ing simila
C(s)
dis ibu ions as hose
measu ed expe imen ally wi h he Cu mic oc ys als. O e all, he DDD simula ions indica e ha when
s→sc
,
he indi idual slip e en s a e comp ised o a o al o
N=
2 o 3 mobile disloca ions, whe e
L
0.5μmacco d-
ing o Eq.(2). A he cu -o slip size
sc≈
1
b
, we hen ind ha
Nc≈
3 so ha
Lc≈0.7
μm(Eq. (2)), which is in
excellen acco d wi h he de e minis ic alue o he mean swep dis ance es ima ed in he abo e sec ion. Wi h
inc easing slip size (
s>sc
), single-ended sou ce ope a ion accoun s o an inc easing ac ion o he o al slip
magni ude in he ees anding mic oc ys al. All o he obse ed slip e en s p oducing
s>sc
hen consis o
N≥
3 mobile disloca ions, eaching he maximum alue o
N=
17 wi h
L
≈1.2
μm o he la ges measu ed
a alanche e en . The DDD simula ions inally show ha while disloca ion s o age and disloca ion annihila ion
p ocesses de elop wi hin he incipien slip egime, la ge a alanche emissions (
s>sc
) a e cha ac e ized by ne
disloca ion s o age (Fig.4j).
The gene al slip dis ibu ion unc ion in Eq.(3) is main ained o [100]-o ien ed Al mic oc ys als wi h
D=
8µm, whe e
sc
inc eases om
≈
1
b
o
≈3.5b
indica ing he onse o mode a e disloca ion in e ac ions along wi h
a u he inc ease in
Lc
om
≈0.7
μm o
≈
2.3μm (as desc ibed abo e). These mic oc ys als de o m h ough
bulk-like plas ici y, which is e idenced by he occu ence o non-localized slip aces and sample ba eling
(Fig.1a). Since he s ain ha dening esponse is u he in luenced by he disloca ion s acking aul wid h,
d
, i
is concluded ha Al mic oc ys als wi h
d≈
1nm, hus exhibi ing enhanced c oss-slip and educed in e ac ions
be ween mobile and o es disloca ions, will end o p oduce g ea e slip e en s han Cu mic oc ys als wi h
d≈3
nm. Acco ding o he disloca ion s o age- eco e y model, c oss-slip would inc ease he es ima ed
Lc
by
≈
15% in he Al c ys als.
The ollowing discussion conce ns he scaling o he slip dis ibu ions as a unc ion o he mic osample size. In
his sense, bulk-like mic oc ys al plas ici y is known o ul ill a classic con inuum mechanics desc ip ion whe e
he de o med s a e is p esc ibed by he shea s ain
γ
in he ac i e slip sys ems. This implies ha o a gi en slip
e en occu ing in a hypo he ical, ep esen a i e ma e ial olume elemen , he numbe o mobile disloca ions,
N
,
inc ease in p opo ion wi h he elemen ’s diame e ,
D
, so ha he same slip magni ude,
s
, is p oduced (Eq.(2)).
Hence, he shea s ain
γ=s/l
is p ese ed i espec i e o sample diame e
D
. Since a a he homogeneous slip
dis ibu ion de elops along he sample heigh
l
(as e idenced om he di use slip aces in Fig.1a), inc easing
mic oc ys al heigh gi es ise o a p opo ional inc ease in he a alanche emission a e. In his sense, i is no ed
ha in mic oc ys als de o med wi h he same
˙ε
, a h ee- old inc ease in
l
aises
ν(s)
by he same h ee- old ac-
o (inse o Fig.2). The shea s ain
γ
is consequen ly main ained i espec i e o
l
. The a o emen ioned scaling
be ween
N
and
D
ob iously b eaks down in su icien ly small samples whe e an incipien slip e en (
s<sc
) is
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Figu e4. Mechanisms o he indi idual ul a as slip e en s om he MD and DDD simula ions in FCCs. (a–d) MD
simula ions o FCC Al de o ming unde bulk-like plas ici y (see Suppo ing Mo ie S1). The slip e en ma ked in g ey in (b)
unde lies sudden disloca ion expansions in (a), whe e segmen 1 i s e ol es abou junc ion J ( = 0–30ps) while segmen
2 subsequen ly glides owa ds he back o he compu a ional cell ( = 240ps). Segmen 3 inally expands owa ds he on
( = 310–360ps). The o es disloca ion ne wo k is shown in (c) and (d), whe e glissile disloca ions dissocia e in o leading and
ailing pa ials, ma ked in edin (d). Sessile junc ion- ype segmen s, highligh edin bluein (d), and isola ed s acking aul
e aed a (SFT) a e also obse ed. (e– ) DDD simula ions o a Ni mic oc ys al ha de o ms unde se e e disloca ion sou ce
s a a ion (N = 1). The composi e plo in (e) o e lies he e olu ion o he single mobile segmen o e ime (in ed) as i expands
and e ol es a ound he pinning poin s. The mobile segmen e en ually c oss-slips in o he in e cep ing sys ems while smalle ,
seconda y disloca ion segmen s (highligh ed in di e en colo s) bow unde s ess. The esul ing s ess − s ain cu e whe e
plas ici y a ains unde a cons an s ess le el is shown in ( ). (g–j) DDD simula ions o Ni mic opilla s de o ming unde
in ense disloca ion in e ac ions (sc ≈ 1b; Lc ≈ 0.5μm). (g) shows s ess d ops and disloca ion densi y e olu ions wi hin a small
s ain inc emen . A speci ic a alanche, ma ked in g ey in (i), bi u ca es in o wo ac i e {111} <1
1
0> slip sys ems in (e). ( )
illus a es ha he a alanche ini ia ed h ough F ank–Read sou ce ope a ion (1 − 3), p oducing a single-ended sou ce (in ed)
which leads o slip ace o ma ion (4–6) and he s o age o disloca ion segmen (7) wi hin he immobile o es . Expansion o
segmen (3), in ed, hen p oduces mul iple c oss-slip e en s and a su ace slip ace (3 − 6). The segmen is s o ed a (8), inally
igge ing ac i a ion o a seconda y segmen ma ked in blue (1 − 6). (h) shows he a ia ion in disloca ion densi y Δρ o he
incipien slip (s < sc) and la ge a alanche domain (s > sc).
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al eady comp ised o a single disloca ion. This is inconsequen ial o ou s a is ical analyses since he incipien
slip domain adhe es o a scale in a ian powe -law dis ibu ion.
Equa ions(2) and (3) p esc ibe ha he a alanche dis ibu ions scale as
whe e he cu -o , mean disloca ion glide dis ance
Lc
can be es ima ed h ough he measu ed s ess–s ain
cu es, as desc ibed abo e. As illus a ed h ough he scaling o he slip dis ibu ions o Cu and Al mic oc ys als
wi h di e en diame e
D
ia Eq.(4), see Fig.2, i is ound ha he linea densi y,
Nc/D
, o mobile disloca ions
ca ying a slip magni ude
sc
emains cons an wi hin he cu en ange o
ρ≈
1014 o 1015m-2 whe e bulk-like
mic oc ys al plas ici y de elops.
S a is ical ansi ion owa ds con ined plas ici y in FCC mic oc ys als. Figu e2 con ains he slip
dis ibu ion om Cu mic oc ys als wi h he [100] o ien a ion exhibi ing a he e ogeneous a angemen o local-
ized slip aces along he sample heigh
l
. The s ess–s ain cu es unde such con ined plas ici y esponses
whe e disloca ion ne wo k de elopmen is coun e balanced by su ace annihila ion a e hen cha ac e ized by
he onse o la ge s ess d ops,
�σ
, and
θ=0
. Unde hese condi ions, ou esul s show ha he cu -o slip
inc eases om he alue o
sc≈
1
b
, measu ed o he abo e bulk-like s ain ha dening mic oc ys als, o he cu -
en
sc≈
12
b
in con ined plas ici y. As discussed nex , his is consis en wi h he concep ion in ha when s ain
ha dening is hinde ed, plas ici y is inc easingly sus ained by he ac i a ion o a limi ed numbe o single-ended
sou ces in a s eady, o es disloca ion a angemen . The measu ed unc ion
ν(s)
hen shi s owa ds la ge le els
while disloca ion glide becomes highly localized along he sample heigh (inse o Fig.2).
The slip dis ibu ions om a second se o expe imen s in Cu mic oc ys als, es ed as a unc ion o he cha -
ac e is ic sample size
D
and empe a u e (Fig.3), a e key o he s a is ical analysis o con ined plas ici y. These
mic oc ys als also exhibi la ge
�σ
and he he e ogeneous slip ace a angemen in Fig.1b. The measu ed slip
dis ibu ions indica e ha when
D
is educed om 3.5
µ
m o 1
µ
m, a wo- old dec ease in
sc
is a ained (see
Fig.3), which may only occu i he p oduc
(NL)c
exhibi s a se en- old dec ease, as p esc ibed by Eq.(2). E en
unde he assump ion ha he e olu ion o single-ended sou ces abou hei pinning poin s in smalle mic o-
c ys als e en ually leads o a smalle c i ical dis ance
Lc
, he e y signi ican se en- old dec ease in
(NL)c
sug-
ges s ha he numbe o mobile sou ces,
Nc
, would also ha e o dec ease when
D
is educed o
1µ
m. Along he
lineso he disloca ion mechanisms, he DDD simula ions indica e ha wi hin he la ge a alanche domain, he
o al disloca ion swep dis ance ha is sus ained h ough single-ended sou ce ope a ion becomes se e al imes
g ea e han he sample diame e (implying ha
NL ≫D
). This is illus a ed in Fig.4g–i, whe e he pi o ing o
single-ended sou ces abou mul iple junc ions wi h immobile disloca ions esul s in he ac i a ion o disloca ion
segmen s a he in e cep ing c oss-slip sys ems. Finally, ollowing he he mally-ac i a ed na u e o c oss-slip,
an inc ease in he o al disloca ion swep dis ance
(NL)c
is an icipa ed o occu a ele a ed empe a u es, as
disloca ion annihila ion p ecludes s able junc ion o ma ions and he mobile segmen s a e dissemina ed ac oss
he ac i e slip sys ems h oughou he sample heigh . This is suppo ed by he inc ease o he measu ed
sc
and
he a endan ise in
ν(s)
(see Fig.3 o
D=
3.5µm) along he onse o mo e no iceable localized slip (Fig.1b).
Inc eases in empe a u e hus acili a e he a ainmen o con ined plas ici y.
As he sample diame e dec eases u he owa ds submic ome e sizes (i.e.
D≈0.6
µm in Fig.3), he slip
dis ibu ions exhibi d as ic educ ions in
sc
and
ν(s)
which a e a ibu ed o se e e disloca ion sou ce s a a ion.
The e o e, he ma ked dec ease in he numbe o mobile disloca ions,
Nc
, in he ansi ion om mic ome e o
submic ome e sizes signi ican ly educes he p oduc
(NL)c
, which esul s in he cu en ly measu ed smalle
sc
alues. This is suppo ed by he DDD simula ions pe o med wi h
D=1
µm, whe e only one single-ended sou ce
ecu si ely ope a es (
ρ=
1012
m-2;
N=1
) unde a ixed alue o he shea s ess, esul ing in ma kedly se a ed
s ess–s ain cu es and a simila
sc≈4b
as in he abo e expe imen s wi h
D≈0.6
µm (see Fig.4e, ). These
simula ions e en ually show he onse o c oss-slip e en s o he ac i e sou ce and he occu ence o collinea
in e ac ions due o he p esence o a ew seconda y disloca ion segmen s. The la e leads o he segmen a ion
o he slip e en s, which educes
sc
.
Assessmen o he co ela ion be ween he magni ude o he slip e en s and he applied shea s esses in he
ac i e slip sys ems is inally illus a ed in Fig.2b,c. We choose o e alua e his ea u e h ough Pea son co ela ion
coe icien ,
, ising owa ds uni y when an inc easingly linea ela ionship is eached be ween he sca e ed slip
popula ion and he s ess, and he
p
- alue p esc ibing he likelihood in ha his ela ionship is me ely coinciden-
al. In he FCC mic oc ys als in Fig.2c, i is ound ha he same sca e ed slip dis ibu ion de elops i espec i e
o he applied s ess le el, a ea u e p esc ibed by small
<
0.2 o by la ge
p>
20% alues. This cha ac e izes he
de elopmen o bulk-like plas ici y bo h unde in ense and mode a e disloca ion in e ac ions, whe e he size o
he indi idual a alanche e en s is s a is ically una ec ed by he applied s ess, along he lines o SOC. I is hen
no ed ha he g ea es
alues om ou analysis pe ain o he smalle mic oc ys als de o ming well wi hin he
con ined plas ici y egime, whe e
lies in he 0.3 o 0.4 ange while
p<
5%. These alues sugges he onse o
s ess dependen slip dis ibu ions, whe e he likelihood o la ge a alanche emissions (
s>sc
) co ela es wi h
he applica ion o la ge shea s esses ha a e g ea e han a c i ical
τc
alue (see Fig.3b). The e o e, he
C(s)
dis ibu ions measu ed a la ge s ess le els would en a i ely exhibi a highe
sc
, in suppo o STC.
Along he abo e analyses, i is no ed ha a single displacemen -con olled expe imen does no p o ide a
signi ican numbe o slip e en s wi hin a window o s ess so as o enable asse ion o he possible s ess-binning
o unc ion
C(s)
. The e o e, i is no s a is ically sound o e alua e whe he he a alanches adhe e o STC o SOC
by compa ing be ween he a e aged slip magni udes a ained wi hin di e en windows o s ess. The a iabili y in
(4)
C
(s)=As−κexp

−

sD
bNcLcn
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he s ess le el leading o sus ained plas ic low in di e en (nominally iden ical) mic osamples u he p e en s
such asse ions om a single da ase con aining all slip e en s om hese expe imen s.
Mechanisms and s a is ics o in e mi en plas ici y in BCC mic oc ys als. The slip dis ibu ions
in BCC mic oc ys als, shown in Fig.5, adhe e o he gene al unc ion in Eq.(3), whe e he Kolmogo o –Smi no
es con i ms he powe -law hypo hesis wi h unique
κ
exponen
≈
0.4
±0.06
(1≈
1.4) o
s<sc
. In he ligh
o Fig.1c and he disloca ion in e ac ion and slip ace o ma ion mechanisms e ealed in Fig.6, i is a gued in
he ollowing discussion ha he measu emen o di e en
sc
alues in BCC mic oc ys als is he signa u e o he
dis inc mobili y o he sc ew disloca ions as a unc ion o empe a u e, sample size and he applied s ess le el.
In submic ome e -sized pilla s de o med a RT, plas ic de o ma ion p oceeds h ough single-ended sou ce
ope a ion26,27 a applied s esses eaching he dis inc i e la ge Peie ls ba ie 48–50. Plas ici y hus se s-in a
σ
2000MPa, as shown in Fig.1c o D
≤0.5
µm, whe e he cha ac e o he mobilized single ended-sou ces al e -
na es om pu e edge o pu e sc ew (inse o Fig.1c). I is hen a gued ha he ecu en c oss-kinking o sc ew
segmen s igge ed unde such la ge applied s esses51 leads o he wa y slip ace emissions obse ed in he W
mic oc ys als (see Fig.1c whe e simila slip pa e ns a ise in he RT es s).
When
D
3μm, ull disloca ion loops can be e ec i ely accommoda ed wi hin he mic osample. A RT, he
MD simula ions sugges ha he plas ic in e mi encies a e p oduced unde he smalle applied s esses yielding
aniso opic loop expansions along he highly mobile edge segmen s o he loops (Fig.6d,e). Al hough he slip
aces emain globally s aigh (Fig.1c), a omis ically wa y pa e ns a ise h ough c oss-kinking mechanisms
o he su ace-in e cep ed sc ew segmen s (Fig.6a). I is no ed ha while he enhancemen o c oss-kinking a
la ge applied s esses51 is consis en wi h he obse ed wa y glide in he W mic oc ys als (3μm
<D<
5μm)
de o med a RT, whe e
σ≈
1500MPa, plana glide was ound o p e ail in he Ta mic oc ys als o simila size
ha exhibi ed plas ic de o ma ion a he smalle s esses,
σ≈
300MPa, whe e c oss-kinking is hinde ed.
A highly dense disloca ion ne wo k may also de elop in he la ge mic oc ys als (
D≈
5μm), as modelled
h ough he MD simula ions wi h pe iodic bounda ies ha mimic bulk-like esponses. This RT disloca ion
ne wo k is cha ac e ized by long sc ew segmen s (Fig.6c) in acco d wi h expe imen al indings in mac oscale
samples48,52. The MD simula ions wi h ee-s anding cells ha con ain simila en angled disloca ion ne wo ks
hen p edic he p edominance o nanoscale, wa y slip ea u es as he meande ing mobile disloca ion segmen s
Figu e5. Mas e dis ibu ions o he indi idual slip e en s om he expe imen s and MD simula ions in
BCC Ta (a) and BCC W (b), whe e he cu -o slip size sc ma ks he ansi ion om incipien slip o he la ge
a alanche domains p esc ibed h ough he C(s) unc ion in Eq.(3). A RT, educ ions in mic opilla diame e
D lead o g ea e sc. The e e sed end is ound in (b) a inc easing empe a u es. This is associa ed wi h he
onse o disloca ion c oss-kinking along wi h a ansi ion om plana o wa y glide. The inse s p o ide emission
equency unc ion ν(s) om he expe imen s in he main igu es. E olu ion om STC o SOC is illus a ed in
(c,d,e). See ex o de ailed discussions.