1

MODELING OF COMMERCIAL AlMODELING OF COMMERCIAL Al--MgMg--SiSiALLOYS AGEALLOYS AGE--HARDENING BEHAIVIORHARDENING BEHAIVIOR

A.A. A.A. VasilyevVasilyev, N.L. , N.L. KuzminKuzmin, A.S. , A.S. GruzdevGruzdev

St. Petersburg State Polytechnic UniversitySt. Petersburg State Polytechnic UniversityDept. of Metal Physics and Computer Technology in Dept. of Metal Physics and Computer Technology in

Materials ScienceMaterials Science

2

Chemical composition of the alloysChemical composition of the alloys(wt. %)(wt. %)

0.170.171.171.171.171.1744

0.170.171.151.150.730.7333

0.170.170.960.960.760.7622

0.160.160.720.720.710.7111

FeFeSiSiMgMgAlloyAlloy

MaterialMaterial::industrially extruded rods (industrially extruded rods (AMAG, AustriaAMAG, Austria))

HT before agingHT before aging::

NANA(T4) (T4) →→ SSTSST (1 hour at 540 (1 hour at 540 00C) C) →→ WQWQ

3

Some results of isothermal aging kinetics Some results of isothermal aging kinetics investigation by investigation by YoungYoung’’s modulus (YM)s modulus (YM)

measurements.measurements.

((∆∆E/EE/E00)*10)*10+3+3

((∆∆E/EE/E00)*10)*10+3+3

4

Peculiarities of Peculiarities of YM YM behavior under behavior under isothermal aging.isothermal aging.

Alloy 1 Alloy 1 (Al (Al -- 0.71Mg0.71Mg -- 0.72Si)0.72Si)

1. 1. Stages of the modulus incrementStages of the modulus increment..

0 1 2 3 4 5 60

2

4

6

8

Time (*10 ), min

E

/Eo

(*10

+3)

Avrami equation

Stage 1Stage 2

Stage 3

160 Co

-3

5

0 1 2 3 4 5 60

2

4

6

8

Stage 1

Stage 2Stage 3

180 C

Avrami equation

o

Time (*10 ), min-3

E

/Eo

(*10

+3)

0 1 2 3 4 5 60

2

4

6

8

Stage 2

Stage 3

Avrami equation

200 Co

Time (*10 ), min-3

E

/Eo

(*10

+3)

ConclusionConclusion:: three stagesthree stages of the aging process and of the aging process and two types of inflectionstwo types of inflections on on YMYM increment curves are increment curves are observed for all temperatures and alloys under observed for all temperatures and alloys under investigation.investigation.

6

1 2 3 4 5 6

2

4

6

8

10

0

50

100

150

200

250

300

350

0

Stage 1Stage 2

Stage 3

160 C

Yie

ld S

tres

s, M

Pa

Time (*10 ), min-3

E

/Eo

(*10

+3)

o

0 1 2 3 4 5 60

2

4

6

8

10

0

50

100

150

200

250

300

350

Stage 1Stage 2

Stage 3

180 C

Yie

ld S

tres

s, M

Pa

Time (*10 ), min

E

/Eo

(*10

+3)

o

-3

2. 2. Correlation between Correlation between YM YM and and yield stress yield stress (YS)(YS) changes.changes.

Beginning of the second stage correlates well with Beginning of the second stage correlates well with YS YS peak achievement. peak achievement.

7

MODELING OF COMMERCIAL MODELING OF COMMERCIAL AlAl--MgMg--SiSi ALLOYS AGE HARDENINGALLOYS AGE HARDENING

MODEL OF THE PRECIPITATION PROCESSMODEL OF THE PRECIPITATION PROCESS

ModelingModeling of precipitate particles nucleation and growth of precipitate particles nucleation and growth is is realizedrealized in frames of in frames of thethe KampmannKampmann and Wagnerand Wagner’’s s approachapproach ((KampmannKampmann R., Wagner R. R., Wagner R. -- In: Decomposition of In: Decomposition of Alloys: the early stages, p.91. Oxford, Alloys: the early stages, p.91. Oxford, PergamonPergamon Press Press (1984)(1984))) based on classical nucleation theory.based on classical nucleation theory.

MAIN ASSUMPTIONS OF THE MODELMAIN ASSUMPTIONS OF THE MODEL::

•• Two types of the particlesTwo types of the particles are considered in the model are considered in the model ((GPZGPZ and and ''''-- phase particlesphase particles))..

•• Nucleation and growth of the precipitates are described Nucleation and growth of the precipitates are described by operating with by operating with spherical shape particles.spherical shape particles.

•• Solid solution of the ternary alloy is considered as Solid solution of the ternary alloy is considered as binary onebinary one with with MgMg--SiSi ''''molecularmolecular'''' playing a role of the playing a role of the alloying component.alloying component.

•• Elastic strains,Elastic strains, associated withassociated with GPZ GPZ and semiand semi--coherent coherent ''''-- particles, particles, are neglected.are neglected.

8

THE MAIN EQUATIONSTHE MAIN EQUATIONS

Nucleation rate,Nucleation rate, JJCC(t;T(t;T):):

JJсс((t;Tt;T) = ) = N(t;TN(t;T)) Z(t;TZ(t;T)) сс((t;Tt;T)) exp(exp(FFсс((t;T)t;T)kTkT),),

wherewhere tt and and TT are time and absolute temperature;are time and absolute temperature;

Z(t;TZ(t;T)) is is ZeldovichZeldovich factor;factor;

сс ((t;Tt;T)=4)=4RRсс22(t;T)(t;T) D(T)D(T) C(t;T)C(t;T)aa44 is the rate at which is the rate at which

impurity atoms join the critical nucleus;impurity atoms join the critical nucleus;

D(T)=DD(T)=D00exp(exp(--QQi i //kTkT)) is is effective effective diffusion coefficient; diffusion coefficient;

C(t;TC(t;T)) is impurity concentration in the matrix;is impurity concentration in the matrix;

N(t;TN(t;T)) is volume density of potential nucleation sites.is volume density of potential nucleation sites.

Driving force,Driving force, FF11(t,T)(t,T), , for precipitation:for precipitation:

PP is volume per one structural unit of the precipitate is volume per one structural unit of the precipitate particle; particle; CCее(T)(T) is equilibrium impurity concentration:is equilibrium impurity concentration:

CCее(T) = C(T) = C00exp{exp{QQssk(1k(1TT11TTss)} ,)} ,

CC00 is the overall impurity concentration;is the overall impurity concentration; QQss is the energy is the energy of impurity atom solution in the aluminium matrix; of impurity atom solution in the aluminium matrix; TTss is is metameta--stable stable solvoussolvous temperature of the phase.temperature of the phase.

,)T(C)T;t(ClnkT)T;t(F

ep1

9

R)T(D

)T(CC)T(C)T;t(C

dtdR

RP

R

Growth rate of a spherical particle with radiusGrowth rate of a spherical particle with radius R:R:

wherewhere CCPP==PP--11 is volume density of the impurityis volume density of the impurity

(Mg(Mg--SiSi ““molecularmolecular””)) in precipitate particle.in precipitate particle.

Volume density of the nucleation sites,Volume density of the nucleation sites, N(t;TN(t;T):):

N(t;TN(t;T) = N) = N00{1 {1 -- ff33 f(t;Tf(t;T))},))},

herehere NN00 is the number of possible nucleation sites per is the number of possible nucleation sites per unit volume of the unit volume of the FCCFCC matrixmatrix (N(N00 = 4/a= 4/a33););

f f = = RRff / R,/ R, wherewhere RRff is is thethe radius of radius of depleted zone depleted zone

around a particle, inside which further nucleation is around a particle, inside which further nucleation is

suppressed (suppressed ( ff is a fitting parameter of the model);is a fitting parameter of the model);

f(t;Tf(t;T) ) is precipitate particles volume fraction.is precipitate particles volume fraction.

10

PROCEDURE FOR THE STRUCTURAL MODEL PROCEDURE FOR THE STRUCTURAL MODEL

CALIBRATIONCALIBRATION

Set of the main parametersSet of the main parameters::

QQss (effective energy of impurity solution); (effective energy of impurity solution);

TTs1s1, T, Ts2s2 (meta(meta--stable stable solvoussolvous temperatures for temperatures for GPZGPZ and and ''''-- particles);particles);

11, , 22 (specific interface energies);(specific interface energies);

QQii (effective energy of impurity diffusion); (effective energy of impurity diffusion);

DD00 (diffusion coefficient factor); (diffusion coefficient factor);

PP (volume per structural unit of precipitates); (volume per structural unit of precipitates);

zz (parameter of particle influence sphere).(parameter of particle influence sphere).

SelfSelf--consistent consistent set of set of effective values of the effective values of the main main physical parametersphysical parameters (T(Tss, , , , QQii)) is obtained on the is obtained on the bases of bases of experimental data on experimental data on YoungYoung’’s modulus s modulus increment under isothermal aging.increment under isothermal aging.

11

STEPS OF THE MODEL CALIBRATION STEPS OF THE MODEL CALIBRATION

1. 1. Calculation of aging process activation energiesCalculation of aging process activation energies..

AvramyAvramy equation for Youngequation for Young’’s modulus increment:s modulus increment:

EE11(t; T) = (t; T) = EEm1m1(T){1 (T){1 -- exp (exp (--(t/(t/11(T))(T))PP)},)},

where where EEm1m1(T)(T) is the maximal level of modulus increment is the maximal level of modulus increment

at the first stage; at the first stage; 11(T)(T) is characteristic duration of the stage.is characteristic duration of the stage.

First stagesFirst stages for Alloy 4for Alloy 4(1.17Mg (1.17Mg -- 1.17Si)1.17Si)

12

GPZGPZ-- stagesstages for Alloy 4for Alloy 4

Activation energy at theActivation energy at the first stagefirst stage

13

Activation energy at the Activation energy at the GPZGPZ-- stagestage

14

2. Self2. Self--consistent procedure of consistent procedure of

TTs s , , and and QQii values calculation.values calculation.

Precipitates volume fraction at the first stage, Precipitates volume fraction at the first stage, ff11(t;T):(t;T):

ff11(t;T) (t;T) EE11(t;T) (t;T)

dfdf11/dt/dtt=0;T t=0;T d {d {EE11(t;T)}/dt(t;T)}/dtt=0;Tt=0;T (1)(1)

d {d {EE11(t;T)}/dt(t;T)}/dtt=0;Tt=0;T may be calculated on the basesmay be calculated on the bases

of Youngof Young’’s modulus increment experimental curves.s modulus increment experimental curves.

dfdf11/dt/dtt=0;T t=0;T = = JJCCtt=0;T=0;T*V*VC C t=0;T t=0;T = = ((TTSS; ; ; ; QQii) (2)) (2)

QQii values were assumed to be equal to the corresponding values were assumed to be equal to the corresponding QQ11 values. values.

{(1) + (2)} {(1) + (2)} TTSS; ;

2

s

i

s2/1

1

1m

TT1T

XkTQ

TT1ln3'Aln

T)T(Eln

15

Results for Results for ””-- phase particlesphase particles..

Results for Results for GPZGPZ..

16

3. 3. Evaluation of Evaluation of z z andand DD00 values.values.

YoungYoung’’s modulus increment:s modulus increment:

E(tE(t;;T) = T) = E E f(tf(t;;T) ,T) ,wherewhere EE is a fitting parameter of the model for is a fitting parameter of the model for YoungYoung’’s modulus increment simulations modulus increment simulation. .

Results of YoungResults of Young’’s modulus s modulus

increment simulationincrement simulation..

17

18

MODEL FOR AGEMODEL FOR AGE––HARDENING SIMULATIONHARDENING SIMULATION

Macroscopic yield stress of an alloy, Macroscopic yield stress of an alloy, ((t;Tt;T): ):

((t;Tt;T) = ) = 00 + + ssss(t;T(t;T) + ) + ptpt(t;T(t;T),),

wherewhere 00 is yield stress of the pure aluminium; is yield stress of the pure aluminium;

ssss(t;T(t;T)) and and ptpt(t;T(t;T)) are solid solution and precipitate are solid solution and precipitate particles particles contributionscontributions. .

ssss(t;T(t;T) = ) = ssss CC1/21/2(t;T)(t;T)

ssss = (= (q q -- 00)/C)/C001/21/2,,

here here q q is the as quenched yield stress value.is the as quenched yield stress value.

pt pt ((t;Tt;T) = {() = {(ptptzz (t;T))(t;T))2 2 ++ ((ptpt

phph (t;T))(t;T))22}}1/2 1/2 ,,

ptptzz ((t;Tt;T), ), ptpt

phph are contributions of are contributions of GPZGPZ and and

'''' -- particles.particles.

ptptzz = = zz ff zz

1/21/2, ,

zz is strengthening parameter; is strengthening parameter; ff z z is is GPZGPZ volume volume

fraction.fraction.

19

There are two alternative mechanisms of There are two alternative mechanisms of ''''-- phase phase particles particles oovercomingvercoming by moving dislocations: by moving dislocations: particles particles shearingshearing and and bowing.bowing.

At low values of particles volume fractions (in our case At low values of particles volume fractions (in our case the the maximum volume fraction is close to maximum volume fraction is close to 0.010.01), when the ), when the spacing between particles is sufficiently larger then their spacing between particles is sufficiently larger then their characteristic sizes, stress for shearing, characteristic sizes, stress for shearing, pt pt

ss(t;T(t;T),), and and bowing, bowing, pt pt

bb(t;T(t;T), ), may be presented as may be presented as ((RepichRepich B. B. -- In: In: Materials Science and Technology. Edited by Materials Science and Technology. Edited by CahnCahn R.W., R.W., HaasenHaasen P. And Kramer E.J., 6, p.311 (1990)):P. And Kramer E.J., 6, p.311 (1990)):

pt pt ss(t;T(t;T) = (F) = (Fmm(t;T)(t;T)2E2E00))3/23/22E2E00(b(b(t;T)) (t;T))

pt pt bb(t;T(t;T) = 2E) = 2E00(b(b(t;T)),(t;T)),

where where FFmm(t;T(t;T)) is a force required for particle shearing; is a force required for particle shearing; ((t;Tt;T) ) is an average distance between is an average distance between ccrossross--sectionssections of of the the particles by dislocation slip planes; particles by dislocation slip planes; b b and and EE00 are the are the modulus of Burgers vector and linear tension of a modulus of Burgers vector and linear tension of a dislocation, respectively.dislocation, respectively.

The main The main contribution into agecontribution into age--hardening of hardening of alloys under consideration caused by needlealloys under consideration caused by needle--shape shape ""-- phasephase particles. particles.

20

According to the data of microscopic investigations,According to the data of microscopic investigations,

cross sizes of cross sizes of ””--particles change particles change uunsignificantlynsignificantly

(Thomas G.J. (Thomas G.J. --J. Inst. of Metals, V.90, p.57 (1961)).J. Inst. of Metals, V.90, p.57 (1961)).

ThereforeTherefore,, shearing force shearing force valuevalue, , FFmm(t,T(t,T),), may be may be acceptaccepted to be constanted to be constant. .

LetLet’’s denote a volume density of needles denote a volume density of needle--shapeshape

particles with average length particles with average length L(t;TL(t;T)) by by n(t;Tn(t;T).).

Average distance between the cross sections of the Average distance between the cross sections of the

particles by particles by {111} {111} planes is proportional to: planes is proportional to:

1/{n(t;T)L(t;T)}1/{n(t;T)L(t;T)}1/21/2. .

Thus, under the condition of:Thus, under the condition of:

L(t,TL(t,T) ) R(t,TR(t,T))the contribution of the contribution of ""-- phase particles may be expressed phase particles may be expressed as: as:

ptptphph(t;T(t;T) = ) = ptpt{n(t;T)R(t;T)}{n(t;T)R(t;T)}1/21/2

where where R(t;TR(t;T)) is the average value of particles radius;is the average value of particles radius;

ptpt is a fitting parameter of the model.is a fitting parameter of the model.

21

EXCESS VACANCIESEXCESS VACANCIES

Temperature dependence of Temperature dependence of quick reaction durationquick reaction duration was was obtained on the bases of obtained on the bases of experimental data on electrical experimental data on electrical resistivityresistivity change under low temperature agingchange under low temperature aging of water of water quenched quenched Alloy 2.Alloy 2.

22

oTqTkvQoPvoP 11exp

)(exp);(

Tq

tvoPTtvP

)exp()( kToQqoTq

));(1)(();( TtvPTDTteffD

As quenched value of As quenched value of vacancy supervacancy super--saturation,saturation, PPV0V0::

where where QQvv is vacancy formation energy; is vacancy formation energy; TTqq and and TT00are initial and final quenching temperatures; are initial and final quenching temperatures; PP00 is is a fitting parameter. a fitting parameter.

TimeTime--dependent factor of vacancy superdependent factor of vacancy super--saturation,saturation, PPvv(t;T(t;T):):

here here qq(T(T)) isis quick reaction quick reaction duration.duration.

Effective value of the diffusion coefficient, Effective value of the diffusion coefficient, DDeffeff(t;T(t;T):):

23

2.710-5P0

0.63Q0, eV0.60.3 [Pа]0.810.790.87pt, Pam3.00.03.03.03.0E

3.66.05.15.05.3f

00.3300.1150CехSi, аt.%1.190.810.840.72C0, аt. %

1.010-152.510-192.510-181.310-167.010-16D0, m2/s0.6950.460.4610.6120.695Qi, eV0.0450.0260.0450.0450.045, J/m2

615488611612608Ts, К2.91028Cp, m-3

3.510-29p, m3

0.31Qs, eV

β”-phase particles

GPZGPZAlloy 4

Alloy 3Alloy 2Alloy 1Model parameters

SET OF THE MAIN PARAMETERS

24

Effective diffusion activation energy vs excess Si content

0,4

0,45

0,5

0,55

0,6

0,65

0,7

0 0,1 0,2 0,3 0,4

Excess Si, at%

Qi,

eV

1,00E-18

1,00E-17

1,00E-16

1,00E-15

0 0,1 0,2 0,3 0,4

Excess Si, at%

D0,

m^2

/s

Empirical relations between main physical model Empirical relations between main physical model parameters and alloyparameters and alloy’’s compositions composition.

25

0.6 0.8 1 1.2600

610

620

Ts, K

MgSi content, at%

MgSi content, at%0.6 0.8 1 1.2

3

4

5

6

Dz

6,0E-07

6,5E-07

7,0E-07

7,5E-07

8,0E-07

8,5E-07

9,0E-07

0,7 0,8 0,9 1 1,1 1,2

MgSi content, at%

Alfa

_p, M

Pa*m

26

RESULTS OF AGE HARDENING SIMULATIONRESULTS OF AGE HARDENING SIMULATION

1. 1. ISOTHERMAL AGINGISOTHERMAL AGING

Alloy 1Alloy 1

27

Alloy 2Alloy 2

28

Alloy 3Alloy 3

29

Alloy 4Alloy 4

30

2.2. MULTI MULTI -- STEP AGINSTEP AGINGG

2.1. 2.1. Effect of natural preEffect of natural pre--agingaging..

Alloy 4Alloy 4

31

Effect of natural preEffect of natural pre--aging duration on aging duration on YSYS peak peak value and time to reach the peak. value and time to reach the peak.

102 103 104 105

260

280

300

320

340

2000

4000

6000

8000

10000

NA duration, min

YS

peak

, MPa

Peak time, m

in

32

2.2. 2.2. Effect of 100 Effect of 100 00C stabilizationC stabilization..

Alloy 4Alloy 4

33

Effect of Effect of 100 100 00CC--stabilizationstabilization duration on duration on YSYS peak peak value and time to reach the peak. value and time to reach the peak.

0 10 20 30

260

280

300

320

340

2000

4000

6000

8000

10000

YS

peak

, MPa

Peak time, m

in

100 C - stabilization duration, min

34

3. 3. Effect of Effect of the the heating rateheating rate..

Alloy 4Alloy 4

35

10-1 100 101 102280

290

300

310

320

330

340

1000

2000

3000

4000

5000

YS

peak

, MPa

Peak time, m

in

Heating rate, K/min

Effect of Effect of the the heating rateheating rate..

36

.4

.%)wt(nFe(wt.%)-(wt.%) Si (wt.%)Si free

Effective impurity Effective impurity ((MgSiMgSi –– ““molecularmolecular””) concentration ) concentration

evaluation on the bases of actual chemical compositionevaluation on the bases of actual chemical composition..

First free First free SiSi ((wt.%) wt.%) content is calculated as:content is calculated as:

After that atomic After that atomic MgMg andand SiSi concentrations are calculated. concentrations are calculated.

.%);at(CMg .%)at(CSi

Effective Effective MgSiMgSi –– molecularmolecular concentration is obtained as concentration is obtained as follows:follows:

.%)}at(C.%);at(Cmin{C.%)at(C SiMg0MgSi

Atomic concentration of the excess Atomic concentration of the excess SiSi::

.%)at(C.%)at(C.%)at(C MgSiSiexcessSi

37

38

BRIEF DESCRIPTION OF THE BRIEF DESCRIPTION OF THE NUMERICAL NUMERICAL

PROCEDUREPROCEDURE

Continuous evolution of the system of precipitate parContinuous evolution of the system of precipitate par--ticlesticles is simulated as the sequence of changesis simulated as the sequence of changes occur at occur at small time steps,small time steps, tt(k(k))..

Current average concentration of the solid solution,Current average concentration of the solid solution, CC(k(k)),,atat kk--thth step of the calculations is obtained from the mass step of the calculations is obtained from the mass balance equation:balance equation:

(C(CPPCC(k(k--1)1)))VV(k(k--1)1) = (C= (C(k(k--1)1)CC(k)(k))(1)(1VV(k(k--1)1)), ),

wherewhere VV(k(k--1)1) is an increment of particles total volume at is an increment of particles total volume at (k(k--1)1)--thth step;step; VV(k(k--1)1) is a total volume of particles atis a total volume of particles at (k(k--1)1)--ththstep. step.

Particles sizeParticles size--distribution function is changed with distribution function is changed with account of account of particles nucleation, growth and dissolution.particles nucleation, growth and dissolution.

39

160 160 00CC

160 160 00CC

More precise investigation allowed to detect earlier More precise investigation allowed to detect earlier aging stages related with aging stages related with GunierGunier--PrestonPreston’’ss zones zones

(GPZ)(GPZ) precipitation.precipitation.

40

Nature of the stages and inflectionsNature of the stages and inflections..

Stage 1Stage 1 :: nucleation and growth of nucleation and growth of ””-- phase phase particles.particles.

Stage 2Stage 2 :: coarsening of coarsening of ””-- particles.particles.

Stage 3Stage 3 :: formation of formation of ’’-- particles.particles.

First inflection (peak of strength)First inflection (peak of strength) :: beginningbeginning

of of ””-- particles coarseningparticles coarsening

These conclusions are supported by results of the These conclusions are supported by results of the modeling of precipitate particles volume fraction modeling of precipitate particles volume fraction and volume density behavior under isothermal and volume density behavior under isothermal aging.aging.

41

Modeling of precipitates volume fraction and Modeling of precipitates volume fraction and volume density behavior under isothermal volume density behavior under isothermal agingaging..

Relation between modulus increment and Relation between modulus increment and precipitates volume fraction:precipitates volume fraction:

E(t;T) E(t;T) ≈≈ f(t;T)(Ef(t;T)(EP P –– EEssss),),

where where E(t;T)E(t;T) is modulus increment; is modulus increment; f(t;T) (f<<1) f(t;T) (f<<1) is is volume fraction of the precipitate particles; volume fraction of the precipitate particles; EEPP and and EEssss are the modulus of precipitates and matrix (solid are the modulus of precipitates and matrix (solid solution).solution).

Thus, for Thus, for EEPP; ; EEssss≈≈ constconst one may obtain:one may obtain:

f(t;T) f(t;T) E(t;T) E(t;T)

42

Modeling results for Modeling results for Alloy 1Alloy 1

2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.2

0.4

0.6

0.8

1

0

160 C

Num

ber

Den

sity

(*10

-23)

, m-3

Vol

ume

Fra

ctio

n (*

10+2

)

Stage 1Stage 2

VF

ND

Time (*10 ), min-3

o

2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.2

0.4

0.6

0.8

1

0

180 C

Num

ber

Den

sity

(*10

-23)

, m-3

Vol

ume

Fra

ctio

n (*

10+2

)

Stage 1Stage 2

VF

ND

Time (*10 ), min-3

o