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The morphological evolution of equiaxed dendritic microstructures during coarsening
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Acta Materialia 57 (2009) 2418–2428 www.elsevier.com/locate/actamat
The morphological evolution of equiaxed dendritic microstructures during coarsening J.L. Fife *, P.W. Voorhees Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108, USA Received 21 September 2008; received in revised form 26 January 2009; accepted 27 January 2009 Available online 11 March 2009
Abstract The morphological evolution of equiaxed Al–20 wt:%Cu dendritic microstructures was studied in three dimensions. The microstructure evolved into a highly interconnected structure, where the inverse specific surface area scaled linearly with the cube root of time. As the size scale of the microstructure increased during coarsening, the scaled morphology of the interfaces changed only slightly. The distribution of interface normals indicated that the microstructure was approximately isotropic. These results are in contrast to those found using a directionally solidified Al–Cu alloy of a similar solid volume fraction, where the structure evolved into solid cylinders parallel to the growth direction used to create the sample prior to coarsening. Thus, we find that the initial morphology of a dendritic structure can have a major impact on its morphological evolution. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Equiaxed dendritic microstructure; Aluminum–copper; Coarsening; Interfacial curvature
1. Introduction Solidification of metallic alloys typically occurs by the formation of dendrites as the primary growth morphology. Dendrites develop because of a morphological instability of the solid–liquid interface during solidification and are technologically important because their growth is directly related to the properties of a material. Dendritic growth frequently results from a mushy zone, a two-phase region where the solid phase has a dendritic structure and the liquid phase is the surrounding matrix. When this mixture is held for any length of time, the dendrites undergo coarsening or Ostwald ripening. During coarsening, the morphology and the length scale of the microstructure can change considerably. This change in length scale alters the distribution of chemical components in the solid, and thus coarsening can affect many properties of a material. *
Corresponding author. Tel.: +1 847 467 7089. E-mail addresses: jfi[email protected] (J.L. Fife), p-voorhees@ northwestern.edu (P.W. Voorhees).
The coarsening process is driven by the minimization of the interfacial free energy of the system. In alloys, the coarsening process proceeds by the diffusion of mass from one region of the interface to another. As described by the Gibbs–Thomson equation, Eq. (1), the liquid composition at the solid–liquid interface is directly affected by variations in the mean curvature along the interface: C L ¼ C 1 þ lc H
ð1Þ
where C L is the composition of the liquid at a curved interface, C 1 is the composition of the liquid at a flat interface, lc is the capillary length, which is a function of the solid– liquid interfacial energy, and H is the mean interfacial curvature, given by: H¼
j1 þ j2 2
ð2Þ
where j1 and j2 are the minimum and maximum principal curvatures of the interface, respectively. Concentration gradients form because of variations in interfacial curvature, which in turn lead to solute fluxes in the liquid phase. This diffusion-driven flux moves from areas of high curvature,
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.01.036
J.L. Fife, P.W. Voorhees / Acta Materialia 57 (2009) 2418–2428
where there is dissolution, to areas of low curvature, where there is growth. The net effect of this process is a decrease in the total interfacial area over time, and an increase in the average length scale of the two-phase system, while the volume fraction of the solid phase remains nearly constant. The dynamics of the coarsening process are determined by solving a diffusion equation subject to the boundary conditions given by the Gibbs–Thomson equation. This approach was taken by Lifshitz and Slyozov [1] and Wagner [2], who examined the coarsening process in a system of spherical particles. They showed that the growth rate of the average particle size follows a t1=3 power law: R3 ðtÞ R3 ð0Þ ¼ K LSW t
ð3Þ
where R is the average particle radius at a given time, t, R0 is the average particle radius at the start of self-similar coarsening, and K LSW is a constant that is a function of the diffusion coefficient, interfacial energy and other material parameters. While the radius is an easily identifiable length scale for a spherical particle, there are many possible length scales for objects as complicated as a dendrite. One such length scale is the secondary arm spacing, k2 . This is an easy feature to measure on a two-dimensional cross-section of dendritic microstructures. It increases with coarsening time similarly to the average particle radius [3]: 1=3
k2 t f
ð4Þ
Measuring k2 after substantial coarsening occurs has posed a significant problem since dendritic microstructures can evolve into spheroidal solid particles where secondary arms do not exist [4]. Thus, a length scale that is independent of morphology is necessary to quantify the evolution of these structures. Not only is the transition to spherical solid particles possible, but other topological changes may occur, such as secondary arm detachment and arm coalescence. We thus employ the inverse specific surface area per unit volume, S 1 v , as in Ref. [4]. The relationship between surface area per unit volume, S v , and time, t, is: 3 S 3 v ðtÞ S v ð0Þ ¼ Kt 1=3
ð5Þ
The t power law evident in Eqs. (3)–(5) is observed during coarsening in many systems, from spinodally decomposing alloys to systems with spherical particles. In most cases, it is associated with a microstructural self-similarity [1,2], whereby the morphology of the two-phase mixture is time independent when scaled by a time-dependent characteristic length scale such as S 1 v . However, studies of coarsening of solid–liquid mixtures that were initially direc1=3 even though tionally solidified [5,6] found that S 1 v / t the mixtures were not evolving self-similarly. Surface area per unit volume alone, however, is not an adequate measure to fully characterize the microstructure because it is an average over the system; thus, many details of the microstructure are lost in the averaging process. Therefore, the interfacial curvature must also be measured.
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This can only be done in three dimensions because it is necessary to measure the principal curvatures of the interfaces. Thus, the study of coarsening in dendritic microstructures has moved from two-dimensional quantities to those that require measurement in three dimensions. As shown below, interface shape distributions (ISDs) provide such a measure of the morphology of the interfaces. Additional information on the anisotropy of the structure is provided by interface normal distributions (INDs). In order to explore the influence of the initial structure on the coarsening process in solid–liquid systems, we examine the evolution of initially equiaxed dendritic structures in Al–Cu alloys. This study will thus serve as a direct comparison to the previously mentioned directionally solidified Al–Cu studies. We use ISDs to characterize the morphological changes in the microstructure as coarsening time increases. We use INDs to characterize the anisotropy of the structure and also determine the interconnectivity of the microstructure. 2. Experimental procedure Samples of equiaxed dendritic Al–Cu samples were obtained from Dr. Markus Rettenmayr of the University of Jena in Germany. Three samples of Al–20 wt:%Cu were isothermally coarsened at 826 K, 5 K above the eutectic temperature, for 10, 1060 and 6566 min (over 4 days) and then quenched in water to ensure a fully-formed eutectic. Using a recently developed semi-automated serial sectioning technique [7], the samples are serially sectioned 4:75 lm between sections for the 10 min sample and 9:5 lm between sections for the 1060 and 6566 min samples. The newly cut sample surface is cleaned with ethanol and then photographed. A linear variable differential transformer (LVDT) records the horizontal position of the sample when the picture is taken, and then the sample proceeds back to the start position where the procedure begins again. The position recorded by the LVDT is used during the three-dimensional reconstruction to align the individual cross-sections. The pictures are segmented into binary images, and three-dimensional reconstructions are then used for the surface area per unit volume, interconnectivity, curvature and orientation calculations. The size of the reconstructed volume is chosen to be at least 20 times greater than S 1 v to adequately capture the morphological changes relative to the increasing length scale. Thus, the size of the box used to determine the INDs and ISDs varies with the characteristic length of the structure. 3. Analysis Once the three-dimensional reconstruction is generated, the surface area per volume, S v , is calculated as the sum of the interfacial surface patches divided by the total volume of the reconstruction.
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The principal curvatures, j1 and j2 , are the diagonal elements of the 2 2 curvature tensor matrix. The two important invariants of the curvature tensor are the mean and Gaussian curvatures. The mean curvature, H, is defined as half the trace of the curvature matrix, see Eq. (2). The Gaussian curvature, K, is the determinant of the curvature matrix: K ¼ j1 j2
ð6Þ
The mean curvature is important because it sets the concentration at the solid–liquid interface, see Eq. (1). However, the mean and Gaussian curvatures play essential roles during interface evolution since the rate of change of the curvatures of a patch of interface is related to both [8]. In addition, both are necessary to characterize the shape of a small patch of interface. For example, a flat interface and a saddle-shaped interface can both have a mean curvature of zero but will have different Gaussian curvatures. Once the three-dimensional reconstructions are generated, we extract both the mean and Gaussian curvatures by using the mixed finite-element/finite-volume method of Guillaume et al. [9]. Then, j1 and j2 are calculated using: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ j1 ¼ H H 2 K pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ j2 ¼ H þ H 2 K The curvature data is then represented by a probability density map or ISD [5]. Let P ðj1 ; j2 Þ be a probability density function, where P ðj1 ; j2 Þdj1 dj2 is the probability that any randomly chosen interface point will have principal curvatures, j1 and j2 , between j1 þ dj1 and j2 þ dj2 . We present this probability as a two-dimensional probability contour plot. The plot can be divided into four regions, see Fig. 1, where:
Fig. 1. Map of the local interface shapes possible in an interface shape distribution [5].
Region 1 represents elliptic patches where H > 0; K Region 2 represents hyperbolic patches where H K < 0. Region 3 represents hyperbolic patches where H K < 0. Region 4 represents elliptic patches where H < 0; K
> 0. > 0; < 0; > 0.
There can be no probability below j1 ¼ j2 since, by definition, j2 P j1 . We characterize the directionality of the structures by determining the probability of finding an interfacial normal, n, in a given direction [6]. This probability is determined by projecting a unit interface normal, originating at the center of a sphere and ending on the surface of the sphere, onto a plane tangent to the sphere and perpendicular to the axis along which the projection is made. The data is binned in three dimensions before it is projected onto the sphere, which removes potential artifacts from the projections; these artifacts are seen in twodimensional polar plots when the data is binned after the projection is complete. Thus, each bin encompasses the same three-dimensional area of the unit sphere. We then create a contour plot of the probability distributions of interface normal orientations [6]. A similar approach for grains and interfaces in crystals has been used by Saylor et al. [10] and Rowenhorst et al. [11], respectively, wherein the direction of the normals are referenced to the axes of the crystal. Two views are generated by this technique. The nearhemisphere view presents the normals along the positive axis of choice while the far-hemisphere view shows the normals along the negative axis with the center of the sphere at the origin for both views. The two plots should appear qualitatively similar, but will not be exactly the same due to asymmetry within the experimental data, and the peak will be found on one of the two views. We choose to display the projection that contains this peak, regardless of the direction of the projection. There are also two types of projections used in INDs: equal-area and stereographic projections. Equal-area projections enhance features near the center of the plot, while stereographic projections enhance features along the edges of the plot. In this paper, equal-area projection INDs are used. In order to clarify what one would expect from an equal-area projection, consider the INDs for a single sphere and a hollow cylinder. The probability of finding any normal for a spherical interface would be independent of orientation. Thus, the equal-area IND, in any direction, would have a uniform color. The equal-area IND projected in the z-direction of a hollow cylinder with its axis parallel to the x-direction, for example, would be an infinitesimally thin line spanning the length of the IND through the center of the plot, representing the equal probability of finding a patch of surface with interfacial normals perpendicular to the x-direction or, equivalently, parallel to the y–z plane.
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4. Results and discussion Two-dimensional micrographs of the evolution of the microstructure are shown in Fig. 2. It is evident that the size scale increases dramatically during coarsening and that the dendritic structure appears to be largely gone after 1060 min of coarsening. The Al-rich dendrites also undergo significant morphological changes, from a dendritic morphology to a more spheroidal or globular microstructure, which is reminiscent of observations made by Marsh and Glicksman [4]. The three-dimensional microstructures of the three isothermally coarsened samples are shown in Fig. 3, along with the corresponding plot of the inverse surface area per unit volume, Fig. 3(d). To illustrate the increase in size scale of the structure during coarsening, the dimensions of the reconstruction box shown in Fig. 3 are held fixed; thus, only a portion of a much larger data set are shown. The excellent linear fit of the S v data indicates that the characteristic length of the microstructure scales with the cube increases root of time, following Eq. (5). Moreover, S 1 v by a substantial factor of 4 from 10 to 6566 min. It is important to note the dendrite seen in the two-dimensional micrograph of the 10 min sample, Fig. 2(b), is not apparent in the three-dimensional reconstruction, Fig. 3(a). This is because the dendrite stem is positioned diagonally through the reconstruction box and is hidden by other microstructural features. The presence of this dendrite stem affects both the ISD and the IND for this coarsening time, as will be discussed below. Also, there is no evidence of the initial
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dendritic microstructure after 1060 min of coarsening, as the structure is highly interconnected and globular, encompassing a mix of cylindrical-like bodies with other hyperboloids, each of which have some spherically capped ends. Fig. 4(a)–(c) show the evolution of the ISDs during coarsening. In order to assess changes in the ISDs with coarsening time, the minimum and maximum values on the color bar are time independent, and the principal curvatures are scaled by S v . If the system is coarsening in a self-similar manner, scaling the curvatures by S v should render the ISDs time independent. There are general observations one can draw from these ISDs. First, the distribution, or general shape of the ISD, is broad, and the peak probability is not high, both of which indicate that the fraction of area encompassed by the peak is not a significant fraction of the total interfacial area in the structure. The location of the peak remains at approximately, j2 =S v ¼ 1, which indicates S v is a very accurate measure of the magnitude of the curvatures in the structure. In more detail, almost all the surface patches have at least one positive principal curvature. Further, almost all the surface patches have positive mean curvature, which encompasses regions 1 and 2, see Fig. 1. In region 1, the shapes are elliptic and convex toward the solid, while in region 2, the interfaces are hyperbolic toward the solid, where the absolute value of j2 is greater than the absolute value of j1 . This is consistent with the fact that these are dendritic structures embedded in a liquid matrix, and the definition of the curvatures used to calculate the ISDs implies that most of the structure should have positive
Fig. 2. Two-dimensional micrographs of Al–20 wt:%Cu: (a) as-cast, and after (b) 10 min, (c) 1060 min, and (d) 6566 min of coarsening.
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Fig. 3. Three-dimensional reconstructions of the Al-rich dendrites in Al–20 wt:%Cu during coarsening after: (a) 10 min, (b) 1060 min, (c) 6566 min, where the reconstruction box used in (a)–(c) is 1800 1200 470 lm. The Al–Cu eutectic is transparent, the solid volume fraction for the samples is 46% and (d) S 1 v as a function of the cube root of coarsening time, for the three coarsened samples. Note, the error bars for (d) are of the order of the size of the points at 1:2 lm.
mean curvature. Since the volume fraction of the solid is nearly 50%, this would not be expected based on volume fraction alone. For example, results from Kwon et al. [12] show that bicontinuous structures produced following spinodal decomposition at a 50% volume fraction have an average mean curvature of zero. However, not all patches have both positive j1 and j2 . This is because there are many instances where the solid bodies have undergone coalescence, thus increasing the probability for interfaces with negative Gaussian curvature, which would be found in regions 2 and 3. As coarsening time increases, the peak of each ISD remains centered along the solid cylinder line ðj1 =S v ¼ 0Þ, where Gaussian curvature is zero. Interestingly, this occurs without solid cylinders being present in the microstructures. The microstructure is highly interconnected at all coarsening times and dominated by more globular structures. To clarify, the microstructure is a combination of cylindrical-like bodies and hyperbolic-shaped bodies, each of which have spherically capped ends. Thus, these bodies have regions of interface with zero Gaussian curvature without actually being cylindrical in shape, which accounts for the peak location on j1 =S v ¼ 0. As discussed above, the distribution of interface shapes is broad and the value of
the peak probability is not high, and thus the location of the peak does not necessarily translate to the actual morphology observed in the microstructure. The teardrop-shaped peak region seen in the 10 min structure is caused by the remnants of the microstructure present prior to coarsening. As seen in Fig. 2(b), there is a large dendrite representing approximately one-eighth of the total volume analyzed. The interfacial areas associated with the secondary arms of this dendrite lead to an elongation of the peak toward j1 =S v ¼ j2 =S v ¼ 0 or flat interfaces. A cross-section view of these arms is elliptical, which leads to nearly flat interface patches up to the tips of the arms. The tips of the secondary arms, as well as shapes with similarly capped ends, are spherical and thus represented in region 1. So, as coarsening proceeds, the secondary dendrite arms disappear, and thus the elongation disappears. The scaled ISDs at all coarsening times are very similar. The location and magnitude of the peak, and the general shape, do not change significantly with coarsening time. The largest change is observed going from 10 to 1060 min of coarsening, and this is due to the disappearance of a primary dendrite stem. The ISDs for the two longest coarsening times are very similar. The changes in the ISDs evident
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Fig. 4. Interface shape distributions of samples coarsened for: (a) 10 min, (b) 1060 min, and (c) 6566 min.
in Fig. 4 are much smaller than those observed when the initial structure is produced using directional solidification [5,6,13–15] and analyzed below. It is clear that there is no evidence of the structure breaking up into quasi-spherical particles, as this would imply an increase in probability along the j1 ¼ j2 line. In this case, the apparent spheroidization evident in the two-dimensional micrographs shown in Fig. 2 is misleading. To quantify the small changes seen in the ISDs, the fractions of interfacial area with various curvatures are determined, as shown in Fig. 5. There is an approximate 10% increase in interfacial area with positive mean curvature as coarsening time increases. The fraction of interfacial area with hyperbolic and parabolic shapes, as well as the fraction of interfacial area with positive Gaussian curvature, remain constant at approximately 50%. The small increase in the fraction of interfacial area with positive mean curvature indicates that the solid is becoming more
Fig. 5. The fraction of interface of specified curvatures as a function of coarsening time for the three coarsened samples.
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convex with respect to the liquid as coarsening time increases. This is consistent with the drive of the system toward the surface energy-minimizing shape of a sphere. However, this change in positive mean curvature is quite small given that the overall size scale of the system has increased by a factor of 4 during coarsening. In order to quantify the anisotropy of the structures, we examine the INDs, see Fig. 6. The far-hemisphere (negative), z-axis projections of the INDs are provided because the location of the peak value appears strongest in these projections and because it best displays the 2-fold symmetry of the interfacial structure found after 10 min of coarsening. The color bars in the INDs have fixed minimum and maximum values based on the strongest directionality observed in the three samples, which occurs in the 10 min sample.
0.00E+00
1.08E-04
2.16E-04
The IND of the 10 min sample has a strong peak, implying that the microstructure has 2-fold symmetry, which would typically be observed in a plate-like structure. The three-dimensional reconstruction of the 10 min microstructure was split into two sections, one including the dendrite located in the lower left portion of the microstructure and one not including the dendrite. This yielded one IND with only the 2-fold symmetry peak(s) seen in Fig. 6(a), and one IND with an isotropic distribution of normals showing the remaining features in Fig. 6(a), which is very similar to the INDs for the later coarsening times. Thus, we conclude that the dendrite is responsible for the peak(s) seen in the 10 min IND. As discussed above, the secondary arms of the dendrite elongate the peak region in Fig. 4(a) toward j1 =S v ¼ j2 =S v ¼ 0, or flat interfaces, which would account for the peak representing planar interfaces seen in Fig. 6(a).
3.24E-04
4.32E-04
5.40E-04
Y
X
0.00E+00
1.08E-04
2.16E-04
3.24E-04
Z
4.32E-04
(a) 5.40E-04
0.00E+00
Y
X
1.08E-04
2.16E-04
3.24E-04
4.32E-04
5.40E-04
Y
Z
X
(b)
Z
(c)
Fig. 6. Interface normal distributions of the samples coarsened for: (a) 10 min, (b) 1060 min, (c) 6566 min. The far-hemisphere (negative) z-axis projections are shown.
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Because this dendrite disappears as coarsening time increases, the INDs for the later times are approximately isotropic, and quantitatively very similar. We find that INDs provide a particularly sensitive measure of the heterogeneity of the structure. During coarsening, the heterogeneity decreases and the structure becomes isotropic. The changes in the INDs can be seen clearly by determining the fraction of interface with normals nearly perpendicular to the x-;y- and z-directions [6]. Nearly perpendicular to a given direction is defined as any angle, h, within 10° of perpendicular to the axis in question. These values are then divided by the total corresponding interfacial area of the microstructure to give the fraction of interface in this interval. The fraction of interface nearly perpendicular to each direction for each coarsening time, shown in Fig. 7, is between 16% and 19%. This shows that despite the presence of the peak in the 10 min IND, the structures are, on average, isotropic in nature. This is most likely due to the equiaxed microstructure present prior to coarsening. Finally, to asses whether the structures are breaking up into solid particles, the interconnectivity of the solid phase of the microstructures at each coarsening time is evaluated. Interconnectivity of the solid phase increases from 87% to 95% to 96% after 10, 1060, and 6566 min, respectively. It should be noted that the solid regions not interconnected for the two later coarsening times are located at the edges of the reconstruction box and are most likely interconnected elsewhere in the sample. As stated above, the volume of data used for this calculation increases proportionally to the increase in characteristic length with coarsening time; thus, we can be confident we are capturing any size scale effects that might be present as coarsening time increases. The increase in interconnectivity as coarsen-
Fig. 7. The fraction of interface nearly perpendicular to each direction as a function of coarsening time for the three coarsened samples. Nearly perpendicular in each direction is defined as 10 .
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ing time increases is of the same order as the increase in positive mean curvature seen in Fig. 5. This indicates that the microstructure is decreasing interfacial free energy not by breaking up into solid particles but by increasing the length scale of the microstructure and decreasing the fraction of interfacial area with negative mean curvature. What is important about these results is that the assumptions made in traditional two-dimensional models of dendritic coarsening [4,16–19] do not apply to these equiaxed systems because the morphologies are so different than those assumed in the models. More experiments are necessary to elucidate the actual mechanisms for coarsening in these systems. 4.1. Comparison to an initially directionally solidified Al–Cu ingot We compare these results to an initially directionally solidified Al–26 wt:%Cu microstructure, which corresponds to 42% solid volume fraction. Comparing similar solid volume fractions allows us to directly examine the effects of the initial microstructure on coarsening. In the ISDs (Fig. 8(c) and (d)), the principal curvatures are again scaled by S v to eliminate effects caused by an increase in the characteristic length scale with coarsening time. The color bars in both the ISDs and the INDs (Fig. 8(e) and (f)) also have fixed minimum and maximum values, corresponding to the highest probability and strongest directionality, respectively. The characteristic length scale of the initially directionally solidified microstructures increases in a similar fashion to the equiaxed dendritic samples. In both cases, the characteristic length increases with t1=3 , and the amplitudes of the temporal power law are within 20% of each other. This occurs, however, without any evidence of self-similarity in the samples that are directionally solidified prior to coarsening. The three-dimensional reconstructions, Fig. 8(a) and (b), show the system evolving from a highly complex microstructure to one dominated by solid cylinders and solid cylindrical-like shapes. The change in the ISDs, Fig. 8(c) and (d), are consistent with these observations. The peak moves from region 1 in the 10 min sample and aligns with j1 =S v ¼ 0 in the 3600 min sample. The peak probability also increases by over a factor of 4 between the two coarsening times. This increase, along with the change in dominant shapes in the microstructure from solid convex shapes, such as those present at the tips of dendrites, to solid cylindrical-like shapes verify the lack of self-similarity in the microstructure. As reported by Kammer et al. [15], this volume fraction was ultimately coarsened for over 3 weeks, with solid cylinders remaining the dominant interfacial shape in the microstructure, and there was no evidence of break up into solid particles. These results differ greatly from the initially equiaxed dendritic microstructures. By comparison, no significant change in the morphology of the interfaces, other than
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Fig. 8. Three-dimensional reconstructions of Al-rich dendrites and corresponding scaled ISDs and INDs for an initially directionally solidified Al– 26 wt:%Cu microstructure coarsened for: (a [15], c, and e) 10 min and (b, d, and f) 3600 min.
the size scale increase, is observed in these equiaxed structures. For example, the peak probability in the ISD for the directionally solidified structures increases with comparable magnitude to characteristic length with coarsening time
and is over two times greater than the peak probability observed in any of the initially equiaxed structures. The distribution, Fig. 8(d), is also very compact, which indicates that the peak region is representative of a significant
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portion of the interfacial area in the structure, which is also not true of the ISDs for the initially equiaxed structures shown in Fig. 4. This is evident also in a visual inspection of the three-dimensional reconstructions, as solid cylinders dominate the 3600 min directionally solidified structure. Because the evolution of morphology in the directionally solidified structure occurs in the z-direction, parallel to the initial solidification direction, it is evident that the initial morphology of the structure is the cause of these differences. The far-hemisphere (negative), x-axis equal-area projection INDs, Fig. 8(e) and (f), are provided because the location of the peak value appears strongest in these projections and because it best displays the directionality that develops as coarsening time increases. It should be noted that in the 10 min microstructure, there is a strong 4-fold symmetry, but due to the scale of the color bar chosen for the plot, this is not clear. During coarsening, a 2fold symmetry develops with a significant increase in the strength of the peak. This IND is significant because, although there is a range of intensities, the locations of the probabilities span the length of the projection with the maximum probability located in the center. This indicates a strong preference for normals perpendicular to the x-direction, similar to the example of a hollow cylinder discussed previously. Thus, the IND is an illustration of a structure dominated by solid cylinders parallel to the directional solidification direction (z). The interconnectivity of the Al-rich dendrites is also examined for the directionally solidified microstructure. The interconnectivity of the solid phase decreases from 84.7% to 13.5% to 12.1% after 10 min, 3600 min and 3 weeks of coarsening, respectively. Although the lack of interconnectivity in the initially equiaxed dendritic microstructures could be attributed to sections of the structure intersecting the edges of the reconstruction box, and thus they may be connected elsewhere in the structure, this is not the case in the initially directionally solidified microstructure. There are three or four independent solid bodies after 10 min of coarsening for the directionally solidified structure, and as coarsening time increases, the number of independent solid bodies increases significantly, with at least 30 independent bodies in the 3 week coarsened structure. This is attributed to the formation of independent solid cylinders, discussed above. Thus, in the directionally solidified structure, the decrease in interfacial free energy is achieved by breaking up the structure into solid cylinders, while in the initially equiaxed dendritic structure, this is attained by increasing interconnectivity and decreasing the interfacial area with negative mean curvature. 5. Conclusions Three initially equiaxed dendritic Al–20 wt:%Cu microstructures were isothermally coarsened for 10, 1060 and 6566 min, respectively. The microstructures were analyzed in three dimensions by observing changes in characteristic
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length scale, ISDs and INDs. It was found that the characteristic length, S 1 v , scales linearly with the cube root of time. All evidence of the initially solidified dendritic microstructure disappeared after 10 min of isothermal coarsening, and the structure evolved into a highly interconnected globular-like structure with a combination of cylindrical-like and hyperbolic bodies, each with spherically capped ends. There was no evidence of break up into spherical solid particles. As coarsening time increased, the scaled ISDs showed microstructures dominated by convex and hyperbolic solid shapes (regions 1 and 2 in the ISDs) with only very small changes. There was also no preferential directionality within the microstructures because the interface normal distributions were approximately isotropic. Comparing the evolution of the initially equiaxed dendritic structure with its initially directionally solidified counterpart of a similar solid volume fraction provides further evidence that the initial solidification procedures and the microstructure present prior to coarsening have a significant impact on the evolution of the microstructure during coarsening. Acknowledgement This material is based upon work supported under a National Science Foundation Graduate Research Fellowship, which is gratefully acknowledged. This work is also supported by the Department of Energy Office of Basic Energy Science (Grant CNV0037736). The authors would like to thank Dr. Markus Rettenmayr of the University of Jena in Germany for providing these samples for study. The authors would also like to recognize R. Mendoza for use of his structures and irreplaceable assistance. Lastly, the authors would like to thank D. Kammer, D. Rowenhorst, and Y. Kwon for their invaluable insight and time spent in multiple discussions. References [1] Lifshitz IM, Slyozov VV. The kinetics of precipitation from supersaturated solid solutions. J Phys Chem Solids 1961;19:35–50. [2] Wagner C. Theorie der Alterung von Niederschla¨gen Durch Umlo¨sen. Z Electrochemistry 1961;65:581–91. [3] Bower TF, Brody HD, Flemings MC. Measurements of solute redistribution during dendritic solidification. Trans Metall Soc AIME 1966;236:624–34. [4] Marsh SP, Glicksman ME. Overview of geometric effects on coarsening of mushy zones. Metall Mater Trans A 1996;27A:557–67. [5] Mendoza R, Alkemper J, Voorhees PW. The morphological evolution of dendritic microstructures during coarsening. Metall Mater Trans A 2003;34A:481–9. [6] Kammer D, Voorhees PW. The morphological evolution of dendritic microstructures during coarsening. Acta Mater 2006;54:1549–58. [7] Alkemper J, Voorhees PW. Quantitative serial sectioning analysis. J Microsc 2001;201:388–94. [8] Drew DA. Evolution of geometric statistics. J Appl Math 1990;50:649–66. [9] Guillaume L, Florent D, Atilla B. Constant curvature region decomposition of 3D-meshes by a mixed approach vertex-triangle. WSCG 2004;12:1–3.
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[10] Saylor DM, Adams BL, El-Dasher BS, Rohrer GS. Measuring the five-parameter grain-boundary distribution from observations of planar sections. Metall Mater Trans 2004;35A:1981–9. [11] Rowenhorst DJ, Gupta A, Feng CR, Spanos G. 3D crystallographic and morphological analysis of coarse martensite: combining EBSD and serial sectioning. Scripta Mater 2006;55:11–6. [12] Kwon Y, Thornton K, Voorhees PW. Coarsening of bicontinuous structures via nonconserved and conserved dynamics. Phys Rev E 2007;75:021120. [13] Mendoza R, Savin I, Thornton K, Voorhees PW. Topological complexity and the dynamics of coarsening. Nat Mater 2004;3(6): 385–8. [14] Mendoza R, Thornton K, Savin I, Voorhees PW. The evolution of interfacial topology during coarsening. Acta Mater 2006;54:743–50.
[15] Kammer D, Mendoza R, Voorhees PW. Cylindrical domain formation in topologically complex structures. Scripta Mater 2006;55: 17–22. [16] Kattamis TZ, Coughlin JC, Flemings MC. Influence of coarsening on dendrite arm spacing of aluminum-copper alloys. Trans Metall Soc AIME 1967;239:1504–11. [17] Kahlweit M. On the aging of dendrites. Scripta Metall 1968;2: 251–4. [18] Young KP, Kirkwood DH. The dendrite arm spacings of aluminumcopper alloys solidified under steady-state conditions. Metall Trans A 1975;6A:197–205. [19] Huang SC, Glicksman ME. Fundamentals of dendritic solidification. Acta Metall 1981;29:717–34.
Acta Materialia 57 (2009) 2418–2428 www.elsevier.com/locate/actamat
The morphological evolution of equiaxed dendritic microstructures during coarsening J.L. Fife *, P.W. Voorhees Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108, USA Received 21 September 2008; received in revised form 26 January 2009; accepted 27 January 2009 Available online 11 March 2009
Abstract The morphological evolution of equiaxed Al–20 wt:%Cu dendritic microstructures was studied in three dimensions. The microstructure evolved into a highly interconnected structure, where the inverse specific surface area scaled linearly with the cube root of time. As the size scale of the microstructure increased during coarsening, the scaled morphology of the interfaces changed only slightly. The distribution of interface normals indicated that the microstructure was approximately isotropic. These results are in contrast to those found using a directionally solidified Al–Cu alloy of a similar solid volume fraction, where the structure evolved into solid cylinders parallel to the growth direction used to create the sample prior to coarsening. Thus, we find that the initial morphology of a dendritic structure can have a major impact on its morphological evolution. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Equiaxed dendritic microstructure; Aluminum–copper; Coarsening; Interfacial curvature
1. Introduction Solidification of metallic alloys typically occurs by the formation of dendrites as the primary growth morphology. Dendrites develop because of a morphological instability of the solid–liquid interface during solidification and are technologically important because their growth is directly related to the properties of a material. Dendritic growth frequently results from a mushy zone, a two-phase region where the solid phase has a dendritic structure and the liquid phase is the surrounding matrix. When this mixture is held for any length of time, the dendrites undergo coarsening or Ostwald ripening. During coarsening, the morphology and the length scale of the microstructure can change considerably. This change in length scale alters the distribution of chemical components in the solid, and thus coarsening can affect many properties of a material. *
Corresponding author. Tel.: +1 847 467 7089. E-mail addresses: jfi[email protected] (J.L. Fife), p-voorhees@ northwestern.edu (P.W. Voorhees).
The coarsening process is driven by the minimization of the interfacial free energy of the system. In alloys, the coarsening process proceeds by the diffusion of mass from one region of the interface to another. As described by the Gibbs–Thomson equation, Eq. (1), the liquid composition at the solid–liquid interface is directly affected by variations in the mean curvature along the interface: C L ¼ C 1 þ lc H
ð1Þ
where C L is the composition of the liquid at a curved interface, C 1 is the composition of the liquid at a flat interface, lc is the capillary length, which is a function of the solid– liquid interfacial energy, and H is the mean interfacial curvature, given by: H¼
j1 þ j2 2
ð2Þ
where j1 and j2 are the minimum and maximum principal curvatures of the interface, respectively. Concentration gradients form because of variations in interfacial curvature, which in turn lead to solute fluxes in the liquid phase. This diffusion-driven flux moves from areas of high curvature,
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.01.036
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where there is dissolution, to areas of low curvature, where there is growth. The net effect of this process is a decrease in the total interfacial area over time, and an increase in the average length scale of the two-phase system, while the volume fraction of the solid phase remains nearly constant. The dynamics of the coarsening process are determined by solving a diffusion equation subject to the boundary conditions given by the Gibbs–Thomson equation. This approach was taken by Lifshitz and Slyozov [1] and Wagner [2], who examined the coarsening process in a system of spherical particles. They showed that the growth rate of the average particle size follows a t1=3 power law: R3 ðtÞ R3 ð0Þ ¼ K LSW t
ð3Þ
where R is the average particle radius at a given time, t, R0 is the average particle radius at the start of self-similar coarsening, and K LSW is a constant that is a function of the diffusion coefficient, interfacial energy and other material parameters. While the radius is an easily identifiable length scale for a spherical particle, there are many possible length scales for objects as complicated as a dendrite. One such length scale is the secondary arm spacing, k2 . This is an easy feature to measure on a two-dimensional cross-section of dendritic microstructures. It increases with coarsening time similarly to the average particle radius [3]: 1=3
k2 t f
ð4Þ
Measuring k2 after substantial coarsening occurs has posed a significant problem since dendritic microstructures can evolve into spheroidal solid particles where secondary arms do not exist [4]. Thus, a length scale that is independent of morphology is necessary to quantify the evolution of these structures. Not only is the transition to spherical solid particles possible, but other topological changes may occur, such as secondary arm detachment and arm coalescence. We thus employ the inverse specific surface area per unit volume, S 1 v , as in Ref. [4]. The relationship between surface area per unit volume, S v , and time, t, is: 3 S 3 v ðtÞ S v ð0Þ ¼ Kt 1=3
ð5Þ
The t power law evident in Eqs. (3)–(5) is observed during coarsening in many systems, from spinodally decomposing alloys to systems with spherical particles. In most cases, it is associated with a microstructural self-similarity [1,2], whereby the morphology of the two-phase mixture is time independent when scaled by a time-dependent characteristic length scale such as S 1 v . However, studies of coarsening of solid–liquid mixtures that were initially direc1=3 even though tionally solidified [5,6] found that S 1 v / t the mixtures were not evolving self-similarly. Surface area per unit volume alone, however, is not an adequate measure to fully characterize the microstructure because it is an average over the system; thus, many details of the microstructure are lost in the averaging process. Therefore, the interfacial curvature must also be measured.
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This can only be done in three dimensions because it is necessary to measure the principal curvatures of the interfaces. Thus, the study of coarsening in dendritic microstructures has moved from two-dimensional quantities to those that require measurement in three dimensions. As shown below, interface shape distributions (ISDs) provide such a measure of the morphology of the interfaces. Additional information on the anisotropy of the structure is provided by interface normal distributions (INDs). In order to explore the influence of the initial structure on the coarsening process in solid–liquid systems, we examine the evolution of initially equiaxed dendritic structures in Al–Cu alloys. This study will thus serve as a direct comparison to the previously mentioned directionally solidified Al–Cu studies. We use ISDs to characterize the morphological changes in the microstructure as coarsening time increases. We use INDs to characterize the anisotropy of the structure and also determine the interconnectivity of the microstructure. 2. Experimental procedure Samples of equiaxed dendritic Al–Cu samples were obtained from Dr. Markus Rettenmayr of the University of Jena in Germany. Three samples of Al–20 wt:%Cu were isothermally coarsened at 826 K, 5 K above the eutectic temperature, for 10, 1060 and 6566 min (over 4 days) and then quenched in water to ensure a fully-formed eutectic. Using a recently developed semi-automated serial sectioning technique [7], the samples are serially sectioned 4:75 lm between sections for the 10 min sample and 9:5 lm between sections for the 1060 and 6566 min samples. The newly cut sample surface is cleaned with ethanol and then photographed. A linear variable differential transformer (LVDT) records the horizontal position of the sample when the picture is taken, and then the sample proceeds back to the start position where the procedure begins again. The position recorded by the LVDT is used during the three-dimensional reconstruction to align the individual cross-sections. The pictures are segmented into binary images, and three-dimensional reconstructions are then used for the surface area per unit volume, interconnectivity, curvature and orientation calculations. The size of the reconstructed volume is chosen to be at least 20 times greater than S 1 v to adequately capture the morphological changes relative to the increasing length scale. Thus, the size of the box used to determine the INDs and ISDs varies with the characteristic length of the structure. 3. Analysis Once the three-dimensional reconstruction is generated, the surface area per volume, S v , is calculated as the sum of the interfacial surface patches divided by the total volume of the reconstruction.
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The principal curvatures, j1 and j2 , are the diagonal elements of the 2 2 curvature tensor matrix. The two important invariants of the curvature tensor are the mean and Gaussian curvatures. The mean curvature, H, is defined as half the trace of the curvature matrix, see Eq. (2). The Gaussian curvature, K, is the determinant of the curvature matrix: K ¼ j1 j2
ð6Þ
The mean curvature is important because it sets the concentration at the solid–liquid interface, see Eq. (1). However, the mean and Gaussian curvatures play essential roles during interface evolution since the rate of change of the curvatures of a patch of interface is related to both [8]. In addition, both are necessary to characterize the shape of a small patch of interface. For example, a flat interface and a saddle-shaped interface can both have a mean curvature of zero but will have different Gaussian curvatures. Once the three-dimensional reconstructions are generated, we extract both the mean and Gaussian curvatures by using the mixed finite-element/finite-volume method of Guillaume et al. [9]. Then, j1 and j2 are calculated using: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ j1 ¼ H H 2 K pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ j2 ¼ H þ H 2 K The curvature data is then represented by a probability density map or ISD [5]. Let P ðj1 ; j2 Þ be a probability density function, where P ðj1 ; j2 Þdj1 dj2 is the probability that any randomly chosen interface point will have principal curvatures, j1 and j2 , between j1 þ dj1 and j2 þ dj2 . We present this probability as a two-dimensional probability contour plot. The plot can be divided into four regions, see Fig. 1, where:
Fig. 1. Map of the local interface shapes possible in an interface shape distribution [5].
Region 1 represents elliptic patches where H > 0; K Region 2 represents hyperbolic patches where H K < 0. Region 3 represents hyperbolic patches where H K < 0. Region 4 represents elliptic patches where H < 0; K
> 0. > 0; < 0; > 0.
There can be no probability below j1 ¼ j2 since, by definition, j2 P j1 . We characterize the directionality of the structures by determining the probability of finding an interfacial normal, n, in a given direction [6]. This probability is determined by projecting a unit interface normal, originating at the center of a sphere and ending on the surface of the sphere, onto a plane tangent to the sphere and perpendicular to the axis along which the projection is made. The data is binned in three dimensions before it is projected onto the sphere, which removes potential artifacts from the projections; these artifacts are seen in twodimensional polar plots when the data is binned after the projection is complete. Thus, each bin encompasses the same three-dimensional area of the unit sphere. We then create a contour plot of the probability distributions of interface normal orientations [6]. A similar approach for grains and interfaces in crystals has been used by Saylor et al. [10] and Rowenhorst et al. [11], respectively, wherein the direction of the normals are referenced to the axes of the crystal. Two views are generated by this technique. The nearhemisphere view presents the normals along the positive axis of choice while the far-hemisphere view shows the normals along the negative axis with the center of the sphere at the origin for both views. The two plots should appear qualitatively similar, but will not be exactly the same due to asymmetry within the experimental data, and the peak will be found on one of the two views. We choose to display the projection that contains this peak, regardless of the direction of the projection. There are also two types of projections used in INDs: equal-area and stereographic projections. Equal-area projections enhance features near the center of the plot, while stereographic projections enhance features along the edges of the plot. In this paper, equal-area projection INDs are used. In order to clarify what one would expect from an equal-area projection, consider the INDs for a single sphere and a hollow cylinder. The probability of finding any normal for a spherical interface would be independent of orientation. Thus, the equal-area IND, in any direction, would have a uniform color. The equal-area IND projected in the z-direction of a hollow cylinder with its axis parallel to the x-direction, for example, would be an infinitesimally thin line spanning the length of the IND through the center of the plot, representing the equal probability of finding a patch of surface with interfacial normals perpendicular to the x-direction or, equivalently, parallel to the y–z plane.
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4. Results and discussion Two-dimensional micrographs of the evolution of the microstructure are shown in Fig. 2. It is evident that the size scale increases dramatically during coarsening and that the dendritic structure appears to be largely gone after 1060 min of coarsening. The Al-rich dendrites also undergo significant morphological changes, from a dendritic morphology to a more spheroidal or globular microstructure, which is reminiscent of observations made by Marsh and Glicksman [4]. The three-dimensional microstructures of the three isothermally coarsened samples are shown in Fig. 3, along with the corresponding plot of the inverse surface area per unit volume, Fig. 3(d). To illustrate the increase in size scale of the structure during coarsening, the dimensions of the reconstruction box shown in Fig. 3 are held fixed; thus, only a portion of a much larger data set are shown. The excellent linear fit of the S v data indicates that the characteristic length of the microstructure scales with the cube increases root of time, following Eq. (5). Moreover, S 1 v by a substantial factor of 4 from 10 to 6566 min. It is important to note the dendrite seen in the two-dimensional micrograph of the 10 min sample, Fig. 2(b), is not apparent in the three-dimensional reconstruction, Fig. 3(a). This is because the dendrite stem is positioned diagonally through the reconstruction box and is hidden by other microstructural features. The presence of this dendrite stem affects both the ISD and the IND for this coarsening time, as will be discussed below. Also, there is no evidence of the initial
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dendritic microstructure after 1060 min of coarsening, as the structure is highly interconnected and globular, encompassing a mix of cylindrical-like bodies with other hyperboloids, each of which have some spherically capped ends. Fig. 4(a)–(c) show the evolution of the ISDs during coarsening. In order to assess changes in the ISDs with coarsening time, the minimum and maximum values on the color bar are time independent, and the principal curvatures are scaled by S v . If the system is coarsening in a self-similar manner, scaling the curvatures by S v should render the ISDs time independent. There are general observations one can draw from these ISDs. First, the distribution, or general shape of the ISD, is broad, and the peak probability is not high, both of which indicate that the fraction of area encompassed by the peak is not a significant fraction of the total interfacial area in the structure. The location of the peak remains at approximately, j2 =S v ¼ 1, which indicates S v is a very accurate measure of the magnitude of the curvatures in the structure. In more detail, almost all the surface patches have at least one positive principal curvature. Further, almost all the surface patches have positive mean curvature, which encompasses regions 1 and 2, see Fig. 1. In region 1, the shapes are elliptic and convex toward the solid, while in region 2, the interfaces are hyperbolic toward the solid, where the absolute value of j2 is greater than the absolute value of j1 . This is consistent with the fact that these are dendritic structures embedded in a liquid matrix, and the definition of the curvatures used to calculate the ISDs implies that most of the structure should have positive
Fig. 2. Two-dimensional micrographs of Al–20 wt:%Cu: (a) as-cast, and after (b) 10 min, (c) 1060 min, and (d) 6566 min of coarsening.
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Fig. 3. Three-dimensional reconstructions of the Al-rich dendrites in Al–20 wt:%Cu during coarsening after: (a) 10 min, (b) 1060 min, (c) 6566 min, where the reconstruction box used in (a)–(c) is 1800 1200 470 lm. The Al–Cu eutectic is transparent, the solid volume fraction for the samples is 46% and (d) S 1 v as a function of the cube root of coarsening time, for the three coarsened samples. Note, the error bars for (d) are of the order of the size of the points at 1:2 lm.
mean curvature. Since the volume fraction of the solid is nearly 50%, this would not be expected based on volume fraction alone. For example, results from Kwon et al. [12] show that bicontinuous structures produced following spinodal decomposition at a 50% volume fraction have an average mean curvature of zero. However, not all patches have both positive j1 and j2 . This is because there are many instances where the solid bodies have undergone coalescence, thus increasing the probability for interfaces with negative Gaussian curvature, which would be found in regions 2 and 3. As coarsening time increases, the peak of each ISD remains centered along the solid cylinder line ðj1 =S v ¼ 0Þ, where Gaussian curvature is zero. Interestingly, this occurs without solid cylinders being present in the microstructures. The microstructure is highly interconnected at all coarsening times and dominated by more globular structures. To clarify, the microstructure is a combination of cylindrical-like bodies and hyperbolic-shaped bodies, each of which have spherically capped ends. Thus, these bodies have regions of interface with zero Gaussian curvature without actually being cylindrical in shape, which accounts for the peak location on j1 =S v ¼ 0. As discussed above, the distribution of interface shapes is broad and the value of
the peak probability is not high, and thus the location of the peak does not necessarily translate to the actual morphology observed in the microstructure. The teardrop-shaped peak region seen in the 10 min structure is caused by the remnants of the microstructure present prior to coarsening. As seen in Fig. 2(b), there is a large dendrite representing approximately one-eighth of the total volume analyzed. The interfacial areas associated with the secondary arms of this dendrite lead to an elongation of the peak toward j1 =S v ¼ j2 =S v ¼ 0 or flat interfaces. A cross-section view of these arms is elliptical, which leads to nearly flat interface patches up to the tips of the arms. The tips of the secondary arms, as well as shapes with similarly capped ends, are spherical and thus represented in region 1. So, as coarsening proceeds, the secondary dendrite arms disappear, and thus the elongation disappears. The scaled ISDs at all coarsening times are very similar. The location and magnitude of the peak, and the general shape, do not change significantly with coarsening time. The largest change is observed going from 10 to 1060 min of coarsening, and this is due to the disappearance of a primary dendrite stem. The ISDs for the two longest coarsening times are very similar. The changes in the ISDs evident
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Fig. 4. Interface shape distributions of samples coarsened for: (a) 10 min, (b) 1060 min, and (c) 6566 min.
in Fig. 4 are much smaller than those observed when the initial structure is produced using directional solidification [5,6,13–15] and analyzed below. It is clear that there is no evidence of the structure breaking up into quasi-spherical particles, as this would imply an increase in probability along the j1 ¼ j2 line. In this case, the apparent spheroidization evident in the two-dimensional micrographs shown in Fig. 2 is misleading. To quantify the small changes seen in the ISDs, the fractions of interfacial area with various curvatures are determined, as shown in Fig. 5. There is an approximate 10% increase in interfacial area with positive mean curvature as coarsening time increases. The fraction of interfacial area with hyperbolic and parabolic shapes, as well as the fraction of interfacial area with positive Gaussian curvature, remain constant at approximately 50%. The small increase in the fraction of interfacial area with positive mean curvature indicates that the solid is becoming more
Fig. 5. The fraction of interface of specified curvatures as a function of coarsening time for the three coarsened samples.
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convex with respect to the liquid as coarsening time increases. This is consistent with the drive of the system toward the surface energy-minimizing shape of a sphere. However, this change in positive mean curvature is quite small given that the overall size scale of the system has increased by a factor of 4 during coarsening. In order to quantify the anisotropy of the structures, we examine the INDs, see Fig. 6. The far-hemisphere (negative), z-axis projections of the INDs are provided because the location of the peak value appears strongest in these projections and because it best displays the 2-fold symmetry of the interfacial structure found after 10 min of coarsening. The color bars in the INDs have fixed minimum and maximum values based on the strongest directionality observed in the three samples, which occurs in the 10 min sample.
0.00E+00
1.08E-04
2.16E-04
The IND of the 10 min sample has a strong peak, implying that the microstructure has 2-fold symmetry, which would typically be observed in a plate-like structure. The three-dimensional reconstruction of the 10 min microstructure was split into two sections, one including the dendrite located in the lower left portion of the microstructure and one not including the dendrite. This yielded one IND with only the 2-fold symmetry peak(s) seen in Fig. 6(a), and one IND with an isotropic distribution of normals showing the remaining features in Fig. 6(a), which is very similar to the INDs for the later coarsening times. Thus, we conclude that the dendrite is responsible for the peak(s) seen in the 10 min IND. As discussed above, the secondary arms of the dendrite elongate the peak region in Fig. 4(a) toward j1 =S v ¼ j2 =S v ¼ 0, or flat interfaces, which would account for the peak representing planar interfaces seen in Fig. 6(a).
3.24E-04
4.32E-04
5.40E-04
Y
X
0.00E+00
1.08E-04
2.16E-04
3.24E-04
Z
4.32E-04
(a) 5.40E-04
0.00E+00
Y
X
1.08E-04
2.16E-04
3.24E-04
4.32E-04
5.40E-04
Y
Z
X
(b)
Z
(c)
Fig. 6. Interface normal distributions of the samples coarsened for: (a) 10 min, (b) 1060 min, (c) 6566 min. The far-hemisphere (negative) z-axis projections are shown.
J.L. Fife, P.W. Voorhees / Acta Materialia 57 (2009) 2418–2428
Because this dendrite disappears as coarsening time increases, the INDs for the later times are approximately isotropic, and quantitatively very similar. We find that INDs provide a particularly sensitive measure of the heterogeneity of the structure. During coarsening, the heterogeneity decreases and the structure becomes isotropic. The changes in the INDs can be seen clearly by determining the fraction of interface with normals nearly perpendicular to the x-;y- and z-directions [6]. Nearly perpendicular to a given direction is defined as any angle, h, within 10° of perpendicular to the axis in question. These values are then divided by the total corresponding interfacial area of the microstructure to give the fraction of interface in this interval. The fraction of interface nearly perpendicular to each direction for each coarsening time, shown in Fig. 7, is between 16% and 19%. This shows that despite the presence of the peak in the 10 min IND, the structures are, on average, isotropic in nature. This is most likely due to the equiaxed microstructure present prior to coarsening. Finally, to asses whether the structures are breaking up into solid particles, the interconnectivity of the solid phase of the microstructures at each coarsening time is evaluated. Interconnectivity of the solid phase increases from 87% to 95% to 96% after 10, 1060, and 6566 min, respectively. It should be noted that the solid regions not interconnected for the two later coarsening times are located at the edges of the reconstruction box and are most likely interconnected elsewhere in the sample. As stated above, the volume of data used for this calculation increases proportionally to the increase in characteristic length with coarsening time; thus, we can be confident we are capturing any size scale effects that might be present as coarsening time increases. The increase in interconnectivity as coarsen-
Fig. 7. The fraction of interface nearly perpendicular to each direction as a function of coarsening time for the three coarsened samples. Nearly perpendicular in each direction is defined as 10 .
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ing time increases is of the same order as the increase in positive mean curvature seen in Fig. 5. This indicates that the microstructure is decreasing interfacial free energy not by breaking up into solid particles but by increasing the length scale of the microstructure and decreasing the fraction of interfacial area with negative mean curvature. What is important about these results is that the assumptions made in traditional two-dimensional models of dendritic coarsening [4,16–19] do not apply to these equiaxed systems because the morphologies are so different than those assumed in the models. More experiments are necessary to elucidate the actual mechanisms for coarsening in these systems. 4.1. Comparison to an initially directionally solidified Al–Cu ingot We compare these results to an initially directionally solidified Al–26 wt:%Cu microstructure, which corresponds to 42% solid volume fraction. Comparing similar solid volume fractions allows us to directly examine the effects of the initial microstructure on coarsening. In the ISDs (Fig. 8(c) and (d)), the principal curvatures are again scaled by S v to eliminate effects caused by an increase in the characteristic length scale with coarsening time. The color bars in both the ISDs and the INDs (Fig. 8(e) and (f)) also have fixed minimum and maximum values, corresponding to the highest probability and strongest directionality, respectively. The characteristic length scale of the initially directionally solidified microstructures increases in a similar fashion to the equiaxed dendritic samples. In both cases, the characteristic length increases with t1=3 , and the amplitudes of the temporal power law are within 20% of each other. This occurs, however, without any evidence of self-similarity in the samples that are directionally solidified prior to coarsening. The three-dimensional reconstructions, Fig. 8(a) and (b), show the system evolving from a highly complex microstructure to one dominated by solid cylinders and solid cylindrical-like shapes. The change in the ISDs, Fig. 8(c) and (d), are consistent with these observations. The peak moves from region 1 in the 10 min sample and aligns with j1 =S v ¼ 0 in the 3600 min sample. The peak probability also increases by over a factor of 4 between the two coarsening times. This increase, along with the change in dominant shapes in the microstructure from solid convex shapes, such as those present at the tips of dendrites, to solid cylindrical-like shapes verify the lack of self-similarity in the microstructure. As reported by Kammer et al. [15], this volume fraction was ultimately coarsened for over 3 weeks, with solid cylinders remaining the dominant interfacial shape in the microstructure, and there was no evidence of break up into solid particles. These results differ greatly from the initially equiaxed dendritic microstructures. By comparison, no significant change in the morphology of the interfaces, other than
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Fig. 8. Three-dimensional reconstructions of Al-rich dendrites and corresponding scaled ISDs and INDs for an initially directionally solidified Al– 26 wt:%Cu microstructure coarsened for: (a [15], c, and e) 10 min and (b, d, and f) 3600 min.
the size scale increase, is observed in these equiaxed structures. For example, the peak probability in the ISD for the directionally solidified structures increases with comparable magnitude to characteristic length with coarsening time
and is over two times greater than the peak probability observed in any of the initially equiaxed structures. The distribution, Fig. 8(d), is also very compact, which indicates that the peak region is representative of a significant
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portion of the interfacial area in the structure, which is also not true of the ISDs for the initially equiaxed structures shown in Fig. 4. This is evident also in a visual inspection of the three-dimensional reconstructions, as solid cylinders dominate the 3600 min directionally solidified structure. Because the evolution of morphology in the directionally solidified structure occurs in the z-direction, parallel to the initial solidification direction, it is evident that the initial morphology of the structure is the cause of these differences. The far-hemisphere (negative), x-axis equal-area projection INDs, Fig. 8(e) and (f), are provided because the location of the peak value appears strongest in these projections and because it best displays the directionality that develops as coarsening time increases. It should be noted that in the 10 min microstructure, there is a strong 4-fold symmetry, but due to the scale of the color bar chosen for the plot, this is not clear. During coarsening, a 2fold symmetry develops with a significant increase in the strength of the peak. This IND is significant because, although there is a range of intensities, the locations of the probabilities span the length of the projection with the maximum probability located in the center. This indicates a strong preference for normals perpendicular to the x-direction, similar to the example of a hollow cylinder discussed previously. Thus, the IND is an illustration of a structure dominated by solid cylinders parallel to the directional solidification direction (z). The interconnectivity of the Al-rich dendrites is also examined for the directionally solidified microstructure. The interconnectivity of the solid phase decreases from 84.7% to 13.5% to 12.1% after 10 min, 3600 min and 3 weeks of coarsening, respectively. Although the lack of interconnectivity in the initially equiaxed dendritic microstructures could be attributed to sections of the structure intersecting the edges of the reconstruction box, and thus they may be connected elsewhere in the structure, this is not the case in the initially directionally solidified microstructure. There are three or four independent solid bodies after 10 min of coarsening for the directionally solidified structure, and as coarsening time increases, the number of independent solid bodies increases significantly, with at least 30 independent bodies in the 3 week coarsened structure. This is attributed to the formation of independent solid cylinders, discussed above. Thus, in the directionally solidified structure, the decrease in interfacial free energy is achieved by breaking up the structure into solid cylinders, while in the initially equiaxed dendritic structure, this is attained by increasing interconnectivity and decreasing the interfacial area with negative mean curvature. 5. Conclusions Three initially equiaxed dendritic Al–20 wt:%Cu microstructures were isothermally coarsened for 10, 1060 and 6566 min, respectively. The microstructures were analyzed in three dimensions by observing changes in characteristic
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length scale, ISDs and INDs. It was found that the characteristic length, S 1 v , scales linearly with the cube root of time. All evidence of the initially solidified dendritic microstructure disappeared after 10 min of isothermal coarsening, and the structure evolved into a highly interconnected globular-like structure with a combination of cylindrical-like and hyperbolic bodies, each with spherically capped ends. There was no evidence of break up into spherical solid particles. As coarsening time increased, the scaled ISDs showed microstructures dominated by convex and hyperbolic solid shapes (regions 1 and 2 in the ISDs) with only very small changes. There was also no preferential directionality within the microstructures because the interface normal distributions were approximately isotropic. Comparing the evolution of the initially equiaxed dendritic structure with its initially directionally solidified counterpart of a similar solid volume fraction provides further evidence that the initial solidification procedures and the microstructure present prior to coarsening have a significant impact on the evolution of the microstructure during coarsening. Acknowledgement This material is based upon work supported under a National Science Foundation Graduate Research Fellowship, which is gratefully acknowledged. This work is also supported by the Department of Energy Office of Basic Energy Science (Grant CNV0037736). The authors would like to thank Dr. Markus Rettenmayr of the University of Jena in Germany for providing these samples for study. The authors would also like to recognize R. Mendoza for use of his structures and irreplaceable assistance. Lastly, the authors would like to thank D. Kammer, D. Rowenhorst, and Y. Kwon for their invaluable insight and time spent in multiple discussions. References [1] Lifshitz IM, Slyozov VV. The kinetics of precipitation from supersaturated solid solutions. J Phys Chem Solids 1961;19:35–50. [2] Wagner C. Theorie der Alterung von Niederschla¨gen Durch Umlo¨sen. Z Electrochemistry 1961;65:581–91. [3] Bower TF, Brody HD, Flemings MC. Measurements of solute redistribution during dendritic solidification. Trans Metall Soc AIME 1966;236:624–34. [4] Marsh SP, Glicksman ME. Overview of geometric effects on coarsening of mushy zones. Metall Mater Trans A 1996;27A:557–67. [5] Mendoza R, Alkemper J, Voorhees PW. The morphological evolution of dendritic microstructures during coarsening. Metall Mater Trans A 2003;34A:481–9. [6] Kammer D, Voorhees PW. The morphological evolution of dendritic microstructures during coarsening. Acta Mater 2006;54:1549–58. [7] Alkemper J, Voorhees PW. Quantitative serial sectioning analysis. J Microsc 2001;201:388–94. [8] Drew DA. Evolution of geometric statistics. J Appl Math 1990;50:649–66. [9] Guillaume L, Florent D, Atilla B. Constant curvature region decomposition of 3D-meshes by a mixed approach vertex-triangle. WSCG 2004;12:1–3.
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