Fractal characteristics of dendrite in aluminum alloys

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1 IOP Conference Series: Materials Science and Engineering Fractal characteristics of dendrite in aluminum alloys To cite this article: K Ohsasa et al 2012 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. Related content - Modeling and simulation of dendrite growth in solidification of Al-Si-Mg ternary alloys Yufeng Shi, Yan Zhang, Qingyan Xu et al. - A modified cellular automaton method for polydimensional modelling of dendritic growth and microsegregation in multicomponent alloys S C Michelic, J M Thuswaldner and C Bernhard - Modeling of macrosegregation and solidification microstructure for Al-Si alloy under unidirectional solidification by a coupled cellular automaton finite volume model Hongwei Zhang, Keiji Nakajima, Engang Wang et al. This content was downloaded from IP address on 12/09/2018 at 00:16

2 Fractal characteristics of dendrite in aluminum alloys K Ohsasa, T Katsumi, R Sugawara and Y Natsume Akita University, Tegata Gakuenmachi, Akita, , Japan sasa@gipc.akita-u.ac.jp Abstract. The fractal dimensions of dendrites in Al-Si and Al-Cu binary alloys were measured under furnace cooling and casting experiments. The fractal dimension of the Al-Si alloy increased from to along with increase in Si content. The fractal dimension of the Al-Cu alloy increased from to along with increase in Cu content. The fractal dimension of the binary alloys also increased with increase in cooling rate during solidification. Phase-field simulations for the evolution of the dendrites in the binary aluminium alloys were carried out and a same tendency as the experimental results was obtained. The permeability of an Al-5mass%Si alloy was estimated from the measured fractal dimension of an experimentally observed dendrite structure. The estimated permeability agreed well with reported permeability of an Al-Si alloy. 1. Introduction Control of the macro-segregation in castings and ingots is very important because the macrosegregation has influence on the mechanical properties of the final products. One of the major factors causing the macro-segregation is an interdendritic fluid flow during solidification. The fluidity of the solute rich residual liquid in solidifying dendrite arrays can be restricted by complex dendrite morphology. Conventionally, the dendrite morphology has been expressed by using dendrite arm spacing (SDAS). Several attempts have been carried out for applying the fractal theory to describe the morphology of materials structure 1-4). Yang et al. 4) calculated the fractal characteristic of dendrite and cellular structures of a Ni-based superalloy under various cooling conditions. Sanyal et al. 5) described the mushy zone of an alloy as a network of continuous fractal structure, and they calculated the permeability of the mushy zone based on the fractal dimensions of the dendrite network. Fractal dimension may be an effective parameter for representing the complexity of dendrite array of an alloy in comparison with the SDAS. First aim of this study is to examine the factors affecting the fractal dimension of dendrites based on the experiments and the phase-field simulations for Al-Si and Al-Cu binary alloys The fluidity of the solute rich residual liquid in a solidifying dendrite array can be expressed as the permeability. Several measurements of the permeability have been carried out by experimental methods for some metallic 6-12) and borneol-paraffin organic alloy systems 13-15). However, the experimental measurements of the permeability are time consumption works and high cost would be required to obtain the permeability data of many commercial multi-component alloys. The second aim Published under licence by td 1

3 of this study is to develop a new method to estimate the permeability from an experimentally observed dendrite structure by measuring the fractal dimensions of solidified dendrite structure. 2. Experiment Furnace cooling and casting experiments were carried out for the Al-Si and Al-Cu binary alloys. In the furnace cooling experiment, a sample of 50g in mass was set in an alumina crucible and melted in an electric furnace, and then the furnace was switched off and the sample was cooled to room temperature in the furnace. To examine the effect of the solute content on dendrite morphology, Si content was changed from 1mass% to 9mass% in the Al-Si binary alloy and Cu content was changed from 1mass% to 20mass% in the Al-Cu binary alloy. Solidified samples were cut longitudinally at the center of the samples, then polished, and etched with an acid reagent. In the casting experiment, Al- 5mass%Si and Al-4mass%Cu alloys were used. Samples of 1kg in mass were melted in the electric furnace, and poured into a steel mold with 80mm in inner diameter, 100mm in outer diameter and 120mm in height. The superheat of molten samples at the pouring is 110K. Microstructures at the center, middle and near the mold positions in the longitudinal cross section of the casting were metallographically examined. The fractal dimensions of dendrites in the samples were measured by the box counting method as will describe below. 3. Phase-field simulation A phase-field model for an alloy with the thin interface limit condition proposed by Kim et al. was used (KKS model) 16). The governing equations of the phase-field method for a dilute binary alloy are as follows: φ 2 2 = M ( ε φ fφ ) () 1 t c D( = f c ( 2) t fcc where φ is phase-field, c is mole fraction, M and ε are phase-field parameters, D( is the diffusivity of solute and the subscripts under f denote the first and second derivatives of corresponding variables. The free energy density, f(c,, is defined as the sum of free energies of liquid and solid phases and imposed parabolic potential, g(. The solute composition in the interface region, c, is determined to be a fraction-weighted sum of liquid and solid compositions. S f ( c, = h( f ( cs ) + [ 1 h( ] f ( c ) + Wg( ( 3) c = h( cs + [ 1 h( ] c ( 4) where h( = φ 2 (3-2, g( = φ(1-, W is also the phase-field parameter and the subscripts of S and denote solid and liquid phases, respectively. The phase-field parameters of ε and W are related to the interfacial energy, σ, and interfacial thickness, 2λ, and are given as follows: ε 2W σ = π 8 ε 2λ = π 2W The phase-field mobility, M, derived by the thin interface limit condition depends on the kinetic coefficient and the solid and liquid compositions at the interface. () 5 ( 6) 2

4 M d 3 ε = Dσ 2W f S cc ( c e S ) f cc ( c e )( c e M M e c ) S k = 2 ( M + M ) 2 ε = σ 1 k RT V m 0 [ 1 h( φ )] 1 k0 1 m μk h( φ0 )[ 1 h( φ0 )] S e f ( c ) + h( φ ) f 0 d 1 cc S 0 cc ( c e ) dφ0 φ (1 φ ) where R is gas constant, T is temperature, V m is molar volume, k 0 is equilibrium partition coefficient, m is liquidus slope, μ k is linear kinetic coefficient and the subscript of e denotes the equilibrium -1 condition, respectively. In the vanishing kinetic coefficient condition (μ k = 0), the mobility is rewritten as M = M -1 d.. For numerical calculation, Eqs. (1) and (2) were discretized into uniform square grids and solved by using the explicit finite difference method. The anisotropy of the S/ interface is introduced to the phase-field parameter, ε, which is related to the interfacial energy. Four-fold symmetry at two-dimensional condition is expressed as follows: ε ( θ ) = ε[ 1+ ε 4 cos(4θ )] ( 8) where ε 4 is the magnitude of anisotropy, which have to be ε 4 < 1/15, and θ is the angle between the axis normal to the S/ interface and the x axis. In the simulation, a calculation domain with rectangular shape containing grids was used. The grid size is m. A planar solid was put at the left side of the domain as an initial condition to simulate the evolution of a dendrite. Zero flux condition was set at the left and right sides of the domain and a periodic condition was set at the upper and lower sides of the domain. 4. Evaluation of dendrite morphology 4.1. Fractal dimension In this study, "Fractal dimension" was used to evaluate the complexity of the dendrite morphology of alloys quantitatively. The fractal dimensions of observed and simulated dendrites were evaluated by using the box counting method 17). The procedure of the box counting method is as follows; an area including dendrites is divided into square boxes with the size of r. Then, the number of boxes, N, in which the S/ interface is included, is counted. Then, the size of the box, r, is changed and the same procedure is repeated. If following relationship holds true between the number of the boxes, N, and the size of the box, r, the geometry of the dendrite shows fractal characteristic. D N ( r) = r ( 9) The exponent, D, in the equation (9) is called "Fractal dimension" and its value can be obtained from the slope of the plot line of log r vs. log N. In the measurement of the Fractal dimension, 8 box sizes were used and total number of boxes was changed from 32 to during measurement Dimensionless perimeter of dendrite In addition to the fractal dimension, a new index "Dimensionless perimeter" of a dendrite was introduced for evaluating the morphology of dendrites. "Dimensionless perimeter" was calculated as follows; first, the perimeter of a simulated dendrite 1 is measured. Then the perimeter 2 of a circle whose area is same as the perimeter of the dendrite is measured. The "Dimensionless perimeter" of the dendrite was defined as the ratio of 1/2. The dimensionless perimeter increases when dendrite morphology becomes complex. Therefore, dimensionless perimeter is another index for showing the complexity of dendrite shape. Because the measurement of dimensionless perimeter is difficult for experimentally observed dendrites, dimensionless perimeter was measured only for simulated dendrites. 0 0 (7) 3

5 5. Results and discussion 5.1. Experimental results Effect of solute content Change in the dendrite morphology of the Al-Si binary alloy along with increase in Si content is shown in Fig.1. eft side photographs of the Figure 1 are unretouched structures. When Si content increases, primary α dendrites become fine and eutectic regions become large. Eutectic structures consist of α solid solution and eutectic Si with acicular shape. Direct measurement of the fractal dimension from the unretouched structures in the left side of the Figure 1 by using the box counting method is inaccurate because the boxes which include the interface between α solid solution and eutectic Si phase are also counted. Hence, the outlines of primary α dendrites were traced and eutectic regions were painted out. Furthermore, grey primary α dendrites were modified to a simple image with white and black dichotomous colors. The right side figures of the Figure 1 show retouched photographs. This modification makes it possible to measure the fractal dimensions of the primary α dendrites accurately. Numbers in the right side in the Figure 1 are measured fractal dimensions from the modified photographs. The correlation coefficient for determining the Fractal dimension from the slope of the plot line of log r vs. log N is The fractal dimension of the Al-Si alloy increases with increase in Si content. The morphology of primary dendrites becomes complex with increase in the fractal dimension. Figure 1. Change in the solidified structure of a furnace cooled Al-Si binary alloy along with the increase in Si content. eft side photographs are unretouched structures and right side images are retouched structures for the measurement of the fractal dimension. Numbers at the right side of the Figure show measured fractal dimensions from the retouched dendrite structures. 4

6 Figure 2 shows change in the fractal dimension of primary α dendrites with increase in Si content. The fractal dimension increase until 7 mass%si content, and the fractal dimension becomes almost constant value over 7mass%Si content. Figure 3 shows change in the fractal dimension of the Al-Cu Figure 2. Change in the fractal dimension of dendrites in an Al-Si binary alloy along with the increase in Si content. Figure 3. Change in the fractal dimension of dendrite in an Al-Cu binary alloy along with increase in Cu content. alloy with increase in Cu content. Same tendency as that of the Al-Si alloy is seen. The fractal dimension increase until 10 mass%cu, and the fractal dimension becomes almost constant value at over 10mass%Si content. These results can be explained as follows; The increase in solute content enhances the constitutional super cooling and the instability of a S/ interface increases. Complex S/ inter face produces large amount of the S/ interface energy which restrain the instability of the S/ interface. Finally a balance of both driving forces holds true and the increase in the fractal dimension of dendrites halts Effect of cooling rate Figure 4 shows observed and retouched dendrite structures in an Al-5mass%Si alloy casting. Measured local cooling rate at a position near the mold wall was larger than that at the center of the casting, and fractal dimension of dendrites near the mold wall region is greater than that at the center of the casting. (a) (b) Figure 4. The dendrite morphology in an Al-5mass%Si alloy casting along with the dendrite outline for fractal analysis. Images (a) correspond to a position at the center of the casting while (b) correspond to a position near the mold wall. 5

7 Figure 5 shows the relationship between cooling rate and fractal dimension of dendrites measured from the furnace cooled and casting samples. The fractal dimension increases with increase in the cooling rate. Figure 5. Change in the fractal dimension along with the increase in cooling rate in an Al-5mass%Si alloy. The fractal dimensions were measured from the furnace cooled sample and three positions in the casting sample Simulated dendrite morphology Figure 6 shows simulated dendrites of Al-1mass%Si and Al-3mass%Si binary alloys by the phasefield method. It is shown that the dendrite becomes more complex when Si content increases from 1mass% to 3mass%. Comparison between the fractal dimensions of simulated and observed dendrites for Al-Si and Al-Cu binary alloys is shown in Fig. 7. The fractal dimension of simulated dendrite increases with increase in solute content and these results agree well with experimental results. Figure 6. Simulated dendrites of Al-Si binary alloys by using the phase-field method. 6

8 Figure 7. Comparison between the fractal dimensions of simulated and observed dendrites in Al-Si and Al-Cu binary alloys. 6. Permeability Fluid flow through a porous medium is described by the Darcy's law, and mean flow velocity, v, is expressed as follows. K v = ΔP μ where K is permeability (m 2 ), μ is viscosity of liquid (Pa s), is the length of the porous medium (m), ΔP is the pressure drop (Pa). Until now, many studies have been carried out for investigating the interdendritic fluidity of a solidifying alloy. In those studies, the interdendritic flow was analyzed by using the Darcy's law, and the fluidity of a dendrite network is expressed as permeability. Piwonka et al. expressed the permeability, K, of the mushy zone in a solidifying alloy as a following form by regarding the mushy zone as a medium having many flow channels 6) ;. 2 g K = 3 8π nτ where g is volume fraction of liquid, n is number of flow channels per unit area (m 2 ), τ is tortuosity factor. The tortuosity factor was introduced to account for the fact that the flow channels are not straight and symmetrical. If one assumes that the number of channels is equal to the number of regions between dendrite arms, and the spacing between these channels is equal to the dendrite arm spacing, λ 2, then n=1/(λ 2) 2 and 2 2 g λ2 K = 3 8πτ It can be expected that the tortuosity factor, τ corresponds to the complexity of the dendrite morphology, i.e., Fractal dimension. However, the fractal dimension can not be regarded as the tortuosity factor because the value of fractal dimension is relatively small with range from 1 to 2 in 2D figures in comparison with the value of the tortuosity factor. We showed that the dimensionless perimeter is an appropriate parameter to regard as the tortuousity factor in Fe base alloys 18). In this study, permeability of an Al-Si alloy was estimated based on an experimentally observed dendrite structure. Fractal dimension of experimentally observed dendrites can be measured as described above. However, the measurement of the dimensionless perimeter of an observed dendrite is difficult because the observed 2D dendrite is the cross section of a 3D structure. Hence, relationship ( 10) ( 11) ( 12) 7

9 among the fractal dimension, fraction of solid and the dimensionless perimeter was evaluated based on the results of the phase-field simulation; 3.22 = 3.40D f f S 0.27 ( 13) where is dimensionless perimeter, D f is fractal dimension and f s is fraction of solid. The measured value of the fractal dimension of Al-5mass%Si alloy shown in the Fig.1 was converted into a value of dimensionless perimeter through the Eq. (13). A dendrite arm spacing,λ 2 of the Al-5mass%Si alloy shown in the Fig.1 was estimated. Fraction solid of the Al-5mass%Si alloy at the eutectic temperature was obtained from an Al-Si binary phase diagram. The values of the dendrite arm spacing,λ 2, fraction of liquid at the eutectic temperature, g and dimensionless perimeter as the tortuousity factor, τ, were substituted into the Eq.(12), and the value of the permeability of the Al-5mass%Si alloy at the eutectic temperature was calculated. Figure 8 shows the comparison between calculated and reported permeability of the Al-Si alloy. A mark P in the figure denotes the case of parallel flow to primary dendrites, and a mark N denotes the case of normal flow to the primary dendrites. The permeability without marks in the Fig.8 corresponds to the condition with equiaxed dendrite structure. The permeability reported by Apellian et al. 7) was measured in Al-4mass%Si alloy with equiaxed dendrite morphology which is same morphology as present study. Hence, it is found that the value of the estimated permeability in the present study agrees well with the reported value for an Al-Si alloy 7). Figure 8. Comparison between the estimated value of the permeability in present study and the experimentally measured permeability. Conclusions The fractal dimensions of Al-Si and Al-Cu binary alloys were measured under casting and furnace cooling experiments. A phase-field simulation was carried out and the factors affecting dendrite morphology was examined. Obtained results are as follows; 1) Fractal dimension of an Al-Si alloy increased from to along with increase in Si content from 1 to 9mass%. 2) Fractal dimension of an Al-Cu alloy increased from to along with increase in Cu content from 1 to 20mass%. 3) Fractal dimension of Al base binary alloys also increased with increase in cooling rate 4) The phase-field simulation showed that the fractal dimension of dendrites in Al-Si and Al-Cu alloys increase with increase in solute content. 8

10 5) Measured value of the fractal dimension of the Al-5mass%Si alloy was converted into the value of dimensionless perimeter though an evaluated relationship by the phase-field simulation and the permeability of the Al-5mass%Si alloy was estimated by regarding the dimensionless perimeter as the tortuosity factor of interdendritic liquid channels. Obtained permeability agreed well with reported values of an Al-Si alloy. References [1] Kleiser T and Bocek M 1986 Z. Metallkd [2] Hornbogen E 1987 Z. Metallkd [3] Tanaka M J 1922 Mater. Sci [4] Yang Y, Xiong Y and iu 2001 Sci. Tech. Adv. Mat [5] Sanyal D, Ramachandrarao P and Gupta O P 2006 Chem. Eng. Sci [6] Piwonka T S and Flemings M C 1966 Trans. metall. Soc.A.I.M.E [7] Apelian D, Flemings M C and Mehrabian R 1974 Metall. Trans [8] STREAT N and WEINBERG F 1976 Metall. Trans. 7B 417 [9] Takahashi T, Kudoh M and Yodoshi K 1979 Nippon-Kinzoku-Gakkaishi [10] Takahashi T, Kudoh M and Nagai S 1982 Tetsu-to-Hagane [11] NASSER-RAFI R, DESHMUKH R, and POIRIER D R 1985 Metall. Trans. 16A 2263 [12] POIRIER D R 1987 Metall. Trans. 18B 245 [13] MURAKAMI K, SHIRAISHI A and OKAMOTO T 1983 Acta Metall [14] MURAKAMI K, SHIRAISHI A and OKAMOTO T 1984 Acta Metall [15] MURAKAMI K and OKAMOTO T 1984 Acta Metall [16] Kim S G, Kim W T and Suzuki T 1998 Phys. Rev E [17] Matsushita N 2002 "Furakutaru no Buturi" Shokabou 36 [18] Ishida H, Natsume Y and Ohsasa K 2009 ISIJ International