Beyond Classical Strengthening: Eleven Novel Dislocation-Topology Mechanisms in High-Entropy Alloys

Beyond Classical S eng hening: Ele en No el
Disloca ion-Topology Mechanisms in High-En opy Alloys
Edwin Maina
Ma e ials Science P og am, Maseeh College o Enginee ing
Po land S a e Uni e si y, Po land, OR 97201, USA
[email p o ec ed]
Sep embe 2025
Abs ac
Classical s eng hening models o high-en opy alloys
(HEAs) emphasize con igu a ional en opy and a omic
size di e ences bu ail o cap u e he ich opology o
disloca ion dynamics in chemically complex la ices. This
wo k iden i ies and quan i ies ele en p e iously unde -
ecognized mechanisms ha domina e HEA s eng h-
ening: (1) us a ed slip geome y whe e s ain ields
cu e disloca ion pa hs, (2) andom s ess ield supe -
posi ion c ea ing s a is ical ba ie s, (3) he e ogeneous
bond-s i ness landscapes, (4) sho - ange o de ing mo-
saics, (5) local elas ic modulus misma ch, (6) ee en-
e gy landscape oughening, (7) elec onic s uc u e us-
a ion, (8) supp essed dynamic eco e y, (9) disloca-
ion co e sp eading diso de , (10) mosaic g ain bound-
a y cha ac e dis ibu ion, and (11) opological block-
age. Th ough opological analysis and s a is ical me-
chanics, we demons a e ha pa h us a ion alone con-
ibu es 35-45% o o al s eng hening—compa able o
classical Hall-Pe ch e ec s. Bond-s i ness he e ogenei y
adds 15-20%, while elec onic us a ion con ibu es 10-
15%. These mechanisms explain coun e -in ui i e obse -
a ions such as s eng h inc eases om ”so ” elemen
addi ions and non-mono onic composi ion dependencies.
The amewo k p o ides physical in ui ion o HEA de-
sign beyond adi ional desc ip o s.
Keywo ds: Disloca ion opology, us a ed slip, high-
en opy alloys, s a is ical mechanics, bond s i ness, opo-
logical s eng hening
1 In oduc ion
1.1 The Disloca ion Topology P oblem
High-en opy alloys exhibi yield s eng hs 2-3×highe
han ule-o -mix u es p edic ions [1, 2], ye con en ional
explana ions ocus p ima ily on a omic size mis i and
con igu a ional en opy [3]. This pe spec i e misses a un-
damen al ques ion: How does ex eme chemical complex-
i y al e he opology o disloca ion mo ion?
Con en ional Alloy High-En opy Alloy
S aigh pa h
F us a ed pa h
Figu e 1: Disloca ion opology in con en ional s. HEA
la ices. Chemical diso de o ces non-plana slip, in-
c easing line ension ene gy.
In con en ional alloys, disloca ions p opaga e along
well-de ined slip planes wi h p edic able Bu ge s ec o s.
Howe e , in HEAs:
•A omic-scale s ain ields cu e he slip plane locally
•Chemical luc ua ions il he local la ice o ien a ion
•Elas ic modulus a ia ions c ea e di ec ional ba ie s
Resul : Disloca ions canno mo e in s aigh
lines— hey mus na iga e a opologically us a ed land-
scape.
1.2 Why Cu en Models A e Incomple e
Mos heo e ical amewo ks ea HEA s eng hening
h ough classical solid solu ion heo y:
τss ∝Gϵmcn(1)
whe e Gis shea modulus, ϵis size mis i , cis concen a-
ion, and exponen s m, n a e i ed empi ically [4,5].
C i ical limi a ion: This assumes disloca ions expe-
ience a mean ield o obs acles. Bu HEAs exhibi :
1. Co ela ed diso de : Sho - ange o de ing c ea es
chemical mosaics
2. Topological cons ain s: Cu ed slip planes ex-
haus line ene gy
1
3. Elec onic he e ogenei y: Bond s i ness a ies
by 30-50%
These e ec s demand a opology-awa e heo e ical
amewo k.
1.3 No el Con ibu ions
This wo k in oduces ele en mechanisms absen om clas-
sical models:
Geome ic mechanisms:
•F us a ed slip geome y (pa h o uosi y)
•Random s ess ield supe posi ion
•Topological blockage
Elec onic mechanisms:
•Bond-s i ness he e ogenei y
•Elec onic s uc u e us a ion
•Disloca ion co e diso de
Mic os uc u al mechanisms:
•Sho - ange o de ing mosaics
•Local elas ic modulus misma ch
•Mosaic g ain bounda y dis ibu ion
Kine ic mechanisms:
•Supp essed dynamic eco e y
•Ene gy landscape oughening
2 Ele en No el S eng hening
Mechanisms
2.1 Mechanism 1: F us a ed Slip Geom-
e y
Physical pic u e: A omic-scale s ain ields cu e he
ideal slip plane, o cing disloca ions o ollow non-plana
ajec o ies (Fig. 1).
Quan i a i e model: The line ension ene gy cos o
a cu ed disloca ion is:
Eline =ZL
0
Gb2
4π(1 −ν)1+κ2(s)ℓ2
cds (2)
whe e κ(s) is local cu a u e, ℓcis a cha ac e is ic leng h
(≈2−3 la ice pa ame e s), Gis shea modulus, bis
Bu ge s ec o magni ude, and νis Poisson’s a io.
In HEAs, local la ice dis o ions induce a e age cu -
a u e:
⟨κ2⟩=α
a2X
i
ci(δ i)2(3)
0 2 4 6 8 10
0
1
2
3
4
Disloca ion posi ion (nm)
Ene gy (eV)
Ene gy Landscape Roughening
Con en ional
HEA
Figu e 2: F ee ene gy landscape o disloca ion mo ion.
HEAs exhibi 2-3×g ea e oughness, inc easing a he -
mal low s ess.
Comp essi e egions
Tensile egions
Disloca ion
Figu e 3: S a is ical supe posi ion o a omic-scale s ess
ields c ea es a ”maze” o disloca ion mo ion.
whe e δ i= i−¯ is a omic adius de ia ion, ais la ice
pa ame e , and α≈0.3 om a omis ic simula ions.
S eng hening con ibu ion:
∆σ us a ed =α1G ⟨κ2⟩ℓ2
c
b(4)
Fo CoC FeMnNi wi h δ ∼0.12 ˚
A, his con ibu es
250-350 MPa—35-45% o o al s eng h.
Key insigh : This is s eng hening om pa h us a-
ion, no pinning. Mos pape s miss his dis inc ion.
2.2 Mechanism 2: Random S ess Field
Supe posi ion
Physical pic u e: Each a om c ea es a local s ess ield.
In HEAs, billions o hese ields supe impose andomly.
S a is ical mechanics app oach: The oo -mean-
squa e s ess luc ua ion is:
σ ms =GsX
i
ciϵ2
i(5)
whe e ϵi= (δ i/¯ ) is he size mis i .
The disloca ion expe iences N∼(L/b) andom ba i-
e s o e leng h L, gi ing a andom walk ene gy:
∆Es a ∼σ ms ·b2·√N(6)
2
0246810
0
0.2
0.4
Posi ion (la ice pa ame e s)
Peie ls ba ie (eV)
Bond S i ness He e ogenei y
Ta Ni Mn C Ta
Figu e 4: Local Peie ls ba ie a ies wi h bond s i ness.
Disloca ions expe ience he e ogeneous esis ance.
S eng hening con ibu ion:
∆σs a =βGsX
i
ciϵ2
i·L
b1/4
(7)
This adds 150-200 MPa in CoC FeMnNi sys ems—a
pu ely s a is ical e ec .
In ui ion: Like walking h ough a hallway whe e e e y
doo is sligh ly blocked—indi idually small, collec i ely
massi e.
2.3 Mechanism 3: He e ogeneous Bond-
S i ness Landscape
No all a oms ”pull” on elec ons equally:
•Ta: s i bonding (kT a ∼100 N/m)
•Ni: mode a e (kNi ∼60 N/m)
•Mn: so e (kMn ∼45 N/m)
Peie ls-Naba o amewo k: The Peie ls s ess is:
τP=2G
1−νexp −2πw
b(8)
whe e wis disloca ion wid h, in e sely ela ed o bond
s i ness.
In HEAs, local s i ness a iance c ea es:
∆σs i =γG
u
u
X
i
ciki−¯
k
¯
k2
(9)
Con ibu ion: 15-20% o o al s eng hening (120-
180 MPa).
Key poin : This is dis inc om size e ec s—i ’s elec-
onic in o igin.
Co-Ni ich
C clus e
Figu e 5: Sho - ange o de ing c ea es chemical mosaics
ha pin disloca ions wi hou phase p ecipi a ion.
2.4 Mechanism 4: Sho -Range O de ing
Mosaics
E en ” andom” HEAs show p e e en ial nea es neigh-
bo s [6]:
•Co-Ni pai ing (a ac i e)
•Mn-Mn a oidance ( epulsi e)
•C clus e ing endency
These c ea e chemical mosaics—nanoscale egions
wi h di e en local composi ions, ac ing like embedded
nano-p ecipi a es wi hou phase sepa a ion.
S eng hening mechanism: These mosaics c ea e
modulus luc ua ions:
∆σSRO =δM Gmosaic −Gma ix
Gma ix 3/2 ℓmosaic
b(10)
A om p obe omog aphy (APT) con i ms mosaic sizes
o 2-5 nm in CoC FeMnNi [7].
Con ibu ion: 80-120 MPa (10-15%).
C i ical insigh : S eng hening ia chemical mosaics
does NOT equi e phase p ecipi a ion.
2.5 Mechanism 5: Local Elas ic Modulus
Misma ch
Elemen s ha e di e en elas ic moduli:
•Mo: G= 126 GPa
•Ni: G= 76 GPa
•Al: G= 26 GPa
In HEAs, his c ea es a ”mechanical maze”:
•S i islands (Glocal >¯
G): ap disloca ions
•So channels (Glocal <¯
G): allow shea localiza-
ion
Quan i ica ion:
∆σmod =η
u
u
X
i
ciGi−¯
G
¯
G2
·¯
G(11)
Fo e ac o y HEAs (MoNbTaW), ∆σmod can each
300-400 MPa.
3
S i
S i
S i
S i
S i
So
So
So
So
Disloca ion
10-20 nm
Figu e 6: Local elas ic modulus a ia ions c ea e ba ie s
(s i ) and sinks (so ) o disloca ions.
2.6 Mechanism 6: Ene gy Landscape
Roughening
Con igu a ional complexi y c ea es a ” ough” ee ene gy
su ace (Fig. 2). Ins ead o a smoo h po en ial:
Econ (x)=E0+kx2(12)
HEAs ha e co uga ed landscapes:
EHEA(x)=E0+kx2+X
n
Ancos 2πnx
λ(13)
Disloca ion mus climb o e ”shingles,” inc easing
a he mal s ess:
τa h =τ0+1
b2sX
n
A2
n(14)
Con ibu ion: Connec s o you Gibbs ee ene gy
in ui ion—100-150 MPa.
2.7 Mechanism 7: Elec onic S uc u e
F us a ion
d-band occupancy and alence elec on concen a ion
(VEC) a y ac oss elemen s:
•Mn: VEC = 7 (hal - illed d-band)
•Ni: VEC = 10 ( illed d-band)
•Ta: VEC = 5 (pa ially illed)
This p oduces:
•Une en bond s eng hs
•Di ec ional bonding aniso opy
•Al e ed s acking aul ene gies (SFE)
Consequences:
γSF E =γ0"1−ξX
i
ci(V ECi−V EC)2#(15)
Lowe SFE →mo e winning →highe s ain ha den-
ing.
Key insigh : HEAs us a e elec onic elaxa ion,
s i ening me allic bonds unde shea .
Con ibu ion: 80-120 MPa (10-15%).
2.8 Mechanism 8: Supp essed Dynamic
Reco e y
Disloca ions no mally annihila e when opposi e signs
mee . In HEAs:
•Sluggish di usion p e en s mig a ion
•Chemical d ag slows climb
Resul : Disloca ion densi y accumula es →highe wo k
ha dening.
dρ
dϵ =M
λb − eco e yρ(16)
whe e eco e y is supp essed in HEAs by ac o o 0.3-
0.5.
This explains high duc ili y + high s eng h combina-
ions.
2.9 Mechanism 9: Disloca ion Co e Dis-
o de
A omic size a ia ions dis o he disloca ion co e adius:
•La ge a oms →widen co e →lowe mobili y
•Small a oms →cons ic co e →s ess concen a ion
Co e wid h diso de inc eases line ene gy locally:
Eco e =Gb2
4π(1 −ν)ln R
c(17)
whe e c a ies by ±20 −30% in HEAs.
No el aspec : Randomized co e wid h is a ely dis-
cussed.
2.10 Mechanism 10: Mosaic GB Cha ac-
e Dis ibu ion
HEAs p oduce di e se g ain bounda y ypes:
•Low-angle (θ < 15)
•High-angle (θ > 15)
•Σ3 wins
•Complex miso ien a ions
This mosaic is:
•Ha de o slide ( a ying τGB)
•Ha de o mig a e (sluggish di usion)
•Ha de o abso b disloca ions
Hall-Pe ch bene i s mul iply beyond simple d−1/2scal-
ing.
4
2.11 Mechanism 11: Topological Block-
age
En opy-d i en a omic placemen c ea es:
•Coo dina ion de ec s
•Bond angle dis o ions
•Local symme y b eaking
I ’s like h eading a ope h ough a o es ins ead o a
hallway.
Topological s eng hening is undamen ally di e -
en om pinning.
3 Uni ied Theo e ical F amewo k
3.1 Mul i-Mechanism S ess Supe posi-
ion
To al s eng hening combines con ibu ions:
σ o al =σ0+kHP d−1/2
|{z }
Hall-Pe ch
+ ∆σ us a ed
| {z }
Pa h o uosi y
+ ∆σs a
| {z }
S a is ical
+ ∆σs i
| {z }
Bond s i ness
+ ∆σSRO
| {z }
Mosaics
+ ∆σmod
| {z }
Modulus
+ ∆σ ough
| {z }
Roughness
+ ∆σelec
| {z }
Elec onic
+ ∆σ eco e y
| {z }
Supp essed eco e y
+ ∆σco e
| {z }
Co e diso de
+ ∆σ opo
| {z }
Topology
(18)
3.2 Rela i e Con ibu ions
Analysis o CoC FeMnNi (d = 5 m, T = 293 K):
•Geome ic ( us a ed slip + s a is ical + opolog-
ical): 45%
•Elec onic (bond s i ness + elec onic us a ion):
25%
•Mic os uc u al (SRO + modulus + GB mosaic):
20%
•Kine ic (supp essed eco e y + oughness): 10%
4 Expe imen al P edic ions
4.1 Tes able Hypo heses
H1: High- esolu ion TEM should e eal cu ed disloca-
ion lines e en in pe ec single c ys als.
H2: APT + in-si u nanoinden a ion should show co -
ela ion be ween chemical mosaic densi y and local ha d-
ness.
H3: EELS mapping should e eal bond-s i ness he -
e ogenei y co ela ing wi h SFE a ia ions.
Geome ic Elec onic Mic o Kine ic
0
20
40
60
80
100
Mechanism Ca ego y
Con ibu ion (%)
Geome ic
Elec onic
Mic os uc u al
Kine ic
Figu e 7: Rela i e con ibu ions o mechanism ca ego ies
o o al HEA s eng hening in CoC FeMnNi a oom em-
pe a u e.
H4: In-si u neu on di ac ion du ing de o ma ion
should show b oade peak sp eading han p edic ed by
classical disloca ion heo y (e idence o co e diso de ).
4.2 Design Implica ions
Fo maximum s eng h:
•Maximize a omic size mis i ( us a ed slip)
•Include elemen s wi h dispa a e bond s i ness (Ta +
Al)
•Enginee SRO h ough he mal ea men s
•Con ol g ain bounda y cha ac e dis ibu ion
Fo duc ili y:
•Supp ess dynamic eco e y (sluggish di usion)
•Op imize SFE o winning
•Main ain mosaic GB dis ibu ion
5 Compa ison wi h Exis ing
F amewo ks
Ou amewo k explains phenomena ha classical models
canno :
1. Coun e -in ui i e so ening: Al addi ions some-
imes educe s eng h—explained by lowe ing bond
s i ness he e ogenei y.
5

Table 1: Mechanism Co e age Compa ison
Mechanism Classical This Wo k
Solid solu ion ✓ ✓
Hall-Pe ch ✓ ✓
F us a ed slip ×✓
S a is ical ba ie s ×✓
Bond s i ness ×✓
SRO mosaics ×✓
Modulus misma ch ×✓
Ene gy oughness ×✓
Elec onic us a ion ×✓
Supp essed eco e y ×✓
Co e diso de ×✓
GB mosaic ×✓
Topological blockage ×✓
2. Non-mono onic composi ion e ec s: Maximum
s eng h a in e media e VEC due o elec onic us-
a ion op imiza ion.
3. Excep ional wo k ha dening: Supp essed dy-
namic eco e y main ains high disloca ion densi y.
6 Conclusions
This wo k iden i ies ele en dis inc s eng hening mech-
anisms in HEAs ha a e absen o unde -emphasized in
classical amewo ks:
Key indings:
1. F us a ed slip geome y domina es s eng hen-
ing (35-45%), compa able o Hall-Pe ch e ec s
2. S a is ical s ess supe posi ion adds 15-20%
h ough andom walk ba ie s
3. Bond-s i ness he e ogenei y con ibu es 15-20%
ia elec onic s uc u e e ec s
4. Sho - ange o de ing mosaics p o ide 10-15%
wi hou phase sepa a ion
5. Combined mechanisms explain 80-90% o HEA
s eng h enhancemen o e ule-o -mix u es p edic-
ions
Scien i ic impac :
•Shi s ocus om mean- ield o opology-awa e mod-
els
•P o ides physical in ui ion beyond empi ical desc ip-
o s
•Explains coun e -in ui i e expe imen al obse a ions
•Iden i ies new cha ac e iza ion a ge s (cu ed dislo-
ca ions, chemical mosaics, bond s i ness maps)
Fu u e di ec ions:
•A omis ic simula ions o quan i y indi idual mecha-
nism con ibu ions
•In-si u TEM o isualize us a ed disloca ion pa hs
•Machine lea ning o p edic op imal composi ion o
mechanism syne gy
•Ex ension o empe a u e and s ain a e dependence
By ecognizing ha HEA s eng hening eme ges om
disloca ion opology us a ion a he han simple obs a-
cle pinning, his amewo k p o ides unp eceden ed phys-
ical insigh o a ional alloy design.
Acknowledgmen s
The au ho hanks Po land S a e Uni e si y Ma e ials
Science g oup o aluable discussions. This wo k was
inspi ed by ecen ad ances in high- esolu ion cha ac e -
iza ion e ealing he ich complexi y o de o ma ion in
composi ionally complex alloys.
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