University of Groningen Grain boundary phenomena and failure of ... · cohesion and fracture...

$: University of Groningen Grain boundary phenomena and failure of ... · cohesion and fracture mechanism (Chapter 5 of this thesis). The crystallographic structure and chemical composition$
University of Groningen

Grain boundary phenomena and failure of aluminium alloysHaas, Marc-Jan de

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59

Chapter 4

MODELLING OF GRAIN BOUNDARY

SEGREGATION AND PRECIPITATION

4.1 INTRODUCTION Grain boundary segregation has been described extensively during the past decades. However, the techniques to experimentally validate this process were limited. With the advent of the scanning Auger microscope, the solutes at the grain boundary could be inspected, but only after an in-situ fracture process. The probe sizes in transmission electrons microscopes were still orders of magnitude larger than would be desired for scrutinizing the grain boundaries. With the introduction of the field emission gun (FEG) for transmission electron microscopy, this all changed and nowadays it is possible to acquire local information using electron probe sizes of less than a nanometer. Chemical and structural information of the grain boundary can now be obtained simultaneously. Effects of grain boundary misorientation, for example, on the degree of segregation can be studied. In this chapter the basic principles of equilibrium and non-equilibrium segregation to grain boundaries in solids as a function of heat treatment are reviewed. As the latter type of segregation is prevalent in the quenching and subsequent ageing procedures of commercial aluminium alloys, a model based on this type of segregation will be described and applied to predict absolute values for grain boundary solute concentrations. The modelled values will be compared to experimental values after a probe-size correction is made. For AA6061 and AA7050, the elements prone to segregation are Mg, Si, Zn and Cu. These are the elements that are in solid-solution during annealing at elevated

CHAPTER 4

60

temperatures. The other alloying elements like Mn, Fe, Cr and Ti are concentrated in intermetallic compounds already formed during casting [1] and consequently their matrix concentration will be very low. In the second part of this chapter, a detailed description of grain boundary precipitation will be given. As they may have a significant influence on properties as ductility and toughness, prediction of precipitate size and distribution provides important information with respect to grain boundary cohesion and fracture mechanism (Chapter 5 of this thesis). The crystallographic structure and chemical composition of the grain boundary precipitates of AA6061 and AA7050 already investigated in Chapter 3 will be used here to predict the growth and distribution of these phases during prolonged anneals. For this purpose a model, originally developed by [2-8] to predict grain boundary precipitate size and distribution in the AA7XXX-series alloys as a function of heat treatment, was programmed in a computer code and used to predict grain boundary precipitate properties for AA6061. As was described in the previous chapter, the grain boundary precipitates (C-phase) in this alloy were able to form relatively low-energy interfaces with the Al-matrix and it was suggested that due to this, the C-phase would be the nucleation phase (i.e. no precursors). Because these properties have a significant influence on precipitation, they were incorporated into the model for AA6061. Since the growth properties of the grain boundary precipitates during artificial ageing are largely depending on the solute concentration profile across the grain boundary in the as-quenched condition, they can be used as an additional source of information on the amount of solute segregation directly after the quench. 4.2 TYPES OF SEGREGATION The segregation of solute atoms towards grain boundaries is classified into equilibrium and non-equilibrium segregation. To discriminate between these two types, a factor that must be considered is whether or not the segregate is fully equilibrated with respect to both itself and to the adjoining bulk system. Equilibrium segregation is prevalent during annealing for prolonged periods, whereas important forms of non-equilibrium segregation may occur in many situations that arise during industrial processing including, for example, rapid quenching and subsequent annealing, the application of stress or exposure to radiation.

MODELLING OF GRAIN BOUNDARY SEGREGATION AND PRECIPITATION

61

4.2.1 Equilibrium segregation Equilibrium segregation of solute occurs at interfaces such as grain boundaries and surfaces. Misfitting impurity atoms in the parent lattice have their energies reduced by locating at regions of bad crystalline arrangement in the solid. The conditions for the simplest Gibbs-type segregation can be represented by the McLean isotherm [9]

(4.1)

Cb is the concentration of impurity on the boundary, Cb0 is the fraction of grain boundary available for segregated atoms at saturation, Cg is the matrix concentration, ∆Gseg is the free energy of grain boundary segregation (i.e. a negative value) and k is Boltzmann’s constant. The assumptions of the McLean isotherm are most likely to be valid for dilute alloys where the level of segregation is insufficient for solute-solute interactions to become important. Naturally, at very low temperatures it is difficult for the impurity to diffuse to the grain boundary, even though there is a strong driving force to make it do so. At elevated temperatures, the diffusivity will be sufficient but the driving force has decreased. In this case, the profiles will broaden and the solute tends to ‘evaporate’ in the matrix. Therefore, the process of equilibrium segregation can be described by a typical C-curve. The kinetics of the process [9] is given by

(4.2)

where α is the grain boundary enrichment ratio from equation (4.1). Cb(t) is the boundary impurity concentration after time t, Cb(0) is the initial grain boundary concentration, Cb(∞) the grain boundary concentration after ageing for an infinite amount of time, f is related to the atom sizes of the solute and matrix atoms, a and b respectively, by f = b3a-2 and DI the volume diffusion coefficient of solute in the matrix at ageing temperature. Once the solute arrives at the interface, atomic and electronic relaxation occur to lower the energy of the system further [10].

g segb0

gb b

exp1 k

C GCC C C T

∆− = − −

12

b b I I2 2 2 2

b b

( ) (0) 4 41 exp erfc

( ) (0)C t C D t D tC C f fα α

−= − ∞ −

CHAPTER 4

62

4.2.2 Non-equilibrium segregation The theory of non-equilibrium segregation was established by Aust, Anthony and Westbrooke [11,12]. In contrast to equilibrium segregation, non-equilibrium segregation is dependent on the binding energy of the impurity atom to a vacancy. In real crystals, besides free vacancies, also vacancy-impurity complexes may be formed. The physical reason for the formation of such pairs is the lattice relaxation in the vicinity of the impurity atom by the attachment of a vacancy. The amount of non-equilibrium segregation is related to the temperature range over which the fast cooling occurs and to the cooling rate. Furthermore, non-equilibrium segregation becomes more pronounced when cooling from higher temperatures, i.e. temperatures above which vacancy diffusion becomes significant. If the material is subjected to a heat treatment where non-equilibrium conditions prevail, e.g. a very rapid quench, then the vacancy supersaturation varies in different localized regions of microstructure. The grain boundaries are good vacancy sinks and therefore the vacancy concentration becomes rapidly reduced near grain boundaries during quenching. In the grain centres there is no place for the excess vacancies to go and so the grain interior remains supersaturated with vacancies. This results in a concentration gradient for vacancies. Some of the vacancies will be complexed with the impurity atoms and these complexes will be dragged towards the boundary down the vacancy concentration gradient. This produces an accumulation of impurity in regions within a few nanometers of the grain boundary (i.e. the solute-concentrated layer or SCL). The regions adjacent to the grain boundary are depleted of solute (i.e. the solute-depleted-layer or SDL). The situation is not in equilibrium and the segregation will die away if the material is re-annealed to give enough energy for the impurity atoms to diffuse back down the solute concentration gradients that non-equilibrium segregation has produced. The model of Faulkner et al. [6] quantifies the magnitude of non-equilibrium segregation and its concentration profile around the grain boundary as a function of time and temperature. It simply indicates the change in concentration brought about at the vacancy sink during cooling from temperature TSHT to 0.5Tm, where TSHT is the solution heat treatment temperature and 0.5Tm is half the absolute melting temperature of the matrix. This temperature is chosen because it is supposed that relatively little diffusion will occur below 0.5Tm.


63

In this work, this term is however assumed to be equal to 0.65Tm, in order to maintain the grain boundary solute concentrations below 100 %. If during the quench, an equilibrium in the concentration of vacancies is instantaneously established, appropriate to 0.65Tm on the grain boundary and appropriate to TSHT at the grain centre, the excess can be represented by [13]

(4.3)

Ef is the thermodynamic free energy of vacancy formation and Eb that of the vacancy-impurity bond. These values can be taken from literature [14-16]. The factor Eb/Ef is an indication for the absolute concentration of complexes. Without this factor the above equation predicts that the amount of segregation will increase as Eb decreases, which is clearly not correct. For the model presented in this chapter, the grain boundary enrichment of every individual solute element is calculated using equation (4.3) and possible interactions between the solutes (e.g. co-segregation or site-competition [17,18]) are disregarded. The following parameters will lead to a maximum of non-equilibrium segregation and thus a maximal grain boundary concentration of solute:

• High solution heat treatment temperature • No ageing treatment after the quench • Fast cooling rate from the solution heat treatment temperature, so that the

critical time for desegregation is not exceeded • Large grain size

The last two points are correlated and need additional explanation. By cooling slowly, the solute-concentrated layer (SCL) at the grain boundary may widen significantly. When the grain size is sufficiently small, the solute in the interior of the grains may become exhausted and the reverse process, called desegregation, may start after a critical time [6]. The grain boundary solute concentration will then decrease again. Precipitate nucleation and growth also proceed very rapidly, because both supersaturation and diffusivity are significant. It is clear that this will have a decreasing effect upon the grain boundary solute content.

b b b f b f

g SHT mf

expk k0.65

C E E E E EC E T T

− −= −

CHAPTER 4

64

Non-equilibrium segregation profile during quenching The solute concentration profile during the quench is schematically depicted in Fig 4.1. Here, w0i and wi are the instantaneous widths of the SCL and SDL, respectively. Segregation is modelled by assuming formation of vacancy-impurity complexes during the rapid quench by which solute is driven from the grain interior towards the grain boundary. Therefore, the quench-time tq is divided into intervals ∆ such that tq = k⋅∆, where k is an integer. During one time interval ∆, the solute concentration profile is considered constant. In the time interval between (i-1)⋅∆ and i⋅∆, solute atoms, which are separated with a distance of c2 D

�from the SCL-SDL interface, are drawn to

the SCL by the diffusion of vacancy-impurity complexes. Here Dc is the diffusion coefficient of the complexes. The arrived solute atoms are accumulated in the SCL and its width consequently increases. When x is the distance from the grain boundary so that the point x = 0 is located at the centre of the solute-concentrated layer, the instantaneous solute concentration profile C(x,i∆) at the ith time-stage is

(4.4)

When i = 1, C(x,(i-1) ∆) = Cg and woi-1 = 0. Variation of amount of segregation along the grain boundary plane is not accounted for in the above model (i.e. all boundary sites are considered equivalent).

( )= ≤ 1o2b( , ) if iC x i

� � � �

( )−= < ≤ +o 1 1

g o o2 2

2( , ) if

2i

i i ii

x wC x i

� � � � � �

w

( )= > + 1g o2( , ) if i iC x i

� � � � �

oc

o

12

21o o 1

2b

2where ( ,( 1) )

ii

i

wD �

wi iw C x i dx wC

∆−

+

− −= − +∫

o gb

g

( )and i

i

w C Cw

C

−=


65

An iteration is carried out from i=1 until i=k and C(x,k∆) is the final solute concentration profile resulting from the quench. However, during this interval the temperature is decreasing and consequently the diffusion coefficient is changing. This is accounted for by using an average diffusivity that is valid for the temperature range of the quench. Non-equilibrium segregation profile during artificial ageing Depending on the ageing temperature, there are two different mechanisms that can be valid during artificial ageing: 1. The artificial ageing temperature Ta > 0.65Tm; the segregation behaviour is again described by (4.3) and (4.4), but with the difference that the effective time is now the sum of the effective quenching time and the effective ageing time. Under these conditions, desegregation (which occurs when the solute supply from the grain interior is exhausted) may become prevalent. 2. Ta < 0.65Tm; after the quench has stopped, the width of the SCL will not increase further. Precipitates nucleate at the grain boundary, decreasing the local solute concentration. However, the concentration gradient between the grain interior and the grain boundary formed during the quench still enables diffusion of vacancy-impurity complexes towards the grain boundary.

Fig 4.1: Solute concentration profile versus distance from the grain boundary in as-quenched condition.

(SCL: Solute-concentrated-layer SDL: Solute-depleted-layer)

Cg

GB x = 0

Cb

wi

w0i

x

SDL

SCL

Distance from GB

c o n c

CHAPTER 4

66

As the second situation corresponds to the conditions that are prevalent during industrial processing (i.e. artificial ageing at 160 °C – 180 °C), it will be incorporated in the model. Fig 4.2 schematically presents the solute concentration profiles for different artificial ageing times t1 and t2. The expressions describing the solute concentration profile C(x,ta) just at the onset of ageing (ta = 0) are

(4.5) where w0 and w are the widths of the SCL and SDL directly after the quench.

−= < ≤0 1 1

g 0 02 2

2( ,0) ( + )

2x w

C x C w x w ww

= > 1g 02( ,0) ( + ) C x C x w w

= ≤ 102b( ,0) () C x C x w

Fig 4.2: Solute concentration profile versus distance from the grain boundary during artificial ageing.

c o n c

GB x = 0

Cb

Cg

x

Distance from GB

w0

w

t2

t1

=

t1 < t2


67

When Fick’s second law of diffusion is solved for the boundary conditions of equation (4.5), the solute concentration with time and distance from the grain boundary during artificial ageing (i.e. ta > 0) can be described by

(4.6)

Equation (4.6) describes the solute concentration profile in the absence of grain boundary precipitation, which will be described in the next section.

π − + − −= − +

2 2g c

IIIc c

( ) ( )( , ) exp exp

4 4

C D t w x w xC x t

D t D tw

+ −= − −

gIV

c c

( , ) erf erf 2 2 2

C x w x w xC x t

w D t D t

π−= + +

2g g c

I gcc

2( , ) erf exp

42

xC C D tx xC x t C

w D tD t w

+ −= − +

gII

c c

( , ) erf erf 2 2 2

C w x w xC x t

D t D t

= + + + > 1I II III IV 02( , ) ( , ) ( , ) ( , ) ( , ) ()

where

C x t C x t C x t C x t C x t x w

= ≤ 102b( , ) ()C x t C x w

CHAPTER 4

68

4.3 GRAIN BOUNDARY PRECIPITATION 4.3.1 Nucleation The number of the critical precipitate nuclei N0 per unit area of grain boundary is given by [3]

(4.7)

where xβ is the atomic fraction of solute in the nucleus phase β. It should be noted here that this is the fraction of the rate-controlling element (i.e. the element which controls the precipitate growth rate). k is Boltzmann’s constant and Tnuc is the nucleation temperature, which in this case is equal to the temperature of artificial ageing. In other words, the quench is assumed to be so fast, that there is no precipitate nucleation until ageing starts. N is the total number of atoms per unit area of grain boundary, given by

(4.8)

where d0 is the width of the interfacial diffusion layer (i.e. the width in which the grain boundary diffusion rate is prevalent), ρα is the molar density (mol m-3) of the matrix phase α and NA is Avogadro’s constant. The factor d0 is dependent on the grain boundary structure [19] and may be small for low-angle grain boundaries and particular CSL-boundaries. It should not be confused with w0, which is the width of the solute-enriched layer. For the barrier of nucleation ∆G* can be written [20]

(4.9)

Here, γαβ is the interfacial energy between the precipitate nucleus and the matrix. K(ψ) is a geometric factor that reduces the barrier of nucleation with respect to homogeneous nucleation. It depends on nucleus geometry and will be discussed in the next section. From equation (4.9), it can be seen that the barrier of nucleation can be reduced by decreasing the precipitate-matrix interfacial energy (e.g. by faceting).

*

0nuc

�exp

kN

��

Nx T

−=

0 ��NN d ρ=

3��*2v

16( )

3

��πγ

ψ=G KG


69

∆Gv represents the volume free energy of the nuc leus and can be expressed as

(4.10)

where Vβ is the molar volume of the precipitate phase, R is the gas constant, Cb is the concentration of rate-controlling solute at the grain boundary at the moment of precipitate nucleation and nucTxαβ

α is the concentration of the rate-controlling element in equilibrium with the nucleated precipitate phase and the matrix at temperature Tnuc. This parameter can be calculated from the solubility product equation [21,22]

(4.11)

Cc is the total matrix concentration of the non rate-controlling elements, ∆H and ∆S are the enthalpy and entropy of solution and XE is the ratio of the concentration of the non-rate controlling elements to that of the rate-controlling element. The second term is the Gibbs-Thomson factor which allows for the effect of precipitate-matrix interface curvature r’ on solute solubility near the advancing interface. The nucleation- or incubation time τ can be envisaged as the time needed to achieve steady-state nucleation and is expressed by [23]

(4.12)

Here a is the lattice parameter of the matrix, Db is the grain boundary diffusivity at the temperature of nucleation and K(ψ) and L(ψ) are the geometric factors depending on the nucleus shape. They will be explained in the next section. 4.3.2 Geometry of the precipitate nucleus One of the most common precipitate morphologies formed at the grain boundary is that of the allotriomorphs; crystals that nucleate at grain boundaries and grow preferentially along them. Two different geometries for the precipitate nucleus are schematically depicted in Fig 4.3. Here, ψ is the dihedral angle between the grain boundary plane and the cap-shaped face of the precipitate. It is determined by the grain boundary energy γββ and the

4 2 2nuc A�

�

2 30 vb b

�

2k ( )( )

T a N KD d C V

�� γ ψτ

ψ= ⋅

E

��

��

�c

21exp exp

R R '

X

T

VH T SxC T Tr

γ −∆ + ∆ =

nuc bv �

�

��

Rln

T

CT��

V x

=

CHAPTER 4

70

interfacial energy γαβ. When these particles are projected from the top, a circle with radius R is obtained. A single cap-shaped nucleus as presented in Fig 4.3a, prevails when the curved interface is of a substantially higher energy than the planar interface [3]. When formation of a low-energy facet is not possible, a geometry as depicted in Fig 4.3b may be prevalent. Below the figures, the corresponding geometrical factors are presented as well. Next to K(ψ) and L(ψ), a third geometrical factor f(ψ), necessary to predict growth, is given. This factor is the ratio between thickness S of the particle and its radius R [4,24]. The precipitate volume is then calculated according to

(4.13)

Fig 4.3: Geometries of the single-cap (a) or double-cap (b) shaped nucleus and corresponding factors ψ, K(ψ), L(ψ) and f(ψ) used in the model.

32 3cos cos( )

2K ψ ψψ − +=

( ) 1 cosL ψ ψ= −

ψ ψψ

ψ

− + =

3

3

2 1cos cos

3 3( ) 2sin

f

1 cos( )

2L ψψ −=

ψ

ψψψ 3

3

sin

cos3

1cos

3

2

)(+−

=f

32 3 cos cos( )

4K ψ ψψ − +=

(a)

(b)

ψψψψ γγγγββββββββ

γγγγααααββββ

R S

R

γγγγββββββββ

γγγγααααββββc

R

ψ

S

ψψψψ

γγγγααααββββ

Top-view

-1c

-cos

γ γψ

γββ αβ

αβ

=

-1cos2γ

ψγ

ββ

αβ

=

3f( )V Rψ≈ π


71

4.3.3 Growth For the grain boundary precipitates in AA6061, the shape depicted in Fig 4.4 is a representative geometry during growth. This is supported by experimental observations (Fig 3.11). In order to model growth, the artificial ageing time ta is divided into intervals δt such that ta = nδt, where n is an integer. Within the interval δt, the solute concentration profile, the mean solute collector plate area and the mean length of the precipitate can be considered constant. At the ith stage (1 ≤ i ≤ n), the instantaneous length Li of a precipitate with the shape of Fig 4.4 is given by

(4.14)

Or equivalently

(4.15)

Insight into the derivation of expressions for precipitate growth is given in [5]. From experimental observations, the thickness S was found to be on average 1/5th of the length L, which accounts for the factor 25 in equation (4.14). Strictly speaking, the lengthening and thickening rates of the growing precipitate are not equal and therefore this factor is time dependent. D is the diffusion coefficient of the rate controlling solute element at the ageing temperature. It is equal to the grain boundary diffusivity during the first stages of growth, when the precipitate draws solute from the grain boundary. During later stages of ageing, it is equal to the matrix diffusivity, which then becomes rate-controlling.

131 1�

�2 2

3m 1 � � �11 �

�( 1) 2

��

25 ( )

( )

i t i i Ti ii t

T

A D x x tL dt L

x x

δ

δ

ρπ ρ

−−

−−

− = + − ∫

131 1 1�

�2 2 2

3m 1 � � �11 �

�2

��

50 ( )[( ) (( 1) ) ]

( )i i T

i i

T

A D x x i t i tL L

x x

δ δ ρπ ρ

−−

− − − = + −

Fig 4.4: Geometry used for modelling precipitate growth in AA6061.

L

S

S

CHAPTER 4

72

Ami-1 is the mean collector plate area at the (i-1)th stage (this factor will be discussed below) and ρβ is the molar density of the precipitate phase. xαi is the concentration of rate-controlling solute at the grain boundary during the ith time interval. This factor can be taken as the average concentration of solute in the region from the centre of the grain boundary to a distance δl into the grain, where δl is the diffusion distance of the solute within the amount of ageing time ta’ = iδt. The term thus accounts for the altering supply of rate-controlling solute to the precipitate during the artificial ageing treatment. This supply, schematically illustrated in Fig 4.5, initially is constant and equal to the grain boundary concentration. As ageing proceeds, the grain boundary content of solute decreases more than can be compensated for by solute diffusion from the grain interior and consequently the supply of rate-controlling solute to the growing precipitates will decay according to

(4.16)

where C(x,ta’) is the solute concentration profile at the time ta’ = iδt, given by equation (4.6).

Iteration of equation (4.15) from i = 1 to i = n gives the precipitate length L after ageing for a time t at temperature Ta

(4.17)

The term L0 is the precipitate length at ta = 0. As nucleation is assumed to start at the onset of artificial ageing, this term can be taken equal to zero in the model for water-quenched alloys, because it represents the precipitate length directly after the quench.

131 1 1�

�n 2 2 2

3m 1 � � �01 �

�21 ��

50 ( ){( ) [( 1) ] }

( )

ii i T

i T

A D x x i t i tL L

x x

δ δ ρπ ρ

=−

=

− − − = + −

∑

δ

δδ

= =∫� � �

0

1( , ') with 2'

l

ix C x t dx l Dtl

Fig 4.5: Schematic representation of decreasing solute supply xαi to the growing grain boundary precipitate.

Ageing time

xααααi (at%)

Cb


73

For the AA7050 alloy, the observed precipitate geometry during growth is almost identical to the double-cap shape of Fig 4.3b, with the exception that the radius R is converted to a half-length, so that now the projection along the length resembles a circle with the thickness S as diameter. This geometry is schematically depicted in Fig 4.6.

The mean precipitate length versus time is described by

(4.18)

The factor f(ψ) was already explained in Fig 4.3. As soon as all precipitates have nucleated, the entire grain boundary area is divided into square collector plates with a precipitate nucleus at the centre of each plate. The collector plate size thus initially is determined by the number of nuclei per unit area. Solutes originating from the grain interior first diffuse by matrix diffusion (diffusivity DI) towards the collector plate and subsequently by grain boundary diffusion (diffusivity Db) along the collector plate towards the growing precipitate. The mechanism is illustrated schematically in Fig 4.7.

By taking the square root of the instantaneous mean collector plate area during artificial ageing, the distance between the precipitates can be determined.

Fig 4.7: Route of solute diffusion towards a growing grain boundary precipitate.

DI

Db Db

Db

Db

[ ]δ δ ρ

ψ π ρ

=−

=

− − − = + −

∑1

31 1 1��

n 2 2 23m 1 � � �032 �

�21 ��

( ){( ) [( 1) ] }2

( ) ( )

ii i T

i T

A D x x i t i tL L

f x x

S

L

Fig 4.6: Geometry used for modelling precipitate growth in AA7050.

CHAPTER 4

74

The distribution density function of the grain boundary precipitate nuclei n0 can be described by a Gaussian

(4.19)

where Av is defined as an area of grain boundary with a precipitate nucleus at its centre. The mean collector plate area Am0 just before the onset of growth (i = 0) can be derived from N0 using

(4.20)

σ is the standard deviation of the Gaussian distribution according to

(4.21)

where Amin is the smallest possible collector plate area. When a square collector plate is assumed and the length of one side is equal to two times the diffusion distance during the nucleation time, its area is equal to

(4.22)

During precipitate-growth the mean collector plate area will not change with time unless coalescence between neighbouring precipitates occurs. An increase in mean collector plate area is then caused by a reduction in the number of precipitates per unit area. This reduction in number is given by

(4.23)

Here, the first term on the right-hand side is the number of precipitates that have collector plate areas smaller than Ln2 (i.e. the ones that are susceptible to coalescence). The second term is the fraction of these precipitates with respect to the total number of precipitates present after ageing for a time (i-1)δt. This term can be regarded as the probability of coalescence happening.

2n2

n

( )( ) AminAmin

1

LL

ii

N��

N −

= ⋅

σσ π −

= −

2v m00

0 2

( )exp

22A AN

n

σ −= m0 min( )3

A A

τ=min b16A D

=m00

1AN


75

The instantaneous mean collector plate area after ageing for a time ta = nδt can now be described as

(4.24)

where Li is the instantaneous precipitate length after ageing for a time iδt, which for AA6061 is described by equation (4.15). Although modelling the collector plate by a square may not be entirely correct, it is used here because then the total grain boundary plane area can be tiled with squares, which simplifies the computer code. At the centre of each square, a precipitate may nucleate, grow and coalesce with neighbouring precipitates. 4.4 MODEL VARIABLES The values for all variables used to model grain boundary precipitation in AA6061 and AA7050 are given in Appendix 4.A. The most important parameters that need additional explanation will be discussed briefly in this section. Atomic fraction of rate-controlling element xβ Although the matrix diffusivity of Cu in Al is smaller than that of the other solute elements present in AA6061 or AA7050, it will not act as rate-controlling element during precipitation. As will be demonstrated later, the relative Mg-concentration in the precipitates is significantly larger than its grain boundary concentration in the as-quenched state. Moreover, for grain boundary precipitation in AA7050, Cu may simply be substituted by Zn or by Al. In Chapter 3, the absolute Mg-concentration was determined to be 65.3 at% and 33.3 at% for the grain boundary precipitates of AA6061 and AA7050, respectively. These values will be used in the models.

mn 22nm 1 min m 1

01

1

1 0.25 erf erf2 2

ii i i

i

AA A A LN

σ σ

=− −

=

= − − − −

∏

CHAPTER 4

76

Grain boundary energy γββ

The grain boundary energy γββ used as input parameter for the model is equal to 0.40 J m-2, which is a typical value for random high-angle grain boundaries in aluminium [14]. It is used for calculation of the dihedral angle ψ. The precipitate-matrix interfacial energy γαβ Although it is possible to measure the dihedral angle ψ (Fig 4.3) from electron micrographs and to abstract the interfacial energies, it is not certain whether the shape of the precipitate after prolonged times of ageing is representative for the shape of the nucleus [3]. Using the high-resolution transmission electron microscope, precipitate-shapes in AA6061 after 15 minutes of ageing were studied and an angle ψ ~ 20 ° was measured. Based on this angle, possible values of interfacial energies γαβ and γαβc for precipitates are 0.11 J m-2 and 0.31 J m-2, respectively (Fig 4.3a). The former value is a reasonable approximation for the energy of a coherent interface (< 0.2 J m-2) [20], whereas the latter is located in the energy-regime for semi-coherent interfaces (0.2-0.5 J m-2). However, other combinations of γαβ and γαβc yielding the same angle ψ are also possible. For AA7050, a reported [8] value of 0.25 J m-2 is taken for γαβ. Precipitate molar volume Vβ From literature [25], molar volumes of 6.8· 10-4 m3 and 3.0·10 -5 m3 are taken for the grain boundary phases in AA6061 and AA7050, respectively. Precipitate geometry during nucleation and growth For the grain boundary precipitate nuclei in AA6061, a single cap-shaped geometry (as in Fig 4.3a) is used in the model. This is because the C-phase most probably is the nucleation-phase and able to form a low-energy interface with the matrix (Chapter 3). During growth, on the other hand, the shape depicted in Fig 4.4 was observed (Fig 3.11b, Chapter 3). This geometry is incorporated in the model just after nucleation. For the grain boundary precipitate nuclei of AA7050, a double cap-shaped geometry (Fig 4.3b) is used. According to experimental observations (Fig 3.6b, Chapter 3), the precipitate shape during growth is similar to that of Fig 4.6.


77

Diffusivities of solute elements Only the diffusivities of Mg, Cu and Si (for AA6061) or Zn (for AA7050) are needed, because other alloying elements like Fe, Cr and Mn combine as intermetallic phases already during the casting process and are therefore not prone to segregation or precipitation. To predict the grain boundary enrichments in the as-quenched state, the diffusivities of all complexed solutes are needed. Because Mg acts as rate-controlling element for both alloys (Chapter 3), its matrix- and grain boundary diffusivity at the ageing temperature are also needed in order to model growth. All values are given in Appendix 4.A. 4.5 PROCESSING ROUTES AND EXPERIMENTAL PROCEDURES The composition of the AA6061 alloy is 1.0 at% Mg, 0.7 at% Si and 0.15 at% Cu [26]. The alloy was homogenised for 6 hours at 565 °C and cold-rolled to 200 µm. Discs with a diameter of 3 mm were punched and a solution heat treatment at 560 °C for 25 minutes was given. Subsequently, a water quench was applied. A couple of discs were left in this ‘as-quenched’ condition whereas the other discs were aged for 1 h, 3 h, 7 h, 21.5 h, 50 h and 100 h at temperatures of 160 °C or 180 °C. The composition of the AA7050 alloy is 2.7 at% Mg, 5.8 at% Zn and 2.3 at% Cu [26]. The alloy was homogenised for 6 hours at 480 °C and cold-rolled to 200 µm. Discs with a diameter of 3 mm were punched and a solution heat treatment at 480 °C for 25 minutes was given. Subsequently, a water quench was applied. A couple of discs were left in this ‘as-quenched’ condition whereas the other discs were aged for 1 h, 3 h, 7 h, 23.5 h, 51 h and 100 h at 160 °C. Thin foils for transmission electron microscopy were prepared by using a TENUPOL electropolisher containing an electrolyte of 30 % nitric acid in methanol at –20 °C. The applied voltage was 20 V. Afterwards, the samples were milled with 4 keV Ar+-ions under an incidence angle of 6 º for several minutes in order to remove the bent edges and to clean the sample.

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78

4.6 COOLING RATE DURING WATER-QUENCHING After the samples were solution heat treated, they were taken out of the furnace and quenched in water at room temperature. As the cooling rate is very rapid, it is difficult to determine this parameter experimentally and therefore it is calculated using the method described in Chapter 2. The thickness and diameter of the quenched disc are 0.2 mm and 3.0 mm respectively, so that its volume is 1.4· 10-3 cm3 and its surface area is 1.6· 10-1 cm2. Using equation (2.1), together with ρ = 2.7 g cm-3 and C = 0.88 J g-1 °K for aluminium, the temperature curve versus time can be calculated for different heat transfer coefficients. For water, the heat transfer coefficient h is in the range 2-40 kW m-2 °K-1 [26]. Fig 4.8 depicts the calculated temperature curves against time during a water-quench of a 0.2 mm-thick AA6061 sample using for h two different values in this range. As can be seen from the figure, the quench-time tq through the range of 560 °C - 330 °C (i.e. TSHT – 0.65Tm, the range in which non-equilibrium segregation is most pronounced) varies from 3.7 ms for h = 35 kW °K-1 m-2 to 13.0 ms for h = 10 kW °K-1 m-2. As h may be reduced significantly, for instance by the formation of a small vapour layer during the quench, it cannot be determined in a straightforward manner and an uncertainty in the time of cooling is introduced. The cooling-rates determined here will be needed to model non-equilibrium grain boundary segregation during quenching and subsequent precipitate growth during artificial ageing. For AA7050, the temperature of solution heat treatment is 480 °C. The time of cooling to 0.65Tm = 330 °C in this case is 2.7 ms for h = 35 kW °K-1 m-2 and increases to 9.5 ms for h = 10 kW °K-1 m-2.


79

4.7 EXPERIMENTAL DETERMINATION OF GRAIN BOUNDARY ENRICHMENTS IN THE

AS-QUENCHED STATE AND COMPARISON TO THE MODEL After solution heat treatment, the samples were quenched in water at room temperature. Non-equilibrium segregation conditions are prevalent during this process and vacancy-impurity complexes form, which are drawn towards the grain boundaries. The resulting solute concentration profiles of Mg, Si, Zn and Cu can then be determined using equations (4.3) and (4.4). In Table 4.1, the SCL-widths w0 and grain boundary solute concentrations Cb for Mg, Si and Cu in AA6061 are listed, based on a quench-time tq of 3.7 ms (from the previous section). Table 4.2 depicts these values for Mg, Zn and Cu in AA7050 for a quench-time of 2.7 ms. Typical values for the vacancy formation energy Ef and the vacancy-impurity binding energy Eb are taken from [14,27].

Fig 4.8: Temperature versus time of a 0.2 mm-thick sample at 560 ºC quenched in water at 20 ºC. Presented are the calculated curves for

heat transfer coefficients of 35 kW ºK-1 m-2 and 10 kW ºK-1 m-2.

0

100

200

300

400

500

600

0 0.01 0.02Time (s)

Tem

pera

ture

(°C

)

h = 35 kW / K m^2

h = 10 kW / K m^2

CHAPTER 4

80

Table 4.1: Theoretical grain boundary solute concentration profiles in AA6061 after water-quench from 560 °°°°C to 20 °°°°C, Ef = 1.25 eV

Element Cg (at%) Eb (eV) w0 (Å) tq = 3.7 ms

Cb (at%)

Mg 1.00 0.249 4.8 40.5 Si 0.70 0.332 24.0 24.3 Cu 0.15 0.231 12.6 6.2

Table 4.2: Theoretical grain boundary solute concentration profiles in AA7050 after water-quench from 480 °°°°C to 20 °°°°C, Ef = 1.25 eV

Element Cg (at%) Eb (eV) w0 (Å) tq = 2.7 ms

Cb (at%)

Mg 2.70 0.249 6.2 24.9 Zn 5.80 0.164 15.5 48.7 Cu 2.30 0.231 18.7 21.1

Using the JEOL 2010FEG analytical microscope, an electron-probe size of ~ 1.0 nm can be obtained. By orienting the grain boundary plane edge-on with respect to the electron beam, the grain boundary chemistry in the as-quenched state can then be analysed. However, because the width of the segregated layer probably is smaller than the diameter of the electron probe, the latter may only partly cover the region of interest. Beam broadening during transmission of the sample should also be taken into account, because characteristic X-rays emitted from the lower regions of the sample may reach the detector as well. Above considerations are schematically depicted in Fig 4.9. Because only part of the detected X-rays thus originates from the region of interest, the quantification of the spectrum of Fig 4.9b needs correction. An accurate correction procedure was proposed by Faulkner [28]. For a solute distribution of concentration Cb at the grain boundary and Cg in the adjacent matrix, the concentration Cbm measured using an electron probe, which is incident at a distance xd from the layer of width w0, is given by

(4.25) σ σ+ − = + − ⋅ −

0 0d dg gbm b

0.5 0.51( ) erf erf

2 2 2x w x w

C C C C


81

The acquired solute profile at the grain boundary is thus the convolution of the segregation profile and the electron probe current distribution. It can be seen that the experimental segregation profile is determined by the standard deviation σ of the Gaussian probe distribution, which can be expressed in terms of the probe diameter as σ = 0.279φ80 [28]. Here, φ80 defines the diameter that contains 80 % of the total probe current. The actual grain boundary concentration Cb of a certain solute element can thus be determined from the experimental value Cbm if both the probe diameter and the SCL-width w0 are known. xd is then taken equal to zero in equation (4.25).

When the electron beam traverses the sample, it will broaden because of electron scattering. This broadening b can be described by [29,30]

(4.26)

where Z is the atomic number, E0 is the incident electron beam energy (in eV), ρ is the density of the sample (in g cm-3), A is the atomic weight (in a.u.) and d is the sample thickness (in cm). The equation describes the exit diameter, which contains 90% of the scattered electrons.

ρ = ⋅

13

25 2

0

7.21 10Z

b dE A

Fig 4.9: (a) Interaction of electron probe with grain boundary plane. (b) EDS-spectra of matrix and grain boundary enriched in Cu, Mg and Si.

(a)

(b)

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82

Before acquisition of the grain boundary chemistry, the sample thickness was measured by using convergent-beam electron diffraction (CBED), according to the method described in Chapter 2. The average thickness was determined to be ~ 100 nm. Using this value in equation (4.26), an exit beam diameter of ~ 4.7 nm was calculated. The beam diameter at the upper sample surface is ~ 1.0 nm and therefore an effective beam diameter of 3.0 nm will be used in the correction procedure. Table 4.3 and 4.4 present the uncorrected experimental concentrations Cbm determined for random high-angle grain boundaries of AA6061 and AA7050, respectively. The values are the averages of 6 different boundaries. Per grain boundary, an average of 6 measurements was determined. These were performed at different locations, but with the same orientation of the grain boundary plane. During acquisition, the electron probe was shifted slightly along the grain boundary in order to minimize contamination effects. By instantaneous observation in diffraction mode, Kikuchi-lines and lattice plane reflections of both adjacent grains were observed, providing an accurate approach to maintain the electron probe at the grain boundary. In this way, effects of beam- or sample-drift during acquisition could be minimized. The spread between the individual acquisitions did not prove to be significant. Assuming an error of 10 % in the EDS-quantification of elemental concentrations, the final standard deviation in the average of 36 measurements then can be reduced to 1.7 %. Also listed in the tables are the corrected grain boundary concentrations Cb, determined by using equation (4.25) together with a probe diameter of 3.0 nm and the corresponding modelled SCL-widths w0 from Tables 4.1 and 4.2. Because the acquired grain boundary concentration Cbm is the convolution of the segregation profile and the electron probe current distribution, it would theoretically be possible to determine both SCL-width w0 and grain boundary concentration Cb unambiguously by using different electron probe sizes. However, practical difficulties arise as the probe size in this case can only be enlarged, which renders the acquired number of characteristic X-ray quants of segregated elements too small for an accurate peak-to-background ratio.


83

Table 4.3: Experimental grain boundary solute concentrations in AA6061 before and after correction using modelled values for w0

Element Cbm (at%) w0 (Å) tq = 3.7 ms

Cb (at%)

Mg 5.5 4.8 21.1 Si 3.2 24.0 3.7 Cu 0.9 12.6 1.6

Table 4.4: Experimental grain boundary solute concentrations in AA7050 before and after correction using modelled values for w0

Element Cbm (at%) w0 (Å) tq = 2.7 ms

Cb (at%)

Mg 6.1 6.2 14.5 Zn 8.1 15.5 9.4 Cu 3.0 18.7 3.3

Comparing Tables 4.3-4.4 to Tables 4.1-4.2 it can be seen that, although the discrepancies between modelled and experimental values for Cb are large, the values for Mg compare relatively better than those of the other solutes. As the uncertainty in the electron-probe diameter is very small, part the discrepancy probably originates from preferential etching of the grain boundary during sample preparation. The remaining part may be a result of the fact that either the actual SCL-width w0 or the grain boundary concentration Cb is much smaller than the modelled counterpart. This can be explained by the fact that the model calculates SCL-width and grain boundary concentration of a certain solute for the condition that no other solutes are segregating to the grain boundary. Due to solute-solute interactions however, SCL-width and/or grain boundary concentration of one element may be reduced by the preferred presence of another element. The relatively better agreement for the grain boundary concentrations of Mg may then be explainable in terms of its favoured role in site-competition events, which was also observed in Chapter 3.

CHAPTER 4

84

Analogously, the SCL-widths w0 can be determined from the measured concentrations Cbm and the modelled grain boundary concentrations Cb using equation (4.25). These are given in Table 4.5 and 4.6.

Table 4.5: Experimental SCL-widths based on modelled grain boundary concentrations Cb in AA6061

Cbm (at%) Cb (at%) w0 (Å)

Mg 5.5 40.5 2.4 Si 3.2 24.3 2.3 Cu 0.9 6.2 2.6

Table 4.6: Experimental SCL-widths based on modelled grain boundary concentrations Cb in AA7050

Cbm (at%) Cb (at%) w0 (Å)

Mg 6.1 24.9 3.2 Zn 8.1 48.7 1.1 Cu 3.0 21.1 0.8

As already discussed, SCL-width w0 and grain boundary concentration Cb cannot be abstracted unambiguously by using different electron probe sizes. Probably, the actual solute concentration profiles are located within the range imposed by Tables 4.3-4.4 and Tables 4.5-4.6. From these tables, the ranges of relative solute concentrations can be obtained by taking the product of SCL-width and grain boundary concentration for every element and substituting these values in equation (3.1). When these relative concentrations are compared to the relative solute fractions in the precipitates (Tables 3.3-3.4, Chapter 3), conclusions can be drawn about the rate-controlling element during precipitate nucleation and growth. This will be elaborated in section 4.9.


85

4.8 EXPERIMENTAL DETERMINATION OF GRAIN BOUNDARY PRECIPITATE GROWTH

AND COMPARISON TO THE MODEL Precipitate-size measurements were made in the transmission electron microscope with the foil oriented such that the growth direction of the boundary precipitates was perpendicular to the electron beam. In cases where this was not possible (e.g. in cases where the growth direction is inclined with respect to the sample surface as depicted in Fig 4.10), the actual length Lw of the precipitate can be derived from its projected length Lp as demonstrated in the figure. In this case two angles α and δ are needed. The first angle α can be calculated from the projected width of the grain boundary X0 and the sample thickness d, which was determined according to the method described in Chapter 2. The second angle δ can be derived from the angle between the projected precipitate and the grain boundary fringes for different sample orientations. The actual length Lw of the precipitate can then be calculated according to

(4.27)

Because of the distributions for the precipitate size and inter-distances, the average length and inter-distance of 10 precipitates at 3 different random high-angle grain boundaries were determined for each ageing time. Fig 4.11 presents the different stages of precipitation at random high-angle grain boundaries in AA6061 during artificial ageing at 160 °C. No precipitates can be resolved at the grain boundary in the as-quenched condition (Fig 4.11a), which supports the theory that precipitates do not nucleate until the onset of the artificial ageing treatment. With subsequent artificial ageing time, growth of the grain boundary precipitates can be clearly observed (Fig 4.11b-f).

2 2w p (cos ) (sin cos )L L δ δ α= +

Fig 4.10: 2D-projection of precipitate at an inclined grain boundary.

αααα δδδδ

d ~ 100 nm Lw

X0

Lp

CHAPTER 4

86

Experimental and modelled values for the length and inter-distance of the grain boundary precipitates in AA6061 are presented in Figs 4.12a,b for artificial ageing temperatures of 160 °C and 180 °C, respectively. Because it was not possible to determine the actual cooling rates, the modelled values are based on quench-times tq of 3.7 ms and 13 ms, corresponding to heat-transfer coefficients of 35 kW °K-1 m-2 and 10 kW °K-1 m-2, respectively. From the figures it can be seen that the experimental values for both ageing temperatures are located between the modelled values for these two cooling rates. For AA7050, the same comparison between model and experiment is made for an artificial ageing temperature of 160 °C (Fig 4.13). Except for the initial stages of ageing, the correspondence is reasonable. As precipitate nucleation and growth during ageing are largely dependent on the grain boundary concentration profile of the rate-controlling element (i.e. Mg) in the as-quenched state, above results would support accurate modelling of its amount of segregation.

Fig 4.11: Grain boundary precipitation at random high-angle grain boundaries of AA6061 at different stages of ageing at 160° C.

(a) As-quenched; (b) 1 h; (c) 3 h; (d) 21.5 h; (e) 50 h; (f) 100 h


87

Fig 4.12a: Experimental and modelled values for precipitate length and inter-distance vs. ageing time at 160 ºC. System: AA6061

0

20

40

60

80

100

120

0 20 40 60 80 100

Time at 160 °C (h)

Leng

th (n

m)

Experiment

Model tq = 3.7 ms

Model tq = 13.0 ms

0

20

40

60

80

100

120

0 20 40 60 80 100

Time at 160 °C (h)

Dis

tanc

e (n

m)

Experiment

Model tq = 3.7 ms

Model tq = 13.0 ms

CHAPTER 4

88

Fig 4.12b: Experimental and modelled values for precipitate length and inter-distance vs. ageing time at 180 ºC. System: AA6061

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100Time at 180 °C (h)

Dis

tanc

e (n

m)

Experiment

Model tq = 3.7 ms

Model tq = 13.0 ms

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100

Time at 180 °C (h)

Leng

th (n

m)

Experiment

Model tq = 3.7 ms

Model tq = 13.0 ms


89

Fig 4.13: Experimental and modelled values for precipitate length and inter-distance vs. ageing time at 160 ºC. System: AA7050

0

20

40

60

80

100

120

140

0 20 40 60 80 100

Time at 160 °C (h)

Leng

th (n

m)

Experiment

Model tq = 2.7 ms

Model tq = 9.5 ms

0

20

40

60

80

100

120

140

0 20 40 60 80 100

Time at 160 °C (h)

Dis

tanc

e (n

m)

Experiment

Model tq = 2.7 ms

Model tq = 9.5 ms

CHAPTER 4

90

4.9 DISCUSSION & CONCLUSIONS In the model, the assumption is made that non-equilibrium segregation only is prevalent during quenching from TSHT to 0.65⋅Tm. This is in contradiction to Faulkner et al. [6], where 0.50⋅Tm is proposed. However, for this temperature, grain boundary solute concentrations may become larger than 100 at%, which is not physically realistic. Comparison of the measured average relative grain boundary solute concentrations in the as-quenched state to the relative solute concentrations in the precipitates (Tables 3.3-3.4, Chapter 3) strongly supports the idea that during grain boundary precipitate nucleation in both AA6061 and AA7050, Mg is the rate-controlling element. Although Cu is a slower diffuser, for both alloys its average relative concentration at the grain boundary in the as-quenched state is significantly larger than its relative concentration in the grain boundary precipitates during subsequent ageing. The relative concentrations of Si and Zn at the grain boundary and in the precipitates are comparable, but as their diffusivities are significantly larger they are not likely to act as rate-controlling element. For Mg, the relative grain boundary concentration is smaller than its fraction in the precipitates. Regarding also its diffusivity, it can then be concluded that Mg must be the rate-controlling element during precipitate growth. The discrepancies between modelled and experimental grain boundary solute concentration profiles in the as-quenched state are significant. This may be explained by interactions between the different solute elements during segregation, which are not taken into account by the model. Also, preferential etching at the grain boundary during sample preparation may lead to erroneous results. Theoretically, it would be possible to determine both SCL-width w0 and grain boundary concentration Cb unambiguously by using different electron probe sizes. Practical difficulties however arise when the probe size is enlarged, because then the peak-to-background ratio in the energy dispersive spectrum becomes too small for an accurate chemical analysis.


91

A site-competition effect in which Mg is favoured (also observed in Chapter 3), may explain the relatively smaller discrepancy between its experimental and modelled grain boundary concentrations as compared to Si, Cu and Zn. This effect cannot be explained on the basis of vacancy-impurity binding-energy Eb or matrix concentration Cg in equation (4.3). As discussed in [18], the phenomenon of site-competition is very complex and alternative or additional mechanisms must therefore be considered. For instance, the binding energy of the solute elements to the boundary sites may be the determining factor. Although it was suggested by Faulkner et al. [6] that the precipitate geometry remains unaltered during nucleation and growth, this is not in agreement with experimental observations for AA6061 and AA7050. In this work, the geometry of Fig 4.4 is used for modelling precipitate size and distribution in AA6061 during artificial ageing. This geometry may be explained by the fact that the grain boundary precipitates in this alloy are able to share low-energy facets with the matrix (Chapter 3). Similar arguments support the use of a single-cap shape for the precipitate nucleus (Fig 4.3a). For AA7050, in which the precipitate nuclei (η’-phase) probably do not form a particular low-energy facet, the double-cap geometry is used for modelling nucleation (Fig 4.3b). For growth a resembling shape is used, with the exception that the radius is converted to a half-length (Fig 4.6), so that the projection along the length-axis of the precipitate is a circle with the precipitate thickness as diameter. The size and distribution of the grain boundary precipitates during artificial ageing are reasonably predicted for two different alloy systems and for different ageing temperatures. Although the cooling-rate during the water-quench is not exactly known, experimental values for precipitate length and inter-distance are located well within the ranges imposed by this uncertainty. The observed transition from the η’-phase to the η-phase during annealing of AA7050 (Chapter 3), which is not accounted for in the model, may explain the discrepancy between predicted and experimental values for small ageing times (Fig 4.13). For AA6061, the C-phase is already present at nucleation and a phase-transition during artificial ageing is not found to occur. Because precipitate nucleation and growth during artificial ageing are largely determined by the initial grain boundary concentration of the rate-controlling element (i.e. Mg), accurate modelling of its degree of grain boundary segregation during the water-quench is supported.

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92

APPENDIX 4.A Table 4.7: Input parameters for the grain boundary segregation and precipitation model of AA6061 Parameter Value Unit Solution heat treatment temperature TSHT 833 °K Quench time 3.7 or 13 ms Ageing temperature Ta 433 or 453 °K Mg-concentration in the matrix Cg 1.0 at% Vacancy formation energy Ef [8] 1.25 eV Vacancy-impurity binding energy Eb for Mg [14,15] 0.249 eV Vacancy-impurity binding energy Eb for Si [14,15] 0.332 eV Vacancy-impurity binding energy Eb for Cu [14,15] 0.231 eV Molar volume of the matrix phase 1.0⋅10-5 m3 Molar volume of precipitate phase Vβ [25] 6.8⋅10-4 m3 Atomic fraction of rate-controlling solute in the nucleus phase Xβ (i.e. of Mg) (Chapter 3)

0.65

Grain boundary energy γgb [14] 0.40 J m-2 Precipitate-matrix interfacial energy γαβ 0.11 J m-2 Precipitate-matrix interfacial energy γαβc 0.31 J m-2

GB width within which GB diffusion prevails [8] 1⋅10-10 m Enthalpy of solution in sol. prod. equation ∆H [21] 1.67⋅105 J mol-1 Entropy term in solubility product equation S [21] 23.8 Ratio between nr. of moles of the non-rate controlling element and moles of the rate-controlling element in precipitate phase XE ( i.e. (Cu+Si+Al)/Mg )

0.615

Concentration of the non-rate controlling elements in matrix Cc (i.e. Si + Cu)

0.7 at%

Matrix diffusion coefficient DI of Mg [6,8]

D0I = 1.4⋅10-5

QI = 1.54 m2 s-1

eV Grain boundary diffusion coefficient Db of Mg [6,8] D0b = 1.7⋅10-4

Qb = 0.98 m2 s-1 eV

Matrix diffusion coefficient Dc of vacancy-Mg complexes [15]

D0c = 1.4⋅10-5 Qc = 1.50

m2 s-1

eV Matrix diffusion coefficient Dc of vacancy-Si complexes [15]

D0c = 1.4⋅10-5 Qc = 1.29

m2 s-1 eV

Matrix diffusion coefficient Dc of vacancy-Cu complexes [15]

D0c = 1.4⋅10-5 Qc = 1.36

m2 s-1 eV


93

Table 4.8: Input parameters for the grain boundary segregation and precipitation model of AA7050 Parameter Value Unit Solution heat treatment temperature TSHT 753 °K Quench time 2.7 or 9.5 ms Ageing temperature Ta 433 °K Mg-concentration in the matrix Cg 2.70 at% Vacancy formation energy Ef [8] 1.25 eV Vacancy-impurity binding energy Eb for Mg [14,15] 0.249 eV Vacancy-impurity binding energy Eb for Zn [14,15] 0.164 eV Vacancy-impurity binding energy Eb for Cu [14,15] 0.231 eV Molar volume of the matrix phase 1.0⋅10-5 m3 Molar volume of precipitate phase Vβ [25] 3.0⋅10-5 m3 Atomic fraction of solute in the nucleus phase Xβ (i.e. of Mg)

0.33

Grain boundary energy γgb [14] 0.40 J m-2 Precipitate-matrix interfacial energy γαβ [8] 0.25 J m-2

GB width within which GB diffusion prevails [8] 1⋅10-10 m Enthalpy of solution in sol. prod. equation ∆H [6] 1.74⋅104 J mol-1 Entropy term in solubility product equation S [6] -1.2 Ratio between nr. of moles of the non-rate controlling element and moles of the rate-controlling element in precipitate phase XE ( i.e. (Zn+Cu+Al)/Mg )

2.0

Concentration of the non-rate controlling elements in matrix Cc (i.e. Zn + Cu)

8.1 at%

Matrix diffusion coefficient DI of Mg [6,8]

D0I = 1.4⋅10-5

QI = 1.54 m2 s-1

eV Grain boundary diffusion coefficient Db of Mg [6,8] D0b = 1.7⋅10-4

Qb = 0.98 m2 s-1 eV

Matrix diffusion coefficient Dc of vacancy-Mg complexes [15]

D0c = 1.4⋅10-5 Qc = 1.50

m2 s-1

eV Matrix diffusion coefficient Dc of vacancy-Zn complexes [15]

D0c = 1.4⋅10-5 Qc = 1.39

m2 s-1 eV

Matrix diffusion coefficient Dc of vacancy-Cu complexes [15]

D0c = 1.4⋅10-5 Qc = 1.36

m2 s-1 eV

CHAPTER 4

94

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