Grain boundary diffusion in severely deformed Al-based alloy

0.43 MB·

by  Sergiy V. Divinski

Grain boundary diffusion in severely deformed Al-based alloy
Sergiy V. Divinski,1,∗ Vladislav Kulitcki,1,2 Bengu¨ Tas Kavakbasi,1 Ankit Gupta,3
Yulia Buranova,1 Tilmann Hickel,3 J¨org Neugebauer,3 and Gerhard Wilde1
1Institute of Materials Physics, University of Mu¨nster, D-48149 Mu¨nster, Germany
2Belgorod State University, Belgorod, 308015, Russia
3Max-Planck-Institut fu¨r Eisenforschung GmbH, D-40237 Du¨sseldorf, Germany
(Dated: January 25, 2017)
7 Grain boundary diffusion in severely deformed Al-based AA5024 alloy is investigated. Different
1 states are prepared by combination of equal channel angular processing and heat treatments, with
0 the radioisotope 57Co being employed as a sensitive probe of a given grain boundary state. Its
2 diffusion rates near room temperature (320 K) are utilized to quantify the effects of severe plastic
deformationandapresumedformationofapreviouslyreporteddeformation-modifiedstateofgrain
n
boundaries,solutesegregationattheinterfaces,increaseddislocationcontentafterdeformationand
a
J of the precipitation behavior on the transport phenomena along grain boundaries. The dominant
4 effect of nano-sized Al3Sc-based precipitates is evaluated using density functional theory and the
Eshelby model for thedetermination of elastic stresses around theprecipitates.
2
Keywords: Al5024alloy;Severeplasticdeformation;Grainboundarydiffusion;Precipitation
]
i
c
s I. INTRODUCTION meltingpointduetoaknownsemi-empiricalrelationship
-
l betweenthediffusionratesandthemeltingtemperatures
r of various classes of materials13.
t Al-basedalloysareattractivematerialsforvariousindus-
m
trial applications owing to their low density combined As stated above, the deformation-modified state of
t. with sufficient mechanical strength and ductility. The grain boundaries was documented for Ni and Ti with
a
severe plastic deformation (SPD) of metals is a method relativelyhighmelting points deformedatroomtemper-
m
to improve the mechanical properties of the material ature. DoestheSPDprocessingoflow-meltingpointma-
- via a strong microstructure refinement and the forma- terialslike Al orAl-basedalloysinduce the deformation-
d
n tion of nanostructures1,2. Equal channel angular press- modifiedgrainboundarieswithenhanceddiffusionrates,
o ing (ECAP) is one such deformation technique beside of too? Cold plastic deformationof Al has been shown14,15
c high pressure torsion (HPT) or accumulative roll bond- to modify the interface structures in pure Al by intro-
[ ing (ARB) that allows producing materials with ultra- ducing so-called extrinsic grain boundary dislocations.

1 fine grain(UFG) microstructure and supports industrial Still these modifications were limited exclusively to spe-
v up-scaling3. ECAP processing at elevated temperatures cial grain boundaries in Al, whereas the general high-
4 was, e.g., applied to achieve superplastic properties of angle GBs relaxed too fast already at room tempera-
9 Al-based alloys4. ture for any modification to be observed. Addition of
6 The material modification via SPD processing is not MgtoAlstabilizedextrinsicgrainboundarydislocations
6
only limited to grain refinement, but also includes mod- andadeformation-modifiedstateofinterfaceswasstated
0
. ifications of the grain boundary (GB) kinetics, structure to have been observed which then relaxed at elevated
1 and segregation5 simultaneously with the production temperatures16,17. For example, zigzag, highly stepped
0
of abundant point defects and dislocations in the pro- configurationsofgrainboundariesinanUFGAl–3%Mg
7
1 cessedmaterial6,7. Forexample,anenhancementofgrain alloy which recovered after irradiation by the electron
: boundaryself-diffusionratesby ordersofmagnitudewas beam in TEM were reported17. Although these observa-
v established for ECAP-processedNi8 or Ti9, a fact which tionswerediscussedintermsofthedeformation-modified
Xi wasunambiguouslyattributedtoadeformation-modified (’non-equilibrium’ as it was originally introduced14,16)
r (in pioneering works termed as “non-equilibrium”10) state of the interfaces17, it is known that even well-
a stateoftheinterfacesduetodefectincorporationanden- annealedgrainboundariesmayalsobefacetedanddisso-
hancement of the excess free volume9,11. In some cases, ciatedontheatomicscale18. Thus,kineticmeasurements
an extra-ordinary high strength of UFG materials was wouldbehelpfultoelucidateahypotheticaldeformation-
attributed to specific grain boundary segregations12. modifiedcharacterofthegrainboundariesinadditionto
the microstructure observations.
Itiswellknownthatbeyondacriticalstrain,afurther
increaseof the applied totalstraindoes notproduce fur- Thepresentworkisfocusedonstudyingtheeffectofse-
thergrainrefinement,andasaturationofthemicrostruc- vereplasticdeformationonthekineticpropertiesofinter-
turemodificationisobservedwhichcorrespondstoabal- facesinanAl-basedalloyandcontributestoabetterun-
ance between the rates of defect generation and their derstanding of the coupling between the microstructure
annihilation7. For the given deformation temperature evolution, precipitate formation and local strain-stress
and strain rate, the relative rate of recovery processes fields as a function of annealing conditions. For these
will be facilitated in the case of materials with a lower purposes,thetracerdiffusionmethodisappliedtofollow
2
the rate of atomic transport, which is extremely sensi- Local microstructure analysis was performed with a
tive to the state ofthe grainboundaries,grainboundary transmission electron microscope (TEM) using a Libra
segregation and grain boundary precipitation19,20. The 200 FE TEM (Zeiss) and a JEOL JEM-2100 (both with
57Coisotopeischosenasaconvenienttracerelementand anaccelerationvoltageof200kV).Thedisk-shapedsam-
its grain boundary diffusion is measured at about room ples for TEM analysis were cut from sections with the
temperature in Al-alloys which were subjected to dif- foil-normal perpendicular to the extrusion direction in
ferent heat treatments at elevated temperatures. Thus, the centralpart of the processedmaterials. The samples
the determined diffusion coefficients represent a specific fromthe ECAP-processedrodswerecut perpendicularly
probe of the given grain boundary state on a time scale totheextrusiondirection. 1mm-thickdisks,3mmindi-
when bulk diffusion of all substitutional elements in Al ameter, were cut by spark erosionand mechanically pol-
is frozen and the attained state is not modified during ished down to the thickness of about 90 µm. The final
low-temperature diffusion measurements. thinning was done by chemical electropolishing(twin-jet
A complex Al-based AA5024 alloy was chosen for the electro-polishinginasolutionofHNO3:CH3OHwith1:2
investigation,withMg,Sc,andZrasmainalloyingcom- ratio at 253 K).
ponents. Sc was added onpurpose to facilitate the grain The sizes of the equiaxed grains were quantified by
refinement via nano-precipitation21,22. On the other measuring the grain area and calculating an equivalent
hand,astrongsegregationofMginthisalloyistobeex- grain diameter by modeling each grain as a circle in ac-
pected, too23,24. As a result, a sophisticated interplay of cordance with the grain reconstruction method26. The
hypothetical deformation-induced grain boundary mod- sizeoftheprecipitateswasestimatedinbright-fieldTEM
ifications, grain boundary segregation and precipitation (BF-TEM) using at least ten arbitrarily selected micro-
may be expected. graphs,asitwasemployedinRef.27. Additionally,dark
field-TEM (DF-TEM) and high-angle annular dark-field
scanning TEM (HAADF-STEM) micrographs were em-
ployed to measure the size of about 100 precipitates in
II. EXPERIMENTAL PROCEDURE
eachstate. Thetotalnumberofindividualmeasurements
for eachcondition was about 1000,see also Refs. 28 and
A. Material
29. The dislocation density was calculated in TEM by
measuringthenumberofdislocationsforthe giveninter-
The Al-based AA5024 alloy with the nominal com- section length and foil thickness.
position Al–4.6Mg–0.35Mn–0.2Sc–0.09Zr–0.2Ti–0.08Fe– Thedeformedmicrostructuresweredescribedindetail
0.02Si (in wt.%) was prepared in the research group of in previous publications4,22,30. Here we will present the
Prof. R. Kaibyshev (Belgorod State University, Rus- most relevant details and the results for the microstruc-
sia). It was produced by continuous casting and the in- tures relevant for subsequent diffusion measurements.
gots were homogenized by annealing at (643 10) K for
±
12 h. The ingots were then extruded at 653 K impos-
ing about70% strain. The resulting rods were subjected
C. Grain boundary tracer diffusion of 57Co
to ECAP processing with application of about 100 MPa
back-pressure at a temperature of 573 K25. The final
Before the diffusion measurements, the surface of the
grain size was about 500 nm.
samples was polished to a mirror-like quality. Three
For a reference, coarse grained high-purity Al (99.999
to five microliters of the 57Co tracer solution (272 days
wt.%)wasused. Thematerialwasannealedat873Kfor
half-life and 122 keV γ-radiation), was dropped on the
16handslowlycooled. Thegrainsizewasabout0.5mm.
prepared surface and dried. The samples were sealed in
silica tubes under a purified argon atmosphere and an-
nealed. The annealing temperature was set at 320 K
B. Sample preparation and characterization tobeslightlyaboveroomtemperatureandtheannealing
timewasfixedat3days. Inordertoeliminatetheeffects
Discsof10mmindiameterand1.5mminheightwerecut of surface and lateral diffusion, the sample diameter was
by spark erosion. Some samples were subjected to pre- reduced by about 1 mm by grinding after the diffusion
diffusionheattreatmentsatselectedtemperaturesforthe annealing treatment.
given times listed in Table I. TheparallelserialsectioningwasperformedusingMy-
The microstructures were characterized by scanning larfoils (15µmparticle size)ona custom-builtprecision
electronmicroscopy(SEM) usinga Nova Nano SEM230 grinding machine. After removing a section, the sample
(FEI).Orientationimagingmicroscopywasappliedusing was weighed by a microbalance. The thickness of each
electron back-scatter diffraction (EBSD). For the EBSD section was determined by the mass difference from the
analysis,thesamplesweremechanicallyground,polished known density and radius of the sample. The relative
using diamond suspensions and finished with colloidal specific radioactivity of each section (which is propor-
silica (0.04 µm) until a mirror-like surface finish was tional to the tracer concentration) was determined by
achieved. measuringthe radioactivedecaysofthe 57Cotracer by a
3
TABLE I. The results of the diffusion experiments for Co GB diffusion in pure Al and the Al-based AA5024 alloy at 320 K
for 3 days, Dgb. Before the diffusion experiments some samples were annealed at Tan for the time tan. The mean grain size,
∗
d, dislocation density, ρd, and the average particle size, Rav, are specified. The parameter α is calculated using Eq. (4).
The value of the GB diffusion coefficient measured for the ECAP-processed state (with α∗ < 1) has to be corrected and the
corrected value is given, Dcorr (see also text).
gb
Sample Tan (K) tan (h) Dgb (10−15 m2/s) d (µm) ρd (1013 m−2) Rav (nm) α∗ Dgcobrr (10−15 m2/s)
PureAl – – 17.5 500 10−2−10−1 – 6000 –
as-cast – – 7.15 40 0.8 5−6 7.5 –
– – 7.21 0.8 7 11 0.85 9.61
ECAP- 723 1 4.16 1.2 <1 12−14 >6 –
processed 773 20 1.41 7 <1 20 >6 –
at 573 K 823 50 3.88 10 <1 30−40 >6 –
823 200 9.0 >20 <1 >50 >6 –
well-type intrinsic Ge γ-detector. afacecenteredcubic (fcc)Almatrixis obtainedemploy-
The grain boundary diffusion conditions were set to ing the climbing image nudged elastic band (CI-NEB)
fulfill Harrison’s type C kinetic regime31. Therefore, method34,35. The diffusion barrier is calculated as the
the concentration profiles were plotted as the loga- difference between the energies of the saddle point and
rithm of the layer tracer concentration, lnc¯, vs. the the initial equilibrium state along the minimum energy
depth squared, y2, and the corresponding diffusion coef- path in the configuration space along the [110] direction
ficients, D , were determined using the standard Gaus- connecting the initial and the final transition state (see
gb
sian solution32, Fig. 5b). For the present case, the saddle point lies mid-
wayalongthis minimumenergypath. The straindepen-
dence of the diffusion barrier is obtained by calculating
1 ∂lnc¯ −1 the latter in a hydrostaticallystrained Al matrix for dif-
D = (1)
gb 4t(cid:18)− ∂y2 (cid:19) ferent values of the lattice parameterin orderto address
theexperimentally-relevanteffectofhydrostaticpressure
on diffusion.
Here t is the diffusion time.
III. DENSITY FUNCTIONAL THEORY (DFT) A. Computational details
METHODOLOGY
We have used 2 2 2 and 3 3 3 fcc-supercells
× × × ×
Thepresentpaperisfocusedontheeffectofelasticstrain (32 and 108 atoms respectively) with one Co atom and
in the Al-based matrix on GB diffusion of Co. How- one vacancy on a nearest-neighbor site of Co to com-
ever, direct DFT computations for general high-angle pute the minimum energy path employing the CI-NEB
GBs (as they are addressed in the experimental part) method34,35 as implemented in the VTST package36.
represent an extremely involved task and a simplified Both the supercells were generated using the calcu-
approach is used here. We will make use of the exper- lated equilibrium lattice parameter (4.04 ˚A) of pure Al.
imentally established correlation between the activation The total energy calculations were performed employing
energies of GB diffusion, Qgb, (measured directly in the the projector augmented wave (PAW) method as imple-
correspondingC-typeregimeforsoluteGBdiffusion)and mented in the VASP37,38 package with energetics based
those for bulk diffusion in the same matrix, Qv, in met- ondensityfunctionaltheory(DFT).Thegeneralizedgra-
als, Qgb (0.4 0.6)Qv13,32. Moreover, the pressure- dient approximation (GGA) within the Perdew-Burke-
≈ −
dependent measurements of Zn GB diffusion in Al pro- Ernzerhof(PBE)parameterizationscheme39 wasusedto
vide similar values of the activation volume which could describe the electronic exchange and correlation effects.
beexpectedforbulkdiffusioninAl,about0.8Ωand0.9Ω, The integration over the Brillouin zone was performed
respectively33 (Ω is the atomic volume). For the present usingthe Monkhorst-Pack40schemewithan12 12 12
× ×
analysis, the influence of elastic strains on diffusion bar- (for 32 atoms) and 8 8 8 (for 108 atoms) reciprocal-
× ×
riers will rigorously be calculated for the Co atoms in space k-mesh centered around the Γ point and utilizing
the Al lattice and these values will be used to further theMethfessel-Paxtonscheme41withathermalsmearing
estimate Co GB diffusion. width of 0.15 eV to account for the smoothening of par-
The minimum energy path and the diffusion barrier tial occupancies of the electronic states. The plane-wave
for a vacancy-mediated nearest neighbor jump of Co in energy cutoff was set to 400 eV. An energy of 10−6 eV
4
was used as a convergence criteria for the self-consistent y (µm)
0 20 40 60
electronic loop. A tolerancevalue of 0.05eV/˚A wascho-
sen for the force components along and perpendicular
s)s)
to the tangent to the reaction path. A total of 5 im- nitnit
uu
ages/configurations (in addition to the initial and the b. b.
finalstate)wereconsideredtogeneratetheminimumen- arar
ergy path in the configuration space. n (n (
yy
The straindependence of the diffusion barrierwas ob- vitvit
cticti
tained by performing the total energy calculations for aa
c c
13 equally spaced hydrostatic strain (ǫ) values ranging cificifi
ee
between 1.5% (-1.5, -1.25, -1, ..., 1, 1.25, 1.5) where pp
ǫ = (a ±aeq)/aeq, aeq being the computed equilibrium ve sve s
lattice p−arameter of pure Al (4.04 ˚A) as obtained within elatielati
RR
GGA.Withthisdefinitionofǫ,thenegativeandpositive
valuescorrespondtocompressiveanddilatationalstrains
0 1 2 3 4
respectively. The internal atomic coordinates were fully y2 (10-10 m2)
relaxed during the total energy calculations maintaining
the strained volume. FIG. 1. Examples of the penetration profiles for 57Co tracer
diffusion at room temperature in the AA5024 alloy in the
following states: as-cast(squares),ECAP-processed (circles),
IV. RESULTS ECAP-processedandannealedat623Kfor1h(trianglesup)
and at 773 K for 20 h (triangles down).
Examplesofthe penetrationprofilesof57Cotracerdiffu-
sionintheAA5024alloyforthedifferentstatesareshown
in Fig. 1. The first near-surface points are affected by nearly equilibrium conditions for grain boundary segre-
the grindingprocedureandomittedfromthe subsequent gation of residual impurities.
analysis. The bulk diffusion coefficient of Co in pure Al
at 320 K is about 10−30 m2/s as it follows from the Ar-
rhenius parameters reported by Hood et al.42. Similar V. DISCUSSION
values – at least, not extremely enhanced – can safely
be assumed for Co volume diffusion in the Al-alloy, too, In the present work, the tracer diffusion coefficients of
taking into account the relatively low amount of the al- Co atoms in Al-based AA5024 alloy and pure Al are
loying components and the typical values of the solute measured for different material states (Table I). Several
enhancement factors43. striking features have to be highlighted.
The exact values of the volume diffusion coefficients
arenotrelevant,sincetheeffectiveGBwidth, s δ,issig- 1. Co GB diffusion is faster in well-annealed coarse-
nificantly largerthan the effective diffusionleng·th in the grainedpureAlwithrespecttotheas-castAl-based
bulk (condition for C-type diffusion32), i.e. the crucial AA5024 alloy;
parameter
2. Co GB diffusion reveals almost similar rates in as-
cast and ECAP-processed states of the Al-based
s δ AA5024 alloy;
α= · , (2)
2√D t
v 3. There is an unexpected trend with a strongly non-
is significantly larger than unity. In this expression, D monotonousdependenceoftheCoGBdiffusionco-
v
is the volume diffusion coefficient of Co in Al, and t is efficientsintheultrafinegrainedAlalloyonthean-
the diffusion time. The GB width δ was measured for nealing parameters, in fact they first decrease and
fcc metals to be about 0.5 nm44,45 and the segregation thenincreaseagainreachingalmostthelevelwhich
factorforCoatAlgrainboundariesscanbeexpectedto is characteristic for the Co GB diffusion rates in
be 1. Note that simple estimates prove that α > 100 as-cast or ECAP-processes states.
≥
already for s=1.
The determined diffusion coefficients, D , of Co dif-
gb
fusion in the Al-based AA5024 alloy in different investi- A. Effect of segregation on GB diffusion
gated states are listed in Table I.
For a reference, Co GB diffusion in pure Al was mea- The feature 1 may be explained by the alloying effect on
sured, too (Table I). For this aim, after polishing, the GB diffusion, since it follows a general trend observed
polycrystallineAlsample wasannealedat823K forsev- experimentally that the purer is the material,the higher
eral hours and slowly cooled to room temperature to is the GB diffusion rate; as it was found, e.g., for GB
remove all effects of preparation procedure and assure self-diffusion in pure Cu46 or Ni45,47 or solute (Ni) GB
5
diffusion in pure Cu48. It is known that Mg does seg- C. Relaxation of the deformation-modified state
regate to grain boundaries in this alloy after annealing
at slightly elevated temperatures22 as it does in other Nazarov and co-workers10 described a deformation-
Al–Mg alloys49. modified (‘non-equilibrium’) state of grain boundaries
in terms of arrays of extrinsic grain boundary disloca-
In the present case of the ECAP-processing at 573 K
tions. These defects were proposed to relax by disloca-
and post-annealing at higher temperatures, the chemi-
tion climb and annihilate at elevated temperatures. The
calanalysisdidnotrevealanymeasurablesegregationat
corresponding relaxation time, τ, was estimated as51
most of the GBs (about 80%) in the limits of the uncer-
tainty of the EDX measurements. In some of the GBs,
a slight increase of the local Mg concentration by about k Td3
B
τ = (3)
0.5%couldbedetected. Still,asignificantincreaseofthe AδDsdGΩ
gb
Mg concentration was observed at triple junctions.
wherek andT arethe Boltzmannconstantandthe ab-
B
Mg segregation was observed in the as-cast AA5024
solute temperature, respectively, d is the average grain
alloy22, however, Co GB diffusion proceeds with simi- size, δ the GB width, Dsd the self-diffusion coefficient
gb
lar rates in as-cast and annealed states with quite dif-
for general relaxed high-angle grain boundaries, G the
ferent levels of the Mg segregation. Thus it is not Mg
shear modulus, Ω the atomic volume, and A is a ge-
segregation which mainly affects Co GB diffusion in the
ometrical factor. The value of A was suggested to lie
AA5024 alloy and some minority components in the al-
between 100and 500,depending on the specific disclina-
loyareprobablyresponsible forthe diffusionretardation tion model of the ultrafine grained materials used51 and
in the alloy with respect to the pure Al grain bound-
it was shown that this value gives reasonable estimates
aries. Moreover, GB reconstruction in the AA5025 alloy for SPD-processed Ni8. As stated above, the diffusional
(following, e.g., the GB complexion theory50) might be GB width δ 0.5 nm45. The GB self-diffusion coeffi-
responsible, too. cient in Al, D≈sd, is estimated at about 2.8 10−16 m2/s
gb ×
at 320 K based on the Arrhenius parameters reported
in Ref. 52. Using the material parameters of Al (G =
26 GPa and Ω = 17.2 ˚A3), the relaxation time of the
deformation-modifiedstate ofthe grainboundariesinAl
is on the order of seconds at the measurement tempera-
B. Effect of SPD-processing on GB diffusion tureof320K.Takingintoaccountafinitetime,required
foradeformedbillettobecooleddowntotheroomtem-
peraturefromthedeformationtemperatureof573K,one
The present results substantiate that the severe plastic
may safely conclude that a hypothetical enhancement of
deformation of the Al-based alloy, alternatively to the
the diffusion rates due to a deformation-modified state
caseofECAP-processingofNiorTi, doesnotproduce a
of GBs, if it does exist during deformation, relaxes and
deformation-modifiedstateofthegrainboundarieswhich
cannot be measured in the present post-mortem experi-
wouldbecharacterizedbyenhanceddiffusioncoefficients
ments.
(feature2). AdecreaseoftheGBdiffusioncoefficientsas
Thisconclusionfullyagreeswiththepresentresultson
aresultoftheECAP-processingwasalreadyobservedfor
GB diffusion of Co after ECAP-processing.
the case of Co diffusion in ultrafine grainedTi. However
this represents a very special case in view of the inter-
stitial diffusion mechanism of Co atoms in the hexag-
D. Effect of crystal dislocations
onal close-packed (hcp) lattice of α-Ti9. On the other
hand, vacancy-mediated GB diffusion of Ag atoms was
In addition to the excess GB dislocations and the GB
extremely enhanced in ECAP-processed α-Ti9. These
modifications,SPDprocessingis knownto induce ahigh
facts correspond to the generation of excess free volume
density of crystal dislocations which may approach the
atdeformation-modifiedGBswhichserveastrapsforin-
values on the order of 1015 m−2 for Ni or Cu53. Dislo-
terstitially diffusing Co and as vehicles for substitution-
cations in metals are known to represent generally fur-
allydiffusingAgatoms. Sincethesubstitutionaldiffusion
ther short-circuits for enhanced diffusion13,32 and may
mechanismholdsdefinitelyforCoinAl42,onemaysafely
modify diffusion transport during measurements of GB
assume substitutional solubility and a vacancy-mediated
diffusion54. Notethatdeformation-inducedvacanciesan-
diffusionmechanismfor Coatoms ingrainboundariesof
nihilate below roomtemperature in plastically deformed
Al and Al-alloys, too.
Al55 and cannot affect the present diffusion measure-
The absence of an enhancement of Co GB diffusion ments.
after SPD-processing of Al-based alloy is most probably At least two kinds of models have to be analyzed:
related to a relatively high homologous temperature of Dislocation model 1. Since the C-type conditions
the diffusion measurements, T/T , and need a clarifica- are fulfilled, the crystal dislocations represent short-
m
tion(hereT isthecorrespondingmeltingtemperature). circuitsandprovideadiffusionenhancementwithrespect
m
6
to bulk diffusion in addition to grain boundaries. How- dislocation pipes as D = 1/10D 32, the value of the
d gb
∗
ever, the rates of dislocation diffusion correspond typi- parameter α can be determined (Table I).
cally to about 1/10th of the diffusion rate along general An analysis reveals that α∗ > 1 for most of the con-
high-angle GBs32,56 and, thus, the dislocation diffusion ditions and the effect of dislocations can safely be ne-
contribution should not be observed in the present ex- glected, thereby confirming that Eq. (1) was correctly
periments apart from several first near-surface points of used. The only result of the diffusion measurements for
the concentrationprofiles in Fig. 1). Although the dislo- Co GB diffusion in ECAP-processed Al alloy with the
cation density is larger in ECAP-processed material by highestdislocationdensity(withoutannealing)hastobe
orders of magnitude, GB diffusion is slower in the ul- revisited,since the correspondingvalue ofα∗ islessthan
trafine grained state with respect to that in pure Al. unity(TableI).FromtheGBdiffusiontheory32,itiswell
Furthermore,post-deformationannealingtreatmenteven knownthatGBdiffusionunderconditionsof0.1<α<1
enhances GB diffusion, while a dramatic decrease of the correspondstoatransitionregimebetweentheB-andC-
dislocation density is observed and therefore opposite to types kinetics and the determined diffusion coefficients
whatwouldbe expectedwithinthis model. We conclude underestimate the real values60. Following the approach
that Model 1 is not applicable to the present case. suggestedinRef.60andoutlinedinRef. 32,acorrection
Dislocation model 2. The crystalline dislocations factor can be determined and the corrected value of the
are attached not only to the free surface of the sample, corresponding diffusion coefficient is listed in Table I.
but they cross the GBs, too. In the latter case, these
dislocations represent the paths for enhanced leakage of
the tracer atoms from grain boundaries. In the present E. Effect of particles on GB diffusion
case with the absence of bulk diffusion, the GB diffusion
problem with out-diffusion via dislocation corresponds
In Fig. 2a the measured diffusion coefficients D for Co
totheso-calledB-C-typeregimeafterDivinskietal.57,58 gb
intheECAP-processedAA5024alloyareplottedasfunc-
or D -type regime after Klinger and Rabkin54,59. The
1 tion of the post-deformation annealing temperature T
an
tracer leakage from GBs is controlled by the parameter
(symbols) and are compared with the value measured
∗
α , which is (compare with Eq. (2))
in the as-deformed alloy (dot-dashed line). It is obvi-
ous that the dependence cannotbe attributed to a usual
relaxation-likebehavior,sinceannealingat823Kforlong
sδ
∗
α = . (4) times increases the diffusion coefficients, although an-
2A ρ √D t
d d d nealingtreatmentsatlowertemperaturesdecreasethem,
Fig. 2a. This unusual temperature dependence needs
Here g =A ρ is the volume fraction of sites belonging
d d d
clarification.
to the dislocation pipes, with A being the cross-section
d
Figures 3b and c show a typical contrast (including
of the dislocationpipe andρ the dislocationdensity. In
d
the case of α∗ > 1, the C-type regime continues to hold the appearance of Moir´e patterns) at GBs in different
states. It is obvious that the interfaces are straight and
and Eq. (1) has to be applied to determine the corre-
spondingGBdiffusioncoefficients. Thesituationchanges flatandappearasrelaxedinterfaces. Thisfinding agrees
ifα∗ <0.1(andαremainstobelarge,α>1)54,inwhich perfectly with the results ofGB diffusion measurements,
which substantiate an absence of deformation-enhanced
case the concentration profiles have to be analyzed in
GBdiffusionintheAl-basedAA5024alloyunderconsid-
the coordinatesofthe logarithmofconcentrationvs. the
eration.
penetrationdepthy (i.e. forthequasiB-typeconditions)
and the triple product Al6Mnprecipitates arefoundatinterfacesas itissub-
stantiated by the HAADF-STEM image, Fig. 3a. The
bright-appearing particles in these conditions are char-
D ∂lnc¯ −2 acterized by larger Z numbers of scattering atoms with
∗ eff
P =s δ D =1.128 (5)
· · gb r t (cid:18) ∂y (cid:19) respecttothematrix(Al)andcorrespondtoAl6Mnpre-
cipitates.
has to be determined instead of the diffusion coefficient Figure 3a shows that in the as-cast state the Al6Mn
D 54. Here, precipitates are mainly concentrated near GBs. The av-
gb
erage size of these precipitates is about 200 nm with rel-
atively large average distance between them, more than
Deff =gd2Dd, (6) 300 nm. While the nano-sized Al3Sc-based particles are
typically coherent in the as-deformed AA5024 alloy, the
istheeffectivediffusivity,whichdescribesthetracerleak- larger Al6Mn particles are incoherent with the matrix.
agefromGBsintothecrystalvolumebydislocationpipe As discussed below, a relatively small contribution of
diffusion. In Eq. (5), s∗ is the segregation coefficient for these particles on the measured Co GB diffusion can be
the tracer atoms between the grain boundaries and the expected.
dislocation pipes which is probably a value of the order After ECAP processing,the precipitates (both Al Mn
6
of unity. Estimating the diffusion coefficient along the and Al Sc) are more uniformly distributed in the grain
3
7
20
15
15
Co-152D (10 m/s)gb105 Co-152D (10 m/s)gb10
5
0 0
700 750 800 850 900 0 10 20 30 40 50 60
a) Tan (K) b) Rav (nm)
FIG. 2. The dependence of the diffusion coefficient of Co in the ECAP-processed Al alloy on the post-deformation annealing
temperature(a)andtheparticleradiusindifferentstates,circles (b). In(a)theGBdiffusioncoefficient measuredforCoafter
ECAP deformation is shown by the dot-dashed line. In (b) the diffusion coefficient in a particle-free alloy is shown by the
dashed line. The diamonds represent the dependence of the diffusion coefficient (normalized with respect to the particle-free
case) on the particle size, that includes an implicit dependence on the associated elastic fields produced by the particles as
described bythe model outlined in subsection VI. The value of DCo in pureAlis shown, too (square).
gb
a) b) c)
FIG.3. a)HAADF-STEMimageofaGBinAA5024intheas-caststate. ThehighconcentrationofAl6Mnprecipitatesonthe
GBisindicatedbyredarrows. b)TEMimageofasinglegraininAA5024afterECAPat573K.Al6MnandAl3Scprecipitates
areindicatedbyredandgreenarrows, respectively. c)DistributionofAl3Sc-basedprecipitatesingrainsandaroundinterfaces
in AA5024 after ECAP at 573 K and annealing at 723 K.
interiors, Fig. 3b. The Al Mn particles are of a size of we have determined a misfit between the Al Sc precip-
6 3
about 70 nm and are predominantly located at disloca- itates and pure Al matrix of approx. 1.5% at T = 0K,
tions and sub-boundaries. The number fraction of par- whichisdecreasingwithtemperature,inagreementwith
ticles near the GBs in the ECAP-processed sample is the experimental value61 of 1.33% at room temperature.
estimated to be about 102 particles per µm3, which is Thecriticalprecipitatesize,correspondingtotheappear-
lower than the total volume fraction (about 104 µm−3). anceofmisfitdislocations,isthena2/[2(a a )]=15.2
m p− m
Thus the particles are mainly distributed in the grain nm, where 2 correspondsto the (002)-AlBurgersvector.
interiors, keeping the GBs relatively free. The average However,the AA5024alloycontains up to 4.5 wt.% of
distance between the precipitates in the deformed states Mg that increases the lattice parameter of the matrix62
is about 90 nm. thereby decreasing the misfit. Moreover, in the ECAP-
As a result of post-deformation annealing the parti- processedstatetheAl ScparticlescontainZr(andsome-
3
cles grow and the Al Sc precipitates approach an aver- times Ti)22 that decreases the lattice parameter of the
3
age size of 40 nm and more (Table I). It is important precipitates. We performed an analysis using the data
that whereas Al Sc precipitates are fully coherent in the fromRef. 63(see Table II)andthe results showthat the
3
ECAP-processed state (their average size is about 11 Al (Sc,Zr)precipitatesintheAl–4.5wt.%Mgmatrixstart
3
nm), they become semi-coherent after annealing treat- to loose their coherency with the matrix at the sizes of
ment as a result of growth, and finally loose their co- about20 30nm(dependingontheZrcontent). TheTi-
−
herency after annealing at high temperatures. containing Al (Sc,Ti) precipitates will not be analyzed
3
The precipitate/matrix misfit can be determined as further, since their volume fraction is found to be low22.
(a a )/a ,wherea anda arethelatticeparameters Figure 4 shows a dislocation analysis of the interface
p m m m p
−
ofthematrixandaprecipitate,respectively. UsingDFT, structureforanAl Scprecipitate. Thesizeoftheparticle
3
8
F. Elastic field around a precipitate
TABLEII.ThesummaryofcriticalsizesforAl3Sc-basedpre-
cipitates in the case of pure Al and the Al–Mg alloy. The
calculations were performed using thedata from Ref. 63. The stress field caused by the Al3Sc precipitates in the
matrix was calculated according to Eshelby’s solutions
Mismatch criticalparticle
Matrix Particle forsphericalinclusionsinaninfinite isotropicandelastic
(%) size(nm)
body64. The radial dependencies for the strain, ǫ , and
Al3Sc 1.33 15.2 r
Al Al3(Sc,Zr) 1.14 17.7 stress, σr, fields are,
Al3(Sc,Ti) 0.64 31.6
Al3Sc 0.87 23.4 1+ν a 3
Al+4.48%Mg Al3(Sc,Zr) 0.67 30.4 ǫr = (∆a+∆γ∆T) (7)
3(1 ν) r
(cid:16) (cid:17)
Al3(Sc,Ti) 0.18 113.0 −
σ =Eǫ . (8)
r r
is about22 nmandthe appearanceofmisfitdislocations
Here, ν is the Poisson ratio, a is the radius of a precipi-
has to be expected according to the analysis (Table II).
tate,rtheradialdistancefromtheprecipitatecenter,∆a
The precipitate is not perfectly spherical, which corre-
thelatticemisfitbetweentheprecipitateandthematrix,
lates with a relatively low content of Zr atoms22. A fast
E the Young modulus of the matrix, and ∆γ is the dif-
Fourier transformation (FFT) image made in the (002)-
ferenceofthe thermalexpansioncoefficientsbetweenthe
AlBraggmaximaindeedverifiestheexistenceofamisfit
precipitate and the matrix. We have taken into account
dislocationbetweenthe matrixandthe particles(redar-
that the precipitates are formed and relaxedduring hot-
row in Fig. 4).
deformation or annealing, while diffusion measurements
areperformedatnearlyroomtemperature. The value of
∆γ is taken from DFT-based calculations65 which were
performed for both, pure Al and Al Sc including vibra-
3
tional and electronic contributions, and it was further
shown that anharmonicity hardly affects the values.
The elastic strains (compressive) are maximum at the
particle surface and decay with increased distance from
the particle. Typically, the strain becomes small (less
than 0.1%) at the double particle radius. Elastic
−
stressesbehavequalitativelysimilarly,Eq.(8). Notethat
the yield stress of the AA5024 alloy is 530 GPa67.
Inthecaseofvacancy-mediateddiffusion,thediffusion
FIG. 4. Left: HR-TEM image of an Al3(Sc,Zr) precipitate coefficient of a solute, D, is generally,
in the AA5024 alloy after ECAP processing at 573 K and
subsequent annealing at 673 K for 1 h. Right: FFT-analysis
of theprecipitateinterface in the(002)-AlBragg maxima. A 1 ∆Gf +∆Gb+∆Gm
misfit dislocation is indicated by thearrow. D = fλ2ν0exp . (9)
6 (cid:18)− k T (cid:19)
B
In our previous work on Ni and Ti GB diffusion in Here,λisthejumpdistance,ν theattempt(Debye)fre-
0
NiTi alloy, a retardation of the GB diffusion rate was quency, f the correlationfactor (temperature-dependent
measured at high temperatures and was related to the for solute diffusion), and ∆Gf, ∆Gb and ∆Gm are the
precipitation of Ni Ti particles at the interfaces20. In free energies of vacancy formation, solute–vacancy bind-
4 3
thepresentwork,wewillanalyzetheeffectofprecipitates ing,andofvacancymigrationrespectively. Notethatthe
on GB diffusion in the Al-based severely deformed alloy. present tracer diffusion experiments are performed un-
Figure 2b re-plots the measured diffusion coefficients dernearlyequilibriumconditions,sincethedeformation-
D as a function of the averaged radius of the Al Sc- inducedvacanciesindeformedAlmatrixrelaxalreadyat
gb 3
based precipitates, R . A strongly non-monotonous de- room temperature55.
av
pendence is seen where the Co GB diffusion coefficient In the following we will neglect the effect of strain
first decreases with increasing radius, reaching a critical on the solute–vacancy binding energy as well as on the
valueataboutR 20 30nm,followedbyanincrease chemical part of the vacancy formation energy. Direct
av
≈ −
and approaching the value of D for small precipitates. DFT-based calculations, see below, verify this assump-
gb
Note that this is exactly the critical size for Al Sc and tion. The elastic part of the vacancy formation energy
3
Al (Sc,Zr) precipitates when appearance of misfit dis- dependsonthehydrostaticpressurepinducedbyprecip-
3
locations and a gradual loose of the precipitate–matrix itates via a pΩ = 1σ Ω term. Here, Ω is the vacancy
v 3 ii v v
coherency have to be expected (Table II). formation volume. It is known that the migration vol-
In following subsections, we will analyze the effect of ume in the fcc lattice is small13 and we will compute the
these precipitates on the GB diffusion. effect of pressure on the formation volume.
9
FIG. 5. (a) Strain dependence of the vacancy formation energy EVac , the Co-vacancy binding energy ECo−Vac, and the Co
form bind
migration energy Em for vacancy mediated nearest neighbor jumps in pure Al. In addition the pressure-volume work pΩf is
provided,whichalsoentersthesumofallenergycontributions. (b)3Drepresentationofthevariationoftheenergyprofile(for
Coatomdiffusingintothenearest neighborvacancy)alongtheminimumenergypathfordifferentstrainrates. Therespective
intial state corresponds to theGibbs energy of a vacancy formation next to a Co atom.
In the present estimates, the correlation factor f, in a nearest-neighbor configuration, it is already in the
which can in principle be calculated in terms of the five- 5th nearestneighborshelldecreasedto10%ofthisvalue.
frequency model66 for the fcc lattice, is considered as a The results of 3 3 3 supercells, plotted in Fig. 5 can
× ×
constant (at the given temperature of the diffusion mea- therefore be considered as sufficiently accurate.
surements). Moreover,the effect of pressureon the total We observe that the vacancy formation energy EVac ,
form
term fλ2ν0 is neglected13. This is a reasonable asump- the Co-vacancybinding energyEbCion−dVac, andthe Comi-
tion, since the strain dependencies in the exponent are gration energy E depend all almost linearly on the ap-
m
more significant than in the prefactor. plied strain. The term EVac (p) has been determined
form
Thus, the change of the GB diffusion coefficient of Co from defect and defect-free structures that are evaluated
atoms due to applied elastic stresses is estimated as at the same pressure, using pure fcc Al as a reference
system for the pressure (upper x axis) corresponding to
acertainstrainstate(lowerxaxis). TheLegendretrans-
DgCbo(σ) =exp Em(0)−Em(σ) exp 31σiiΩv . formation from Helmholtz to Gibbs energies further re-
DCo(0) (cid:18) k T (cid:19) (cid:18)− k T (cid:19) quirestheadditionofthepressure-volumeworkpΩ with
gb B B f
(10) the vacancy formation volume
As an estimate, the value of the vacancy formation
N 1
volumemeasuredforZnGBdiffusioninAl33,Ωv =0.8Ω, Ωf =V(N 1,p) − V(N,p), (11)
can be used for Co GB diffusion in Al. Now, we need to − − N
determine the strain dependence of E (σ).
m where V(N,p) is the volume of a supercell with N
atomsatpressurep. Thelargelystrain-independentterm
EVac (p) justifies the experimental approach, to identify
form
G. Ab initio energetics of Co diffusion in Al ∆G in Eq. (9) with pΩ .
f v
The migration barrier energy cannot be directly ob-
Figure 5 shows the ab initio calculated energy contri- tained from experiment and we therefore use the result
butions entering the exponent in Eq. (9). They have from DFT calculations for the present evaluation. The
initially been determined as Helmholtz energies for hy- fulldiffusionprofileforthedifferentstrainstatesisshown
drostatic strain values up to 1.5%. The maximum in Figure 5b. The respective initial state corresponds to
±
elastic strain of 1.5% corresponds to the elastic misfit the Gibbs energy of a vacancy formation next to a Co
between the pure Al matrix and Al Sc particles, deter- atom as given by the sum EVac (p)+pΩ +ECo−Vac. In
3 form f bind
mined in subsection VE. By performing the calculations thisconfiguration,Costillhasahighercoordination(11-
for 2 2 2 and 3 3 3 supercells we confirmed that fold) than in the saddle point configuration. This could
× × × ×
finite size errors at the maximum strain are below 5 %. explain why the latter is comparatively more sensitive
The convergence is determined by the point defect in- to the compressing/stretching (compressive/dilatational
teraction energies as a function of distance in the fcc Al strain) of the lattice (see Figure 5b). Nevertheless, also
matrix: While the Co-Co interaction energy is 155 meV the migration barrier of Co in Al follows to a good ap-
10
1 k=100 Figure 6 shows that even in the case of a low (about
k=10 1%) fraction of precipitates in which diffusion is forbid-
0.8 k=1 den, D =0, a strong segregationof tracer atoms to the
p
GB/precipitate interfaces (k = 0.01) would slow down
0.6 the GB diffusion rate by a factor of two. This finding
Dgb
/Deff0.4 k=0.1 aplrreesaednytmageraeseusrwemitehntthse(Feixgp.e2rbim).eNntoatleotbhsaetrfvoarttiohnescrinititchael
sizeoftheAl Scprecipitatesofabout20nm,TableI,and
3
k=0.01
0.2 the average distance between them about 200 nm (the
measured precipitate density of 102 µm−1 in the near
GB region), the particle fraction is estimated at about
0 0.02 0.04 0.06 0.08 0.1
precipitate volume fraction 1% in the case of 2D GB diffusion.
FIG. 6. Diffusion retardation in precipitated GBs as a func-
tion of the precipitate volume fraction. The dependencies I. General model of the particle effect on GB
were calculated according to Eq. (13) for different values of diffusion
theparameter k (see text).
Inthissubsection,wewillsummarizetheobtainedresults
anddescribea generalmodelofthe diffusionretardation
proximation a linear dependence on the applied strain,
in the precipitated GBs.
for which the expression
Grain boundaries in ECAP-processed AA5024 alloys
containsomefractionofsmall,sizeofabout11nm,coher-
E =( 0.09ǫ+1.31) eV (12) ent Al Sc precipitates. The Al Mn particles are larger,
m 3 6
−
non-coherent, and sparsely distributed. Their contribu-
will be used in subsequent estimates.
tion to the diffusion retardationis estimated to be small
The calculations further reveal that the maximum
under the assumption that they block GB diffusion lo-
changeinthebindingenergyduetothestrainis0.015eV
cally without introducing long-range strain fields. Thus,
if ǫ 1.5%. This is by an order of magnitude smaller
| | ≤ wearefocusingonAl3Scprecipitates. Stillaslightdiffer-
than the change of the migration barrier (0.12 eV).
ence between the Co GB diffusion coefficients in as-cast
Hence, it is safe to ignore this contribution.
and ECAP-processedstates, Fig. 2b, may be induced by
the Al Mn particles, Fig. 3a. The fraction of Al Sc par-
6 3
ticles at grain boundaries in ECAP-processed AA5024
H. Diffusion coefficient
alloy is estimated at about 0.2%.
The net effect ofprecipitationonCo GB diffusionwas
To estimate the influence of the particles on the net dif-
estimated according to the following scheme:
fusion flux along GBs in the AA5024 alloy, we use the
modified Maxwell-Garnett equation68,69, 1. The elastic strain and stress fields around the co-
herent precipitates are determined according to
Deff k Eqs. (7) and (8);
=
D f +k(1 f )×
gb p p
− 2. The jump barriers, E (σ), at the given radial po-
k(1−fp)+(1+fp)DDgpb. (13) sitions are determinedm, Eq. (12);
k(1+f )+(1 f )Dp
p − p Dgb 3. ThetermσΩv isdeterminedfromEq.(8)andusing
Ω =0.8Ω for grain boundaries;
Here, f is the volume fraction of precipitates which are v
p
characterizedby the diffusion coefficient Dp andthe fac- 4. The resulting variation of the GB diffusion coeffi-
torktakesintoaccountprobablesegregationofCoatoms cientofCoatomsisestimatedbyEq.(10)asafunc-
to the Al/Al3Sc (or Al/Al3(Sc,Zr)) interfaces within Al tion of the radial distance from the precipitates.
grain boundaries and the equation is written for the 2D
case of GB diffusion. Figure 7 illustrates these calculations for the case of a
The effectofthe precipitate volumefractiononthe ef- coherent Al Sc precipitate of the size of 20 nm as func-
3
fectivediffusioncoefficientisshowninFig.6fordifferent tion of the distance from the precipitate center. The
values ofk andthe limiting caseD =0. It is important determined elastic strain is plotted by blue circles (left
p
to recognize that in the originalpapers, Refs. 68 and 69, ordinate),thecorrespondingchangeofthemigrationbar-
effective diffusion in a homogeneous media is analyzed rier∆E (σ)andtheσΩ termareshownbytrianglesup
m v
with an application to a general case of GB diffusion (in anddown,respectively(rightordinate),andtheestimate
a d-dimensional solid68). Then, the segregation was in- variation of the Co GB diffusion coefficient (normalized
troduced as solute enrichment in the grain boundaries on strain-free case) is given by blue squares (left ordi-
with respect to the crystalline bulk. nate).