- Email: [email protected]
Effect of solidification conditions on fractal dimension of dendrites
Journal of Crystal Growth 363 (2013) 49–54
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effect of solidification conditions on fractal dimension of dendrites Amber L. Genau a,n, Alex C. Freedman b, Lorenz Ratke c a b c
Department of Materials Science and Engineering, University of Alabama at Birmingham, United States Department of Materials Science and Engineering, Northwestern University, United States Institut f¨ ur Materialphysik im Weltraum, Deutsches Zentrum f¨ ur Luft- und Raumfahrt (DLR), K¨ oln, Germany
a r t i c l e i n f o
abstract
Article history: Received 1 August 2012 Received in revised form 7 September 2012 Accepted 26 September 2012 Communicated by M. Plapp Available online 4 October 2012
Dendrites are complex, three-dimensional structures that have conventionally been characterized by measuring the secondary dendrite arm spacing or the primary spacing in a dendritic network, but these global measures do not adequately describe the branched appearance of secondary and tertiary arms. This work focuses on the integral measurement of fractal dimension, a measure of complexity relatively unexplored in dendrites, in addition to specific surface area. Measurements were made on aluminum dendrites in directionally solidified Al–Si alloys of varying composition and solidification velocity, with and without induced convection currents. Contrary to expectations, average fractal dimension was found to be relatively insensitive to changes in solidification velocity and fluid flow within the ranges observed, compared to the variation in fractal dimension measured within any individual data set. Specific surface area was found to increase linearly with solidification velocity. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Solidification A1. Dendrites A1. Fluid flows A1. Coarsening A1. Characterization
1. Introduction The dendritic network is conventionally characterized by measuring the primary dendrite spacing on sections perpendicular to the growth direction, and the secondary arm spacing on longitudinal sections. These global measures of the dendrite network do not, however, adequately describe the branched appearance of secondary and tertiary arms. Recent interest has turned to integral parameters such as volume, specific surface area, and contour length [1,2]; fractal dimension is an additional integral measurement. The term fractal was coined by French mathematician Benoˆıt Mandelbrot in 1975 [3] to describe self-similar curves or objects whose roughness cannot be described by traditional Euclidian geometry, but can instead be characterized by a non-integer dimension that measures their ability to fill space. This value is sometimes called a ‘‘broken’’ dimension. A jagged line, for example, will have a fractal dimension between 1 and 2, becoming closer to 2 with increasing jaggedness. A two-dimensional area will have a fractal dimension between 2 and 3. True fractals exhibit this scaling behavior over an infinite range and so can only exist in theory. Some well-known fractals which can be calculated exactly are the Koch curve ðf D ¼ logð4Þ=logð3ÞÞ, the Sierpinski triangle ðf D ¼ logð3Þ=logð2ÞÞ and the Menger sponge ðf D ¼ logð20Þ=logð3ÞÞ. There are also many examples of
n
Corresponding author. Tel.: þ1 205 975 3271; fax: þ1 205 934 8485. E-mail address: [email protected] (A.L. Genau).
0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.09.044
approximate fractals in the natural world, where fractal dimension over some limited range can be estimated. Examples include coastlines, trees and snowflakes, and range in size from galaxy clusters to microscopic aggregates. In each case, a small segment will be at least statistically similar to the whole. While not perfect fractals, these and many other phenomenon exhibit fractal properties over a wide enough scale that fractal dimension is possible and useful to calculate. After the idea of fractal geometry was introduced, fractal dimension was quickly identified as a potential descriptor for particular types of irregular microstructures that resist quantitative description such as dendrite growth, crack branching, particle agglomerates and grain boundaries [4]. The earliest applications of fractal theory to microstructure considered crystal slip patterns [5] and grain boundaries [6]. More recently it has been used to successfully characterize and understand a range of material phenomenon ranging from particle agglomeration during liquid phase sintering [7] to the tensile strength of brittle materials [8]. While fractal analysis works well for characterizing simulated dendrite structures [9,10], analyzing the fractal dimension of real dendrites is more difficult, and attempts to correlate the fractal dimension of experimental results with various experimental and materials parameters has met with varying degrees of success. Bisang and Bilgram found that the projected outline of individual xenon dendrites has a characteristic fractal dimension, fD, of 1:42 7 0:05 regardless of initial undercooling, although fD can be decreased by varying thermal conditions during solidification [1,11]. They obtained the same results using both the box counting method (the Minkowski dimension) and the correlation function method
50
A.L. Genau et al. / Journal of Crystal Growth 363 (2013) 49–54
(called the correlation dimension) developed by Grassberger and Procaccia [12], so the accuracy of their findings is likely high. Yang et al. [13] considered directionally solidified nickel-based superalloys, finding that the fractal dimension varied from 1.228 to 1.418, depending on the solidification velocity. Sanyal et al. [14] took a variety of fractal measurements on Pb–Sn samples, but found the perimeter fractal dimension to vary between 1.026 and 1.094, depending partly on primary dendrite spacing. The large number of significant figures reported in both of these works indicates a misunderstanding of the nature of fractal calculations. Fractal dimension is notoriously difficult to determine with any degree of precision, and even when carefully measured, can produce inconclusive or even misleading data if the structure is not sufficiently fractal [15]. It is generally believed that a structure must exhibit fractal behavior over at least one order of magnitude to be a meaningful measure. Most recently, Ishida et al. [10] calculated the Minkowski dimension of Fe-based alloy dendrites simulated using the phase field method. They measured fD values between 1.266 and 1.464, varying with material properties such as interfacial energy and anisotropy, as well as composition and cooling rate. While it is now possible to create and analyze 3D images of dendrites, the procedure is difficult and time consuming. This work focuses on specific contour length and fractal dimension, both integral measures of complexity which can be measured on 2D cross-sectional images. For this work, fractal dimension is measured via a form of the box counting method.
of diameter 8 mm were machined from the initial ingots. These cylinders were directionally solidified in silica aerogel crucibles using the ARTEMIS facility, described in detail elsewhere [16]. Solidification occurred with a gradient of 3 K/mm with a solidification velocity from 30 to 150 mm=s. Samples were solidified with and without fluid flow induced by rotating magnetic fields of 6 mT. The rotating magnetic field operating at 50 Hz induced a meridional flow with velocities of around 10 mm/s and an axial and radial flow of about the same magnitude. For this configuration, the Taylor number is around 2200 and the Reynolds number is calculated as 0.2, such that even in the liquid above the mush, non-turbulent flow conditions exist (see [17]). After solidification, samples were cut and polished to reveal cross-sections taken perpendicular to the direction of growth. Fig. 1 shows several examples of such cross-sections, highlighting the effects of each parameter (composition, solidification velocity, and fluid flow) on the dendrite morphology. On each crosssection, individual dendrites were identified. Each dendrite was removed from the image and segmented into a binary image. While it is possible to calculate fractal dimension of lines, surfaces or volumes, dendrites are not volume/area fractals [1]. We are therefore interested in the perimeter rather than the area of the dendrites, and the edge-finding algorithm in Adobe Photoshop CS5 was used to produce images with a one-pixel thick outline of the dendrite. The progression is illustrated in Fig. 2.
3. Measurement of fractal dimension 2. Experimental set-up The samples used for analysis are Al–5wt %Si and Al–9wt %Si. After creating alloys of the desired composition, cylindrical samples
There are a variety of ways of defining and measuring fractal dimension. Early workers calculated fractal dimension by measuring the length of a perimeter step-wise, while varying the distance
AI-5%Si, 30µm/s, 0 mT
AI-5%Si, 120µm/s, 0 mT
AI-9%Si, 30µm/s, 0 mT
AI-9%Si, 30µm/s, 6 mT
Fig. 1. Sample cross-sections showing variation in dendrite morphology for different experimental conditions. Sample diameter is 8 mm.
A.L. Genau et al. / Journal of Crystal Growth 363 (2013) 49–54
51
Fig. 2. (a) Dendrite selected from micrograph. (b) Dendrite image converted to binary and rotated. (c) Box grid overlaid on an outline of the dendrite.
between two points of a compass [18], and by computerized versions of the same procedure. One disadvantage of this method is that it cannot be applied to structures with disconnected parts or interior holes, both of which are commonly found in 2D slices of dendrites. This difficulty is avoided by using the box counting method, which does not require a fully connected structure, and in fact most modern work utilizes the box-counting method due to its ease of implementation. The box counting or so-called Minkowski dimension is defined as
dM ¼ lim
log NðEÞ
E-0 logð1=EÞ
ð1Þ
where NðEÞ is the number of boxes of side length E needed to fully cover the set. Analysis was conducted in MATLAB [19] by placing a virtual grid over each image (see Fig. 2c) and determining the number of boxes required to completely cover the perimeter of the dendrite. By varying the grid spacing, a plot such as the one shown in Fig. 3 was produced, graphing the log of the number of boxes required to cover the dendrite versus the log of the side length of the box. The Minkowski dimension ðdM Þ is, by definition, the absolute slope of this line. This value will be equal to or slightly higher than the more general and rigorous Hausdorf dimension [3], because dM varies depending on the orientation of the grid relative to the dendrite. We found dM could be minimized by aligning the axes of the dendrites with the grid. Stalder [20] also rotated images through several angles to obtain an ‘optimized’ box counting dimension. Note that the Hausdorf dimension is not generally used to evaluate experimental structures. Since natural objects can only display fractal qualities over a limited range, the slope was measured over the range where a linear fit could be well executed, assuming that range extended over at least one order of magnitude. Dendrites for which a line could not be fitted in this range with a value of R2 Z 0:990 were not used. The code [21] was tested on several structures with known fractal dimensions and calculated values fell within 0.01 of theoretical values for a circle, an Apollonian gasket, and a Minkowski sausage (theoretical values of d ¼ 1:000, 1.3057 and 1.500, respectively). The Minkowski dimension was calculated for an average of 18 dendrites at each of five solidification velocities, both with and without fluid flow. The results are shown in Fig. 4, where the open points represent individual measurements and the large filled circles the mean value for each condition. The range of values measured for any given sample is very wide, and the average of dM does not show any trend with changing solidification velocity.
Fig. 3. An example of the determination of slope for a dendrite of composition AlSi5, v ¼ 60 mm=s. The fractal dimension for this dendrite is 1.73.
Fig. 5 shows the mean dM values for all velocities of each alloy, with and without induced fluid flow. From this figure, it is seen that the differences between mean values are all within the range of standard deviations, however the average fractal dimension increases slightly with fluid flow and with increased dendrite fraction (decreasing Si content). Ishida et al. also found that fractal dimension increases slightly with increasing volume fraction of dendrite [10]. For all measured dendrites, the average value of the Minkowski dimension was 1.69 with a standard deviation of 0.06. Most previously reported fractal values for dendrites are in the 1.2–1.4 range [1,10,13] (Fig. 6). For some dendrites, a second linear region at small box sizes is present. This region is identified by Kaye [18] as the textural fractal ðdPT Þ, as opposed to the structural fractal ðdS Þ. These two values describe the ruggedness of the boundary at different levels of inspection, where dS gives information about the overall morphology of the object and dT about the surface roughness. For dendrites where a clear inflection point between two linear regions was present, the textural Minkowski dimension was found to be 1.12 with a standard deviation of 0.07. This confirms that the dendrites have relatively smooth interfaces, as one would expect from the action of surface tension.
4. Specific surface area Specific surface area, Sv, also referred to as surface area density, is another integral measure of structure which has the advantage
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Fig. 4. Fractal dimension measured for different experimental conditions. Open points indicate individual measurements, while solid circles indicate mean value for each solidification velocity. Samples were solidified without induced fluid flow (0 mT, left) or with fluid flow induced by 6 mT magnetic fields (right). (a) Al-5Si, 0 mT, (b) Al-5Si, 6 mT, (c) Al-9Si, 0 mT and (d) Al-9Si, 6 mT.
of length L, and LA is the total length of the boundaries per unit area [22]. As computer analysis now makes direct measurement of phase boundary lengths relatively easy, we use the second method, calculating the ratio of the perimeter of a dendrite or group of dendrites to the area of the dendrite(s). Specific surface area measurements were done for the Al–5wt %Si composition only. Micrographs for analysis were taken from cross-sectional faces using optical dark field imaging because a clearer boundary between the primary dendrite phase and the eutectic could be identified by the analysis software. The average Sv values calculated for each velocity, with and without induced fluid flow, are shown in Fig. 7. Values increase in a nearly linear manner with increasing solidification velocity. The effects of added fluid flow are inconclusive.
5. Discussion Fig. 5. Boxplots of fractal dimension for each alloy measured over all solidification velocities, with and without induced fluid flow. Dot indicates mean value, box indicates interquartile range, and whiskers indicate full data range.
of being shape-independent, in that it can be accurately and meaningfully measured for any morphology. Sv can be defined in terms of the entire sample volume or of the volume of the phase in question. For this work, we take the second approach, reporting the surface area per volume of dendrite, because it normalizes any change in dendritic volume fraction. Sv can be measured directly from a 3D reconstruction or from a 2D micrograph with the help of stereology. In the case of the 2D cross-section, Sv ¼ 2P=L ¼ ð4=pÞLA where P is the average number of intercept points along a line
The results show that ramification (branching) as measured by fractal dimension is not significantly affected by either fluid flow or solidification velocity, at least within the ranges being measured. The uniformity of fractal dimension across these samples was initially surprising, since the experimental conditions being investigated produce significant changes in dendrite morphology as shown in Fig. 1 and as measured by such parameters as primary and secondary dendrite arm spacing and specific surface area [23,24]. For dendritic structures it is known that the primary spacing l1 decreases with increasing solidification velocity to the inverse fourth power [25] and the secondary dendrite arm spacing l2 decreases with the inverse cube root as long as solute diffusion
A.L. Genau et al. / Journal of Crystal Growth 363 (2013) 49–54
4 3.5 T
3
= 1.09
2.5 2 S
Log [ Nu u
1.5
= 1.73
1 0.5 0
0
0.5
1
1.5
2
2.5
3
Fig. 6. Data from a dendrite showing both a structural and a textural fractal dimension.
53
for the fractal dimension to be constant. It may be that the scale of the dendrite network is changed by fluid flow, but the form of the network (its branching and compactness) is not. The action of fluid flow has also been reported to decrease the specific surface area in Al–Cu alloys during isothermal coarsening [27]. However, our results do not show any clear effect of fluid flow on Sv for directionally solidified samples. The specific surface area in a sample depends on the primary spacing and the secondary dendrite arm spacing in a complex way [28]. Since it has been established that fluid flow does affect both primary and secondary spacing, it may be that factors with opposing effects are effectively canceling each other out, producing Sv values that appear unchanged with the addition of fluid flow. This is a question that needs to be addressed by experiments with other alloys tested at different velocities and thermal gradients. While these results suggest interesting implications, it is difficult to make definitive statements about the effects of one parameter or another on the fractal dimension because of the relatively small changes in average values between data sets when compared to the large range in values measured within each set. The measured values depend on a number of factors, including the individual dendrites selected for analysis, the segmentation of the images and precise determination of the interface location, and the range of box sizes over which the box counting dimension is calculated. Finding ways to make the analysis process more objective and, ideally, automated, would allow for confident analysis of large experimental data sets. It is possible that changes in fractal dimension due to altered processing parameters may appear if the standard deviations were reduced. If no change is observed with further refined analysis, it will require a number of assumptions to be revisited.
6. Summary
Fig. 7. Effect of solidification velocity and fluid flow on specific surface area, Sv.
determines the microstructural evolution. The ratio of l1 =l2 , which can be considered as a simple measure of the ramification of the structure, has been found not to vary with the solidification velocity within a range of non-extreme velocities [26]. Thus one could imagine that although the solidification velocity increases and the microstructure becomes finer, its ramification might not change. Our results of fractal dimension, a more exact measure of branching, appear to indicate that indeed the dendritic structure remains self-similar . It was also expected that the action of fluid flow would change the ramification or ruggedness, since fluid flow is known to drastically accelerate coarsening, and secondary dendrite arm spacing has been confirmed to increase with the addition of fluid flow in these samples [23]. We therefore expected that a lower fractal dimension would be measured in samples processed with magnetic fields. On the contrary, our results indicate that branching is not significantly affected by fluid flow within the range tested. Note that the stirring induced by the 6 mT rotating magnetic field is still rather mild, with flow velocities in the melt around 10 mm/s and a low Reynolds number such that no turbulence exists. Steinbach and co-workers determined that the secondary dendrite arm spacing increases under conditions of laminar flow as the inverse square root of the solidification velocity [23]. The primary spacing, however, decreases more quickly with increasing fluid flow with almost the same functional dependence on velocity as in diffusional conditions. Thus, both scales in the structure react differently to flow but their ratio depends only slightly on solidification velocity and thus it is not unreasonable
Fractal dimension is an integral measure of microstructural complexity that holds promise for describing the branched nature of dendrites, but it has not been well explored for experimentally obtained structures. In this work, the Minkowski dimension was calculated using the box counting method for two-dimensional cross-sections of aluminum dendrites in Al–Si alloys. Variations due to composition (fraction solid), solidification velocity and magnetically induced fluid flow were investigated. Fractal dimension was found to be relatively constant regardless of processing parameter, with an average value of 1.69 and a large standard deviation. For some dendrites, it was possible to distinguish both a structural and a textural fractal value. Contrary to the results from isothermally coarsened samples, fluid flow was not found to decrease specific surface area in directionally solidified samples.
Acknowledgments ¨ The authors wish to thank Fabian Kasprzyk and Markus Kohler for work with the micrographs and William Warriner for valuable discussion of fractal dimension in dendrites. Al–Si samples were originally created by Sonja Steinbach. References [1] U. Bisang, J. Bilgram, The fractal dimension of xenon dendrites, Journal of Crystal Growth 166 (1996) 207–211. [2] R. Gonza´lez-Cinca, L. Ramı´rez-Piscina, Numerical study of the shape and integral parameters of a dendrite, Physical Review E 70 (2004) 051612. [3] B.B. Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977, published in 1975 in French. [4] E. Hornbogen, A systematic description of microstructure, Journal of Materials Science 21 (1986) 3737–3747.
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¨ [5] T. Kleiser, M. Bocek, The fractal nature of slip in crystals, Zeitschrift fur Metallkunde 77 (1986) 582. [6] E. Hornbogen, Fractal analysis of grain boundaries in hot-worked poly¨ Metallkunde 78 (1987) 622–625. crystals, Zeitschrift fur ¨ [7] W. Bender, L. Ratke, Ostwaldreifung und Koaleszenz in flussig-phasengesin¨ Metallkunde 83 (1992) 541–547. terten Co–Cu -Legierungen, Zeitschrift fur [8] A. Carpinteri, Fractal nature of material microstructure and size effects on apparent mechanical properties, Mechanics of Materials 18 (1994) 89–101. [9] M. Uwaha, Y. Saito, Fractal aggregation and dendritic crystal growth, Journal of Crystal Growth 99 (1990) 175–178. [10] H. Ishida, Y. Natsume, K. Ohsasa, Characterization of dendrite morphology for evaluating interdendritic fluidity based on phase-field simulation, ISIJ International 49 (2009) 37–43. [11] M. Fell, H. Singer, J. Bilgram, Controlling the symmetry of xenon crystal by repetitive heating, Materials Science and Engineering A 413–414 (2005) 452–454. [12] P. Grassberger, I. Procaccia, Characterization of strange attractors, Physical Review Letters 50 (1983) 346–349. [13] A. Yang, Y. Xiong, L. Liu, Fractal characteristics of dendrite and cellular structure in nickel-based superalloy at intermediate cooling rate, Science and Technology of Advanced Materials 2 (2001) 101–103. [14] D. Sanyal, P. Ramachandrarao, O. Gupta, A fractal description of transport phenomena in dendritic porous network, Chemical Engineering Science 61 (2006) 307–315. [15] D. Ruelle, Deterministic chaos: the science and the fiction, Proceedings of the Royal Society of London A 427 (1990) 241–248. ¨ [16] J. Alkemper, S. Sous, C. Stocker, L. Ratke, Directional solidification in an aerogel furnace with high resolution optical temperature measurements, Journal of Crystal Growth 191 (1998) 252–260.
¨ [17] M. Hainke, J. Friedrich, G. Muller, Numerical study on directional solidification of AlSi alloys with rotating magnetic fields under microgravity conditions, Journal of Materials Science 39 (2004) 2011–2015. [18] B. Kaye, A Random Walk Through Fractal Dimensions, VCH Publishers, New York, 1989. [19] MATLAB version 7.11.0.854. Natick, Massachusetts: The MathWorks Inc., 2010. [20] I.D. Stalder, Morphology of Structures in Three-Dimensional Diffusional Growth, Ph.D. in Natural Science, Swiss Federal Institute of Technology, 2000. [21] Algorithm based on ‘Fractal Dimension’ code by Bernd Flemish, IANS, University of Stuttgart, 2004. Available at MatLab Database of TU Munich. /http://m2matlabdb.ma.tum.deS. [22] J. Russ, R. Dehoff, Practical Stereology, 2nd edition, Kluwer Academic/Plenum Publishers, New York, 2000. [23] S. Steinbach, L. Ratke, The influence of fluid flow on the microstructure of directionally solidified AlSi-base alloys, Metallurgical and Materials Transactions A 38 (2007) 1388–1394. [24] S. Steinbach, L. Ratke, The effect of rotaating magnetic fields on the microstructure of directionally solidified Al–Si–Mg alloys, Materials Science and Engineering A 413–414 (2005) 200–204. [25] W. Kurz, D. Fisher, Fundamentals of Solidification, Trans Tech Publications, Ackermannsdorf, Switzerland, 1986. [26] C. Cicutti, R. Boeri, On the relationship between primary and secondary dendrite arm spacing in continuous casting products, Acta Materialia 45 (2001) 1455–1460. [27] G. Kasperovich, A. Genau, L. Ratke, Mushy zone coarsening in an AlCu30 alloy accelerated by a rotating magnetic field, Metallurgical and Materials Transactions A 42 (2011) 1657–1666. [28] R. Santos, M. Melo, Permeability of interdendritic channels, Materials Science and Engineering A 391 (2005) 151–158.
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effect of solidification conditions on fractal dimension of dendrites Amber L. Genau a,n, Alex C. Freedman b, Lorenz Ratke c a b c
Department of Materials Science and Engineering, University of Alabama at Birmingham, United States Department of Materials Science and Engineering, Northwestern University, United States Institut f¨ ur Materialphysik im Weltraum, Deutsches Zentrum f¨ ur Luft- und Raumfahrt (DLR), K¨ oln, Germany
a r t i c l e i n f o
abstract
Article history: Received 1 August 2012 Received in revised form 7 September 2012 Accepted 26 September 2012 Communicated by M. Plapp Available online 4 October 2012
Dendrites are complex, three-dimensional structures that have conventionally been characterized by measuring the secondary dendrite arm spacing or the primary spacing in a dendritic network, but these global measures do not adequately describe the branched appearance of secondary and tertiary arms. This work focuses on the integral measurement of fractal dimension, a measure of complexity relatively unexplored in dendrites, in addition to specific surface area. Measurements were made on aluminum dendrites in directionally solidified Al–Si alloys of varying composition and solidification velocity, with and without induced convection currents. Contrary to expectations, average fractal dimension was found to be relatively insensitive to changes in solidification velocity and fluid flow within the ranges observed, compared to the variation in fractal dimension measured within any individual data set. Specific surface area was found to increase linearly with solidification velocity. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Solidification A1. Dendrites A1. Fluid flows A1. Coarsening A1. Characterization
1. Introduction The dendritic network is conventionally characterized by measuring the primary dendrite spacing on sections perpendicular to the growth direction, and the secondary arm spacing on longitudinal sections. These global measures of the dendrite network do not, however, adequately describe the branched appearance of secondary and tertiary arms. Recent interest has turned to integral parameters such as volume, specific surface area, and contour length [1,2]; fractal dimension is an additional integral measurement. The term fractal was coined by French mathematician Benoˆıt Mandelbrot in 1975 [3] to describe self-similar curves or objects whose roughness cannot be described by traditional Euclidian geometry, but can instead be characterized by a non-integer dimension that measures their ability to fill space. This value is sometimes called a ‘‘broken’’ dimension. A jagged line, for example, will have a fractal dimension between 1 and 2, becoming closer to 2 with increasing jaggedness. A two-dimensional area will have a fractal dimension between 2 and 3. True fractals exhibit this scaling behavior over an infinite range and so can only exist in theory. Some well-known fractals which can be calculated exactly are the Koch curve ðf D ¼ logð4Þ=logð3ÞÞ, the Sierpinski triangle ðf D ¼ logð3Þ=logð2ÞÞ and the Menger sponge ðf D ¼ logð20Þ=logð3ÞÞ. There are also many examples of
n
Corresponding author. Tel.: þ1 205 975 3271; fax: þ1 205 934 8485. E-mail address: [email protected] (A.L. Genau).
0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.09.044
approximate fractals in the natural world, where fractal dimension over some limited range can be estimated. Examples include coastlines, trees and snowflakes, and range in size from galaxy clusters to microscopic aggregates. In each case, a small segment will be at least statistically similar to the whole. While not perfect fractals, these and many other phenomenon exhibit fractal properties over a wide enough scale that fractal dimension is possible and useful to calculate. After the idea of fractal geometry was introduced, fractal dimension was quickly identified as a potential descriptor for particular types of irregular microstructures that resist quantitative description such as dendrite growth, crack branching, particle agglomerates and grain boundaries [4]. The earliest applications of fractal theory to microstructure considered crystal slip patterns [5] and grain boundaries [6]. More recently it has been used to successfully characterize and understand a range of material phenomenon ranging from particle agglomeration during liquid phase sintering [7] to the tensile strength of brittle materials [8]. While fractal analysis works well for characterizing simulated dendrite structures [9,10], analyzing the fractal dimension of real dendrites is more difficult, and attempts to correlate the fractal dimension of experimental results with various experimental and materials parameters has met with varying degrees of success. Bisang and Bilgram found that the projected outline of individual xenon dendrites has a characteristic fractal dimension, fD, of 1:42 7 0:05 regardless of initial undercooling, although fD can be decreased by varying thermal conditions during solidification [1,11]. They obtained the same results using both the box counting method (the Minkowski dimension) and the correlation function method
50
A.L. Genau et al. / Journal of Crystal Growth 363 (2013) 49–54
(called the correlation dimension) developed by Grassberger and Procaccia [12], so the accuracy of their findings is likely high. Yang et al. [13] considered directionally solidified nickel-based superalloys, finding that the fractal dimension varied from 1.228 to 1.418, depending on the solidification velocity. Sanyal et al. [14] took a variety of fractal measurements on Pb–Sn samples, but found the perimeter fractal dimension to vary between 1.026 and 1.094, depending partly on primary dendrite spacing. The large number of significant figures reported in both of these works indicates a misunderstanding of the nature of fractal calculations. Fractal dimension is notoriously difficult to determine with any degree of precision, and even when carefully measured, can produce inconclusive or even misleading data if the structure is not sufficiently fractal [15]. It is generally believed that a structure must exhibit fractal behavior over at least one order of magnitude to be a meaningful measure. Most recently, Ishida et al. [10] calculated the Minkowski dimension of Fe-based alloy dendrites simulated using the phase field method. They measured fD values between 1.266 and 1.464, varying with material properties such as interfacial energy and anisotropy, as well as composition and cooling rate. While it is now possible to create and analyze 3D images of dendrites, the procedure is difficult and time consuming. This work focuses on specific contour length and fractal dimension, both integral measures of complexity which can be measured on 2D cross-sectional images. For this work, fractal dimension is measured via a form of the box counting method.
of diameter 8 mm were machined from the initial ingots. These cylinders were directionally solidified in silica aerogel crucibles using the ARTEMIS facility, described in detail elsewhere [16]. Solidification occurred with a gradient of 3 K/mm with a solidification velocity from 30 to 150 mm=s. Samples were solidified with and without fluid flow induced by rotating magnetic fields of 6 mT. The rotating magnetic field operating at 50 Hz induced a meridional flow with velocities of around 10 mm/s and an axial and radial flow of about the same magnitude. For this configuration, the Taylor number is around 2200 and the Reynolds number is calculated as 0.2, such that even in the liquid above the mush, non-turbulent flow conditions exist (see [17]). After solidification, samples were cut and polished to reveal cross-sections taken perpendicular to the direction of growth. Fig. 1 shows several examples of such cross-sections, highlighting the effects of each parameter (composition, solidification velocity, and fluid flow) on the dendrite morphology. On each crosssection, individual dendrites were identified. Each dendrite was removed from the image and segmented into a binary image. While it is possible to calculate fractal dimension of lines, surfaces or volumes, dendrites are not volume/area fractals [1]. We are therefore interested in the perimeter rather than the area of the dendrites, and the edge-finding algorithm in Adobe Photoshop CS5 was used to produce images with a one-pixel thick outline of the dendrite. The progression is illustrated in Fig. 2.
3. Measurement of fractal dimension 2. Experimental set-up The samples used for analysis are Al–5wt %Si and Al–9wt %Si. After creating alloys of the desired composition, cylindrical samples
There are a variety of ways of defining and measuring fractal dimension. Early workers calculated fractal dimension by measuring the length of a perimeter step-wise, while varying the distance
AI-5%Si, 30µm/s, 0 mT
AI-5%Si, 120µm/s, 0 mT
AI-9%Si, 30µm/s, 0 mT
AI-9%Si, 30µm/s, 6 mT
Fig. 1. Sample cross-sections showing variation in dendrite morphology for different experimental conditions. Sample diameter is 8 mm.
A.L. Genau et al. / Journal of Crystal Growth 363 (2013) 49–54
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Fig. 2. (a) Dendrite selected from micrograph. (b) Dendrite image converted to binary and rotated. (c) Box grid overlaid on an outline of the dendrite.
between two points of a compass [18], and by computerized versions of the same procedure. One disadvantage of this method is that it cannot be applied to structures with disconnected parts or interior holes, both of which are commonly found in 2D slices of dendrites. This difficulty is avoided by using the box counting method, which does not require a fully connected structure, and in fact most modern work utilizes the box-counting method due to its ease of implementation. The box counting or so-called Minkowski dimension is defined as
dM ¼ lim
log NðEÞ
E-0 logð1=EÞ
ð1Þ
where NðEÞ is the number of boxes of side length E needed to fully cover the set. Analysis was conducted in MATLAB [19] by placing a virtual grid over each image (see Fig. 2c) and determining the number of boxes required to completely cover the perimeter of the dendrite. By varying the grid spacing, a plot such as the one shown in Fig. 3 was produced, graphing the log of the number of boxes required to cover the dendrite versus the log of the side length of the box. The Minkowski dimension ðdM Þ is, by definition, the absolute slope of this line. This value will be equal to or slightly higher than the more general and rigorous Hausdorf dimension [3], because dM varies depending on the orientation of the grid relative to the dendrite. We found dM could be minimized by aligning the axes of the dendrites with the grid. Stalder [20] also rotated images through several angles to obtain an ‘optimized’ box counting dimension. Note that the Hausdorf dimension is not generally used to evaluate experimental structures. Since natural objects can only display fractal qualities over a limited range, the slope was measured over the range where a linear fit could be well executed, assuming that range extended over at least one order of magnitude. Dendrites for which a line could not be fitted in this range with a value of R2 Z 0:990 were not used. The code [21] was tested on several structures with known fractal dimensions and calculated values fell within 0.01 of theoretical values for a circle, an Apollonian gasket, and a Minkowski sausage (theoretical values of d ¼ 1:000, 1.3057 and 1.500, respectively). The Minkowski dimension was calculated for an average of 18 dendrites at each of five solidification velocities, both with and without fluid flow. The results are shown in Fig. 4, where the open points represent individual measurements and the large filled circles the mean value for each condition. The range of values measured for any given sample is very wide, and the average of dM does not show any trend with changing solidification velocity.
Fig. 3. An example of the determination of slope for a dendrite of composition AlSi5, v ¼ 60 mm=s. The fractal dimension for this dendrite is 1.73.
Fig. 5 shows the mean dM values for all velocities of each alloy, with and without induced fluid flow. From this figure, it is seen that the differences between mean values are all within the range of standard deviations, however the average fractal dimension increases slightly with fluid flow and with increased dendrite fraction (decreasing Si content). Ishida et al. also found that fractal dimension increases slightly with increasing volume fraction of dendrite [10]. For all measured dendrites, the average value of the Minkowski dimension was 1.69 with a standard deviation of 0.06. Most previously reported fractal values for dendrites are in the 1.2–1.4 range [1,10,13] (Fig. 6). For some dendrites, a second linear region at small box sizes is present. This region is identified by Kaye [18] as the textural fractal ðdPT Þ, as opposed to the structural fractal ðdS Þ. These two values describe the ruggedness of the boundary at different levels of inspection, where dS gives information about the overall morphology of the object and dT about the surface roughness. For dendrites where a clear inflection point between two linear regions was present, the textural Minkowski dimension was found to be 1.12 with a standard deviation of 0.07. This confirms that the dendrites have relatively smooth interfaces, as one would expect from the action of surface tension.
4. Specific surface area Specific surface area, Sv, also referred to as surface area density, is another integral measure of structure which has the advantage
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Fig. 4. Fractal dimension measured for different experimental conditions. Open points indicate individual measurements, while solid circles indicate mean value for each solidification velocity. Samples were solidified without induced fluid flow (0 mT, left) or with fluid flow induced by 6 mT magnetic fields (right). (a) Al-5Si, 0 mT, (b) Al-5Si, 6 mT, (c) Al-9Si, 0 mT and (d) Al-9Si, 6 mT.
of length L, and LA is the total length of the boundaries per unit area [22]. As computer analysis now makes direct measurement of phase boundary lengths relatively easy, we use the second method, calculating the ratio of the perimeter of a dendrite or group of dendrites to the area of the dendrite(s). Specific surface area measurements were done for the Al–5wt %Si composition only. Micrographs for analysis were taken from cross-sectional faces using optical dark field imaging because a clearer boundary between the primary dendrite phase and the eutectic could be identified by the analysis software. The average Sv values calculated for each velocity, with and without induced fluid flow, are shown in Fig. 7. Values increase in a nearly linear manner with increasing solidification velocity. The effects of added fluid flow are inconclusive.
5. Discussion Fig. 5. Boxplots of fractal dimension for each alloy measured over all solidification velocities, with and without induced fluid flow. Dot indicates mean value, box indicates interquartile range, and whiskers indicate full data range.
of being shape-independent, in that it can be accurately and meaningfully measured for any morphology. Sv can be defined in terms of the entire sample volume or of the volume of the phase in question. For this work, we take the second approach, reporting the surface area per volume of dendrite, because it normalizes any change in dendritic volume fraction. Sv can be measured directly from a 3D reconstruction or from a 2D micrograph with the help of stereology. In the case of the 2D cross-section, Sv ¼ 2P=L ¼ ð4=pÞLA where P is the average number of intercept points along a line
The results show that ramification (branching) as measured by fractal dimension is not significantly affected by either fluid flow or solidification velocity, at least within the ranges being measured. The uniformity of fractal dimension across these samples was initially surprising, since the experimental conditions being investigated produce significant changes in dendrite morphology as shown in Fig. 1 and as measured by such parameters as primary and secondary dendrite arm spacing and specific surface area [23,24]. For dendritic structures it is known that the primary spacing l1 decreases with increasing solidification velocity to the inverse fourth power [25] and the secondary dendrite arm spacing l2 decreases with the inverse cube root as long as solute diffusion
A.L. Genau et al. / Journal of Crystal Growth 363 (2013) 49–54
4 3.5 T
3
= 1.09
2.5 2 S
Log [ Nu u
1.5
= 1.73
1 0.5 0
0
0.5
1
1.5
2
2.5
3
Fig. 6. Data from a dendrite showing both a structural and a textural fractal dimension.
53
for the fractal dimension to be constant. It may be that the scale of the dendrite network is changed by fluid flow, but the form of the network (its branching and compactness) is not. The action of fluid flow has also been reported to decrease the specific surface area in Al–Cu alloys during isothermal coarsening [27]. However, our results do not show any clear effect of fluid flow on Sv for directionally solidified samples. The specific surface area in a sample depends on the primary spacing and the secondary dendrite arm spacing in a complex way [28]. Since it has been established that fluid flow does affect both primary and secondary spacing, it may be that factors with opposing effects are effectively canceling each other out, producing Sv values that appear unchanged with the addition of fluid flow. This is a question that needs to be addressed by experiments with other alloys tested at different velocities and thermal gradients. While these results suggest interesting implications, it is difficult to make definitive statements about the effects of one parameter or another on the fractal dimension because of the relatively small changes in average values between data sets when compared to the large range in values measured within each set. The measured values depend on a number of factors, including the individual dendrites selected for analysis, the segmentation of the images and precise determination of the interface location, and the range of box sizes over which the box counting dimension is calculated. Finding ways to make the analysis process more objective and, ideally, automated, would allow for confident analysis of large experimental data sets. It is possible that changes in fractal dimension due to altered processing parameters may appear if the standard deviations were reduced. If no change is observed with further refined analysis, it will require a number of assumptions to be revisited.
6. Summary
Fig. 7. Effect of solidification velocity and fluid flow on specific surface area, Sv.
determines the microstructural evolution. The ratio of l1 =l2 , which can be considered as a simple measure of the ramification of the structure, has been found not to vary with the solidification velocity within a range of non-extreme velocities [26]. Thus one could imagine that although the solidification velocity increases and the microstructure becomes finer, its ramification might not change. Our results of fractal dimension, a more exact measure of branching, appear to indicate that indeed the dendritic structure remains self-similar . It was also expected that the action of fluid flow would change the ramification or ruggedness, since fluid flow is known to drastically accelerate coarsening, and secondary dendrite arm spacing has been confirmed to increase with the addition of fluid flow in these samples [23]. We therefore expected that a lower fractal dimension would be measured in samples processed with magnetic fields. On the contrary, our results indicate that branching is not significantly affected by fluid flow within the range tested. Note that the stirring induced by the 6 mT rotating magnetic field is still rather mild, with flow velocities in the melt around 10 mm/s and a low Reynolds number such that no turbulence exists. Steinbach and co-workers determined that the secondary dendrite arm spacing increases under conditions of laminar flow as the inverse square root of the solidification velocity [23]. The primary spacing, however, decreases more quickly with increasing fluid flow with almost the same functional dependence on velocity as in diffusional conditions. Thus, both scales in the structure react differently to flow but their ratio depends only slightly on solidification velocity and thus it is not unreasonable
Fractal dimension is an integral measure of microstructural complexity that holds promise for describing the branched nature of dendrites, but it has not been well explored for experimentally obtained structures. In this work, the Minkowski dimension was calculated using the box counting method for two-dimensional cross-sections of aluminum dendrites in Al–Si alloys. Variations due to composition (fraction solid), solidification velocity and magnetically induced fluid flow were investigated. Fractal dimension was found to be relatively constant regardless of processing parameter, with an average value of 1.69 and a large standard deviation. For some dendrites, it was possible to distinguish both a structural and a textural fractal value. Contrary to the results from isothermally coarsened samples, fluid flow was not found to decrease specific surface area in directionally solidified samples.
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