95
8. Summary
Solute clustering and precipitation during multi-stage ageing in a commercial AA6014 alloy are systematically investigated in this study, covering 1) the role of pre-ageing (PA) in multi-stage ageing treatments, 2) the influence of quench rate on multi-multi-stage ageing treatments, 3) the clustering and precipitation behaviour during linear heating, all by means of various characterisation methods including hardness testing, tranmission electron microscopy, electrical resistivity measurements, thermoanalysis, and positron lifetime measurements.
Conclusions corresponding to each specific study are stated at the end of chapters 4 – 7. Here, the most general findings pointed out in various chapters are summarised:
1) Direct vacancy annihilation alone cannot explain the excess vacancy evolution during natural ageing (NA or NSA). This has been demonstrated by the NA kinetics after various quenches or by the NSA kinetics after various PA treatments. Other mechanisms such as vacancy clustering and vacancy trapping by clusters must play a role in the reduction of vacancies.
2) During multi-stage ageing, clusters formed in early stages play a very important role in later stages, not only in a direct manner by transforming to precipitates, causing e.g.
different PB hardening effects after PA at different temperatures but also indirectly by influencing the vacancy evolution by vacancy trapping, which regulates the mobile vacancy site fraction by repartitioning the total vacancies (quick process) and buffering vacancy annihilation and generation (slow process).
Besides, the current study shows a way to improve industrial PA strategies and demonstrates that the behaviour of samples during multi-stage ageing under simulated industrial conditions is similar to that of samples processed by water quenches with just slightly inferior PB hardening. Industry might benefit from the current study to design better processes and improve the alloy properties.
96
Appendix A
Supplemental material to the effect of PA to NSA and PB hardening
A1. Additional hardness data
Fig. SA1 – SA3 show the hardening of the alloy during NSA and after subsequent PB for alloys that have been pre-aged under different conditions.
The curves in Fig. SA1a–h are analogous to those in Fig. 4.3a,e, just that PA conditions differ.
Fig. SA1i-n correspond to Fig. 3c,g.
Appendix A
97
98
Fig. SA1. (a, c, e, g) Hardening curves during NSA after PA at various temperatures for various times.
(b, d, f, h) Hardnesses after additional PB. (i, k, m) Comparison of NSA after PA to other hardnesses than those given in Fig. 4.3(c) of Chap. 4 (60 HBW), (j, l, n) displays hardnesses after additional PB.
Fig. SA2 displays hardness after PA and NSA as a function of PA hardness, A continuous evolution from short NSA time to long NSA time is seen. NSA hardness increases first in the low PA hardness regime and at high PA temperature regime.
Fig. SA3 corresponds to Fig.SA2, just that a final PB was carried out. The final PB hardness changes most after the first week of NSA in the low PA hardness regime.
Appendix A
99 Fig. SA2. Hardness after NSA as a function of PA hardness and temperature. Different graphs correspond to different NSA times. f) is identical to Fig. 4.3d in Chap. 4.
100
Fig. SA3. Similar to Fig. SA2 but with additional PB. f) is identical to Fig. 4.3h in Chap. 4.
A2. Additional DSC measurements
Fig. SA4a – c shows DSC traces of alloys aged at 100 °C for 60 min, 240 min, and pre-aged at 120 °C for 60 min and then NSA for various times. The trends of the DSC curve evolution for increasing NSA time are similar, just that the influence of the same NSA time on the DSC curve is dependent on PA condition. The heat content for clustering during ageing (PA and/or NSA) can be estimated by ∆𝐻ageing= 𝐻AQ− 𝐻after ageing, where 𝐻 is the
Appendix A
101 integrated DSC heat flow from 𝑇𝑎 to 𝑇𝑒, 𝐻 = ∫ 𝑑𝑄
𝑑𝑡𝑑𝑇
𝑇𝑒
𝑇𝑎 . 𝑇𝑎 is the temperature before the first reaction begins, and 𝑇𝑒 the temperature at which equilibrium is reached, e.g. above the solution heat treatment temperature. However, as pointed out by Esmaeili et al. [19, 144], due to a high baseline drift in the higher temperature regime, a lower 𝑇𝑒 can also be chosen as long as all relevant precipitation/dissolution peaks are covered within the temperature range. We choose 𝑇𝑎 = 50 °C and 𝑇𝑒 = 315 °C because DSC curves before 𝑇𝑎 and after 𝑇𝑒 are converged. 𝐻AQ is constant, thus 𝐻after ageing = ∆𝐻ageing+ 𝐻AQ. Fig. 4.8 shows hardness plotted as a function of the DSC integral for samples pre-aged and subsequently naturally aged (in cases where DSC and hardness were both measured). A linear relationship is seen although with some fluctuations possibly due to different samples and possibly some baseline drift. If the heat signal is assumed to be proportional to the precipitated volume the latter is also proportional to hardness after NSA.
(a)
102
(b)
(c)
Fig. SA4. DSC traces of samples PA a) at 100 °C for 60 min, (b) 240 min, (c) PA at 120 °C for 60 min with and without ensuing NSA.
Appendix A
103 A3. Retardation factor from resistivity measurements
Retardation factors were determined using hardness curves and collapsing them into one master curve by normalising the NSA time with Θ−1. As electrical resistivity is more sensitive to clustering than hardness, it might detect earlier changes during NSA. Fig. SA5 shows the data of Fig. 4.4d on a linear time scale. A linear increase is found at the beginning of NSA with almost no incubation time. Thus the retardation factor can be calculated from the slope of the initial resistivity increase [71]. Here relative retardation of PA 80/480 to PA 160/10 is 12, which is larger than the value obtained from hardness (ℛθ ≈ 3) but still smaller than the ratio between PA equilibrium vacancy site fractions (ℛc ≈ 58). This underlines the importance of PA clusters in determining the mobile vacancy site fraction and strengthen the arguments in Sec. 4.2.2.2. The main conclusions remain the same.
(a) (b)
Fig. SA5. (a) Same as Fig. 4.4d but plotted with linear NSA time. (b) the initial linear increase stage of PA 80/480 and PA 160/10. Curves in (b) have been smoothed by filtering artefact data points (sudden drops [145]) and averaging every 5 points.
A4. Possible other cluster strengthening mechanism
PA clusters formed at different temperatures but yielding the same hardness were assumed in Sec. 4.2.2 to consume the same fraction of solute supersaturation. Now we discuss if clusters formed at different temperatures have a different strengthening effect. The ‘Weak obstacle’
model [34, 146, 147] assumes a relationship between hardness increase and fraction of solute clusters, ∆ℋ ∝ √𝛼 ∙ √𝑟 with 𝑟 the radius of solute clusters. Since clusters formed at higher temperature have on average a larger radius [53], a smaller fraction of solutes is consumed when the same hardness is reached by PA at higher temperature. As a consequence, the
104
retardation factor ratio will be smaller than the one following from Fig. 4.10 if the same solute fraction has to be consumed at different PA temperatures. This does not affect the main conclusions of the work.
A5. Fitting hardness data by various kinetic models
We attempted to obtain kinetic parameters by fitting the hardness data during NSA, ℋ(𝑡), by two known functions. We present them here although none of these attempts yielded consistent kinetic parameters to illustrate the difficulties involved. The problems encountered are:
The function does not represent the data well
Representation is good but the use of too many parameters makes usage of function questionable.
Function parameters function ℋ(𝑡) =
1 Single Avrami [148] ℋ0, ℋ1, 𝑘, 𝑛 ℋ0+ (ℋ1− ℋ0)[1 − 𝑒−(𝑘𝑡)𝑛] 2 Starink-Zahra [117] ℋ0, ℋ1, 𝑘, 𝑛, 𝜂 ℋ0+ (ℋ1− ℋ0) {1 − [(𝑘𝑡)𝑛
𝜂 + 1]−𝜂}
Function #1 does not yield satisfactory fit results and the values for the Avrami-coefficient 𝑛 are unrealistically low in some cases (𝑛 = 0.5 for 10 min NA), see Fig. SA6a. For longer PA times 𝑛 ≈ 1 is found. The conditions for JMAK are not fulfilled in our case because the vacancy fraction is continuously decreasing, which is why use of the JMAK model is questionable.
By enforcing 𝑛 = 1 for all the fits we obtain a rate constant 𝑘 that can be compared to the retardation factor −1 (Figs. 7 and 8 in [118]). We see that the general course is the same but 𝑘 tends to be larger than −1 by a factor up to 5.
Appendix A
105 (a)
(b)
Fig. SA6. (a) Fit with Function #1. (b) comparison of rate constant 𝑘 obtained by fitting with constant Avrami index 𝑛 = 1 and retardation factor obtained in [118].
Fig. SA7 shows that function #2 fits the NSA curves quite well. However, the parameters obtained do not vary in a continuous way, indicating over-determination of the function (too many parameters).
106
Fig. SA7. Fitting using Function #2.
The same applies to a double-stage JMAK function, i.e. a generalisation of Function #1 with two 𝑘 and 𝑛 parameters and a weight factor for the two contributions, leading to 7 free parameters. This allows for a good fit but not to derive meaningful parameters (fit not shown).
A6. Modelling of NSA clustering consider excess vacancy annihilation and vacancy trapping by NSA clusters
In the main text of the current work (Sec. 4.2.2.2), we showed that NSA hardening is delayed after applying PA, and especially, to different extents when PA temperature varies while keeping PA hardness constant (as long as >55 HBW). NSA kinetics is faster after PA at higher temperature (for a shorter time) than at lower temperature (for a longer time), with a factor of
~3 for PA at 160 °C and 80 °C, whereas the equilibrium mobile vacancy site fraction at 160 °C is ~58 times higher than at 80 °C.
Here we discuss whether such discrepancy can be explained by vacancy losses due to annihilation at sinks or vacancy trapping in NSA clusters (Zurob’s model [93]). The kinetic master equation (Eq. (4.1) in the main text) is simplified here to Eq. (SA1) using 𝛼𝑁𝑆𝐴 to
Appendix A
107 represent the relative fraction of total clustered solutes during NSA, and (1 − 𝛼𝑁𝑆𝐴) as the simple function related to supersaturation. i.e. 𝜑 ≡ 1. Since we have a non-equilibrium excess vacancy site fraction, which is influenced concurrently by annihilation kinetics and trapping, we solve the master equation numerically instead of analytically.
𝑑𝛼𝑁𝑆𝐴
𝑑𝑡 = 𝑎 × (1 − 𝛼𝑁𝑆𝐴) × 𝑐vac(𝑡). (SA1) Vacancy annihilation during NSA
Two excess vacancy annihilation models were found in the literature, proposed by Schulze et al. [87, 88] and Fischer et al. [78] respectively. Schulze’s model assumes that the annihilation rate of vacancies is proportional to the excess vacancy site fraction to a power of 𝑞 (Eq. (SA2), where 𝑞 = 1 is the most simple case [87]), while Fischer’s model gives an annihilation rate in Eq. (SA3) if we consider only annihilation at dislocation jogs and no hydrostatic stress. 𝑏1, 𝑏2 are both constants governing kinetics. For both models, the difference between the vacancy site fractions after PA at 160 °C and 80 °C starts with ~58 at the beginning of NSA and gradually goes to 1 during NSA, as illustrated in Fig. SA8. The difference between the two models is Fischer’s model eliminates the vacancy discrepancy faster than Schulze’s model (𝑞 = 1). Thus, Fischer’s model will be used in the following discussion.
𝑑𝑐vac(𝑡)
𝑑𝑡 = 𝑏1× (𝑐vac(𝑡) − 𝑐vaceq)𝑞 (SA2)
𝑑𝑐vac(𝑡)
𝑑𝑡 = 𝑏2× 𝑐vac(𝑡) × ln (𝑐vac(𝑡)
𝑐vaceq ) (SA3)
(a) (b)
Fig. SA8. (a) Schematic evolution of vacancy fraction during NSA after PA at different temperatures.
𝑏1 and 𝑏2 are adjusted to produce similar annihilation kinetics after PA at 80 °C by both models.
Fischer’s model gives much faster annihilation after PA at 160 °C. (b) Ratio of the vacancy site fractions during NSA after PA at different temperatures.
108
Vacancy trapping by NSA clusters
Here we take into account that vacancies are trapped by emerging NSA clusters. One model describing such phenomenon was established by Zurob et al. [93] by assuming that the probability of a trapped vacancy to escape from a cluster is proportional to the Boltzmann weight of the interaction energy between the cluster and the vacancy, which is proportional to the number of solute atoms in the cluster due to the number of successful jumps required to escape from the cluster. If the number of clusters remains constant and just their size increases during NSA, then we can express the mobile vacancy site fraction by Eq. (SA4), where 𝑚 and 𝑛 are adjustable constants, and 𝑛 is negative. Note that this model is based on kinetic arguments, which is different from the thermodynamic approach proposed by Pogatscher et al. [83], although they result in similar equations. Since only mobile vacancies can annihilate, 𝑐vac(𝑡) in Eq. (SA3) has to be replaced by 𝑐vac,mob(𝑡) when vacancy trapping is considered.
𝑐vac,mob(𝑡)
𝑐vac(𝑡) = 𝑚 × exp(𝑛 × 𝛼𝑁𝑆𝐴). (SA4) Using the above mentioned models, i.e. Eq. (SA3) and (SA4) with 𝑐vac(𝑡) in Eq. (SA3) replaced by 𝑐vac,mob(𝑡), Fig. SA9 demonstrates the simulated kinetics during NSA after PA at two representative temperatures (160 °C and 80 °C) under various vacancy annihilation and trapping effects by changing the adjustable constant parameters. The initial kinetic difference stems from different equilibrium vacancy site fractions at 160 °C and 80 °C (factor 58). This difference remains (weak annihilation or strong trapping) or gets larger, but is not reduced to 3 as obtained from the experiments.
Appendix A
109 Fig. SA9. Schematic representation of 𝛼𝑁𝑆𝐴 as a function of 𝑡𝑁𝑆𝐴 after PA at 160 °C and 80 °C under various vacancy annihilation and trapping conditions. Stronger trapping: 𝑛 decreases (more negative).
Faster annihilation: 𝑏2 increases.
110
Appendix B
Supplemental material regarding effect of quench rate on multi-stage ageing
B1. Additional TEM images and EDX analysis
Appendix B
111 Fig. SB1. (a – d) Bright-field TEM images of samples after (a) AC, (b) AC and AA for 4 h, (c) IWQ, (d) IWQ and AA for 4 h. (e – f) EDX spectrum of the particles shown in (a).
Two tpyes of particles (P1 and P2) can be observed in the as-quenched state AC sample but only onpe type (P1) can be seen in IWQ sample (Fig. SB1a and c). EDX analysis (Fig. SB1e and f) show that P1 contains Si, Mn, and Fe, which is typical intermetallic particles (dispersoids) in 6xxx alloys. P2 contains mainly Mg and Si, which should be quenched-in precipitates heterogeneously grown on dispersoids. After AA for 4 h (Fig. SB1b and d), needle precipitates can be seen in the matrix. Around dispersoids there are precipitate free zones. The width of these zones is seen larger if quench is slower.
B2. PALS measurements of T2-type quenched samples during NA
Fig. SB2 shows how positron lifetimes evolve at ‘room temperature’ after slow quenches (VC and AC) from 540 °C that were interrupted at a given temperatures (100 °C to 400 °C) by a fast IWQ quench to preserve the configuration. Most curves feature an initially low value of 𝜏1c, a subsequent increase and a merger into a curve that is almost the same for all the experiments including those in Fig. 6.7a. The insets show the data used for the extrapolation to zero NA time on a linear time scale. Fig. 6.8b shows the initial and these extrapolated values.
The mechanism of NSA is not discussed in this work because of the lack of a consistent theory.
112
(a)
(b)
Fig. SB2. Positron lifetimes 𝜏1𝑐 measured during NA of samples quenched in a T2 fashion as shown in Fig. 3.3 for (a) VC, and (b) AC.
Appendix B
113 B3. Vacancy site fraction calculation
Calculating vacancy fractions from positron lifetime data usually requires decompositions of lifetime spectra into 2 or more components. In this work, we measure 𝜏1c only, which, however, is very close to the average positron lifetime that one would calculate from individual components:
𝜏1C ≈ 𝜏̅ = 𝐼0𝜏0+ 𝐼𝑣𝑎𝑐𝜏𝑣𝑎𝑐, 𝐼0+ 𝐼𝑣𝑎𝑐 = 1, (SB1) where the index 0 refers to the reduced bulk lifetime and ′𝑣𝑎𝑐′ refers to annihilation in vacancy-related defects. 𝜏𝑣𝑎𝑐= 245 ps assumed [149].
By applying an equation of the two-state trapping model [150] we obtain:
𝜏0 = 1 1
𝜏𝐵 + 𝜅𝑣𝑎𝑐 (SB2)
𝐼𝑣𝑎𝑐 = 1 𝜅𝑣𝑎𝑐 𝜏𝐵 − 1
𝜏𝑣𝑎𝑐 + 𝜅𝑣𝑎𝑐, (SB3)
where 𝜏𝐵 ≈160 ps is the lifetime in defect-free aluminium, and 𝜅𝑣𝑎𝑐 is the positron trapping rate of the vacancy-related defect. A relationship applies between 𝜅𝑣𝑎𝑐 and the site fraction of the vacancies 𝑥𝑣𝑎𝑐:
𝑥𝑣𝑎𝑐= 𝜅𝑣𝑎𝑐
𝜇 , (SB4)
where 𝜇 is the positron trapping coefficient of a vacancy in aluminium. Combining Eq.(SB1) – (SB4) eventually leads us to the site fraction of vacancies:
𝑥𝑣𝑎𝑐= 𝜏𝐵−𝜏̅
𝜇𝜏𝐵(𝜏̅−𝜏𝑣𝑎𝑐). (SB5)
Using a commonly accepted value for the specific trapping rate in aluminium, = 250 ps-1 [151], we can calculate the vacancy site fraction from the one-component positron lifetime.
Fig. SB3 shows the results, which, however, should not been taken too literally due to the approximations used (mainly the absence of clusters and usage of 𝜏1C ≈ 𝜏̅). The presence of clusters would contribute with a lifetime component around 215 ps and would require fewer vacancies to explain a given 𝜏̅.
114
Fig. SB3. Vacancy site fraction calculated using Eqs. (SB1-SB4) as a function of positron lifetime assuming validity of the positron trapping model for one trap related to vacancies characterised by a lifetime of 245 ps. The upper three arrows correspond to 𝜏1𝑐 (extrapolated to zero NSA) as measured after three quenches, the lower one marks the equilibrium vacancy fraction at 180 °C as calculated from the formation enthalpy and vibrational entropy of mono-vacancies in Al [114].
B4. Positron lifetimes of Al-Mg alloy after T1-type quenches
Fig. SB4 displays the positron lifetime evolution in a binary Al-Mg alloy after quenching. It is the analogue to the measurements in Fig. 6.7. Obviously, very little evolution of 𝜏1c is observed.
Appendix B
115 Fig. SB4. Positron lifetimes 𝜏1𝑐 during NA of binary Al-0.5Mg (wt.%) alloy after various T1-type quenches.
B5. AA hardening after NA
Alloy 6014 shows the well-known ‘negative effect’ of NA on AA as AA directly after solutionising and quenching leads to a much faster and eventually higher degree of hardening as AA after 1 week of intermediary NA, see Fig. SB5.
Fig. SB5. Hardening curves during AA directly after IWQ and after NA for 1 week.
116
Appendix C
Supplemental material to linear heating experiments
C1. Normalisation of electrical resistivity
Electrical resistivity calculated using Eq. (3.1) assumes the thickness of the sample ℎ = 0.3 mm, the width of the meander 𝑏 = 1 mm, the length of the measured part 𝐿 = 349 mm, and the cross section 𝐴 = ℎ × 𝑏. All these quantities are nominal values as manufactured by cold rolling and laser cutting, and do not account for manufacturing errors and differences in individual samples. This error can be minimised by more precise measurements of the geometries for each sample, or alternatively without measuring the geometry but assuming that the resistivity measured in Sec.4.1.2 (3911 nΩ·cm at 20 °C) applies to all samples. The resistivity at any temperature can be represented as Eq. (SC1)
𝜌 = 𝑈
𝑈20°𝐶× 3911 𝑛𝛺 · 𝑐𝑚 , (SC1) where 𝑈20°C is the voltage of the sample at 20 °C. If room temperature before start is higher than 20 °C, Eq. (SC2) was used to extrapolate the curve to 20 °C.
𝑈20°𝐶 = 𝑈0 + 𝑈′(20 − 𝑇0) , (SC2) with 𝑈0 being the voltage before heating up and 𝑇0 the corresponding temperature. 𝑈′ is the temperature coefficient of the voltage which can be obtained by linear fitting of the voltage during the second heating up to 300 °C.
Fig. SC1 shows resistivity during the second ramp obtained without (Fig. SC1a) and with (Fig. SC1b) such normalisation. It is clearly shown that by employing this method differences between various samples are minimised.
Appendix C
117 Fig. SC1. Resistivity change during the second ramp LH at various heating rates calculated (a) by Eq. (3.1) and nominal geometries of the sample, or (b) by fixing the resistivity at 20 °C and apply Eqs. (SC1, SC2).
C2. PALS lifetime offsets caused by different source corrections
Positron lifetimes during NA in AQ samples were measured many times in this work during 3 years. The lifetimes measured in Fig. 7.5 is observed to be lower than those presented before in Fig. 4.6 and Fig. 6.7. All these curves are now replotted together in Fig. SC2 for a better visuallisation. It is seen that the difference between the curves is a systematic offset. It is speculated that this offset is caused by different source corrections as examplified in Ref. [63]
or spectrometer offsets caused by variations of calibrations that are renewed regularly.
Although all source corrections were aimed to yield ~160 ps for pure Al, small differences might still occur. As all lifetime analyses in Sec. 7.1.4 were conducted using the same source correction, this does not change the relative positions of the curves on the figures, and thus also the discussions based on the lifetime evolution, just the absolute lifetime values might be slightly different to other experiments batches. Thus, comparison of lifetimes between different experiments with different source corrections should be done with care.
118
Fig. SC2. Single-component lifetimes during NA in three AQ samples appeared in this thesis. SC1 corresponds to the curve in Fig. 7.5. SC2 corresponds to the curve in Fig. 6.7a, and SC3 corresponds to the curve in Fig. 4.6.
C3. DSC measurements starting from different temperatures
In the main text, DSC curves and the integrals of samples heated from 0 °C and even lower were used to compare the properties obtained by other measurements such as resistivity and hardness.
Here we show in Fig. SC3 the DSC traces of two AQ samples started from 0 °C and 20 °C respectively. Heating rate was chosen to be 3 K/min because the time spent between 0 °C and 20 °C is the longest. It shows that the starting temeprature has very limited influence on the DSC curve, especially below 300 °C where the features are of most interest. The small differences can be caused different samples or baseline correction, which can be ignored.
Moreover, the instability of DSC device apparently has influenced the first peak in the measurement with starting temperature 20 °C.
Appendix C
119 Fig. SC3. DSC measurements of the AQ samples with different starting temperatures.
C4. Smoothing of resistivity curve for 50 K/min
The resistivity change as a function of temperature during linear heating at 50 K/min is much noisier than other two heating rates. The reason for that is the temperature control of our heating device at high heating rate is less precise than other two. In order to minimise the noise, Savitzky-Golay smoothing using 2 order polynomial function with 10 points in each bin was applied, as demonstrated in Fig. SC4. Smoothed curve has a much smaller variation and has kept the main features of the curve.
Fig. SC4. Smoothing of the resistivity change curve using Savitzky-Golay method with 2 order polynomial function and 10 points per fitting bin.
120
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