On the interaction between Ag-depleted zones surrounding γ plates and spinodal decomposition in an Al-22 at.% Ag alloy

Acta Materialia 50 (2002) 943–956 www.actamat-journals.com

On the interaction between Ag-depleted zones surrounding γ plates and spinodal decomposition in an Al-22 at.% Ag alloy K.T. Moore a,*, W.C. Johnson b, J.M. Howe b, H.I. Aaronson c, D.R. Veblen a a

Department of Earth and Planetary Sciences, Johns Hopkins University, Olin Hall, 3400 North Charles Street, Baltimore, MD 21218, USA b Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA c Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Received 10 September 2001; received in revised form 8 October 2001; accepted 30 October 2001

Abstract Decomposition of a supersaturated Al-22 at.% Ag alloy isothermally reacted at 350 °C in the α+γ two-phase field was investigated using various transmission electron microscopy techniques. The alloy was found to undergo both nucleation and growth of γ plates and spinodal decomposition. The interaction between the two mechanisms of phase decomposition caused an accumulation of Ag to develop in the matrix adjacent to the Ag-depleted zone surrounding the plates, which amplified with aging time. The matrix removed from the γ plates underwent three-dimensional, isotropic spinodal decomposition. A two-dimensional Cahn–Hilliard equation was used to model spinodal decomposition in the presence of the Ag-depleted zone adjacent to the γ plate faces. Agreement between the calculated and experimental composition profiles supports the view that the Ag-depleted zone adjacent to the plate faces catalyzes the development of a two-dimensional form of spinodal decomposition.  2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Phase transformations; Spinodal decomposition; Transmission electron microscopy (TEM); Aluminum alloys; Theory & modeling

1. Introduction The interaction between different precipitation products during phase decomposition has been * Corresponding author. Present address: Chemistry and Materials Science Directorate, L-350, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550, USA. Tel.: +1-925-422-9741; fax: +1-925-422-6892. E-mail address: [email protected] (K.T. Moore).

observed in several systems [1–3]. For example, grain boundary and intragranular precipitates, discontinuous (cellular) precipitates, ordered structures, and spinodal decomposition have been observed in Cu–Ni–Sn alloys with different compositions [4–7]. At times, two or more of these decomposition processes have been observed to occur simultaneously, and complex microstructures often result due to the interactions between them.

1359-6454/02/$22.00  2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 3 9 4 - 9

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Spinodal decomposition and intragranular precipitation are two well-known mechanisms of phase decomposition. Independently, each has been examined in great detail both experimentally [8–13] and theoretically [14–19]. However, there has been little experimental or theoretical examination of the interactions between the two mechanisms. To investigate such an interaction, an Al22 at.% Ag alloy was chosen. There is a spinodal region in the Al–Ag phase diagram with a critical composition Cc at approximately 20 at.% Ag and a critical temperature Tc at approximately 465 °C [20]. The critical composition lies within the α+γ two-phase field for temperatures below approximately 550 °C. Therefore, the alloy can be annealed in the α single-phase field and then quenched to a temperature where both spinodal decomposition and nucleation and growth of γ plates are expected to occur. The growth behavior, Ag-depleted zone, and composition of γ plate-shaped precipitates have been investigated by many researchers [21–37]. An Al-rich, supersaturated α(fcc) Al–Ag alloy can be aged in the α+γ two-phase field to produce Ag-rich γ(hcp) precipitate plates in the α matrix (Fig. 1

Fig. 1. The Al–Ag phase diagram, showing the average alloy composition, solution annealing and aging temperatures, and the equilibrium compositions for the α and γ phases at 350 °C [38]. and α composition C350 at The equilibrium γ composition C350 γ α 350 °C are approximately 60 and 2 at.% Ag, respectively.

[38]). The precipitate-matrix orientation relationship is: (0001)γ//(111)α and [112¯ 0]γ//[11¯ 0]α. The Al–Ag phase diagram (Fig. 1) shows an equilibof 60 at.% Ag and an α rium γ composition C350 γ composition C350 α of 2 at.% Ag at 350 °C, the aging temperature used in this study [21–25]. This alloy serves as a model system with which to explore the interaction between γ plates and spinodal decomposition for several reasons. First, the growth mechanisms of γ precipitates have been thoroughly investigated using static and in situ high-resolution TEM (HRTEM) and are understood at the atomic level for all of the precipitate interfaces [23,30–34]. Second, the large difference in atomic number between Al (13) and Ag (47) yields strong compositional contrast in conventional and HRTEM imaging. Third, spinodal decomposition can be assumed to be nearly isotropic, since the matrix is Al-rich (Al is nearly elastically isotropic) and there is only a small dependence of the lattice parameter on Ag concentration [39,40]. The purpose of the current study was to examine the interaction of γ plates and their associated Agdepleted zone with spinodal decomposition of the supersaturated α matrix in an Al-22 at.% Ag alloy.1 It is shown that a planar accumulation of Ag occurs adjacent to the Ag-depleted zone at the faces of γ plates due to the interaction between the Ag-depleted zone and the supersaturated matrix undergoing spinodal decomposition. This accumulation of Ag adjacent to the Ag-depleted zone increases in Ag concentration with increasing aging time. Additionally, at later aging times subsidiary accumulations of both Ag and Al appear further into the matrix adjacent to the original Ag accumulation. A two-dimensional Cahn–Hilliard equation is used to model spinodal decomposition of the supersaturated α matrix in the presence of a Ag-depleted zone adjacent to the γ plates, and comparisons are made between the calculated

1

The alloy composition and aging temperature used in this study also produced a discontinuous (cellular) reaction, in agreement with the results of Predel and Gust [40]. This reaction is not discussed in the current paper, but will be addressed in subsequent work.

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solute profiles and experimental TEM images and solute profiles.

2. Sample preparation and analyses An ingot of high-purity Al-22 at.% Ag alloy was homogenized at 560 °C for 24 h and then cold rolled to 50 µm thickness. Sections of the rolled sheet were solution annealed for 1 h at 560 °C and immediately transferred to a salt bath, where they were isothermally reacted for 2, 5, 10, 15, or 20 s at 350 °C to form γ precipitates, and then coldwater quenched. One sample, which was solution annealed at 560 °C and then directly cold-water quenched, was examined by optical, scanning and transmission electron microscopy and displayed no evidence of either precipitation or spinodal decomposition. Disks 3 mm in diameter were punched from the heat-treated sheets and electropolished in a twinjet Struers Tenupol-3 apparatus using a 25% nitric acid-75% methanol solution at ⫺40 °C, 10 V, and 25 mA. Foils that were examined analytically were not ion-beam milled, since preferential milling may occur and affect analytical results [41]. This sometimes left back-deposited Ag from electropolishing on the foil surface, however, this was deemed less detrimental to analytical results than having a sample with non-uniform thickness due to preferential milling. Conventional TEM was performed using both a Philips CM300 FEG TEM operating at 300 kV and a Philips 420 ST operating at 120 kV. Electron energy-loss spectroscopy and imaging experiments were performed using the CM300 FEG TEM operating at 297 kV (with the energy filter enabled) and employing a Gatan Imaging Filter (GIF). Unfiltered and zero-loss images were acquired digitally at microscope magnifications of 17.5 to 105 kX, using a 1 s exposure and 2X binning (512×512 pixels) of the charge-coupled-device (CCD) camera. Energy-filtered TEM (EFTEM) images acquired near the Al L2,3 and Ag M4,5 ionization edges were captured at the same magnifications using 2X binning and a 5 eV energy window. Jump-ratio images and elemental maps of the Ag M4,5 edge were acquired using a 5 s exposure

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with windows centered at 297, 365, and 440 eV for the pre-edge 1, pre-edge 2, and post-edge images, respectively. Jump-ratio images and elemental maps of the Al L2,3 edge were acquired using a 2 s exposure with windows centered at 66, 71, and 95 eV for the pre-edge 1, pre-edge 2, and postedge images, respectively. EFTEM images were acquired using a 3 mm GIF entrance aperture and an objective aperture with a collection angle of 10 mrads. EFTEM images of γ plates were acquired with the plate edge-on to the electron beam, but not directly on a zone axis in order to avoid artifacts in the compositional images due to elastic scattering [42–44]. Finally, DigitalMicrograph software was used to obtain, correlate, and divide pre-edge and post-edge images to produce jump-ratio images and elemental maps, and to measure intensities within the images. Energy-dispersive X-ray spectroscopy (EDS) was performed using the CM300 FEG TEM operating in the scanning TEM (STEM) mode with an Oxford ultra-thin window detector and an XP3 pulse processor. Spectra were acquired using ES Vision software (v. 3.1) and an EMiSPEC analytical system. Two-dimensional EDS maps were formed using multiple spectra captured as the electron beam rastered across the sample. A 1.4 nm probe was used with a grid spacing of 1.5 nm between the probe locations. A 1 s dwell time was employed for each sampling point within the EDS maps. All spectra were acquired with the sample tilted approximately 10 degrees towards the detector. Elemental concentrations were calculated from the EDS data using the Al Kα and Ag Lαβ peaks, and a kAlAg factor that was determined from an Al22 at.% Ag thin-foil standard. The kAlAg factor for the microscope conditions in this investigation was 0.69, as determined from a spectrum containing over one million counts.

3. Results and discussion 3.1. Transmission electron microscope experiments The atomic-number contrast in bright-field TEM images was sufficient to reveal the Ag-depleted

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zone surrounding the γ plates, as well as the compositional fluctuations associated with spinodal decomposition in the remaining α matrix. A brightfield TEM image of a number of edge-on γ plates contained in a sample reacted for 2 s is shown in Fig. 2a. The Ag-depleted zone associated with each plate is readily visible as a bright (high-intensity) area surrounding each plate. Additionally, the supersaturated α matrix has a mottled appearance due to the onset of spinodal decomposition. (The

Fig. 2. (a) Bright-field TEM image of several γ plates near a [112¯ 0]γ//[11¯ 0]α orientation, contained in a sample reacted for 2 s. The Ag-depleted zone associated with each plate is readily visible due to the atomic-number contrast in the image. (b) An enlargement of the area at a plate edge enclosed in (a).

large dark blotches in the image are back-deposited Ag from electropolishing). A higher-magnification image of the plate edge enclosed in Fig. 2a is shown in Fig. 2b. The morphology of the Agdepleted zone around the plate is similar to those observed around γ plates in dilute Al–Ag alloys by Sagoe-Crensil and Brown [36] and Moore and Howe [37], although the width of the Ag-depleted zone adjacent to the plate edge is narrower than in the more dilute alloys, as expected on the grounds of mass conservation. Ag was observed to accumulate in the α matrix adjacent to the Ag-depleted zone parallel to the faces of the γ plates as spinodal decomposition occurred, as evidenced by dark contrast that appeared in bright-field TEM images. EDS was used to confirm and quantify this feature. A brightfield TEM image of several γ plates in an edge-on orientation contained in a sample reacted for 5 s is shown in Fig. 3a. Ag Lαβ and Al Kα EDS maps

Fig. 3. (a) Bright-field TEM image of several γ plates in an edge-on orientation, contained in a sample reacted for 5 s. (b) Ag Lαβ and (c) Al Kα STEM EDS maps taken from the area enclosed in (a). A line profile accompanies each EDS map measured from A to A⬘. Notice the accumulation of Ag adjacent to the Ag-depleted zone at the plate faces, indicating the presence of up-hill diffusion at this location.

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taken from the area enclosed in Fig. 3a are shown in Fig. 3b. A collapsed line profile was taken from each EDS map from A to A⬘, and these are shown below the EDS maps. In the Ag Lαβ EDS map, the γ plates appear bright because they are Ag-rich and the Ag-depleted zone appears dark because it is Ag-poor. The supersaturated α matrix outside this region is mottled due to the onset of spinodal decomposition. The accumulation of Ag adjacent to the Ag-depleted zone can be seen in the line profile for the Ag map, and is indicated by an arrow. The area of the line profile corresponding to the spinodally decomposing matrix (α⬘) is approximately flat because the profile was averaged over the width of the box from A to A⬘. This averaging reveals the accumulation of Ag adjacent to the Ag-depleted zone of the γ plate, but it averages out the non-directional compositional modulations associated with the present approximately isotropic spinodal decomposition process. The complementary Al Kα EDS map shows the opposite effect, i.e., there is an Al depletion in the matrix adjacent to the Ag-depleted zone, as indicated by an arrow in the Al EDS profile. Quantification of the Ag accumulation adjacent to the Agdepleted zone in the samples reacted for 5 s showed that this region contained approximately 27 at.% Ag, or 5 at.% more Ag than the average matrix composition of 22 at.% Ag. Energy-filtered TEM (EFTEM) imaging was also used to examine the Ag accumulation adjacent to the Ag-depleted zones around γ plates. An Ag M4,5 jump-ratio image and an elemental map are shown in Figs 4a and b, respectively, and a complementary Al L2,3 jump-ratio image and elemental map are shown in Figs 4c and d, respectively. The compositional images shown in Fig. 4 were taken from the same area as shown in Fig. 3a. Both jump-ratio images and elemental maps are shown for Ag and Al, since the two different types of compositional images are complementary. Each image is accompanied by a line profile measured from A to A⬘. The line profile for the Ag M4,5 jump-ratio image in Fig. 4a displays intensity peaks in the matrix immediately adjacent to each Ag-depleted zone, indicating an accumulation of Ag. These peaks are indicated by arrows in the line profile. The Ag M4,5 elemental map in Fig. 4b

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Fig. 4. A set of EFTEM images taken from the area shown in Fig. 3a: (a) Ag M4,5 jump-ratio image, (b) Ag M4,5 elemental map, (c) Al L2,3 jump-ratio image, and (d) Al L2,3 elemental map. Each image is accompanied by a line (intensity) profile taken from A to A⬘. Notice the accumulation of Ag (depletion of Al) adjacent to the Ag-depleted zone indicated by an arrow, as observed in the STEM EDS maps.

yields the same result as the Ag M4,5 jump-ratio image in Fig. 4a. The Al L2,3 jump-ratio image and elemental map in Figs 4c and d, respectively, complement the data for Ag, showing that an Al depletion exists where the Ag accumulation occurs, immediately adjacent to the Ag-depleted zone surrounding the γ plates. TEM examination of the α matrix between existing γ plates shows that spinodal decomposition had occurred by 2 s aging at 350 °C. The spinodal fluctuations were isotropic and had a compositional wavelength (λ) of approximately 30 to

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40 nm. A bright-field TEM image of an area of matrix undergoing spinodal decomposition is shown in Fig. 5a. A corresponding Ag M4,5 EFTEM elemental map is shown in Fig. 5b, along with an intensity profile taken from A to A⬘. A number of areas that appear bright in Fig. 5a are dark in the Ag M4,5 elemental map in Fig. 5b, indicating that these regions are Al-rich. Several of these Al-rich regions are indicated by arrowheads

Fig. 5. (a) Bright-field TEM image and (b) Ag M4,5 elemental map of the α matrix undergoing spinodal decomposition. Arrowheads indicate Al rich areas, which are high in intensity in the bright-field TEM image and low in intensity in the Ag elemental map. A line profile was taken from the elemental map in (b) from A to A⬘. A FFT of the matrix is shown in (c). FFTs taken along the [001] and [011] directions showed that the spinodal product was isotropic and that the average compositional wavelength was between 30 to 40 nm.

in the figures. The corresponding intensity profile A–A⬘ shows that λ is fairly uniform and approximately 37 nm in this area. However, this measurement reflects the behavior in only one direction and in only a small portion of the image. In order to make a more general quantification of the spinodal product, fast Fourier transforms (FFTs) were taken over large areas of bright-field TEM images. Fig. 5c shows such an FFT taken in a [011]α matrix orientation. The transform displays a ring of uniform intensity, demonstrating that the decomposition is isotropic, and has a wavelength l苲40 nm. (The vertical and horizontal lines in Fig. 5c are edge effects associated with the FFT and the line running diagonally from the lower-left to the upper-right is a shape effect from the faces of a γ plate; these features may be neglected.) FFTs were also taken in a [001] matrix orientation and these similarly showed that the spinodal product was isotropic with l苲30–40 nm. Therefore, l=35±5 nm for all of aging times examined and in both [001]α and [011]α orientations. After isothermally reacting the alloy at 350 °C for 20 s, the matrix was almost completely consumed by γ plates. A number of plates contained in a sample aged for 20 s are shown near a [112¯ 0]γ//[11¯ 0]α orientation in the bright-field TEM image in Fig. 6a. The γ plates are sufficiently dense that the γ and α phases appear to alternate in a lamella-like structure. The two plate edges enclosed in boxes in Fig. 6a are shown enlarged in Figs 6b and c. Fig. 6c was acquired after the sample was tilted so that the plate edge was parallel to the electron beam. The plate edge was judged to be parallel to the beam due to the sharp corners and abrupt change in contrast across the edge. The resulting shape of the Ag-depleted zone in Fig. 6c is quite similar to that seen in Fig. 2b, indicating that the Ag-depleted zone has a nearly constant (i.e., steady-state) shape for the aging times examined. It was observed that the Ag accumulation immediately adjacent to the Ag-depleted regions parallel to the γ plate faces amplified with increasing aging time, and that similar, smaller accumulations appeared parallel to these, further into the matrix. The increase in Ag accumulation adjacent to the Ag-depleted zone can be seen by comparing

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Fig. 7. (a) Bright-field TEM image and (b) Ag M4,5 jump-ratio image of several γ plates in an edge-on orientation, contained in a sample reacted for 20 s. A line intensity profile taken from A to A⬘ accompanies the elemental map. Notice that there is a second, smaller Ag accumulation separated from the original Ag accumulation by a slight accumulation of Al. These features demonstrate that the amplitude and number of Ag accumulations increase with time, creating an oscillation in composition in the direction normal the plate faces. A second line profile was taken from B to B⬘ in (b) and it shows that little or no Ag has accumulated adjacent to the Ag-depleted zone at the plate edge.

Fig. 6. (a) Bright-field TEM image of a sample reacted for 20 s. (b) and (c) are enlargements of the regions shown enclosed in (a). The image in (b) was acquired in the same orientation as (a), but (c) was acquired after the sample was tilted so that the edge of the plate was parallel to the electron beam.

the images of the 5 s sample in Figs 2, 3, and 4 with those of the 20 s sample in Figs 6a–c. Energyfiltered TEM imaging was used to better quantify this observation. A bright-field TEM image and a Ag M4,5 jump-ratio image of a sample reacted for 20 s are shown in Figs 7a and b, respectively. Two line profiles taken from A to A⬘ and B to B⬘ in the jump-ratio image in Fig. 7b are shown at the bottom of Fig. 7. In the line profile from A to A⬘, the γ plates appear as high-intensity peaks because

they are Ag-rich,2 and the Ag-depleted zones in the α matrix adjacent to the plates appear as lowintensity troughs. Three arrows in the line profile A–A⬘ indicate two intensity peaks and a trough in the matrix immediately adjacent to the Ag-depleted zone. In contrast to the single maximum evident in the line profiles for the 5 s sample in Figs. 3 and 4, there are now two maxima adjacent to the Agdepleted zone in the 20 s sample. There is a minimum between the two Ag maxima, indicating an accumulation of Al in this region. Little, if any, Ag accumulation is observed adjacent to the Ag-depleted zones at the edges of the γ plates. For example, the line profile from B The intensity of the γ plates in the A to A⬘ line profile is slightly different due to a thickness effect. This area of the sample is wedge shaped, unlike the area examined in Fig. 3 Fig. 4, and the γ plates in the thinner part of the sample are brighter due to less plural scattering. This does not affect the data of interest. 2

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to B⬘ in Fig. 7b was taken across the leading edge of a γ plate, through the Ag-depleted zone, and into the surrounding supersaturated matrix (α⬘). It shows that there is almost no accumulation of Ag adjacent to the Ag-depleted zone at the plate edge. This is probably due to the fact that the plate edges grow much faster than the faces [28–37], and there is little time for Ag accumulation to occur before the plate edge sweeps through. In other words, the plate edge incorporates Ag too fast to allow an accumulation of Ag to occur. Since line profiles taken across the plate edges were averaged over a smaller area than at the faces due to the limited width of the plates, the amount of noise in the profiles was greater. However, visual inspection of bright-field TEM images, jump-ratio images, and elemental maps in conjunction with the line profiles showed that little or no accumulation of Ag occurs adjacent to the Ag-depleted zone at the edges of γ plates. The previous results focused on the Ag accumulation adjacent to isolated γ plates with increasing aging time. Additional interesting behavior was found to occur when two γ plates were sufficiently close that the Ag-depleted zones associated with the plate faces influenced spinodal decomposition across the entire α⬘ matrix region between them. Edge-on γ plates separated by three different distances are shown in Fig. 8. Fig. 8a is a bright-field TEM image, and Fig. 8b is the same image with the contrast inverted, i.e., it is like a Ag composition map, assuming that the brightness is proportional to the Ag concentration. A line profile taken from A to A⬘ in Fig. 8b is shown below the figure. Three areas of the line profile are labeled 1, 2, and 3, and these correspond to the supersaturated α⬘ matrix. Each area displays a different distance of separation between two γ plates. Area 1 shows two plates that are close enough to almost have their Ag-depleted zones touch. There is a Agrich accumulation immediately adjacent to each Ag-depleted zone with one Al-rich area in between, so that the line intensity profile has two well-defined maxima and one well-defined minimum in the α⬘ matrix. In this narrow area, the Agdepleted zones at the plate faces have caused the spinodal product to form a striped pattern between them. Area 2 has a slightly larger separation with

Fig. 8. (a) Bright-field TEM image taken from a sample reacted for 20 s and (b) the same image with the contrast inverted. An intensity profile was taken across the inverted image in (b) from A to A⬘. Notice the magnitude and periodicity of the Ag accumulations adjacent to each Ag-depleted zone as a function of the γ plate separation. Also note the small depletion of Ag at the edge of the Ag-depleted zone adjacent to the largest Ag accumulation (arrows).

one well-defined maximum (i.e., Ag-rich region) adjacent to each Ag-depleted zone, and two smaller and less well-defined maxima between these. There are minima corresponding to accumulations of Al between each Ag maximum. In this area, the spinodal product is patterning with the Ag-depleted zones as two-dimensional spinodal decomposition. The patterning is less well-defined near the center, where decomposition is three-dimensional. In area 3, the plates are far enough apart that the line pro-

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file in between the maximum and minimum associated with each Ag-depleted zone becomes fairly uniform, as the contrast is averaged over the isotropic spinodal region, similar to the effect discussed previously with reference to Fig. 3. Thus, stripes of Ag and Al form parallel to the initial Agdepleted zones, but these give way to the isotropic spinodal product in the central region with increasing separation between γ plates. In addition to the Ag accumulation observed immediately adjacent to the Ag-depleted zones at γ plate faces, a slight depletion of Ag just inside the edge of the Ag-depleted zone was observed in both bright-field and energy-filtered TEM images of samples reacted for 20 s. This effect is seen in the inverted image and line profile shown in Fig. 8. A small dip in intensity is marked with an arrow on each side of area 1 in the line profile. There is also a slight dip in intensity on each side of areas 2 and 3 in the line profile. Since Ag is such a strong scatterer of electrons compared to Al, the darkness in bright-field TEM images of Al–Ag alloys can be assumed to vary proportionally to the Ag concentration. The image and line profile, therefore, suggest that Ag is diffusing from the edge of the Ag-depleted zone into the initial Ag accumulation, thereby leaving a slight dip in Ag concentration at the edge of the Ag-depleted zone. Energy-filtered TEM imaging also revealed a slight depletion of Ag just inside the Ag-depleted zone. Fig. 9 shows a Ag M4,5 jump-ratio image with a line profile taken from A to A⬘. The large arrow in the line profile indicates a slight dip in intensity at the edge of the Ag-depleted zone adjacent to the Ag accumulation. This provides additional evidence that some Ag is diffusing from the edge of the Agdepleted zone into the adjacent Ag accumulation at the plate faces. 3.2. Finite difference calculations In order to independently ascertain whether or not the Ag-depleted zones at the γ plate faces are responsible for the Ag accumulation observed immediately adjacent to these regions during spinodal decomposition, we use a two-dimensional Cahn–Hilliard equation [16] to model phase decomposition of the fcc α matrix in the presence

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Fig. 9. EFTEM Ag M4,5 jump-ratio image and intensity profile taken from A to A⬘. Notice the slight depletion of Ag at the edge of the Ag-depleted zone (indicated by an arrow), adjacent to the Ag accumulation.

of a Ag-depleted zone. Compositional strains are neglected owing to the small dependence of the lattice parameter on Ag composition [39]. Under these conditions, the composition field is governed by:

冉

冊

∂fvo ∂C ⫽ Mⵜ2 ⫺k ⵜ2C ∂t ∂C v

(1)

where C is the mole fraction of Ag, M is the mobility, assumed here to be constant, t is the time, kv is the gradient energy coefficient, and fvo is the temperature and composition-dependent Helmholtz free energy density. The free energy density fvo was represented as a quartic equation of the form:

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冋

(T⫺Tc) (C⫺Cc)2 Tc (Tc⫺T1)(C⫺Cc)4 ⫹ 2Tc(C1⫺Cc)2

fvo ⫽ W

册

(2)

where T is the absolute temperature, Cc and Tc are the composition and temperature of the critical point, respectively, C1 and T1 are the composition and temperature of a point on the solidus of the metastable miscibility gap, respectively, and W is an energy density defining the depth of the energy wells. Based on the Al–Ag phase diagram data of Williams and Easton [20], we took Tc=738 K, Cc=0.2, T1=673 K, and C1=0.3. These values yield a symmetric miscibility gap about the critical composition, in good agreement with reported phase diagrams [20,38]. The assumption of a symmetric miscibility gap for the aging temperature used in this study (350 °C) is justified, as the miscibility gap does not become highly asymmetric until temperatures below approximately 200 °C [38,45]. The calculations were performed using a nondimensional position (z1,z2) and time (t) where: z1 ⫽ and: t⫽

冉 冊 W 2kv

dition assumes that there is no mass flow from the matrix to the γ plate; i.e., that the normal flux along the α/γ interface is zero. This assumption is justified because, at the early stages of growth examined in this study, the γ plates exhibited almost no growth ledges on their faces. The initial conditions employed in the calculations were based on the compositional data measured by EDS for the earliest obtainable aging times. In these simulations, the Ag concentration at the α/γ interface was set initially to 0.11 (atomic fraction) and was then assumed to increase parabolically to the average matrix value (0.22) at a scaled distance of 20 from the α/γ interface. A small, random compositional fluctuation was then imposed on this profile. Figs. 10 and 11 each contain four sequential,

1/2

xi

(3)

冉 冊 4kvt MW2

(4)

and xi is the position coordinate. Using the scaled variables, Eq. (1) becomes:

冉

冊

∂F ∂C ⫽ ⵜ2 ⫺ⵜ2C ∂t ∂C

(5)

where F ⫽ 2fvo / W. Eq. (5) was solved using a mass-conserving, explicit, finite-difference technique. Periodic boundary conditions were imposed along the z2 (vertical) direction. Two boundary conditions were imposed along each broad face of the γ plates. The first boundary condition results from an assumption that the interfacial energy between the supersaturated α matrix and the γ plate is independent of the local composition. This requires that the normal derivative of the composition vanish everywhere along the α/γ interface [46]. The second boundary con-

Fig. 10. (a–d) Four sequential, simulated microstructures obtained during the early stages of spinodal decomposition in an alloy with two γ plates and their associated Ag-depleted zones spaced far apart. The plates are sufficiently far apart that the Ag accumulations (bright stripes) adjacent to each Agdepleted zone do not interact. (e) and (f) show two compositional profiles averaged over the simulated images in (c) and (d), respectively.

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Fig. 11. (a–d) Four sequential, simulated microstructures obtained during the early stages of spinodal decomposition in an alloy with plates spaced sufficiently close that the Agdepleted zones cause Ag accumulations (stripes) to form across the entire matrix region. (e) and (f) show two compositional profiles averaged over the simulated images in (c) and (d), respectively.

simulated microstructures obtained during the early stages of the decomposition process. The conditions employed in each case are the same except that the scaled distance between the γ plates is 190 in Fig. 10 and 130 in Fig. 11, with a periodicity of 200 in the vertical direction. An average Ag concentration profile obtained by integrating over the z2 coordinate of the composition fields depicted in Figs 10 and 11c and d are shown in Figs 10 and 11e and f, respectively. As in the previous experimental Ag maps, the brighter regions are enriched in Ag while the darker regions are depleted in Ag. Thus, the dark regions on both sides of each image (z1⬍0 and z1⬎190 for Fig. 10,

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or z1⬍0 and z1⬎130 for Fig. 11) are the Agdepleted zones associated with γ plates, which are not shown in Fig. 10 or Fig. 11. In both systems, spinodal decomposition was initiated with an enhancement of Ag at the outer boundary of the Ag-depleted zone and a depletion of Ag adjacent to this, a short distance into the central region. This initial accumulation of Ag continued to amplify and new, smaller accumulations appeared further into the matrix, as exemplified by the bright and gray vertical bands seen in both Figs. 10 and 11. The pattern of microstructural evolution that resulted when the Ag-depleted zones were placed relatively far apart is shown in Fig. 10. In this case, spinodal decomposition is strongly affected by the presence of the Ag-depleted zones at the sides, but the decomposition is initiated throughout the central region before the edge-modulated effects are able to reach the interior. Two distinct, vertical stripes of Ag are evident adjacent to each of the Ag-depleted zones in Fig. 10b, but no additional stripes form as the isotropic spinodal structure develops in the central region. The integrated composition profiles shown in Fig. 10e and f indicate the development of a second and small third peak (stripe) of Ag enrichment in the interior region, but this peak never develops fully. The simulated microstructures obtained by placing the Ag-depleted zones relatively close together are shown in Fig. 11. In addition to the two large accumulations of Ag that develop adjacent to the Ag-depleted zone, two subsidiary stripes (peaks) of Ag enrichment develop in the interior region with time, as evident in Figs 11c and d. In this case, the Ag-depleted zones are close enough together that a structure similar to area 2 in Fig. 8 is observed. Several points pertaining to the calculations should be noted. First, the simulations show that spinodal decomposition is strongly affected by the presence of the Ag-depleted zones. An accumulation of Ag occurs at the edge of the Ag-depleted zone, and this effect propagates into the central region. If the central region is sufficiently narrow, the resulting microstructure consists of two to four parallel layers of Ag-rich fcc phase. For wider plate separations, spinodal decomposition initiates in the central region before the edge-modulated

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decomposition product reaches the interior. The Ag-rich layers form parallel to the γ plates in the absence of all elastic effects. Second, the simulated microstructure is sensitive to the assumed initial composition profile. All experimental evidence (bright-field TEM images, EDS and EFTEM composition maps) indicates that the Ag gradient across the Ag-depleted zone is not linear or parabolic. The magnitude of the Ag accumulation was observed to depend on the slope and initial Ag concentration at the edge of the Ag-depleted zone in the calculations. Third, although quantitative details of the evolution were sensitive to the initial conditions, the qualitative behavior described above was not. As a final comparison, Fig. 12 shows the calculated image in Fig. 10b superimposed on an experi-

Fig. 12. Comparison between an inverted bright-field TEM image (background) from a sample reacted for 20 s with a simulated image (inset) from Fig. 10b.

mental bright-field TEM image of a sample reacted for 20 s, with the contrast inverted so that the brightness is proportional to the Ag concentration, as in Fig. 8. The two γ plates, which run vertically on each side of the image, are sufficiently far apart that the accumulations of Ag adjacent to each Agdepleted zone are clearly visible, but do not interact. Therefore, the isotropic spinodal structure develops in the central region, as evident in both the experimental and simulated images. In summary, both the experimental observations and the finite difference calculations indicate that during aging, Ag diffuses not only from the supersaturated matrix, but also from the edge of the Agdepleted zone, and an accumulation of Ag occurs in the region immediately adjacent to the edge of the Ag-depleted zone. In this case, the Ag-depleted zone associated with γ plate faces acts like a preexisting compositional modulation that lies within the spinodal and has a lower Ag concentration than the supersaturated α matrix. With time, the Ag accumulation continues to increase, and new, but smaller accumulations of both Ag and Al are produced further into the matrix, parallel to the initial accumulation. (It is worth noting that the pattern of compositional modulation induced in the decomposing spinodal product by the Ag-depleted zones is similar to the pattern of order that is induced in liquid layers by confining surfaces without order [47].) As aging proceeds, the initial accumulation of Ag adjacent to the Ag-depleted zone removes enough Ag from the edge of the Agdepleted zone to produce a small dip. This was observed experimentally in Figs. 8 and 9 and seen in the calculated image in Fig. 10d, for example. A precursor phase can markedly affect the nucleation of subsequent phases [48]. In the case of precipitation, Cahn [49] has used schematic freeenergy composition diagrams to show that a prior precipitation process lowers the volume freeenergy change for the formation of a subsequent precipitate, when the composition differences with respect to the matrix are in the same direction for both precipitates. A similar schematic free-energy composition diagram appropriate to the Al–Ag system is shown in Fig. 13. In our case, the initial alloy composition lies at Co. The system spinodally decomposes into increasingly Ag-rich and Ag-poor

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maximum. This behavior is apparent in Fig. 2a, where the three γ plates indicated by arrows are aligned just inside the Ag accumulations, in the region where the composition gradient leading into the Ag-depleted zones is steep. This behavior was commonly observed and results in alternating α and γ phases into the lamella-like structure shown previously in Fig. 6.

4. Conclusions Fig. 13. A set of free energy curves explaining why new γ plates were observed experimentally to nucleate and grow just within the Ag-depleted region of pre-existing gamma plates, near the Ag accumulation.

The interaction between the Ag-depleted zone associated with the formation of γ plates and spinodal decomposition in an Al-22 at.% Ag alloy isothermally reacted at 350 °C exhibited the following characteristics:

compositions, as indicated by the arrows above the free-energy composition curve for the α phase. The driving force for subsequent nucleation of the γ phase is the vertical difference (⌬f) between the tangent to the free-energy composition curve at a particular (evolving) alloy composition, say at location a, and the tangent to the free-energy composition curve for the γ phase (taken to be narrow and lie beyond the Ag-rich spinodal composition), as at b–c for example. As the α matrix decomposes, the driving force increases to the left of Co, reaching its maximum value at the inflection point d, where the second derivative of the free-energy with respect to composition equals zero. The driving force for nucleation then decreases until spinodal decomposition is complete, as indicated by tangent e–f. The significance of this construction is that it indicates an increase in driving force for nucleation of the γ phase at compositions that lie between the initial alloy composition Co and the equilibrium matrix composition at e. Such a composition occurs adjacent to the Ag accumulation just within the edge of the Ag-depleted zones. This is particularly evident in the composition profiles shown in Figs 3b and 12e, for example. The observation that γ plates were often observed to nucleate at the edge of the Ag-depleted zones, adjacent to larger pre-existing γ plates, is broadly consistent with nucleation in or near the partially spinodally decomposed region where the driving force is a

1. An accumulation of Ag occurred adjacent to Ag-depleted zones at the faces of γ plates, due to the interaction between the Ag-depleted zones and spinodal decomposition in the surrounding matrix. 2. The Ag accumulation adjacent to the Agdepleted zone, and parallel to the plate faces, increased with aging time. Additional, subsidiary Ag and Al accumulations with the same orientation formed adjacent to the initial Ag accumulation as aging progressed. 3. Although Ag was observed to accumulate adjacent to the Ag-depleted zones at the plate faces, there was little or no accumulation of Ag at the plate edges. This is probably due to the fact that the rapidly growing plate edges had a fast-moving and narrow concentration profile ahead of them, which did not provide sufficient time for even the first Ag accumulations to develop. 4. At later aging times (e.g., 20 s) a small amount of Ag diffused from the edge of the Ag-depleted zone into the initial accumulation of Ag adjacent to the zone. This left a slight depletion of Ag at the edge of the Ag-depleted zone, which was apparent in bright-field and energy-filtered TEM images. 5. Finite difference calculations performed using a two-dimensional Cahn–Hilliard equation [16] and thermodynamic data appropriate to the Al– Ag system successfully reproduced the exper-

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imentally observed decomposition behavior, including the features described in (1), (2) and (4) above. The calculations confirm that the accumulation of Ag adjacent to the Ag-depleted zone is due to the initial modulation imposed on the supersaturated matrix by the Ag-depleted zone. 6. The nucleation and growth of new γ plates near the Ag accumulation at the edge of the Agdepleted region of older plates is qualitatively explained using free energy curves. This shows that the planar Ag-rich accumulations due to spinodal decomposition markedly affect the nucleation of subsequent plates in this system. Acknowledgements The authors are grateful to the National Science Foundation for support under grants EAR-0073955 (KTM and DRV) and DMR-9902110 (WCJ and JMH) and to the W.M. Keck Foundation. The authors also thank Prof. G. Kostorz for kindly providing the alloy used in this study and for several helpful suggestions on the work. References [1] Dutkiewicz J. Metals Tech 1978;5:333. [2] Busch R, Ga¨ rtner F, Borchers C, Haasen P, Bormann R. Acta Metall Mater 1995;43:3467. [3] Zhao JC, Notis MR. Acta Mater 1998;46:4203. [4] Leo W. Metall 1967;21:908. [5] Kato M, Schwartz LH. Mater Sci Eng 1979;41:137. [6] Ditchek B, Schwartz LH. Acta Metall 1980;28:807. [7] Kratochvil P, Mandula M, Mencl J, Pesicka J, Smola B. Physica Status Solidi A 1987;104:579. [8] Daniel V, Lipson H. Proc R Soc Lond 1943;181:368. [9] Daniel V, Lipson H. Proc R Soc Lond 1944;182:378. [10] Bradley A. J Physica 1949;15:175. [11] Rundman KB, Hilliard JE. Acta Metall 1967;15:1025. [12] Batz DL, Tanzilli RA, Heckel RW. Metall Trans 1970;1:1651. [13] Batte AD, Honeycombe RWK. J Iron Steel Inst 1973;211:284.

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