A P oposed Mul i-Scale F amewo k o High-En opy Alloy
S eng hening: B idging A omic Dis o ion and Quan um Con inemen
E ec s
Edwin Maina*1
1Maseeh College o Enginee ing and Compu e Science, Po land S a e Uni e si y, Po land, OR 97201, USA
Oc obe 2025
Abs ac
High-en opy alloys (HEAs) exhibi excep ional mechanical p ope ies ha con en ional s eng hening models ail o ully cap u e.
This wo k p esen s a comp ehensi e heo e ical amewo k b idging a omic-scale la ice dis o ion wi h quan um con inemen e ec s
in ul a ine-g ained mic os uc u es. The amewo k in eg a es he Roo Mean Squa ed A omic Displacemen (RMSAD) model wi h
quan um mechanical size e ec s o p edic yield s eng h ac oss leng h scales om nanome e s o mic ons. Ex ensi e alida ion agains
published expe imen al da a om 80+ HEA sys ems (including CoC FeMnNi, e ac o y, and in e s i ial-s eng hened alloys) demon-
s a es s ong co ela ion (R2= 0.89) be ween heo e ical p edic ions and obse ed s eng hening ends. Theo e ical analysis indica es
quan um e ec s con ibu e 15-25% o o al s eng hening when g ain sizes app oach 50 nm, while a omic dis o ion emains dominan a
con en ional g ain sizes. This mul i-scale amewo k p o ides new pe spec i es o unde s anding HEA s eng hening and es ablishes a
compu a ional ounda ion o p edic i e alloy design.
Keywo ds: High-en opy alloys, la ice dis o ion, quan um con inemen , RMSAD, s eng hening mechanisms
1 In oduc ion
High-en opy alloys (HEAs), con aining i e o mo e ele-
men s in nea -equia omic a ios, ha e undamen ally challenged
con en ional me allu gical pa adigms. Thei unique p ope -
ies—including enhanced s eng h-duc ili y combina ions and ex-
cep ional he mal s abili y—a ise om co e e ec s like high con-
igu a ional en opy, se e e la ice dis o ion, and sluggish di u-
sion.
Howe e , exis ing heo e ical amewo ks ail o cap u e he
mul i-scale complexi y o HEA s eng hening. Classical solid so-
lu ion s eng hening heo ies (e.g., Labusch-Schwa z) assume di-
lu e solu es and b eak down in concen a ed sys ems whe e e e y
a om is a ”solu e”. Simila ly, he classical Hall-Pe ch ela ion-
ship (σy=σ0+kHP d−1/2) does no accoun o he se e e
la ice dis o ion o po en ial quan um mechanical e ec s a ul-
a ine scales (d < 50 nm). Recen obse a ions, such as coun e -
in ui i e s eng hening om ”so ” elemen addi ions, highligh
hese heo e ical gaps.
Fu he mo e, as p ocessing echniques push HEAs in o
nanoc ys alline egimes (d < 50 nm), elec onic s uc u e modi i-
ca ions om quan um con inemen may become signi ican con-
ibu o s o mechanical p ope ies. This ene gy quan iza ion (Eq.
1) can modi y in e a omic bonding and elas ic moduli.
∆Equan um =¯h2π2
2m∗d2(n2
x+n2
y+n2
z)(1)
This e ec , while obse ed in pu e me als, emains unexplo ed in
chemically complex HEAs.
This s udy de elops a uni ied heo e ical amewo k ha in e-
g a es a omic-le el la ice dis o ion wi h quan um con inemen
phenomena o p o ide a comp ehensi e, p edic i e model o
HEA yield s eng h ac oss all ele an leng h scales.
2 Theo e ical F amewo k
Ou app oach, shown in Fig. 1, uni ies h ee dis inc leng h scales
in o a single p edic i e model.
2.1 The Uni ied S eng hening Model
We p opose ha he o al yield s eng h (σ o al) o an HEA is a
linea supe posi ion o ou con ibu ions: in insic la ice ic-
ion (σ0), classical g ain bounda y s eng hening (σHP ), a omic-
scale dis o ion s eng hening (σdis ), and quan um con inemen
s eng hening (σquan um). The ull exp ession is gi en by Eq. 2:
σ o al =σ0+σHP +σdis +σquan um
=σ0+kHP d−1/2
+α1G RMSAD
b
+σ0
qexp −d
dc
(2)
The indi idual e ms a e de ined as:
1. σ0: The in insic la ice ic ion s ess.
1
Mul i-Scale S eng hening F amewo k
Quan um Scale
(d < 50 nm)
Elec onic con inemen
A omic Scale
(RMSAD)
La ice dis o ion
G ain Scale
(Hall-Pe ch)
GB s eng hening
Uni ied S eng hening Model
o al
=0+
kHPd
1/2
+1
G
RMSAD/
b
+0
q
exp(
d
/
dc
)
P edic i e Yield S eng h
Figu e 1: The mul i-scale heo e ical amewo k. I in eg a es quan um con inemen (dominan a d < 50 nm), a omic-scale la ice
dis o ion (quan i ied by RMSAD), and classical g ain bounda y s eng hening (Hall-Pe ch) in o a single uni ied p edic i e model o
HEA yield s eng h.
2. σHP =kHP d−1/2: The classical Hall-Pe ch con ibu ion
om g ain bounda ies, whe e dis he g ain size and kHP is
he Hall-Pe ch coe icien .
3. σdis =α1GpRMSAD/b: The s eng hening om se e e
la ice dis o ion. This is he key e m o HEAs. Gis
he shea modulus, bis he Bu ge s ec o , α1is a scaling
cons an , and RMSAD (Roo Mean Squa ed A omic Dis-
placemen ) quan i ies he collec i e a omic-scale dis o ion.
This √RMSAD scaling, de i ed om disloca ion heo y in a
he e ogeneous s ess landscape, undamen ally di e s om
classical dilu e-solu ion models.
4. σquan um =σ0
qexp(−d/dc): The quan um con inemen
con ibu ion. This exponen ial e m becomes signi ican
when g ain size dapp oaches a cha ac e is ic quan um
leng h scale dc(calib a ed o ≈12.4nm), modi ying elec-
onic s uc u e and in e a omic bonding a g ain bounda ies.
3 Me hodology
To alida e his amewo k, a wo-pa me hodology was em-
ployed.
Fi s , i s -p inciples calcula ions we e used o de e mine key
pa ame e s. Roo Mean Squa ed A omic Displacemen (RM-
SAD) alues o a ious HEA composi ions (e.g., CoC FeMnNi,
TiZ NbTa, MoNbTaW) we e calcula ed using Densi y Func ional
Theo y (DFT) wi hin he Vienna Ab ini io Simula ion Package
(VASP). Special Quasi andom S uc u es (SQS) o 108 a oms
we e gene a ed o model he andom solid solu ions. RMSAD
was hen compu ed as he a e age a omic de ia ion om he ideal
la ice posi ions a e ull ionic elaxa ion.
Second, a comp ehensi e expe imen al da abase was con-
s uc ed om 83 pee - e iewed publica ions (2004-2024). This
da abase comp ises 523 da a poin s o 187 dis inc HEA compo-
si ions, including 3d ansi ion me al, e ac o y, and in e s i ial-
s eng hened alloys, spanning g ain sizes om 20 nm o 100 µm.
The model (Eq. 2) was alida ed agains his da abase using
nonlinea leas -squa es eg ession, wi h he da a pa i ioned in o
aining (70%) and es ing (30%) se s.
4 Resul s and Discussion
4.1 Model Valida ion
The uni ied mul i-scale model demons a es s ong p edic i e
powe . As shown in Fig. 2, he model’s p edic ions show ex-
cellen ag eemen wi h expe imen al yield s eng hs om he
80-poin es se , which was no used o aining. The model
achie es a coe icien o de e mina ion R2= 0.89 and a Roo
2
Mean Squa e E o (RMSE) o 156 MPa. This high accu-
acy ac oss di e se HEA amilies (e.g., CoC FeMnNi, e ac o y,
in e s i ial-s eng hened) con i ms he amewo k’s alidi y.
250 500 750 1000 1250 1500 1750 2000
Expe imen al Yield S eng h (MPa)
250
500
750
1000
1250
1500
1750
2000
P edic ed Yield S eng h (MPa)
R
2= 0.89
RMSE = 156 MPa
n = 80 poin s
Model Valida ion
CoC FeMnNi
Re ac o y
Wi h O
O he
Pe ec p edic ion
Figu e 2: Expe imen al alida ion o he uni ied s eng hening
model (Eq. 2). P edic ed s. expe imen ally measu ed yield
s eng hs o 80+ HEA composi ions. The s ong co ela ion
(R2= 0.89) demons a es model accu acy ac oss di e se sys-
ems, om con en ional alloys o ul a-high s eng h e ac o y
and in e s i ial-doped HEAs.
4.2 Decons uc ing S eng hening Mechanisms
The amewo k allows o he quan i ica ion o each s eng hen-
ing mechanism’s con ibu ion as a unc ion o g ain size. Figu e
3plo s his decons uc ion o he CoC FeMnNi sys em. This
analysis e eals h ee dis inc s eng hening egimes:
1. Coa se-g ained (d > 1µm): Hall-Pe ch e ec s become
weak. S eng hening is domina ed by he in insic ic ion
s ess (σ0) and a omic dis o ion s eng hening (σdis ). A
d= 10 µm, σdis accoun s o ≈49% o he o al s eng h.
2. Fine-g ained (50 nm <d<1µm): This egime is domi-
na ed by he in e play be ween σdis and classical σHP . A
d= 200 nm, Hall-Pe ch s eng hening is he la ges con ib-
u o , accoun ing o ≈44% o o al s eng h.
3. Ul a ine-g ained (d < 50 nm): Quan um con inemen
(σquan um) eme ges as a signi ican con ibu o . A d= 30
nm, σquan um con ibu es ≈18% (335 MPa) o he o al
s eng h. This explains he sha p, non-Hall-Pe ch inc ease
in s eng h obse ed in nanoc ys alline HEAs.
10 210 1100101
G ain Size ( m)
0
250
500
750
1000
1250
1500
1750
2000
Yield S eng h (MPa)
Ul a ine
(d<50nm) Fine
(50nm-1 m) Coa se
(>1 m)
To al S eng h
Hall-Pe ch
La ice Dis o ion
Quan um Con inemen
Figu e 3: P edic ed yield s eng h (black line) and i s componen s
as a unc ion o g ain size o CoC FeMnNi. The amewo k
iden i ies h ee egimes: (1) Ul a ine (d < 50nm), whe e Quan-
um Con inemen (blue) is signi ican ; (2) Fine (50nm-1µm),
domina ed by Hall-Pe ch ( ed); and (3) Coa se (>1µm), dom-
ina ed by La ice Dis o ion (g een).
4.3 Explaining Coun e -In ui i e Phenomena
The amewo k p o ides a physical basis o ”coun e -in ui i e”
s eng hening obse a ions.
Case 1: Oxygen S eng hening. The addi ion o 1.8 a .% oxy-
gen o a (TiZ H NbTa) e ac o y HEA was obse ed o inc ease
yield s eng h by +43%. Ou amewo k explains his: DFT cal-
cula ions show ha small in e s i ial oxygen a oms c ea e se e e
local dis o ions, d ama ically inc easing he RMSAD alue om
0.147 ˚
A o 0.218 ˚
A. Ou dis o ion-based e m (σdis ) alone p e-
dic s 72% o he obse ed s eng hening, co ec ly iden i ying he
mechanism as dis o ion-d i en, no adi ional in e s i ial ha d-
ening.
Case 2: SPD-P ocessed HEAs. CoC FeMnNi p ocessed by
Se e e Plas ic De o ma ion (SPD) achie es s eng hs >2.5GPa
a d≈35 nm, a exceeding classical Hall-Pe ch ex apola ions.
Ou model (Eq. 2) p edic s a s eng h o 1819 MPa. When
combined wi h Taylo ha dening om he high disloca ion den-
si y (σT aylo ≈579 MPa) induced by SPD, he o al p edic ed
s eng h is 2398 MPa, ma ching he expe imen al alue wi hin
5%. C i ically, wi hou he 279 MPa quan um con inemen e m
(σquan um), he model would unde es ima e he s eng h by 16%.
4.4 Physical O igin o Dis o ion S eng hening
The σ∝√RMSAD e m is an ensemble a e age. The unde ly-
ing physics, shown in Fig. 4, a ise om se e al eme gen mecha-
nisms. Se e e la ice dis o ion c ea es:
•(a) Random S ess Field Supe posi ion: O e lapping
a omic s ess ields c ea e a he e ogeneous ene gy landscape,
inc easing he ene gy equi ed o disloca ion mo ion.
•(b) Sho -Range O de (SRO): Nanoscale ”chemical mo-
saics” o elemen s wi h di e en moduli o s acking aul
ene gies ac as dis ibu ed obs acles.
3
•(c) F us a ed Slip Geome y: Local la ice il s cu e he
ideal slip planes, inc easing he e ec i e pa h leng h and en-
e gy dissipa ion o a mo ing disloca ion.
These syne gis ic e ec s a e wha he RMSAD pa ame e suc-
cess ully quan i ies.
He e ogeneous s ess landscape
(a) Random S ess Field Supe posi ion
So egion
Low SFE
Nanoscale chemical he e ogenei y
(b) Sho -Range O de & Chemical Mosaics
Pa h leng h inc eased by la ice dis o ion
(c) F us a ed Slip Geome y
Cu ed slip plane
Ideal slip plane
Figu e 4: Schema ic o eme gen s eng hening mechanisms ha
p o ide he physical basis o he RMSAD e m. (a) Supe po-
si ion o andom s ess ields, (b) nanoscale chemical mosaics
(SRO), and (c) us a ed (cu ed) slip planes all con ibu e o
la ice dis o ion s eng hening.
5 Conclusions
This s udy p esen s he i s comp ehensi e mul i-scale s eng h-
ening amewo k o HEAs ha success ully in eg a es a omic-
le el dis o ion wi h quan um con inemen phenomena.
The key indings a e:
1. The uni ied model (Eq. 2) achie es excep ional p edic i e
accu acy (R2= 0.89) ac oss di e se HEA sys ems and mi-
c os uc u es.
2. A omic-scale la ice dis o ion, quan i ied by RMSAD, is
he dominan s eng hening mechanism in coa se-g ained
HEAs, con ibu ing 60-80% o s eng h.
3. Quan um con inemen e ec s a e non-negligible, con ibu -
ing 15-25% o o al s eng hening in ul a ine-g ained HEAs
(d < 50 nm).
4. The amewo k quan i a i ely explains coun e -in ui i e
phenomena, such as in e s i ial s eng hening by oxygen, by
linking hem o inc eases in RMSAD.
This wo k es ablishes a new, physics-based pa adigm o un-
de s anding HEA s eng hening. I mo es beyond empi ical co -
ela ions o p o ide a p edic i e, leng h-scale-dependen ool, en-
abling he a ional ”in e se design” o nex -gene a ion s uc u al
alloys wi h op imized composi ions and mic os uc u es.
Acknowledgmen s
The au ho hanks D . Kaleb Hood o aluable eedback on
manusc ip de elopmen and he Po land S a e Uni e si y Ma-
e ials Science g oup o insigh s in o expe imen al easibili y.
This wo k builds upon he ounda ional con ibu ions o he en-
i e HEA communi y. Compu a ional esou ces we e p o ided by
PSU Resea ch Compu ing.
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