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The morphological evolution of dendritic microstructures during coarsening
Acta Materialia 54 (2006) 1549–1558 www.actamat-journals.com
The morphological evolution of dendritic microstructures during coarsening D. Kammer *, P.W. Voorhees Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Cook Hall, Evanston, IL 60208-3108, USA Received 17 March 2005; received in revised form 14 November 2005; accepted 20 November 2005 Available online 23 January 2006
Abstract The coarsening process of dendritic microstructures is studied in the Pb–Sn system using three-dimensional reconstructions. We analyze the morphology of the microstructure by determining the interfacial shape distribution, the probability of finding a patch of surface with a given pair of principal curvatures and its anisotropy through measurements of the probability of finding an interfacial normal in a certain direction. We find that the cube of the inverse surface area per unit volume increases linearly with time, despite the apparent lack of microstructural self-similarity. Interfacial normal distributions demonstrate a strong preferential directionality, specifically an evolution to twofold symmetry, as coarsening proceeds. During coarsening, the fraction of interface with normals perpendicular to the directional solidification direction increases dramatically. This preferred direction is a result of the existence of interfaces along the directional solidification direction that have a lower absolute value of the mean curvature than the surrounding interfaces. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dendritic microstructure; Three-dimensional reconstruction; Coarsening; Mean curvature
1. Introduction Dendritic microstructures are very commonly encountered following solidification of metallic alloys. A twophase region is created, consisting of the dendritic (solid) phase, embedded within the matrix (liquid) phase. This region is called the mushy zone and it evolves by a coarsening or Ostwald ripening process. Understanding coarsening and being able to study the morphology of the dendritic structure is of technological importance. Dendrites constitute the primary growth morphology during the early stages of solidification. Many properties of cast materials are intimately related to the dendritic morphology that is largely set by coarsening. Even if the effects of the dendritic microstructure are altered by subsequent heat treatments, they rarely fully disappear.
*
Corresponding author. Tel.: +1 847 491 3425. E-mail address: [email protected] (D. Kammer).
The driving force for coarsening is the variation in the mean solid–liquid interfacial curvature H. This is because there is an excess free energy associated with the presence of the interface. The system wants to minimize its free energy and does so through a mass transfer process, in which the total interfacial area is reduced over time. The importance of the variation in the mean curvature for this mass transfer to occur is illustrated in the Gibbs–Thomson equation for a binary alloy: C L ¼ C 1 þ lc H ;
ð1Þ
where CL is the composition in the liquid at the solid–liquid interface, C1 is the composition in 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 curvature of the dendritic surface, given as follows: 1 1 1 H¼ þ ; ð2Þ 2 R1 R2 where R1 and R2 are the principal radii of curvature.
1359-6454/$30.00 Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.031
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It can be seen from Eq. (1) that the concentration in the liquid at the interface depends on the mean interfacial curvature. Therefore, variations in curvature with position result in concentration gradients, which lead to fluxes of solute through the liquid phase. These diffusional fluxes can occur from regions of high interfacial mean curvature to regions of low interfacial mean curvature. It is important to note that these solute fluxes that are responsible for coarsening occur at essentially a constant volume fraction. Through this dissolution and reprecipitation process, the microstructure coarsens and the average length scale of the system increases and the total interfacial area decreases. Since the mass fluxes are a result of a spatially varying interfacial curvature, in order to understand fully the coarsening process, characterization of the interfacial curvature as a function of coarsening time is required. Much of the initial research devoted to dendritic coarsening was focused on measuring the secondary dendritic arm spacing k2 and proposing mechanisms aiming to describe the evolution of this secondary dendritic arm spacing [1–8]. This was partly because of the difficulty in measuring the three-dimensional microstructure, since k2 is easy to measure on a plane of section and was found to increase during coarsening in a similar way that the average particle radius R does for a system of spherical particles. In particular, the time dependence of k2 is described as [9]: 1=3
k2 t f ;
ð3Þ
1=3 tf
is the local solidification time, namely the time where over which the solid and liquid have coexisted. Although Eq. (3) may be useful in engineering applications, it does not provide information on the complexity of the dendritic structure. Moreover, measuring k2 after significant morphological changes have occurred poses problems, since very often k2 cannot be measured. A very good example of the above is provided by Marsh and Glicksman [10]. In their paper it is shown how a dendritic structure of a Sn–Bi alloy evolves to a structure with spheroidal particles over long coarsening times. Another length scale is, therefore, needed to characterize the coarsening process, a length scale that is shape-independent, so that regardless of how the microstructure evolves, it can still be used to describe the coarsening process. The parameter that will be used here is the surface area per unit volume, Sv. It has units of reciprocal length and as the microstructures coarsens, Sv decreases. The way that this length scale parameter is related to coarsening time is given as follows: 3 S 3 v ðtÞ S v ð0Þ ¼ Kt.
ð4Þ
This equation is analogous to the equation that was proposed by Lifshitz and Slyozov [11] and by Wagner [12] for a system with spherical particles. It was predicted then that the microstructure and specifically R will evolve with time as follows: 3
3
R ðtÞ R ð0Þ ¼ K LSW t;
ð5Þ
where RðtÞ is the average particle radius at a time t, Rð0Þ is the average particle radius at the beginning of coarsening and KLSW is the coarsening constant. In both Eqs. (4) and (5), the expressions contain the cube of a characteristic length scale measure (S 1 and R, v respectively) that increases linearly with time. Marsh and Glicksman [10] found that despite the drastic morphological changes from a dendritic microstructure to a spheroidal one, the time dependence of the specific interfacial area, Sv, was always described as 1=3 S 1 . v t
ð6Þ
The goal of our work is to analyze quantitatively the evolution of topologically complex dendritic structures during coarsening. As stated before, early research on dendritic structures [1–8] was limited to analysis in two dimensions. However, many morphological measures are three-dimensional properties, with the interfacial curvature being the most important one, since it is the driving force for coarsening. Therefore, by reconstructing the structure in three dimensions, we are able to measure the evolution of curvature over coarsening time. Quantitative representation of the curvature distribution allows one to better understand the effect of the interfacial shapes on the coarsening process. Moreover, with three-dimensional reconstructions, topological features can be revealed that are not apparent otherwise during coarsening, such as the transition from a typical dendrite to a cylindrical particle. Mendoza et al. [13] recently studied the evolution of Al dendrites in Al–Cu during coarsening. The present paper expands on that work, using a different system (Pb–Sn) and revealing some very different findings, as well as using a new method for presenting orientation distributions. 2. Experimental An ingot of Pb–80 wt.% Sn was produced by melting 99.99% Pb and 99.99% Sn and then casting in a mould. The cast rod was then swaged and directionally solidified in a Bridgman furnace to produce a uniform dendritic microstructure, consisting of 58% Sn-rich dendrites and 42% Pb–Sn eutectic phase. The rod was solidified using a temperature gradient of 1.22 K/mm and a velocity of 0.02 mm/s. In contrast to the typical utilization of a Bridgman furnace, in our case the rod was held immobile and the furnace was moved. This prevented any sort of contact and vibrations that could have possibly affected the solidification process and consequently the desired microstructure. Cylindrical samples (6 mm height, 12 mm diameter) were cut from the rod, using a wire-cutter in order to preserve the microstructure. The samples were then coarsened at 185 °C (2 °C above the eutectic temperature for the Pb–Sn system) for different coarsening times (3, 24, 158, 2880, 4320 and 5760 min). Subsequently, the samples were quenched in water, converting any liquid present during coarsening into eutectic solid. During quenching, additional material can be deposited on the interfaces. In order
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to avoid this, the samples were processed very close to the eutectic temperature, so that only a small degree of cooling is needed to nucleate the eutectic. This near-eutectic coarsening procedure that eliminates growth during quenching has been used before successfully for the Pb–Sn system [14] and by Kattamis et al. [2] and Mendoza et al. [13] for the Al–Cu system. Using a recently developed semi-automated serial sectioning technique [15], the samples were serial-sectioned with a distance of 4.75 lm between sections. For each section the sample was automatically etched, a digital image was obtained and the position of the sample was recorded using a linear variable differential transformer [15]. The images were thresholded into binary images and, using the recorded positioning data, they were aligned with respect to each other. This series of thresholded images was stacked, making it possible to reconstruct the microstructure in three dimensions. Having these threedimensional volumes, we were able to conduct surface area, curvature and orientation measurements of the microstructure. 3. Analysis The complex interfacial morphologies in dendritic solid– liquid mixtures are quantified by determining the probability of finding a patch of interface with a certain pair of principal curvatures, j1, j2. To determine these curvatures we measure the two invariants of the curvature tensor, jij, namely the mean curvature H and the Gaussian curvature K, given as follows [16]: H ¼ trfjij g ¼ 12ðj1 þ j2 Þ;
ð7Þ
K ¼ detfjij g ¼ j1 j2 .
ð8Þ
As discussed before, the variation of the mean curvature along the dendritic surface is responsible for a diffusion flux, which leads to coarsening. The mean curvature may be the driving force for coarsening but the importance of the Gaussian curvature is not to be neglected. There is a definite link between the evolution of the mean curvature and the Gaussian curvature [17]. A good example for a better understanding of this correlation is a saddle-shaped surface, where by definition j1 = j2. In this case H = 0 and according to Eq. (1), CL = C1, which means that the concentration in the liquid is equal to that of a planar interface, although the interface is obviously not planar. Therefore, both the mean and Gaussian curvatures are needed in order to describe the coarsening process. The mean and Gaussian curvatures can be determined using a variety of methods. Stokely and Wu [18] describe five different methods for this purpose. In most of their methods they approximate the surface patch by continuous mathematical functions. Another more recent method that is of great interest to us for future use is described by Guillaume et al. [19]. For our purposes in this paper, we use a method employed by Jinnai et al. [20], called the parallel
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surface method. Using the mean and Gaussian measured curvatures it is possible to determine the two principal curvatures j1 and j2 (see Eqs. (7) and (8)) and compute the probability of finding a patch of interface with a certain pair of principal curvatures. We call such a probability function the interfacial shape distribution (ISD). The ISD is presented using contour plots that are generated from three-dimensional probability plots. The ISD is our main tool for the representation of our curvature data. A map of the different regions and the corresponding interfacial shapes for this plot is shown in Fig. 1. Since by the definition that we employ j2 is the maximum principal curvature, j2 P j1 and all the contours must lie to the left of the j1 = j2 line. This map provides us with the following information: Interface patches with curvatures that lie on the j1 = j2 line correspond to spherical interfacial shapes, with liquid on the inside if j1 = j2 < 0 and solid on the inside if j1 = j2 > 0. Interface patches have a cylindrical shape when one principal curvature is zero. If j1 = 0 solid is inside the patch and if j2 = 0 liquid is inside the patch. Interface patches with j1 < 0 and j2 > 0 are saddleshaped. Interface patches with j1 > 0 and j2 > 0 are defined as convex towards the solid. Interface patches with j2 < 0 are defined as concave towards the solid. Interface patches with j1 = j2 = 0 are planar. Interface patches in region 1 of the map have H > 0, K > 0. Interface patches in region 2 of the map have H > 0, K < 0. Interface patches in region 3 of the map have H < 0, K < 0.
Fig. 1. Map of the local interfacial shapes for the interfacial shape distribution contour plots [13].
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Interface patches in region 4 of the map have H < 0, K > 0. In order to determine the orientation of the interfaces and to see if there is any preferential directionality in the microstructure we make use of the interface normals, n. The data are represented via spherical projections. All the interface normals are contained within a unit reference sphere with their origins at the center of the sphere and their ends on its surface. The sphere is then projected onto the projection plane, a plane tangent to the sphere and perpendicular to the axis along which the projection is made. In our case an equal-area projection is used and the projection is taken to be along the x-axis. The reason we use an equal-area projection is that it expands any features that exist near the center of the plot, which is where our data are primarily concentrated. Since our microstructures are not symmetrical, in order to represent all the data, two projections are needed for every structure. The plots correspond to the near-hemisphere projection (projecting along positive x) and the far-hemisphere projection (projecting along negative x), while maintaining a righthand coordinate system. However, it needs to be pointed out that the two plots look qualitatively similar. The binning of our data is done in three dimensions, on the reference sphere, and each bin stretches across the same area of the sphere. Often, in polar plots the projection precedes the binning of the data, which is consequently done in two dimensions. However, binning in three dimensions, before projecting, has the advantage of eliminating any potential projection-related effects, concerning the area of the bins when binning in two dimensions. In essence, the equal-area projection is a contour plot showing the probability distribution of the various interfacial normal orientations. In order to provide a better understanding of the interfacial normal distributions (INDs), two examples of an equal-area projection are described, that of a sphere and that of a cylinder. The distribution of interfacial normals of a single spherical object is completely isotropic and thus one would expect interfacial normals evenly distributed along all spatial orientations. The IND of a sphere shows a uniform probability distribution, telling us that the probability of finding an interfacial patch with a specific orientation is the same for any given orientation. Another example is that of a hollow cylinder, aligned along the z-axis. As we mentioned, we are using the convention that the projections will be taken along the x-axis. Compared to the sphere, this is an anisotropic shape, with all the interface normals perpendicular to z or equivalently parallel to the x–y plane. On the IND, this preferred orientation gives rise to an infinitesimally thin line of the same intensity (representing the equal probability) from one end of the projection to the other passing through the center of the projection. As mentioned earlier, the length scale being used to characterize the dendritic microstructures is the inverse surface area per unit volume S 1 v . Measuring this microstruc-
tural parameter is easily done, given the threedimensional reconstruction. Once the interfacial area of the three-dimensional reconstruction is triangulated, the sum of all these interfacial patches is divided by the volume of our reconstruction to give us Sv. 4. Results and discussion Three-dimensional reconstructions of the microstructures of four isothermally coarsened samples are shown in Figs. 2(a)–(d) for coarsening times of 3, 158, 2880 and 5760 min, respectively. The solid volume fraction of the samples is 58%. The reconstructions shown are based on serial sections taken 4.75 lm apart with the growth direction being parallel to the z-axis. The solid denotes the Sn dendrites while the voids in the structure represent the liquid (Pb–Sn) phase. What is obvious by looking at the microstructures of the samples is that they have coarsened over time. It can be seen that along with coarsening of course, comes an evident increase in the length scale. For the 3 min sample, the microstructure appears very dendritic and complex and there does not seem to be any preferential directionality. Although not clearly seen just by looking at the reconstruction but confirmed by our ISDs, solid cylindrical patches dispersed throughout the microstructure are the prevalent shape. Looking at the 158 min sample, the interfacial shapes start becoming more defined. Compared to the earlier time, a higher fraction of the interface seems to be aligned along the z-axis. The structure is definitely coarser and has thicker dendritic arms. This is greatly enhanced if one looks at the 2880 min sample. After having coarsened a sample for 2 days, it is these round-edged plate-like shapes that dominate the structure, along with a few solid cylinders. An even higher fraction of interface is aligned along the z-axis and the length scale of the microstructure has significantly increased. Finally, the sample coarsened for 5760 min shows the evolution of these plate-like shapes towards rod-like shapes or in many cases to well defined solid cylinders. Moreover, a further increase in the length scale can be seen. In order to quantify the increase in the length scale of the microstructures, the inverse surface area per unit volume was calculated for each of six different samples. These samples consist of the four samples discussed above, as well as samples coarsened for 24 min and 3 days. The results are shown in Fig. 3, where the characteristic length scale S 1 v is plotted as a function of the cube root of the coarsening time. A linear fit agrees quite well with the data, which means that the time dependence of our characteristic length scale is described by Eq. (4). What is very interesting is the existence of such a linear dependence, despite the lack of any self-similarity in our microstructures. The linear increase of our length scale measure with time proves also that the microstructure coarsened over time. In fact, if we take the least coarsened sample (3 min) and the most coarsened
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Fig. 2. Three-dimensional reconstructions of the coarsened samples: (a) 3 min, 50 sections; (b) 158 min, 43 sections; (c) 2880 min, 100 sections; (d) 5760 min, 100 sections.
Fig. 3. S 1 v as a function of the cube root of coarsening time, for the six samples coarsened. The samples have a volume fraction of solid of 58%.
sample (5760 min), we see that S 1 goes from 35.81 to v 125.69 lm, the length scale increasing by a factor of 3.5. Equivalently, the surface area per unit volume Sv decreases from 0.0279 to 0.00795 lm1, respectively. The ISDs are shown in Figs. 4(a)–(d). These ISDs represent the curvature data of the microstructures. They are read with the help of Fig. 1, which allows for the determination of the various interfacial shapes. Each plot is scaled by the inverse surface area per unit volume that corresponds to that particular microstructure. It should also
be noted that the range of probabilities represented by the color bar is independent of coarsening time. The area of nonzero probability of the non-scaled ISDs decreases with coarsening time, which agrees with the increase in the length scale of the microstructures. This decrease in area can be associated with the decrease in Sv. This explains why the area of nonzero probability in the scaled plots, Figs. 4(a)–(d), remains almost constant. The ISDs, however, do not scale, indicating the system’s lack of self-similarity. A general observation from the ISDs is that nearly all the interface patches have at least one positive principal curvature and only a very small percentage have both principal curvatures negative. Actually a significant percentage of the surface has both principal curvatures positive. In terms of interfacial shapes, this translates to dominance of convex shapes towards the solid over concave shapes. Also it can be seen that the majority of the interfaces have positive mean curvature H (regions 1 and 2 of Fig. 1). Specifically, taking the sample coarsened for 4 days as an example, we calculated that 93.84% of the interface patches have at least one positive principal curvature, 5.91% have both principal curvatures negative, 37.75% have both principal curvatures positive and 81.47% have positive mean curvature H. Monitoring the peak movement of the ISDs, a transition from solid cylindrical patches to saddle-shaped surfaces and then back towards solid cylinders can be observed. After 3 min of coarsening, the peak (blue) is centered along the j1/Sv = 0 line that corresponds to solid cylindrical patches at j2/Sv 0.8. The value of j2 is sufficiently large that it is unlikely for these patches to survive as coarsening progresses and the microstructure evolves. After 158 min of
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Fig. 4. ISDs of the coarsened samples: (a) 3 min; (b) 158 min; (c) 2880 min (2 days); (d) 5760 min (4 days).
coarsening, the peak (blue) has drifted to the saddle-shaped region of the ISD. The drift of the peak to this region corresponds to the creation of the round-edged plate-like shapes mentioned in the description of the three-dimensional reconstructions, and specifically to the sides of these shapes. As the microstructure changes significantly from 158 min of coarsening time to 2 days of coarsening time, so does the ISD. The distribution is narrower and has two main characteristics. First, the peak position remains in the saddle-shaped region for the sample coarsened for 2 days but the peak is now higher (red). This is in accord with the three-dimensional reconstruction, where the plate-like shapes have clearly become the dominant shape. Second, the creation of a secondary peak (turquoise color), part of a tongue-like region stretching along the j1/Sv = 0 line, can also be observed. The tongue-like region is the result of the solid cylindrical-like shapes and the round edges of the dominant round-edged plate-like shapes shown in the three-dimensional reconstruction of the sample coarsened for 2 days. With the microstructure evolving even more towards solid cylinders, the primary peak drifts
from the saddle-shaped region to the solid cylinder line for the sample coarsened for 4 days, making the tongue-like region the main region of the ISD. The peak (yellow) is now situated along the j1/Sv = 0 line, consistent with this sample’s microstructure, characterized by the coexistence of cylindrical-like shapes and well defined solid cylinders of various diameters. While the scaled value of j2 at the peak after 4 days of coarsening is similar to that after 3 min of coarsening, the unscaled value is considerably smaller. After 4 days of coarsening j2 0.005 lm1 and after 3 min of coarsening j2 0.022 lm1. We are also interested in determining the existence of any preferential directionality in our microstructures. The three-dimensional reconstructions give us a rough idea of what to expect and prepare us for the presence of a twofold symmetry, especially for the late coarsening times. Figs. 5(a)–(d) show the INDs for the four samples studied. The near-hemisphere projection of every microstructure is shown. However, as mentioned before, the far-hemisphere projections yield quite similar results, especially for late coarsening times. The projections were chosen to be along
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Fig. 5. INDs of the coarsened samples: (a) 3 min; (b) 158 min; (c) 2880 min (2 days); (d) 5760 min (4 days). The near-hemisphere projection for every sample is shown.
the x-axis, since the majority of the interface normals are nearly parallel to the x-axis. This is clearly evident if one looks at the reconstructions of the late coarsening times. Similarly to the ISDs, the color bar of the plots is fixed, based on the microstructure with the strongest directionality, namely the sample coarsened for 2 days. It should also be noted that before performing our calculations, the interface normals were rotated in the x–y plane. This does not affect our results in any way, since it corresponds to a rigid body rotation of the sample about the z-axis, and was simply done so that a better comparison between the INDs could be obtained. The IND of the 3 min sample is not an isotropic distribution but it is also hard to claim that there is a strong preferential directionality. This statement agrees with the fact that we are dealing with a very complex microstructure, at a very early coarsening stage, that has not had the time yet to evolve and develop specific orientations. This is not the case though with the IND of the 158 min sample. The primary peak is due to the preferred orientation of the thicker dendritic arms that appear during coars-
ening. The nonzero probability area, though, at and near the poles reveals that the microstructure still largely consists of interfaces with various orientations. It is only by looking at the IND of the 2880 min sample that a strong preferential directionality can be observed. This preferred orientation, in conjunction with the far-hemisphere projection, reveals a strong twofold symmetry. Now the majority of the interface has interface normals on the x–y plane and specifically along the x-axis. The round-edged plate-like structures are the dominant shape at this coarsening time and their normals are responsible for this peak. The IND of the 5760 min sample is similar to the IND of the 2880 min sample, indicating a definite twofold symmetry. However, the stripe is now thinner and the primary peak is lower. This is due to the microstructure’s evolution towards cylinders. The transition from the plate-like shapes to cylindrical and cylindrical-like shapes takes place during coarsening from 2880 to 5760 min. As described earlier, the IND of a cylinder is a line along the center of the projection. It is expected, therefore, to see the stripe getting thinner and of more uniform intensity.
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A striking result of these plots is that the microstructure starts as relatively isotropic and becomes more aligned during coarsening. The fact that the microstructure aligns itself along the z-axis as coarsening time proceeds is a result of the directional solidification process used to create the samples. We examine the mechanism that is behind this phenomenon. The fraction of interface with normals nearly perpendicular to the three directions was calculated for all six samples and is shown in Fig. 6 as a function of the cube root of coarsening time. The range that defines nearly perpendicular was chosen to be an angle of ±10° about a certain direction. For example, in order to assess the fraction of interface with normals nearly perpendicular to the z-axis, the percentage of normals within h = ±10° from perpendicularity was calculated, for all /. This is shown schematically in Fig. 7, in order to better illustrate the above calculation. The percentage of normals is then scaled by the corresponding total interfacial area to give the fraction of interface. The fraction of interface is similarly calculated for normals nearly perpendicular to the other two directions. Fig. 6 shows that interface normals that are nearly perpendicular to the z-direction dominate as coarsening time increases. Eventually, for sample coarsened for 4 days the fraction of interface with normals nearly perpendicular to z is 79%, a number that is consistent with the three-dimensional reconstruction of this sample (Fig. 2(d)). What is very interesting to note from the plot and not obvious from the three-dimensional reconstructions is that interfaces with normals nearly perpendicular to z do not compose the largest fraction of the interface at the two earliest coars-
Fig. 6. Fraction of interface area as a function of the cube root of coarsening time for the six samples coarsened from the Pb–80 wt.% Sn ingot. Shown is the fraction of interface with normals that are perpendicular to each of the three directions, ±10°.
Fig. 7. Schematic of the spherical segment that is covered by interface normals that are nearly perpendicular to the z-axis (h = ±10°), for all /.
ening times. On the contrary, interfaces with interface normals nearly perpendicular to x compose the largest fraction of interface. After 3 min of coarsening time the interface normals that are nearly perpendicular to x compose 26% of the surface area, as opposed to interface normals that are nearly perpendicular to z with 19% of the surface. After 24 min of coarsening the percentage becomes about 20% for both directions. It is only until after 158 min of coarsening that the dominance of interfaces with normals nearly perpendicular to z becomes clear. Therefore, the answer as to why the microstructure becomes aligned with normals perpendicular to z clearly is not due to the interfaces with normals perpendicular to z having the largest interfacial area. The reason why interfaces along the z-axis dominate over time is provided if one takes a closer look at the three-dimensional reconstructions. Fig. 8 shows the three-dimensional reconstructions of four samples but now the liquid is shown, while the solid is transparent. Also, the view is such that the z-axis is out of the plane of the page. For better viewing, only a portion from the reconstructions of samples coarsened for shorter times is shown. Specifically, Figs. 8(a) and (b) share a common microstructural feature, namely the existence of dendrites bearing the same characteristic shape. The voids along the z-axis comprise the primary stem of these dendrites, while the parts that look like dimples on the surface of the reconstruction are the side-branches of these dendrites (see the arrows in Figs. 8(a) and (b)). At the onset of coarsening (Fig. 8(a)), there are a number of these types of stems. Impressively enough they maintain their shape, while also growing, as coarsening proceeds. This is the case for the 24 min sample (not shown here) and for the 158 min sample (Fig. 8(b)). Eventually, these dendrites evolve to the round-edged plate-like dendrites of Fig. 8(c) and these in turn evolve to cylinders or cylindrical-like shapes shown in Fig. 8(d), always along the directional solidification direction.
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Fig. 8. Three-dimensional reconstructions of the coarsened samples: (a) 3 min; (b) 158 min; (c) 2880 min; (d) 5760 min. The liquid is shown, while the solid is transparent. The z-axis is out of the plane of the page. A void (stem) and a dimple (side-branch) are shown in (a) and (b).
Our next task then is to explain what is it that allows these dendrite stems to persist in their shape and why they maintain their directionality along the z-axis as coarsening proceeds. This persistence leads to the dramatic increase of the fraction of interface with normals nearly perpendicular to z over time. Fig. 9 illustrates why. The interface is shown here (both liquid and solid are transparent), colored by the mean curvature H, for two coarsened samples: the 24 min sample (Fig. 9(a)) and the 158 min sample (Fig. 9(b)). For better viewing, a small region of each structure is only
presented. It should be noted, though, that the normals of the interface that encapsulates the solid, or voids shown in Fig. 8, are nearly perpendicular to the z-axis. The key feature that Fig. 9 allows us to see is that the dendrite stems that have interface normals perpendicular to the z-axis have a lower mean curvature than their surroundings. As coarsening proceeds, solute diffuses from regions of high absolute value of the mean curvature to regions of low absolute value of the mean curvature. In our case, the interface that forms the stems, which is along z, has lower mean
Fig. 9. Interface colored by the mean curvature H. Shown are regions from the three-dimensional structures of two samples: (a) 24 min; (b) 158 min. Both liquid and solid are transparent. Some dendrite stems are shown with an arrow.
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curvature (blue and turquoise) than the interface that forms the side-branches (green, yellow, orange and red). Therefore, the regions near the stem coarsen at the expense of the surrounding interface. It can be seen from Fig. 9(a) that the mean curvature along a stem varies with position. One would thus expect that stem to evolve into a cylinder over time. However, at shorter coarsening times this is not the case. The reason is that the stem is surrounded by interfaces (side-branches) with curvatures that are larger in absolute value than the variation along the stem’s interface. As a result, the interface that forms these stems becomes coarser, while maintaining its shape. When the magnitude of the mean curvature along the stem becomes comparable to the magnitude of the mean curvature of the surrounding interfaces, then the stem cannot maintain its shape and eventually evolves to the round-edged plate-like shapes seen in the sample coarsened for 2 days. This process continues and these round-edged plate-like shapes evolve to the cylindrical-like shapes and finally to the cylinders seen in the sample coarsened for 4 days. Most of the interface that is along the z-axis is characterized by the smallest absolute value of the mean curvature throughout each structure. Therefore, that interface maintains its directionality over time and, due to growth of these structures, leads to a microstructure with a very high fraction of interface with normals nearly perpendicular to z for long coarsening times. 5. Conclusions The microstructures of four Pb–80 wt.% Sn dendritic samples were studied. The samples were isothermally coarsened for 3, 158, 2880 and 5760 min. The three-dimensional reconstructions show the drastic morphological changes that occur as the microstructures evolve over coarsening time. Surface area per unit volume measurements allow us to quantify the increase in the length scale of the structures. Probability density contour plots of the principal curvatures (ISDs) were used to quantify the microstructural evolution. The most probable interfacial shapes were determined and a transition from small solid cylindrical shapes to saddle-shaped surfaces and finally back to large, well defined solid cylinders was observed. Equal-area projections of the interface normals permit
the determination of the existence of any preferential directionality in our microstructure. Strong preferential directionality, namely twofold symmetry, was noticed for late coarsening times, but not at earlier times. The microstructures are characterized by interfaces with normals perpendicular to the solidification direction with lower curvature than their surroundings. This leads to an increased fraction of interface with normals perpendicular to the z-axis during later coarsening times and the strong alignment that is observed experimentally. Acknowledgements The financial support from NASA, Grant No. NAG81660 is gratefully acknowledged. Also, discussions with D. Rowenhorst and R. Mendoza were invaluable. References [1] Kattamis TZ, Flemings MC. Trans Metall Soc AIME 1966;236:1523–32. [2] Kattamis TZ, Coughlin JC, Flemings MC. Trans Metall Soc AIME 1967;239:1504–11. [3] Kattamis TZ, Holmberg UT, Flemings MC. J Inst Met 1967;95:343–7. [4] Chernov AA. Kristallografiya 1956;1:583–7. [5] Klia MO. Kristallografiya 1956;65:576–81. [6] Kahlweit M. Scripta Metall 1968;2:251–4. [7] Kirkwood DH. J Mater Sci Eng 1985;73:L1–4. [8] Young KP, Kirkwood DH. Metall Trans A 1975;6A:197–205. [9] Bower TF, Brody HD, Flemings MC. Trans Metall Soc AIME 1966;236:624–34. [10] Marsh SP, Glicksman ME. Metall Mater Trans A 1996;27A:557–67. [11] Lifshitz IM, Slyozov VV. J Phys Chem Solids 1961;19:35–50. [12] Wagner CZ. Elektrochem 1961;65:581–91. [13] Mendoza R, Alkemper J, Voorhees PW. Metall Mater Trans A 2003;27A:481–9. [14] Alkemper J, Snyder VA, Akaiwa N, Voorhees PW. Metall Trans 1999;11:77–91. [15] Alkemper J, Voorhees PW. J Microsc 2001;201:388–94. [16] Alkemper J, Voorhees PW. Acta Mater 2001;49:897–902. [17] Drew DA. J Appl Math 1990;50:649–66. [18] Stokely EM, Wu SY. IEEE Trans Pattern Anal Mach Intell 1992;14:833–40. [19] Lavoue´ G, Dupont F, Baskurt A. WSCG 2004;12:245–52. [20] Jinnai H, Koga T, Nishikawa Y, Hashimoto T, Hyde S. Phys Rev Lett 1997;78:2248–51.
The morphological evolution of dendritic microstructures during coarsening D. Kammer *, P.W. Voorhees Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Cook Hall, Evanston, IL 60208-3108, USA Received 17 March 2005; received in revised form 14 November 2005; accepted 20 November 2005 Available online 23 January 2006
Abstract The coarsening process of dendritic microstructures is studied in the Pb–Sn system using three-dimensional reconstructions. We analyze the morphology of the microstructure by determining the interfacial shape distribution, the probability of finding a patch of surface with a given pair of principal curvatures and its anisotropy through measurements of the probability of finding an interfacial normal in a certain direction. We find that the cube of the inverse surface area per unit volume increases linearly with time, despite the apparent lack of microstructural self-similarity. Interfacial normal distributions demonstrate a strong preferential directionality, specifically an evolution to twofold symmetry, as coarsening proceeds. During coarsening, the fraction of interface with normals perpendicular to the directional solidification direction increases dramatically. This preferred direction is a result of the existence of interfaces along the directional solidification direction that have a lower absolute value of the mean curvature than the surrounding interfaces. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dendritic microstructure; Three-dimensional reconstruction; Coarsening; Mean curvature
1. Introduction Dendritic microstructures are very commonly encountered following solidification of metallic alloys. A twophase region is created, consisting of the dendritic (solid) phase, embedded within the matrix (liquid) phase. This region is called the mushy zone and it evolves by a coarsening or Ostwald ripening process. Understanding coarsening and being able to study the morphology of the dendritic structure is of technological importance. Dendrites constitute the primary growth morphology during the early stages of solidification. Many properties of cast materials are intimately related to the dendritic morphology that is largely set by coarsening. Even if the effects of the dendritic microstructure are altered by subsequent heat treatments, they rarely fully disappear.
*
Corresponding author. Tel.: +1 847 491 3425. E-mail address: [email protected] (D. Kammer).
The driving force for coarsening is the variation in the mean solid–liquid interfacial curvature H. This is because there is an excess free energy associated with the presence of the interface. The system wants to minimize its free energy and does so through a mass transfer process, in which the total interfacial area is reduced over time. The importance of the variation in the mean curvature for this mass transfer to occur is illustrated in the Gibbs–Thomson equation for a binary alloy: C L ¼ C 1 þ lc H ;
ð1Þ
where CL is the composition in the liquid at the solid–liquid interface, C1 is the composition in 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 curvature of the dendritic surface, given as follows: 1 1 1 H¼ þ ; ð2Þ 2 R1 R2 where R1 and R2 are the principal radii of curvature.
1359-6454/$30.00 Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.031
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It can be seen from Eq. (1) that the concentration in the liquid at the interface depends on the mean interfacial curvature. Therefore, variations in curvature with position result in concentration gradients, which lead to fluxes of solute through the liquid phase. These diffusional fluxes can occur from regions of high interfacial mean curvature to regions of low interfacial mean curvature. It is important to note that these solute fluxes that are responsible for coarsening occur at essentially a constant volume fraction. Through this dissolution and reprecipitation process, the microstructure coarsens and the average length scale of the system increases and the total interfacial area decreases. Since the mass fluxes are a result of a spatially varying interfacial curvature, in order to understand fully the coarsening process, characterization of the interfacial curvature as a function of coarsening time is required. Much of the initial research devoted to dendritic coarsening was focused on measuring the secondary dendritic arm spacing k2 and proposing mechanisms aiming to describe the evolution of this secondary dendritic arm spacing [1–8]. This was partly because of the difficulty in measuring the three-dimensional microstructure, since k2 is easy to measure on a plane of section and was found to increase during coarsening in a similar way that the average particle radius R does for a system of spherical particles. In particular, the time dependence of k2 is described as [9]: 1=3
k2 t f ;
ð3Þ
1=3 tf
is the local solidification time, namely the time where over which the solid and liquid have coexisted. Although Eq. (3) may be useful in engineering applications, it does not provide information on the complexity of the dendritic structure. Moreover, measuring k2 after significant morphological changes have occurred poses problems, since very often k2 cannot be measured. A very good example of the above is provided by Marsh and Glicksman [10]. In their paper it is shown how a dendritic structure of a Sn–Bi alloy evolves to a structure with spheroidal particles over long coarsening times. Another length scale is, therefore, needed to characterize the coarsening process, a length scale that is shape-independent, so that regardless of how the microstructure evolves, it can still be used to describe the coarsening process. The parameter that will be used here is the surface area per unit volume, Sv. It has units of reciprocal length and as the microstructures coarsens, Sv decreases. The way that this length scale parameter is related to coarsening time is given as follows: 3 S 3 v ðtÞ S v ð0Þ ¼ Kt.
ð4Þ
This equation is analogous to the equation that was proposed by Lifshitz and Slyozov [11] and by Wagner [12] for a system with spherical particles. It was predicted then that the microstructure and specifically R will evolve with time as follows: 3
3
R ðtÞ R ð0Þ ¼ K LSW t;
ð5Þ
where RðtÞ is the average particle radius at a time t, Rð0Þ is the average particle radius at the beginning of coarsening and KLSW is the coarsening constant. In both Eqs. (4) and (5), the expressions contain the cube of a characteristic length scale measure (S 1 and R, v respectively) that increases linearly with time. Marsh and Glicksman [10] found that despite the drastic morphological changes from a dendritic microstructure to a spheroidal one, the time dependence of the specific interfacial area, Sv, was always described as 1=3 S 1 . v t
ð6Þ
The goal of our work is to analyze quantitatively the evolution of topologically complex dendritic structures during coarsening. As stated before, early research on dendritic structures [1–8] was limited to analysis in two dimensions. However, many morphological measures are three-dimensional properties, with the interfacial curvature being the most important one, since it is the driving force for coarsening. Therefore, by reconstructing the structure in three dimensions, we are able to measure the evolution of curvature over coarsening time. Quantitative representation of the curvature distribution allows one to better understand the effect of the interfacial shapes on the coarsening process. Moreover, with three-dimensional reconstructions, topological features can be revealed that are not apparent otherwise during coarsening, such as the transition from a typical dendrite to a cylindrical particle. Mendoza et al. [13] recently studied the evolution of Al dendrites in Al–Cu during coarsening. The present paper expands on that work, using a different system (Pb–Sn) and revealing some very different findings, as well as using a new method for presenting orientation distributions. 2. Experimental An ingot of Pb–80 wt.% Sn was produced by melting 99.99% Pb and 99.99% Sn and then casting in a mould. The cast rod was then swaged and directionally solidified in a Bridgman furnace to produce a uniform dendritic microstructure, consisting of 58% Sn-rich dendrites and 42% Pb–Sn eutectic phase. The rod was solidified using a temperature gradient of 1.22 K/mm and a velocity of 0.02 mm/s. In contrast to the typical utilization of a Bridgman furnace, in our case the rod was held immobile and the furnace was moved. This prevented any sort of contact and vibrations that could have possibly affected the solidification process and consequently the desired microstructure. Cylindrical samples (6 mm height, 12 mm diameter) were cut from the rod, using a wire-cutter in order to preserve the microstructure. The samples were then coarsened at 185 °C (2 °C above the eutectic temperature for the Pb–Sn system) for different coarsening times (3, 24, 158, 2880, 4320 and 5760 min). Subsequently, the samples were quenched in water, converting any liquid present during coarsening into eutectic solid. During quenching, additional material can be deposited on the interfaces. In order
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to avoid this, the samples were processed very close to the eutectic temperature, so that only a small degree of cooling is needed to nucleate the eutectic. This near-eutectic coarsening procedure that eliminates growth during quenching has been used before successfully for the Pb–Sn system [14] and by Kattamis et al. [2] and Mendoza et al. [13] for the Al–Cu system. Using a recently developed semi-automated serial sectioning technique [15], the samples were serial-sectioned with a distance of 4.75 lm between sections. For each section the sample was automatically etched, a digital image was obtained and the position of the sample was recorded using a linear variable differential transformer [15]. The images were thresholded into binary images and, using the recorded positioning data, they were aligned with respect to each other. This series of thresholded images was stacked, making it possible to reconstruct the microstructure in three dimensions. Having these threedimensional volumes, we were able to conduct surface area, curvature and orientation measurements of the microstructure. 3. Analysis The complex interfacial morphologies in dendritic solid– liquid mixtures are quantified by determining the probability of finding a patch of interface with a certain pair of principal curvatures, j1, j2. To determine these curvatures we measure the two invariants of the curvature tensor, jij, namely the mean curvature H and the Gaussian curvature K, given as follows [16]: H ¼ trfjij g ¼ 12ðj1 þ j2 Þ;
ð7Þ
K ¼ detfjij g ¼ j1 j2 .
ð8Þ
As discussed before, the variation of the mean curvature along the dendritic surface is responsible for a diffusion flux, which leads to coarsening. The mean curvature may be the driving force for coarsening but the importance of the Gaussian curvature is not to be neglected. There is a definite link between the evolution of the mean curvature and the Gaussian curvature [17]. A good example for a better understanding of this correlation is a saddle-shaped surface, where by definition j1 = j2. In this case H = 0 and according to Eq. (1), CL = C1, which means that the concentration in the liquid is equal to that of a planar interface, although the interface is obviously not planar. Therefore, both the mean and Gaussian curvatures are needed in order to describe the coarsening process. The mean and Gaussian curvatures can be determined using a variety of methods. Stokely and Wu [18] describe five different methods for this purpose. In most of their methods they approximate the surface patch by continuous mathematical functions. Another more recent method that is of great interest to us for future use is described by Guillaume et al. [19]. For our purposes in this paper, we use a method employed by Jinnai et al. [20], called the parallel
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surface method. Using the mean and Gaussian measured curvatures it is possible to determine the two principal curvatures j1 and j2 (see Eqs. (7) and (8)) and compute the probability of finding a patch of interface with a certain pair of principal curvatures. We call such a probability function the interfacial shape distribution (ISD). The ISD is presented using contour plots that are generated from three-dimensional probability plots. The ISD is our main tool for the representation of our curvature data. A map of the different regions and the corresponding interfacial shapes for this plot is shown in Fig. 1. Since by the definition that we employ j2 is the maximum principal curvature, j2 P j1 and all the contours must lie to the left of the j1 = j2 line. This map provides us with the following information: Interface patches with curvatures that lie on the j1 = j2 line correspond to spherical interfacial shapes, with liquid on the inside if j1 = j2 < 0 and solid on the inside if j1 = j2 > 0. Interface patches have a cylindrical shape when one principal curvature is zero. If j1 = 0 solid is inside the patch and if j2 = 0 liquid is inside the patch. Interface patches with j1 < 0 and j2 > 0 are saddleshaped. Interface patches with j1 > 0 and j2 > 0 are defined as convex towards the solid. Interface patches with j2 < 0 are defined as concave towards the solid. Interface patches with j1 = j2 = 0 are planar. Interface patches in region 1 of the map have H > 0, K > 0. Interface patches in region 2 of the map have H > 0, K < 0. Interface patches in region 3 of the map have H < 0, K < 0.
Fig. 1. Map of the local interfacial shapes for the interfacial shape distribution contour plots [13].
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Interface patches in region 4 of the map have H < 0, K > 0. In order to determine the orientation of the interfaces and to see if there is any preferential directionality in the microstructure we make use of the interface normals, n. The data are represented via spherical projections. All the interface normals are contained within a unit reference sphere with their origins at the center of the sphere and their ends on its surface. The sphere is then projected onto the projection plane, a plane tangent to the sphere and perpendicular to the axis along which the projection is made. In our case an equal-area projection is used and the projection is taken to be along the x-axis. The reason we use an equal-area projection is that it expands any features that exist near the center of the plot, which is where our data are primarily concentrated. Since our microstructures are not symmetrical, in order to represent all the data, two projections are needed for every structure. The plots correspond to the near-hemisphere projection (projecting along positive x) and the far-hemisphere projection (projecting along negative x), while maintaining a righthand coordinate system. However, it needs to be pointed out that the two plots look qualitatively similar. The binning of our data is done in three dimensions, on the reference sphere, and each bin stretches across the same area of the sphere. Often, in polar plots the projection precedes the binning of the data, which is consequently done in two dimensions. However, binning in three dimensions, before projecting, has the advantage of eliminating any potential projection-related effects, concerning the area of the bins when binning in two dimensions. In essence, the equal-area projection is a contour plot showing the probability distribution of the various interfacial normal orientations. In order to provide a better understanding of the interfacial normal distributions (INDs), two examples of an equal-area projection are described, that of a sphere and that of a cylinder. The distribution of interfacial normals of a single spherical object is completely isotropic and thus one would expect interfacial normals evenly distributed along all spatial orientations. The IND of a sphere shows a uniform probability distribution, telling us that the probability of finding an interfacial patch with a specific orientation is the same for any given orientation. Another example is that of a hollow cylinder, aligned along the z-axis. As we mentioned, we are using the convention that the projections will be taken along the x-axis. Compared to the sphere, this is an anisotropic shape, with all the interface normals perpendicular to z or equivalently parallel to the x–y plane. On the IND, this preferred orientation gives rise to an infinitesimally thin line of the same intensity (representing the equal probability) from one end of the projection to the other passing through the center of the projection. As mentioned earlier, the length scale being used to characterize the dendritic microstructures is the inverse surface area per unit volume S 1 v . Measuring this microstruc-
tural parameter is easily done, given the threedimensional reconstruction. Once the interfacial area of the three-dimensional reconstruction is triangulated, the sum of all these interfacial patches is divided by the volume of our reconstruction to give us Sv. 4. Results and discussion Three-dimensional reconstructions of the microstructures of four isothermally coarsened samples are shown in Figs. 2(a)–(d) for coarsening times of 3, 158, 2880 and 5760 min, respectively. The solid volume fraction of the samples is 58%. The reconstructions shown are based on serial sections taken 4.75 lm apart with the growth direction being parallel to the z-axis. The solid denotes the Sn dendrites while the voids in the structure represent the liquid (Pb–Sn) phase. What is obvious by looking at the microstructures of the samples is that they have coarsened over time. It can be seen that along with coarsening of course, comes an evident increase in the length scale. For the 3 min sample, the microstructure appears very dendritic and complex and there does not seem to be any preferential directionality. Although not clearly seen just by looking at the reconstruction but confirmed by our ISDs, solid cylindrical patches dispersed throughout the microstructure are the prevalent shape. Looking at the 158 min sample, the interfacial shapes start becoming more defined. Compared to the earlier time, a higher fraction of the interface seems to be aligned along the z-axis. The structure is definitely coarser and has thicker dendritic arms. This is greatly enhanced if one looks at the 2880 min sample. After having coarsened a sample for 2 days, it is these round-edged plate-like shapes that dominate the structure, along with a few solid cylinders. An even higher fraction of interface is aligned along the z-axis and the length scale of the microstructure has significantly increased. Finally, the sample coarsened for 5760 min shows the evolution of these plate-like shapes towards rod-like shapes or in many cases to well defined solid cylinders. Moreover, a further increase in the length scale can be seen. In order to quantify the increase in the length scale of the microstructures, the inverse surface area per unit volume was calculated for each of six different samples. These samples consist of the four samples discussed above, as well as samples coarsened for 24 min and 3 days. The results are shown in Fig. 3, where the characteristic length scale S 1 v is plotted as a function of the cube root of the coarsening time. A linear fit agrees quite well with the data, which means that the time dependence of our characteristic length scale is described by Eq. (4). What is very interesting is the existence of such a linear dependence, despite the lack of any self-similarity in our microstructures. The linear increase of our length scale measure with time proves also that the microstructure coarsened over time. In fact, if we take the least coarsened sample (3 min) and the most coarsened
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Fig. 2. Three-dimensional reconstructions of the coarsened samples: (a) 3 min, 50 sections; (b) 158 min, 43 sections; (c) 2880 min, 100 sections; (d) 5760 min, 100 sections.
Fig. 3. S 1 v as a function of the cube root of coarsening time, for the six samples coarsened. The samples have a volume fraction of solid of 58%.
sample (5760 min), we see that S 1 goes from 35.81 to v 125.69 lm, the length scale increasing by a factor of 3.5. Equivalently, the surface area per unit volume Sv decreases from 0.0279 to 0.00795 lm1, respectively. The ISDs are shown in Figs. 4(a)–(d). These ISDs represent the curvature data of the microstructures. They are read with the help of Fig. 1, which allows for the determination of the various interfacial shapes. Each plot is scaled by the inverse surface area per unit volume that corresponds to that particular microstructure. It should also
be noted that the range of probabilities represented by the color bar is independent of coarsening time. The area of nonzero probability of the non-scaled ISDs decreases with coarsening time, which agrees with the increase in the length scale of the microstructures. This decrease in area can be associated with the decrease in Sv. This explains why the area of nonzero probability in the scaled plots, Figs. 4(a)–(d), remains almost constant. The ISDs, however, do not scale, indicating the system’s lack of self-similarity. A general observation from the ISDs is that nearly all the interface patches have at least one positive principal curvature and only a very small percentage have both principal curvatures negative. Actually a significant percentage of the surface has both principal curvatures positive. In terms of interfacial shapes, this translates to dominance of convex shapes towards the solid over concave shapes. Also it can be seen that the majority of the interfaces have positive mean curvature H (regions 1 and 2 of Fig. 1). Specifically, taking the sample coarsened for 4 days as an example, we calculated that 93.84% of the interface patches have at least one positive principal curvature, 5.91% have both principal curvatures negative, 37.75% have both principal curvatures positive and 81.47% have positive mean curvature H. Monitoring the peak movement of the ISDs, a transition from solid cylindrical patches to saddle-shaped surfaces and then back towards solid cylinders can be observed. After 3 min of coarsening, the peak (blue) is centered along the j1/Sv = 0 line that corresponds to solid cylindrical patches at j2/Sv 0.8. The value of j2 is sufficiently large that it is unlikely for these patches to survive as coarsening progresses and the microstructure evolves. After 158 min of
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Fig. 4. ISDs of the coarsened samples: (a) 3 min; (b) 158 min; (c) 2880 min (2 days); (d) 5760 min (4 days).
coarsening, the peak (blue) has drifted to the saddle-shaped region of the ISD. The drift of the peak to this region corresponds to the creation of the round-edged plate-like shapes mentioned in the description of the three-dimensional reconstructions, and specifically to the sides of these shapes. As the microstructure changes significantly from 158 min of coarsening time to 2 days of coarsening time, so does the ISD. The distribution is narrower and has two main characteristics. First, the peak position remains in the saddle-shaped region for the sample coarsened for 2 days but the peak is now higher (red). This is in accord with the three-dimensional reconstruction, where the plate-like shapes have clearly become the dominant shape. Second, the creation of a secondary peak (turquoise color), part of a tongue-like region stretching along the j1/Sv = 0 line, can also be observed. The tongue-like region is the result of the solid cylindrical-like shapes and the round edges of the dominant round-edged plate-like shapes shown in the three-dimensional reconstruction of the sample coarsened for 2 days. With the microstructure evolving even more towards solid cylinders, the primary peak drifts
from the saddle-shaped region to the solid cylinder line for the sample coarsened for 4 days, making the tongue-like region the main region of the ISD. The peak (yellow) is now situated along the j1/Sv = 0 line, consistent with this sample’s microstructure, characterized by the coexistence of cylindrical-like shapes and well defined solid cylinders of various diameters. While the scaled value of j2 at the peak after 4 days of coarsening is similar to that after 3 min of coarsening, the unscaled value is considerably smaller. After 4 days of coarsening j2 0.005 lm1 and after 3 min of coarsening j2 0.022 lm1. We are also interested in determining the existence of any preferential directionality in our microstructures. The three-dimensional reconstructions give us a rough idea of what to expect and prepare us for the presence of a twofold symmetry, especially for the late coarsening times. Figs. 5(a)–(d) show the INDs for the four samples studied. The near-hemisphere projection of every microstructure is shown. However, as mentioned before, the far-hemisphere projections yield quite similar results, especially for late coarsening times. The projections were chosen to be along
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Fig. 5. INDs of the coarsened samples: (a) 3 min; (b) 158 min; (c) 2880 min (2 days); (d) 5760 min (4 days). The near-hemisphere projection for every sample is shown.
the x-axis, since the majority of the interface normals are nearly parallel to the x-axis. This is clearly evident if one looks at the reconstructions of the late coarsening times. Similarly to the ISDs, the color bar of the plots is fixed, based on the microstructure with the strongest directionality, namely the sample coarsened for 2 days. It should also be noted that before performing our calculations, the interface normals were rotated in the x–y plane. This does not affect our results in any way, since it corresponds to a rigid body rotation of the sample about the z-axis, and was simply done so that a better comparison between the INDs could be obtained. The IND of the 3 min sample is not an isotropic distribution but it is also hard to claim that there is a strong preferential directionality. This statement agrees with the fact that we are dealing with a very complex microstructure, at a very early coarsening stage, that has not had the time yet to evolve and develop specific orientations. This is not the case though with the IND of the 158 min sample. The primary peak is due to the preferred orientation of the thicker dendritic arms that appear during coars-
ening. The nonzero probability area, though, at and near the poles reveals that the microstructure still largely consists of interfaces with various orientations. It is only by looking at the IND of the 2880 min sample that a strong preferential directionality can be observed. This preferred orientation, in conjunction with the far-hemisphere projection, reveals a strong twofold symmetry. Now the majority of the interface has interface normals on the x–y plane and specifically along the x-axis. The round-edged plate-like structures are the dominant shape at this coarsening time and their normals are responsible for this peak. The IND of the 5760 min sample is similar to the IND of the 2880 min sample, indicating a definite twofold symmetry. However, the stripe is now thinner and the primary peak is lower. This is due to the microstructure’s evolution towards cylinders. The transition from the plate-like shapes to cylindrical and cylindrical-like shapes takes place during coarsening from 2880 to 5760 min. As described earlier, the IND of a cylinder is a line along the center of the projection. It is expected, therefore, to see the stripe getting thinner and of more uniform intensity.
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A striking result of these plots is that the microstructure starts as relatively isotropic and becomes more aligned during coarsening. The fact that the microstructure aligns itself along the z-axis as coarsening time proceeds is a result of the directional solidification process used to create the samples. We examine the mechanism that is behind this phenomenon. The fraction of interface with normals nearly perpendicular to the three directions was calculated for all six samples and is shown in Fig. 6 as a function of the cube root of coarsening time. The range that defines nearly perpendicular was chosen to be an angle of ±10° about a certain direction. For example, in order to assess the fraction of interface with normals nearly perpendicular to the z-axis, the percentage of normals within h = ±10° from perpendicularity was calculated, for all /. This is shown schematically in Fig. 7, in order to better illustrate the above calculation. The percentage of normals is then scaled by the corresponding total interfacial area to give the fraction of interface. The fraction of interface is similarly calculated for normals nearly perpendicular to the other two directions. Fig. 6 shows that interface normals that are nearly perpendicular to the z-direction dominate as coarsening time increases. Eventually, for sample coarsened for 4 days the fraction of interface with normals nearly perpendicular to z is 79%, a number that is consistent with the three-dimensional reconstruction of this sample (Fig. 2(d)). What is very interesting to note from the plot and not obvious from the three-dimensional reconstructions is that interfaces with normals nearly perpendicular to z do not compose the largest fraction of the interface at the two earliest coars-
Fig. 6. Fraction of interface area as a function of the cube root of coarsening time for the six samples coarsened from the Pb–80 wt.% Sn ingot. Shown is the fraction of interface with normals that are perpendicular to each of the three directions, ±10°.
Fig. 7. Schematic of the spherical segment that is covered by interface normals that are nearly perpendicular to the z-axis (h = ±10°), for all /.
ening times. On the contrary, interfaces with interface normals nearly perpendicular to x compose the largest fraction of interface. After 3 min of coarsening time the interface normals that are nearly perpendicular to x compose 26% of the surface area, as opposed to interface normals that are nearly perpendicular to z with 19% of the surface. After 24 min of coarsening the percentage becomes about 20% for both directions. It is only until after 158 min of coarsening that the dominance of interfaces with normals nearly perpendicular to z becomes clear. Therefore, the answer as to why the microstructure becomes aligned with normals perpendicular to z clearly is not due to the interfaces with normals perpendicular to z having the largest interfacial area. The reason why interfaces along the z-axis dominate over time is provided if one takes a closer look at the three-dimensional reconstructions. Fig. 8 shows the three-dimensional reconstructions of four samples but now the liquid is shown, while the solid is transparent. Also, the view is such that the z-axis is out of the plane of the page. For better viewing, only a portion from the reconstructions of samples coarsened for shorter times is shown. Specifically, Figs. 8(a) and (b) share a common microstructural feature, namely the existence of dendrites bearing the same characteristic shape. The voids along the z-axis comprise the primary stem of these dendrites, while the parts that look like dimples on the surface of the reconstruction are the side-branches of these dendrites (see the arrows in Figs. 8(a) and (b)). At the onset of coarsening (Fig. 8(a)), there are a number of these types of stems. Impressively enough they maintain their shape, while also growing, as coarsening proceeds. This is the case for the 24 min sample (not shown here) and for the 158 min sample (Fig. 8(b)). Eventually, these dendrites evolve to the round-edged plate-like dendrites of Fig. 8(c) and these in turn evolve to cylinders or cylindrical-like shapes shown in Fig. 8(d), always along the directional solidification direction.
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Fig. 8. Three-dimensional reconstructions of the coarsened samples: (a) 3 min; (b) 158 min; (c) 2880 min; (d) 5760 min. The liquid is shown, while the solid is transparent. The z-axis is out of the plane of the page. A void (stem) and a dimple (side-branch) are shown in (a) and (b).
Our next task then is to explain what is it that allows these dendrite stems to persist in their shape and why they maintain their directionality along the z-axis as coarsening proceeds. This persistence leads to the dramatic increase of the fraction of interface with normals nearly perpendicular to z over time. Fig. 9 illustrates why. The interface is shown here (both liquid and solid are transparent), colored by the mean curvature H, for two coarsened samples: the 24 min sample (Fig. 9(a)) and the 158 min sample (Fig. 9(b)). For better viewing, a small region of each structure is only
presented. It should be noted, though, that the normals of the interface that encapsulates the solid, or voids shown in Fig. 8, are nearly perpendicular to the z-axis. The key feature that Fig. 9 allows us to see is that the dendrite stems that have interface normals perpendicular to the z-axis have a lower mean curvature than their surroundings. As coarsening proceeds, solute diffuses from regions of high absolute value of the mean curvature to regions of low absolute value of the mean curvature. In our case, the interface that forms the stems, which is along z, has lower mean
Fig. 9. Interface colored by the mean curvature H. Shown are regions from the three-dimensional structures of two samples: (a) 24 min; (b) 158 min. Both liquid and solid are transparent. Some dendrite stems are shown with an arrow.
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curvature (blue and turquoise) than the interface that forms the side-branches (green, yellow, orange and red). Therefore, the regions near the stem coarsen at the expense of the surrounding interface. It can be seen from Fig. 9(a) that the mean curvature along a stem varies with position. One would thus expect that stem to evolve into a cylinder over time. However, at shorter coarsening times this is not the case. The reason is that the stem is surrounded by interfaces (side-branches) with curvatures that are larger in absolute value than the variation along the stem’s interface. As a result, the interface that forms these stems becomes coarser, while maintaining its shape. When the magnitude of the mean curvature along the stem becomes comparable to the magnitude of the mean curvature of the surrounding interfaces, then the stem cannot maintain its shape and eventually evolves to the round-edged plate-like shapes seen in the sample coarsened for 2 days. This process continues and these round-edged plate-like shapes evolve to the cylindrical-like shapes and finally to the cylinders seen in the sample coarsened for 4 days. Most of the interface that is along the z-axis is characterized by the smallest absolute value of the mean curvature throughout each structure. Therefore, that interface maintains its directionality over time and, due to growth of these structures, leads to a microstructure with a very high fraction of interface with normals nearly perpendicular to z for long coarsening times. 5. Conclusions The microstructures of four Pb–80 wt.% Sn dendritic samples were studied. The samples were isothermally coarsened for 3, 158, 2880 and 5760 min. The three-dimensional reconstructions show the drastic morphological changes that occur as the microstructures evolve over coarsening time. Surface area per unit volume measurements allow us to quantify the increase in the length scale of the structures. Probability density contour plots of the principal curvatures (ISDs) were used to quantify the microstructural evolution. The most probable interfacial shapes were determined and a transition from small solid cylindrical shapes to saddle-shaped surfaces and finally back to large, well defined solid cylinders was observed. Equal-area projections of the interface normals permit
the determination of the existence of any preferential directionality in our microstructure. Strong preferential directionality, namely twofold symmetry, was noticed for late coarsening times, but not at earlier times. The microstructures are characterized by interfaces with normals perpendicular to the solidification direction with lower curvature than their surroundings. This leads to an increased fraction of interface with normals perpendicular to the z-axis during later coarsening times and the strong alignment that is observed experimentally. Acknowledgements The financial support from NASA, Grant No. NAG81660 is gratefully acknowledged. Also, discussions with D. Rowenhorst and R. Mendoza were invaluable. References [1] Kattamis TZ, Flemings MC. Trans Metall Soc AIME 1966;236:1523–32. [2] Kattamis TZ, Coughlin JC, Flemings MC. Trans Metall Soc AIME 1967;239:1504–11. [3] Kattamis TZ, Holmberg UT, Flemings MC. J Inst Met 1967;95:343–7. [4] Chernov AA. Kristallografiya 1956;1:583–7. [5] Klia MO. Kristallografiya 1956;65:576–81. [6] Kahlweit M. Scripta Metall 1968;2:251–4. [7] Kirkwood DH. J Mater Sci Eng 1985;73:L1–4. [8] Young KP, Kirkwood DH. Metall Trans A 1975;6A:197–205. [9] Bower TF, Brody HD, Flemings MC. Trans Metall Soc AIME 1966;236:624–34. [10] Marsh SP, Glicksman ME. Metall Mater Trans A 1996;27A:557–67. [11] Lifshitz IM, Slyozov VV. J Phys Chem Solids 1961;19:35–50. [12] Wagner CZ. Elektrochem 1961;65:581–91. [13] Mendoza R, Alkemper J, Voorhees PW. Metall Mater Trans A 2003;27A:481–9. [14] Alkemper J, Snyder VA, Akaiwa N, Voorhees PW. Metall Trans 1999;11:77–91. [15] Alkemper J, Voorhees PW. J Microsc 2001;201:388–94. [16] Alkemper J, Voorhees PW. Acta Mater 2001;49:897–902. [17] Drew DA. J Appl Math 1990;50:649–66. [18] Stokely EM, Wu SY. IEEE Trans Pattern Anal Mach Intell 1992;14:833–40. [19] Lavoue´ G, Dupont F, Baskurt A. WSCG 2004;12:245–52. [20] Jinnai H, Koga T, Nishikawa Y, Hashimoto T, Hyde S. Phys Rev Lett 1997;78:2248–51.