Modelling Mg2Si Dissolution in an AA6063 Alloy During Pre-heating to theExtrusion Temperature

J. van de Langkruis1, N. C. W. Kuijpers2, W. H. Kool1, F. J. Vermolen3, S. van der Zwaag1,2

1Lab. for Materials Science, Delft University of Technology, Delft, The Netherlands,2Netherlands Institute for Metals Research, Delft, The Netherlands, 3CWI, Amsterdam, TheNetherlands.

ABSTRACT

A finite volume model was used to calculate Mg2Si particle dissolution and the solute Mg content in an AA6063extrusion alloy during pre-heating to the extrusion temperature. The model contains a limited number of physicalparameters, which are obtained from the literature. Firstly, the model was validated with DSC experiments.Secondly, dissolution diagrams were constructed, which show a strong dependence of solute Mg concentrationafter pre-heating on heating rate and initial Mg2Si particle size. The solute content can be used to predict thepeak hardness after artificial ageing. When the initial Mg2Si particle size and volume fraction are known, thedissolution diagrams can be used to determine the maximum allowable pre-heating rate before extrusion leadingto a maximum peak hardness in the final product.

INTRODUCTION

Quantitative information concerning the presence orabsence of Mg2Si precipitates and the Mg and Sisolute content in an extrusion billet is valuable forassessing extrusion behaviour and peak hardness ofthe extrudate as a function of prior thermal historyand composition. In the past, Mg2Si dissolution duringpre-heating to extrusion temperatures has beenmodelled with classical phenomenological phasetransformation theory [1-5]. However, this approachalways incorporates numerous fitting parameters,which have to be optimised depending on the alloyand its thermal history. Also, some important effects,such as the contact surface between particles andaluminium matrix, and the saturation of the matrix atthe particle interface, both affecting the reaction rate,cannot be calculated directly with this approach.

In recent years physical methods have beendeveloped [6-8], which are capable of modellingparticle dissolution and growth in binary and morecomplex systems. These models do not need fittingparameters, but depend on physical parameters,such as solubility products and diffusion coefficientsfor the alloying elements in the matrix.

Vermolen et al. [8] developed a finite volumemodel for dissolution of a multi-component particle,and calculated the dissolution of Mg2Si-precipitates inan industrial AlMgSi alloy. Vermolen’s finite volumemodel was applied to estimate the solute contentafter pre-heating in a Mg2Si-containing AA6063 alloy,and to correlate the Mg and Si solute content to the

hot flow stress and the peak hardness after artificialageing [9].

In the present paper the finite volume model wasvalidated experimentally by calculating andmeasuring Mg2Si dissolution during DifferentialScanning Calorimetry (DSC) experiments. DSCexperiments are very suitable for studying particledissolution in metals, as the heat flow is directlyproportional to the dissolution rate. Furthermore, themodel was used to construct a diagram of the Mgsolute content after pre-heating as a function of theinitial Mg2Si particle radius and the heating rate. Thediagram can be used to predict the obtainable peakhardness.

THE PARTICLE DISSOLUTION MODEL

The model [8] considers the dissolution process of aspherical Mg2Si particle of initial radius r=r0,surrounded by an aluminium matrix cell with cut-offradius r=RAl, as depicted in Figure 1. The Al cell sizeR is chosen such that (r0

3/RAl3) is equal to the volume

fraction of the Mg2Si particles in the alloy.During pre-heating the temperature T(t) increases

from room temperature to the extrusion temperature.As a result of dissolution, the particle radius, rp(t), willdecrease with time which is calculated by the model(rP(0)=r0).

Al-matrix

Mg Si particles2

Al-matrix

Mg Si2RAl

(b)

(a)r0

Figure 1 (a) Modelled configuration, indicating r0 andRAl. (b) Schematic display of the microstructure of thealuminium alloy with Mg2Si precipitates.

It is assumed that at the start of the preheatingprocess (t=0) the initial Mg- and Si- concentrations inthe matrix are uniform with cMg

init and cSiinit (wt%). The

diffusive flux of Mg or Si in the matrix is taken to beindependent of the presence of Si or Mg in the matrix,respectively. For the concentrations of both elements(cMg and cSi) the diffusion may be represented byFick’s second law in spherical co-ordinates:

∂∂

∂∂

=∂∂

rc

rrr

Dt

c iii 22

(1)

where i=Mg,Si.For each element Mg and Si the diffusion

coefficient Di in the aluminium matrix is given by anArrhenius relationship:

−=

)(exp0

tRTQ

DDd

iii (2)

where R is the molar gas constant. The diffusionparameters Di

0 and Qi

d have been determined

experimentally by Fujikawa [10], and are listed inTable 1.

In the model three boundary conditions areimposed. The first one is for the concentrations in theAl-matrix on the interface of the precipitate, where forboth the Mg and Si concentrations the Dirichletboundary condition is used:

( ) solipi crrc == . (3)

The concentrations cMgsol and cSi

sol are not constant,but change in time. Assuming local equilibrium at theinterface of the precipitate, these concentrations areconnected by the following hyperbolic relation [8]:

−=⋅

)(exp)()( 0

3/23/1tRT

QKcc solsol

Mgsol

Si (4)

where the powers of the concentrations on the lefthand side of the equation correspond to thestoichiometric Mg2Si-phase. The right hand side ofeq. (4) is the solubility product, where K0 is a pre-exponential factor, and Qsol is the activation energyfor the solubility. The solubility parameters are takenfrom literature [11] and are given in Table 1.

The second boundary condition applies to thematrix-boundary at r=RAl where there is no fluxthrough the boundary, for both Mg and Siconcentrations:

0=

∂∂

= AlRr

ir

c. (5)

The third boundary condition arises when theMg2Si particle is completely dissolved. In that case,from symmetry considerations there can be no fluxthrough the centre of the Al cell, and therefore, forboth Mg and Si concentrations a new Neumannboundary condition must be imposed:

00

==

∂∂

rrc i . (6)

In the model it is assumed that the composition ofthe Mg2Si precipitates remains stoichiometric at alltimes, and that there is no volume change of the cellwhen the Mg2Si-particle dissolves in the Al-matrix.Therefore the density of Mg2Si is taken equal to thatof the aluminium matrix.

From conservation of mass and the stoichiometryof the Mg2Si particle, the interfacial particle-matrixvelocity can be derived from the interfacial gradientsof the Mg- or the Si-concentration in the Al-matrix:

0

0

)(

)(

)(

rr

Sisol

Sip

Si

Si

rr

Mgsol

Mgp

Mg

Mgp

rc

tcc

D

r

c

tcc

D

dt

tdr

=

=

∂

∂

−=

∂

∂

−=

(7)

where cMgp and cSi

p are the stoichiometricconcentrations of the particle.

Table 1 Diffusion and solubility parameters withliterature source.

Parameter Value

DMg0

QMgd

DSi0

QSid

K0

Qsol

0.49⋅10-4 m2/s

124 kJ/mol

2.02⋅10-4 m2/s

136 kJ/mol

89 wt%

31.97 kJ/mol

[10]

[10]

[10]

[10]

[11]

[11]

EXPERIMENTAL DETAILS

An AA6063 billet with a composition given in Table 2,was industrially extruded to a rectangular profile withdimensions 1000×100×30 mm3. SEM specimenswere taken from the central part, and homogenised at853 K (580 ºC) for 6 hours in an air furnace andsubsequently down-quenched in a salt bath to 640 K(367 ºC), pre-aged for 13.3 hours to induceprecipitation of β Mg2Si particles, and then waterquenched. This material condition is designated as β-fine.

Table 2 Alloy composition in wt%.

Mg Si Fe Cu, Mn, Cr,Zr, Pb

Ti

0.45 0.40 0.19 <0.01 0.01

From SEM images the effective Mg2Si particle radius,r0, for this material condition, was estimated. Theinitial solute concentrations after pre-ageing (cMg

init

and cSiinit) and the Mg2Si volume fraction were

calculated with the software package MTDATA [12].With these values and using mass conservation, theAl-cell radius, R, was calculated for the β-finecondition. These parameters of the β-fine conditionare given in Table 3.

Table 3 Parameters values for β-fine.

parameter valuer0Mg2Si volume fractionRcMg

init

cSiinit

0.40 ± 0.2 µm0.65 vol%2.14± 0.5 µm0.067 wt%0.19 wt%

For the DSC experiments, disc-shaped polishedsamples with a diameter of ∅ 5 mm and a thicknessof about 0.6 mm were obtained from the centre of theextrusion profile. DSC experiments were carried out,using a Perkin & Elmer DSC 7. The DSC-sampleswere treated in situ in the DSC equipment with thesame temperature scheme as used for the SEMspecimens [9, 13]. Scans are taken from 273 K (0 ºC)to 853 K (580 ºC) with heating rates of 10 or170 K/min.

The experimental heat flow (J/K) measured byDSC is directly compared to the dissolution rate of theparticle (vol%/K), as calculated by the finite volumemodel, using the data in Table 3.

Further, the solute Mg content after preheatingwas calculated. The pre-heating process, used inthese calculations consists of linear heating (with aheating rate dT/dt) to 773 K and holding at the sametemperature for a certain time. For the calculationsthe initial matrix concentrations and Mg2Si volumefraction of β-fine are used (Table 3), where theheating rate (dT/dt), initial partial radius (r0) andholding time are varied. As a result, examples ofdissolution diagrams are constructed which depict thecalculated solute Mg content after preheating as afunction of heating rate and initial particle radius r0.The Mg solute content in the dissolution diagrams iscalculated by averaging the Mg concentration (cMg(r))in the matrix.

RESULTS

Heat Effects Due to Mg2Si Dissolution

In Figure 2 the modelled dissolving volume per Kelvinand the measured DSC heat flow for the β-finecondition are given. It can be observed that in theexperimental 10 K/min curve the onset-, peak- andend temperatures are lower than the correspondingtemperatures of the experimental 170 K/min curve,which is a result of kinetic effects. The peak positionsof the experimental curves are predicted well by thenumerical calculations. In contrast to the experimentalcurves, where the dissolution speed displays a tail athigher temperature, the calculated curves show a

sharp decrease in dissolution speed. Chen et al. [14]showed that this tail can be explained byincorporating a complete particle distribution.However, taking one effective particle radius insteadof the complete distribution does not significantlyaffect the calculated peak positions.

To assess the sensitivity of the model to the valuetaken for the initial particle radius, a calculated170 K/min-curve for r0 = 0.22 µm, with the sameparticle volume fraction as β-fine, is also added inFigure 2. It can be observed that there is a significantshift of the dissolution peak to lower temperatures,indicating that the results of the calculations arelargely influenced by the particle size.

0

2

4

6

8

10

12

600 650 700 750 800 850Temperature (K)

Hea

t flo

w (

µJ/K

)

(a) (c)(b)

Dissolution rate (vol.%

/K)

10

5

500 ºC400 ºC

0

Figure 2 Experimental DSC curve (solid line) andcalculated dissolution rate (dashed line). (a) dT/dt = 10K/min, r0=0.40 µm (b) dT/dt= 170 K/min, r0 = 0.22 µm. (c)dT/dt = 170 K/min, r0=0.40 µm.

The Effect of Initial Particle Size and HeatingRate

Having established the satisfying accuracy of themodel for some specific cases it can now be used topredict the effect of initial Mg2Si particle size andheating rate upon the final solute content at the endof a pre-heating procedure.

In Figure 3 Mg dissolution diagrams are given,constructed for two holding times: (a) 4s;(b) 5 minutes. The dissolution diagrams show the iso-solute Mg contours. The numbers in the figureindicate the Mg solute content (wt%). The shadedareas represent the conditions for which the particlesare completely dissolved after preheating, and thesolute content is at its maximum. The diagramsindicate that the solute content after pre-heatingdecreases with increasing initial particle size andincreasing heating rate. Furthermore, it can beobserved that the final solute Mg content issignificantly influenced by the holding time.

Figure 3 Mg solute content after pre-heating, as afunction of heating rate and initial particle radius (a) holdingtime 4 s; (b) holding time 5 minutes.

DISCUSSION

The comparison of DSC experiments and numericalcalculations shows good agreement over a widerange of heating rates, even when using standardliterature values for the physical parameters. Underconditions of high heating rates and relatively smallparticles (<0.5 µm) the results of the calculations arevery sensitive to the value of the initial particlediameter.

The dissolution diagrams show that for longerholding times and heating rates á 0.3 K/s the heatingrate does not strongly influence the final solute Mgcontent, because most of the dissolution takes placeduring holding.

To illustrate how the model can be used to predictobtainable mechanical properties, a simple method toestimate the peak hardness after a standard peakageing treatment in an AA6063 alloy is described,following the approach of Bratland et al. [2]. Forexcess Si alloys, the peak hardness, Hsample, after agiven artificial ageing treatment, is related to its soluteMg content, cMg,matrix, after pre-heating, but before

ageing, as well as to the maximum peak hardnessincrease obtainable with the given ageing treatment,through

Hc

cHH

alloyMg,

matrixMg,oversample ∆×+= . (8)

Here ∆H is the hardness difference between a peakhardened fully solutionised sample and anextensively over-aged sample Hover, and cMg,alloy is thenominal Mg alloy content. When using eq. (8) it isassumed that the increase during peak ageing can beattributed completely to the formation of β’’, and thatall other effects on the hardness, such as the effect ofdislocations and the effect of solute Si, areincorporated in Hover or do not influence the peakageing behaviour significantly. In previous work [9] itwas found that for the same alloy as in this work,∆H = 41 HV and Hover = 39 HV. The nominalcomposition cMg,alloy = 0.45 wt%. The finite volumemodel calculates a value of cMg,matrix = 0.3 wt% (seeFigure 3a) for the β-fine condition after heating upwith 170 K/min (2.83 K/s) and holding for 4s. Hsample ispredicted 66 HV, close to the measured value of70 HV, found previously [9], which demonstrates theapplicability of dissolution diagrams as given in Figure3.

Such dissolution diagrams can be used todetermine under which conditions the particlesdissolve completely and maximum peak hardnesscan be expected. Also, the solute dependency of thehot flow stress can be modelled [15-16]. Dependingon the required mechanical properties thecombinations of heating rate and initial particlediameters can optimised. For instance, for a minimalpeak hardness of 70 HV we derive from eq. (8) thatthe necessary Mg solute content before peak ageingis approximately 0.3 wt%, and then the diagramshows that with a typical industrial pre-heating rate of20 K/min, the maximum allowable initial Mg2Siparticle size is 1 µm.

Similar diagrams can be constructed, with iso-contours for the final particle radii after pre-heating. Itis known [17] that large Mg2Si precipitates can induceso-called incipient melting at the surface duringextrusion. Assuming that incipient melting may occurfor particles larger than a specific critical size, acritical range of initial radii and heating rates can bedetermined to prevent incipient melting of Mg2Si atthe extrudate surface.

Further research on the correlation betweenextrusion and extrudate behaviour and solute contentis in progress.

CONCLUSIONS

The dissolution behaviour of Mg2Si precipitates ofvarying sizes in an AA6063 alloy was calculated witha finite volume model for varying particle diametersand heating rates. Some calculations were validatedwith DSC experiments and the agreement betweencalculations and experiments was satisfactory.

It was illustrated how the dissolutiondiagrams, calculated with the model, can be appliedto predict the peak hardness, for a standard artificialageing treatment, as a function of initial structure andpre-heating schedule to the extrusion temperature.

ACKNOWLEDGEMENTS

This research is partially supported by theTechnology Foundation (STW). We thank BoalProfielen BV for supplying the material, andMr. S. P. Chen for valuable discussions, andMr. C. G. Borsboom and Mr. N. Geerlofs for technicalsupport.

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