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Solid–liquid interfacial energy and its anisotropy measurement from double grain boundary grooves
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Scripta Materialia 69 (2013) 1–4 www.elsevier.com/locate/scriptamat
Solid–liquid interfacial energy and its anisotropy measurement from double grain boundary grooves Lilin Wang, Xin Lin,⇑ Meng Wang and Weidong Huang State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, 710072 Xi’an, People’s Republic of China Received 11 January 2013; revised 22 March 2013; accepted 22 March 2013 Available online 1 April 2013
The accurate determination of solid–liquid interfacial energy and its anisotropy is important for solidification research. In previous measurements using the grain boundary groove method, we found that ignoring anisotropy caused significant uncertainty in measurements of the mean interfacial energy. We thus present an improved grain boundary groove method to eliminate this uncertainty and to measure both the mean interfacial energy and its anisotropy accurately. The validity of this method was testified by the measurement results of pivalic acid and succinonitrile. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Solid–liquid interfacial energy; Grain boundaries; Dendritic growth; Solidification microstructure
Solidification occurs commonly during material processing, such as casting, welding and crystal growth. Solid–liquid interfacial energy and its anisotropy play an important role in the entire solidification process from nucleation to subsequent grain growth, which determines the solidified microstructures and final properties of materials [1,2]. Specifically, interfacial energy anisotropy is a key parameter in understanding self-organized solidification patterns, such as dendrites [3–6], doublons [7] and seaweed [8]. However, the accurate measurement of the interfacial energy and its anisotropy still remains a big challenge. One common method for measuring the solid–liquid interfacial energy is based on the grain boundary groove (GBG) shape. Early in the 1960s, the classic analysis of GBG was given by Mullins and Shewman [9,10] and Bolling and Tiller [11]. Since then, the GBG method has been employed to determine the mean solid–liquid interfacial energy in some transparent organic materials [12–16] and opaque metallic materials [17–19]. However, all these experiments did not include the anisotropy from the GBG shape, even though it was considered theoretically as a potential high-precision method for anisotropy measurement [20–22]. Moreover, after our analysis, it was found that it was the neglect of the anisotropy in the previous experiments that caused
⇑ Corresponding nwpu.edu.cn
author. Tel./fax: +86 29 88460510; e-mail: xlin@
significant uncertainty in the measurement of the mean interfacial energy. Therefore, in this study, we first assess the deviation in the measurement of the mean interfacial energy resulting from this neglect. We subsequently present an improved GBG method to accurately measure both the mean solid–liquid interfacial energy and its anisotropy. A two-dimensional grain boundary groove forming at the intersection between the solid–liquid interface and the grain boundary under a linear temperature gradient was considered. Let x-axis be the macroscopic planar part of the solid–liquid interface and y-axis the temperature gradient direction from the triple-point of the GBG, as shown in Figure 1. The equilibrated GBG shape y(x) satisfies Gibbs–Thomson equation, that is: DT ¼
1 ðc þ chh ÞK DS
ð1Þ
where DT is the curvature undercooling, DS is the entropy of fusion, c is h-dependent interfacial energy, h is the crystallographic orientation, chh is the second derivative of c with respect to h and K is the solid–liquid interface curvature. According to the definition of the x–y coordinate, DT = Gy, h = a h0, K = cosada/dy, where G is the temperature gradient, a is the angle between the groove interface normal direction and x-axis, and h0 is the tilting angle of the h1 0 0i crystal axis from x-axis.
1359-6462/$ - see front matter Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2013.03.028
2
L. Wang et al. / Scripta Materialia 69 (2013) 1–4
Figure 1. Schematic diagram of grain boundary groove.
For a cubic crystal, the solid–liquid interfacial energy in {0 0 1} crystal plane can be described by the expression [22,23]: cðhÞ ¼ c0 ½1 þ e4 cosð4hÞ
ð2Þ
Figure 2. The deviation in the measurement of the mean interfacial energy by the traditional GBG method as a function of the anisotropy e4 and the tilting angle h0.
where c0 is the magnitude of the interfacial energy and e4 is anisotropy of the interfacial energy. Substituting Eq. (2) into Eq. (1) and integrating it, we obtain an expression for describing the grain boundary groove shape: 2c0 5 ½1 sin a þ e4 sinð3a 4h0 Þ 2 GDS 3 ð3Þ þ e4 sinð5a 4h0 Þ þ e4 cosð4h0 Þ ¼ c0 F ðaÞ 2 where the function F(a) was just used for clarity. By the traditional GBG method [14–19], the interfacial energy anisotropy was ignored (e4 = 0), with only the mean interfacial energy being determined from 2c0 ð1 sin aÞ ¼ the GBG shape. In this case, y 2 ¼ GDS c0 F ðaÞje4 ¼0 , where y and a were evaluated from the experimental GBG shape y(x), G was measured directly and DS is a known physical parameter. The slope of ðF ðaÞje4 ¼0 ; y 2 Þ could then be solved, and was considered as the mean interfacial energy. However, this slope value deviates from the true interfacial energy due to the nonzero e4. In order to quantify this deviation, the theoretical GBG shape at certain values of c0, e4 and h0 was first calculated by Eq. (3), then the slope of ðF ðaÞje4 ¼0 ; y 2 Þ of this groove shape was obtained by linear fitting. This slope value denotes the interfacial energy obtained by the traditional GBG method without consideration of the anisotropy. We label it as ce to distinguish it from the true interfacial energy c0. A factor D = (ce c0)/c0 was used to describe the deviation of ce from c0, as shown in Figure 2. It can be seen that the deviation fluctuated with h0 from 0 to 90° and the fluctuation range increased with e4. This suggests that uncertainty in the measurement of the interfacial energy is apparent even at a small e4, and becomes larger as e4 increases. Therefore, in order to obtain an accurate measurement of the mean interfacial energy, the interfacial energy anisotropy should be considered. Theoretically, both c0 and e4 can be determined when Eq. (3) is employed to fit a single experimental GBG shape. In practice, however, the results fitted in this way were disturbed significantly by experimental noise because the effect of interfacial energy anisotropy on the GBG shape is very weak. This was the main obstacle blocking the extraction of anisotropy from the GBG
y2 ¼
Figure 3. Double-GBG method: (a) grains in the sample cell; (b) calculation of the interfacial energy and its anisotropy from the groove shapes of the middle grain.
shape in the past. Here we present a double-GBG method to overcome this drawback. Three grains were chosen in a thin sample cell, where the middle grain is oriented with its {0 0 1} crystal plane parallel to the observation plane and the neighboring grains are oriented differently. Two GBGs are formed among them when the sample cell is imposed in a linear temperature gradient, as shown in Figure 3(a). The h0 of the middle grain can be determined by orientation analysis, so c0 and e4 are the only unknowns in Eq. (3). In order to illustrate the double-GBG method in detail, the relationship of c0 and e4 in the fitting groove shape with Eq. (3) was studied. First, the anisotropy e4 was set at an assumed value, e. Then the slope of ðF ðaÞje4 ¼e ; y 2 Þ
L. Wang et al. / Scripta Materialia 69 (2013) 1–4
was determined by linear fitting. This slope denotes the interfacial energy fitted from the groove shape at e4 = e. We label it as ce. Then the deviation factor D = (ce c0)/c0 was calculated. In this way, the relationship between D and e can be obtained. Figure 3(b) shows this relationship as theoretical examples of two cases, h0 = 30 and 70°. The point at D = 0 and e = 0.02 corresponds to the adopted true values of the interfacial energy and anisotropy in this example. The error bars denote the standard deviation of fit (r). It can be seen that r decreases to zero when the fitted values approach the true value. However, r varies only a little while e changes significantly. This suggests that a single left or right groove shape is not sensitive to the anisotropy. That is, the fitted anisotropy from a single groove shape will be disturbed significantly by experimental noise and will hardly converge to the true value. Fortunately, it was found that D changed monotonously with e and the monotonic relationships of (D, e) between the left and right grooves of the middle grain were opposite. The point at which they cross corresponds to the true value, as shown in Figure 3(b). This suggests that the shape difference between the left and the right grooves of the middle grain can be employed to determine both c0 and e4 accurately, and, using this method, the influence of experimental noise on them can be weakened significantly. Accordingly, this cross point was located by finding the optimal anisotropy through an optimization algorithm, in which the fitted interfacial energy between the left and right grooves are consistent. This fitted interfacial energy and the optimal anisotropy are our measurement results. It is worth mentioning that the range of a was chosen as [p/2, 5p/6] for the left GBG and [p/6, p/2] for the right GBG of the middle grain in the above calculation. This is because the groove part of [0, p/6] and [5p/6, p] is either indistinct with regard to edge detection or absent due to the low grain boundary energy in the experiment. In addition, this method becomes invalid when either the h1 0 0i or h1 1 0i crystal axis is parallel to the temperature gradient direction (h0 = 0, 45 or 90°), in which case the shapes of the left and right grooves of the middle grain are symmetrical and there is no cross point between them on the D–e curves. Pivalic acid (PVA) and succinonitrile (SCN), two typical transparent organic metal analogs, were chosen to test the above method. Raw sample materials were purified by rectifying distillation before use [24]. A thin sample cell was made of two 0.15 mm thick circular glass slices with a diameter of 40 mm using epoxy resin adhesive. The space height of the sample cell was 120 lm. The sample filled the thin cell by capillary force and a suitable argon pressure within a specialized filling chamber, which was used to avoid contamination of the sample and bubble formation in the cell. After filling, the sample cell entrance was sealed with the epoxy resin adhesive. After the adhesive had cured for at least 24 h, three crystal grains were selected in the sample cell. The middle one was oriented with its {0 0 1} crystal plane parallel to the observation plane, as illustrated in Figure 3(a). The orientation of the dendritic arms growing from the undercooled melt in the sample cell allowed us to assess the {0 0 1} crystal plane of the middle grain because the dendrite adopts h1 0 0i crystalline axes as the preferred growth direction for a cubic crystal.
3
The sample cell was placed on a temperature gradient stage, which includes a hot copper plate and a cold copper plate. Their temperatures are controlled by two high-precision water thermostats, with the temperature stability being better than ±2 mK. The temperature gradient between the hot part and the cold part was measured accurately by a K-type thermocouple affixed in the sample cell. The sample was annealed under a specific temperature gradient for at least 24 h, then the equilibrated GBG images were captured with an optical microscope and a digital CCD camera. The GBG shapes were extracted from images with the Canny edge detection algorithm supplied by Matlab software. The GBG shapes at different h0 could be easily obtained by rotating the sample cell and repeating the above experimental operations. The value of h0 was determined by measuring the tilting angle of the dendritic arms of the middle grain under high-speed directional solidification. GBGs of PVA under the temperature gradient of G = 0.831 K mm1 were observed and analyzed, as shown in Figure 4(a). h0 of the middle grain was determined as 72° from its dendrite morphology (inset a1) and the misorientation angle from its neighboring grains was measured as 17°. It can be seen that the left groove of
Figure 4. The mean interfacial energy and its anisotropy measurement from the double-GBG shape: (a) PVA, (b) SCN.
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L. Wang et al. / Scripta Materialia 69 (2013) 1–4
Table 1. Comparison of our measured interfacial energy and anisotropy of PVA and SCN with previous experimental results. Material
c0 (103 J m2)
e4 (%)
PVA
5.87 ± 0.3 2.84 [13] 2.67 ± 0.21 [14]
2.2 ± 0.2 2.5 ± 0.2 [26] 5 [3]
SCN
10.35 ± 0.25 8.94 ± 0.5 [12] 7.86 ± 0.79 [15]
0.56 ± 0.15 0.5 [3] 0.55 ± 0.15 [26]
the middle grain (inset a2) was less curved than the right one (inset a3). This shape difference represents the effect of the interfacial energy anisotropy on the GBG shape. If the interfacial energy anisotropy is ignored (e4 = 0), the points ðF ðaÞje4 ¼0 ; y 2 Þ between the left and right grooves of the middle grain are separated from each other significantly, indicating that the fitted slopes between them were different. Based on the optimization algorithm, an optimal anisotropy was found to be 2.2%, at which the fitted slopes of ðF ðaÞje4¼2:2% ; y 2 Þ between the left and right grooves were consistent, as indicated by the straight line in Figure 4(a). The correlation coefficient of fit R is 0.9988. In addition, the entropy of fusion for PVA used in our calculation is 6.80 104 J K1 m3 [25]. GBGs of SCN under the temperature gradient of 0.506 K mm1 were observed and analyzed, as shown in Figure 4(b). h0 of the middle grain was determined as 60° from its dendrite morphology (inset b1) and the misorientation angle from its neighboring grains was measured as 3°. Unlike PVA, the shape difference between the left groove (inset b2) and the right one (inset b3) of SCN cannot be distinguished by the naked eye because the interfacial energy anisotropy of SCN is much weaker than that of PVA. The values of e4 and c0 of SCN were found using the same method as above, and are represented by the fitted straight line in Figure 4(b). The correlation coefficient of fit R is 0.9989. The entropy of fusion for succinonitrile used in our calculation is 1.448 105 J K1 m3 [12]. Our final results for c0 and e4 for PVA and SCN are listed in Table 1. Previous results for c0 from GBG measurement without consideration of the anisotropy and previous results of e4 by the equilibrium shape method are also cited in Table 1. It can be seen that our results of c0 are slightly larger for SCN but one time larger for PVA than others. This difference can be attributed to the anisotropy being ignored in the previous GBG measurements. Also, the anisotropy of PVA is much larger than that of SCN. Our results of e4 are consistent with those obtained by the equilibrium shape method—a sophisticated technique for anisotropy measurement. This proves the validation of the double-GBG method for the measurement of interfacial energy anisotropy. It is worth noting that, although the uncertainty of c0 due to the anisotropy is eliminated in our experiment, our results still include uncertainties from the effect of impurity and from errors in measurement of the temperature gradient, the crystallographic orientation and the groove shape. In summary, when the mean solid–liquid interfacial energy is measured by the GBG method, the interfacial energy anisotropy should be considered because ignoring
it will cause significant measurement uncertainty in the mean interfacial energy. In this letter, an improved GBG method for measuring both the mean interfacial energy and its anisotropy accurately was presented. Two typical transparent metal analogs – PVA with high anisotropy and SCN with weak anisotropy – were also used to demonstrate this improved method. Our results agree with previous experimental results in SCN but not in PVA, which suggests that high anisotropy casts greater uncertainty on the mean interfacial energy as measured by the previous GBG method, which does not consider anisotropy. We thank Jincheng Wang for helpful discussion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 51271213 and 50971102), the National Basic Research Program (“973” Program) of China (No. 2011CB610402) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20116102110016). [1] M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi, Acta Mater. 57 (2009) 941. [2] M. Rappaz, J. Friedli, A. Mariaux, M. Salgado-Ordorica, Scr. Mater. 62 (2010) 904. [3] M.E. Glicksman, N.B. Singh, J. Cryst. Growth 98 (1989) 277. [4] T. Haxhimali, A. Karma, F. Gonzales, M. Rappaz, Nat. Mater. 5 (2006) 660. [5] M. Glicksman, J. Lowengrub, S. Li, X. Li, JOM 59 (2007) 27. [6] Z. Wang, J. Wang, G. Yang, Phys. Rev. E 78 (2008) 042601. [7] B. Utter, E. Bodenschatz, Phys. Rev. E 72 (2005) 011601. [8] B. Utter, E. Bodenschatz, Phys. Rev. E 66 (2002) 051604. [9] W.W. Mullins, Trans. AIME 218 (1960) 354. [10] W.W. Mullins, P.G. Shewmon, Acta Metall. 7 (1959) 163. [11] G.F. Bolling, W.A. Tiller, J. Appl. Phys. 31 (1960) 1345. [12] R.J. Schaefer, M.E. Glicksman, J.D. Ayers, Philos. Mag. 32 (1975) 725. [13] N.B. Singh, M.E. Glicksman, J. Cryst. Growth 98 (1989) 573. [14] B. Bayender, N. Marasli, E. Cadirli, H. Sisman, M. Gu¨ndu¨z, J. Cryst. Growth 194 (1998) 119. [15] N. Marasßlı, K. Kesßlıog˘lu, B. Arslan, J. Cryst. Growth 247 (2003) 613. [16] Y. Ocak, S. Akbulut, U. Bo¨yu¨k, M. Erol, K. Kesßliog˘lu, N. Marasßlı, Scr. Mater. 55 (2006) 235. [17] N. Marasli, J.D. Hunt, Acta Mater. 44 (1996) 1085. [18] M. Erol, N. Marasßlı, K. Kesßliog˘lu, M. Gu¨ndu¨z, Scr. Mater. 51 (2004) 131. [19] S. Engin, U. Bo¨yu¨k, N. Marasßlı, J. Alloys Compd. 488 (2009) 138. [20] E. Arbel, J.W. Cahn, Surf. Sci. 66 (1977) 14. [21] R.E. Napolitano, S. Liu, R. Trivedi, Interface Sci. 10 (2002) 217. [22] J.R. Morris, R.E. Napolitano, JOM 56 (2004) 40. [23] J.J. Hoyt, M. Asta, A. Karma, Phys. Rev. Lett. 86 (2001) 5530. [24] F. Wang, L. Wang, X. Lin, W. Huang, Chem. Eng. Commun. 197 (2010) 1357. [25] N.B. Singh, M.E. Glicksman, Thermochim. Acta 159 (1990) 93. [26] M. Muschol, D. Liu, H.Z. Cummins, Phys. Rev. A 46 (1992) 1038.
Scripta Materialia 69 (2013) 1–4 www.elsevier.com/locate/scriptamat
Solid–liquid interfacial energy and its anisotropy measurement from double grain boundary grooves Lilin Wang, Xin Lin,⇑ Meng Wang and Weidong Huang State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, 710072 Xi’an, People’s Republic of China Received 11 January 2013; revised 22 March 2013; accepted 22 March 2013 Available online 1 April 2013
The accurate determination of solid–liquid interfacial energy and its anisotropy is important for solidification research. In previous measurements using the grain boundary groove method, we found that ignoring anisotropy caused significant uncertainty in measurements of the mean interfacial energy. We thus present an improved grain boundary groove method to eliminate this uncertainty and to measure both the mean interfacial energy and its anisotropy accurately. The validity of this method was testified by the measurement results of pivalic acid and succinonitrile. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Solid–liquid interfacial energy; Grain boundaries; Dendritic growth; Solidification microstructure
Solidification occurs commonly during material processing, such as casting, welding and crystal growth. Solid–liquid interfacial energy and its anisotropy play an important role in the entire solidification process from nucleation to subsequent grain growth, which determines the solidified microstructures and final properties of materials [1,2]. Specifically, interfacial energy anisotropy is a key parameter in understanding self-organized solidification patterns, such as dendrites [3–6], doublons [7] and seaweed [8]. However, the accurate measurement of the interfacial energy and its anisotropy still remains a big challenge. One common method for measuring the solid–liquid interfacial energy is based on the grain boundary groove (GBG) shape. Early in the 1960s, the classic analysis of GBG was given by Mullins and Shewman [9,10] and Bolling and Tiller [11]. Since then, the GBG method has been employed to determine the mean solid–liquid interfacial energy in some transparent organic materials [12–16] and opaque metallic materials [17–19]. However, all these experiments did not include the anisotropy from the GBG shape, even though it was considered theoretically as a potential high-precision method for anisotropy measurement [20–22]. Moreover, after our analysis, it was found that it was the neglect of the anisotropy in the previous experiments that caused
⇑ Corresponding nwpu.edu.cn
author. Tel./fax: +86 29 88460510; e-mail: xlin@
significant uncertainty in the measurement of the mean interfacial energy. Therefore, in this study, we first assess the deviation in the measurement of the mean interfacial energy resulting from this neglect. We subsequently present an improved GBG method to accurately measure both the mean solid–liquid interfacial energy and its anisotropy. A two-dimensional grain boundary groove forming at the intersection between the solid–liquid interface and the grain boundary under a linear temperature gradient was considered. Let x-axis be the macroscopic planar part of the solid–liquid interface and y-axis the temperature gradient direction from the triple-point of the GBG, as shown in Figure 1. The equilibrated GBG shape y(x) satisfies Gibbs–Thomson equation, that is: DT ¼
1 ðc þ chh ÞK DS
ð1Þ
where DT is the curvature undercooling, DS is the entropy of fusion, c is h-dependent interfacial energy, h is the crystallographic orientation, chh is the second derivative of c with respect to h and K is the solid–liquid interface curvature. According to the definition of the x–y coordinate, DT = Gy, h = a h0, K = cosada/dy, where G is the temperature gradient, a is the angle between the groove interface normal direction and x-axis, and h0 is the tilting angle of the h1 0 0i crystal axis from x-axis.
1359-6462/$ - see front matter Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2013.03.028
2
L. Wang et al. / Scripta Materialia 69 (2013) 1–4
Figure 1. Schematic diagram of grain boundary groove.
For a cubic crystal, the solid–liquid interfacial energy in {0 0 1} crystal plane can be described by the expression [22,23]: cðhÞ ¼ c0 ½1 þ e4 cosð4hÞ
ð2Þ
Figure 2. The deviation in the measurement of the mean interfacial energy by the traditional GBG method as a function of the anisotropy e4 and the tilting angle h0.
where c0 is the magnitude of the interfacial energy and e4 is anisotropy of the interfacial energy. Substituting Eq. (2) into Eq. (1) and integrating it, we obtain an expression for describing the grain boundary groove shape: 2c0 5 ½1 sin a þ e4 sinð3a 4h0 Þ 2 GDS 3 ð3Þ þ e4 sinð5a 4h0 Þ þ e4 cosð4h0 Þ ¼ c0 F ðaÞ 2 where the function F(a) was just used for clarity. By the traditional GBG method [14–19], the interfacial energy anisotropy was ignored (e4 = 0), with only the mean interfacial energy being determined from 2c0 ð1 sin aÞ ¼ the GBG shape. In this case, y 2 ¼ GDS c0 F ðaÞje4 ¼0 , where y and a were evaluated from the experimental GBG shape y(x), G was measured directly and DS is a known physical parameter. The slope of ðF ðaÞje4 ¼0 ; y 2 Þ could then be solved, and was considered as the mean interfacial energy. However, this slope value deviates from the true interfacial energy due to the nonzero e4. In order to quantify this deviation, the theoretical GBG shape at certain values of c0, e4 and h0 was first calculated by Eq. (3), then the slope of ðF ðaÞje4 ¼0 ; y 2 Þ of this groove shape was obtained by linear fitting. This slope value denotes the interfacial energy obtained by the traditional GBG method without consideration of the anisotropy. We label it as ce to distinguish it from the true interfacial energy c0. A factor D = (ce c0)/c0 was used to describe the deviation of ce from c0, as shown in Figure 2. It can be seen that the deviation fluctuated with h0 from 0 to 90° and the fluctuation range increased with e4. This suggests that uncertainty in the measurement of the interfacial energy is apparent even at a small e4, and becomes larger as e4 increases. Therefore, in order to obtain an accurate measurement of the mean interfacial energy, the interfacial energy anisotropy should be considered. Theoretically, both c0 and e4 can be determined when Eq. (3) is employed to fit a single experimental GBG shape. In practice, however, the results fitted in this way were disturbed significantly by experimental noise because the effect of interfacial energy anisotropy on the GBG shape is very weak. This was the main obstacle blocking the extraction of anisotropy from the GBG
y2 ¼
Figure 3. Double-GBG method: (a) grains in the sample cell; (b) calculation of the interfacial energy and its anisotropy from the groove shapes of the middle grain.
shape in the past. Here we present a double-GBG method to overcome this drawback. Three grains were chosen in a thin sample cell, where the middle grain is oriented with its {0 0 1} crystal plane parallel to the observation plane and the neighboring grains are oriented differently. Two GBGs are formed among them when the sample cell is imposed in a linear temperature gradient, as shown in Figure 3(a). The h0 of the middle grain can be determined by orientation analysis, so c0 and e4 are the only unknowns in Eq. (3). In order to illustrate the double-GBG method in detail, the relationship of c0 and e4 in the fitting groove shape with Eq. (3) was studied. First, the anisotropy e4 was set at an assumed value, e. Then the slope of ðF ðaÞje4 ¼e ; y 2 Þ
L. Wang et al. / Scripta Materialia 69 (2013) 1–4
was determined by linear fitting. This slope denotes the interfacial energy fitted from the groove shape at e4 = e. We label it as ce. Then the deviation factor D = (ce c0)/c0 was calculated. In this way, the relationship between D and e can be obtained. Figure 3(b) shows this relationship as theoretical examples of two cases, h0 = 30 and 70°. The point at D = 0 and e = 0.02 corresponds to the adopted true values of the interfacial energy and anisotropy in this example. The error bars denote the standard deviation of fit (r). It can be seen that r decreases to zero when the fitted values approach the true value. However, r varies only a little while e changes significantly. This suggests that a single left or right groove shape is not sensitive to the anisotropy. That is, the fitted anisotropy from a single groove shape will be disturbed significantly by experimental noise and will hardly converge to the true value. Fortunately, it was found that D changed monotonously with e and the monotonic relationships of (D, e) between the left and right grooves of the middle grain were opposite. The point at which they cross corresponds to the true value, as shown in Figure 3(b). This suggests that the shape difference between the left and the right grooves of the middle grain can be employed to determine both c0 and e4 accurately, and, using this method, the influence of experimental noise on them can be weakened significantly. Accordingly, this cross point was located by finding the optimal anisotropy through an optimization algorithm, in which the fitted interfacial energy between the left and right grooves are consistent. This fitted interfacial energy and the optimal anisotropy are our measurement results. It is worth mentioning that the range of a was chosen as [p/2, 5p/6] for the left GBG and [p/6, p/2] for the right GBG of the middle grain in the above calculation. This is because the groove part of [0, p/6] and [5p/6, p] is either indistinct with regard to edge detection or absent due to the low grain boundary energy in the experiment. In addition, this method becomes invalid when either the h1 0 0i or h1 1 0i crystal axis is parallel to the temperature gradient direction (h0 = 0, 45 or 90°), in which case the shapes of the left and right grooves of the middle grain are symmetrical and there is no cross point between them on the D–e curves. Pivalic acid (PVA) and succinonitrile (SCN), two typical transparent organic metal analogs, were chosen to test the above method. Raw sample materials were purified by rectifying distillation before use [24]. A thin sample cell was made of two 0.15 mm thick circular glass slices with a diameter of 40 mm using epoxy resin adhesive. The space height of the sample cell was 120 lm. The sample filled the thin cell by capillary force and a suitable argon pressure within a specialized filling chamber, which was used to avoid contamination of the sample and bubble formation in the cell. After filling, the sample cell entrance was sealed with the epoxy resin adhesive. After the adhesive had cured for at least 24 h, three crystal grains were selected in the sample cell. The middle one was oriented with its {0 0 1} crystal plane parallel to the observation plane, as illustrated in Figure 3(a). The orientation of the dendritic arms growing from the undercooled melt in the sample cell allowed us to assess the {0 0 1} crystal plane of the middle grain because the dendrite adopts h1 0 0i crystalline axes as the preferred growth direction for a cubic crystal.
3
The sample cell was placed on a temperature gradient stage, which includes a hot copper plate and a cold copper plate. Their temperatures are controlled by two high-precision water thermostats, with the temperature stability being better than ±2 mK. The temperature gradient between the hot part and the cold part was measured accurately by a K-type thermocouple affixed in the sample cell. The sample was annealed under a specific temperature gradient for at least 24 h, then the equilibrated GBG images were captured with an optical microscope and a digital CCD camera. The GBG shapes were extracted from images with the Canny edge detection algorithm supplied by Matlab software. The GBG shapes at different h0 could be easily obtained by rotating the sample cell and repeating the above experimental operations. The value of h0 was determined by measuring the tilting angle of the dendritic arms of the middle grain under high-speed directional solidification. GBGs of PVA under the temperature gradient of G = 0.831 K mm1 were observed and analyzed, as shown in Figure 4(a). h0 of the middle grain was determined as 72° from its dendrite morphology (inset a1) and the misorientation angle from its neighboring grains was measured as 17°. It can be seen that the left groove of
Figure 4. The mean interfacial energy and its anisotropy measurement from the double-GBG shape: (a) PVA, (b) SCN.
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L. Wang et al. / Scripta Materialia 69 (2013) 1–4
Table 1. Comparison of our measured interfacial energy and anisotropy of PVA and SCN with previous experimental results. Material
c0 (103 J m2)
e4 (%)
PVA
5.87 ± 0.3 2.84 [13] 2.67 ± 0.21 [14]
2.2 ± 0.2 2.5 ± 0.2 [26] 5 [3]
SCN
10.35 ± 0.25 8.94 ± 0.5 [12] 7.86 ± 0.79 [15]
0.56 ± 0.15 0.5 [3] 0.55 ± 0.15 [26]
the middle grain (inset a2) was less curved than the right one (inset a3). This shape difference represents the effect of the interfacial energy anisotropy on the GBG shape. If the interfacial energy anisotropy is ignored (e4 = 0), the points ðF ðaÞje4 ¼0 ; y 2 Þ between the left and right grooves of the middle grain are separated from each other significantly, indicating that the fitted slopes between them were different. Based on the optimization algorithm, an optimal anisotropy was found to be 2.2%, at which the fitted slopes of ðF ðaÞje4¼2:2% ; y 2 Þ between the left and right grooves were consistent, as indicated by the straight line in Figure 4(a). The correlation coefficient of fit R is 0.9988. In addition, the entropy of fusion for PVA used in our calculation is 6.80 104 J K1 m3 [25]. GBGs of SCN under the temperature gradient of 0.506 K mm1 were observed and analyzed, as shown in Figure 4(b). h0 of the middle grain was determined as 60° from its dendrite morphology (inset b1) and the misorientation angle from its neighboring grains was measured as 3°. Unlike PVA, the shape difference between the left groove (inset b2) and the right one (inset b3) of SCN cannot be distinguished by the naked eye because the interfacial energy anisotropy of SCN is much weaker than that of PVA. The values of e4 and c0 of SCN were found using the same method as above, and are represented by the fitted straight line in Figure 4(b). The correlation coefficient of fit R is 0.9989. The entropy of fusion for succinonitrile used in our calculation is 1.448 105 J K1 m3 [12]. Our final results for c0 and e4 for PVA and SCN are listed in Table 1. Previous results for c0 from GBG measurement without consideration of the anisotropy and previous results of e4 by the equilibrium shape method are also cited in Table 1. It can be seen that our results of c0 are slightly larger for SCN but one time larger for PVA than others. This difference can be attributed to the anisotropy being ignored in the previous GBG measurements. Also, the anisotropy of PVA is much larger than that of SCN. Our results of e4 are consistent with those obtained by the equilibrium shape method—a sophisticated technique for anisotropy measurement. This proves the validation of the double-GBG method for the measurement of interfacial energy anisotropy. It is worth noting that, although the uncertainty of c0 due to the anisotropy is eliminated in our experiment, our results still include uncertainties from the effect of impurity and from errors in measurement of the temperature gradient, the crystallographic orientation and the groove shape. In summary, when the mean solid–liquid interfacial energy is measured by the GBG method, the interfacial energy anisotropy should be considered because ignoring
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