PHYSICAL REVIEW E 91, 062205 (2015)
Pattern formation in a sandpile of ternary granular mixtures Michiko Shimokawa,1,* Yuki Suetsugu,1 Ryoma Hiroshige,1 Takeru Hirano,1 and Hidetsugu Sakaguchi2 1
Fukuoka Institute of Technology, Wajiro-higashi, Higashi-ku, Fukuoka 811-0295, Japan 2 Kyushu University, Kasuga-Koen, Kasuga, Fukuoka 816-8580, Japan (Received 10 February 2015; published 9 June 2015)
Pattern formation in a sandpile is investigated by pouring a ternary mixture of grains into a vertical narrow cell. Size segregation in avalanches causes the formation of patterns. Four kinds of patterns emerge: stratification, segregation, upper stratification–lower segregation, and upper segregation–lower stratification. A phase diagram is constructed in a parameter space of θ11 /θ33 and θ22 /θ33 , where θ11 , θ22 , and θ33 are the repose angles of small, intermediate, and large grains, respectively. To qualitatively understand pattern formation, a phenomenological model based on a roll-or-stay rule is proposed. A similar pattern formation is found in a numerical simulation of the phenomenological model. These results suggest that the ratios of the repose angles of three kinds of grains are important for pattern formation in a sandpile. DOI: 10.1103/PhysRevE.91.062205
PACS number(s): 45.70.−n, 05.65.+b, 05.10.−a, 64.60.av
I. INTRODUCTION
Size segregation of granular mixtures is a ubiquitous phenomenon studied in widespread fields, including agricultural science, geophysics, materials science, and engineering [1,2]. The fundamental property of size segregation, however, is not yet fully understood, and it has been studied as a topic of physics. It has been reported that a mechanism called void-filling percolation causes size segregation in granular avalanches [3–6]. The process is as follows: Gaps are opened between grains due to random fluctuation in the avalanche, and smaller grains drop down below larger grains under the action of gravity. This leads to the emergence of layered avalanches [6–9] in which the upper and lower layers consist of larger and smaller grains, respectively. Size segregation in avalanches causes pattern formation in a sandpile; when binary mixtures of grains are poured into a narrow vertical cell, a segregation or a stratification pattern is formed spontaneously [1,2,6–15]. The pattern is determined by the angles of repose and the sizes of the two grains. Experimental results show that the segregation pattern appears when θ11 θ22 , and the stratification pattern appears when θ11 θ22 , where θ11 and θ22 represent the repose angles of the smaller and larger grains, respectively [10]. To understand the experimental results, phenomenological models based on the Bak-Tang-Wiesenfeld model [16] were proposed [2,10,13]. These models could reproduce the transition between the segregation and stratification patterns at θ11 ∼ θ22 . However, the model is too simple to explain the dynamical process of pattern formation. Then, other types of continuum models were proposed based on the Navier-Stokes equation [6–9]. The models focused on a complex dynamics in the avalanche, such as granular convection and granular size segregation. In contrast to binary mixtures, pattern formation in a sandpile of ternary granular mixtures has not yet to be investigated in detail. Lecocq and Vandewalle focused only on the stratification pattern in their experiments [17]. Noguchi numerically studied the classification of various patterns in a sandpile of ternary granular mixture using a roll-or-stay rule [13]. In this
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[email protected]
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article, we first show experimental results with ternary granular mixtures and then compare these results with the numerical results of a phenomenological model for such mixtures. II. EXPERIMENTAL PROCEDURE
We performed experiments using a vertical narrow cell (see Fig. 1). The vertical cell consists of an acrylic board and a wooden board of the same size. The height and width of the vertical cell are 600 and 700 mm, respectively. The two boards are mounted parallel to one another on a horizontal base plate. The space between the two parallel boards is 5.0 mm. Nineteen different types of grains (see Table I) are used to make various kinds of ternary granular mixtures. The diameters are distributed between 0.11 and 1.60 mm, and the repose angles between 21.8° and 42.5°. The values in Table I are obtained by the measurement of 100 samples of each grain type. The grains all have very similar densities, as shown in Table I, and it is realized that the reverse-Brazil-nut effect cannot occur [18]. In our experiments, 100 ml volumes of each of three kinds of grains are uniformly mixed. The granular mixture is poured through a funnel at the left edge of the vertical cell, as shown in Fig. 1. The width d of the funnel is either 2.0 or 2.5 mm. The experiment is performed at a temperature of 24 ± 2 ◦ C and a room humidity of 47.5 ± 2.5%, since special care has to be taken to prevent cohesion due to moisture. III. EXPERIMENTAL RESULTS A. Patterns in a sandpile of ternary granular mixtures
Figure 2 shows the patterns found in experiments on ternary granular mixtures, including a stratification pattern [Fig. 2(a)], a segregation pattern [Fig. 2(b)], an upper stratification–lower segregation (upper stra.-lower seg.) pattern [Fig. 2(c)], and an upper segregation–lower stratification (upper seg.-lower stra.) pattern [Fig. 2(d)]. The stratification pattern in Fig. 2(a) has three layers in which the ternary mixtures segregate vertically. The upper, middle, and lower layers consist of burnt sand, African sand, and glass beads of diameter 0.4 mm. These grains correspond to large, intermediate, and small grains, respectively. Figure 2(b) shows a segregation pattern. The upper, middle, and lower regions in a sandpile consist of
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FIG. 1. Schematic drawing of the experimental setup. The width of the funnel d could be 2.0 or 2.5 mm.
African sand, glass beads of diameter 0.4 mm, and glass beads of diameter 1.0 mm, which are respectively small, intermediate, and large grains. Figure 2(c) shows an upper stra.-lower seg. pattern. This pattern is a characteristic pattern in a sandpile of a ternary mixture and is not observed in experiments with binary mixtures. The stratification pattern at the upper region consists of glass beads of diameter 0.1 mm (small grains) and large brick sand (intermediate grains). The lower region includes glass beads of diameter 1.0 mm (large grains). The upper seg.-lower stra. pattern in Fig. 2(d) is another characteristic pattern in experiments on ternary granular mixtures. The pattern includes African sand (small grains) at the upper region, and zirconia balls (intermediate grains) and glass beads of diameter 1.0 mm (large grains) at the lower region. The patterns in Figs. 2(a)–2(d) are the fundamental patterns observed in our experiments with ternary granular mixtures. Size segregation in avalanches forms the structures of these fundamental patterns. A kink, which is a wave that moves toward the top of the sandpile along the slope,
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FIG. 2. (Color online) Patterns of a sandpile. (a) Stratification pattern formed by glass beads of diameter 0.4 mm, African sand, and burnt sand. (b) Segregation pattern formed by African sand and glass beads of diameters 0.4 and 1.0 mm. (c) Upper stra.-lower seg. pattern formed by glass beads of diameters 0.1 and 1.0 mm and brick sand. (d) Upper seg.-lower stra. pattern formed by African sand, zirconia balls, and glass beads of diameter 0.8 mm. (e), (f) Mixture patterns formed by (e) coral sand, brick sand, and glass beads of diameter 1.0 mm and (f) brick sand, Lakanto, and black sand.
TABLE I. Properties of the grains we used. Material Glass beads 0.l mm Glass beads 0.4 mm Glass beads 0.6 mm Glass beads 0.8 mm Glass beads l.0 mm Zirconia ball Black sand Brick sand Canary sand Silica sand Lakanto sugar Coral sand Australian sand Color sand Natural sand Burnt sand African sand Grain of millet Salt
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0.11 ± 0.01 0.40 ± 0.03 0.59 ± 0.06 0.85 ± 0.08 0.93 ± 0.06 0.66 ± 0.02 0.82 ± 0.15 0.43 ± 0.10 0.71 ± 0.15 0.52 ± 0.31 0.69 ± 0.10 0.35 ± 0.10 0.22 ± 0.06 0.93 ± 0.15 0.85 ± 0.18 0.84 ± 0.16 0.25 ± 0.08 1.60 ± 0.24 0.38 ± 0.09
29.6 ± 0.4 31.2 ± 0.4 36.3 ± 0.8 33.8 ± 0.4 26.9 ± 0.5 21.8 ± 0.3 39.8 ± 0.2 39.2 ± 0.5 41.1 ± 0.2 39.0 ± 0.7 39.2 ± 0.3 38.5 ± 0.3 29.2 ± 0.3 42.5 ± 1.9 40.6 ± 0.2 41.5 ± 0.5 36.5 ± 0.3 39.4 ± 0.3 37.9 ± 0.5
1.49 1.52 1.51 1.57 1.54 1.88 1.88 1.23 1.52 1.31 1.52 1.57 1.49 1.52 1.56 1.53 2.87 0.83 1.28
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develops in the formation of stripe structures as shown in Figs. 2(a), 2(c), and 2(d). The behavior is similar to that observed in ordinary experiments with binary granular mixtures [6–15]. In addition, other ambiguous patterns are observed as shown in Figs. 2(e) and 2(f), which are difficult to classify into the categories of Figs. 2(a)–2(d). Insufficient size segregation in avalanches might make these patterns, as the sizes and repose angles of three kinds of grains are close to one another. B. Phase diagram of the pattern of a sandpile
We perform experiments using 46 different kinds of ternary granular mixtures to construct a phase diagram. Patterns are independently observed in experiments with different funnels of width d = 2.0 and 2.5 mm, although the stripe wavelength in the stratification pattern and the segregation area in the segregation pattern vary. The experimental results of the sandpile patterns are summarized as a phase diagram in a θ11 /θ33 − θ22 /θ33 parameter space, as shown in Fig. 3, where θ11 , θ22 , and θ33 are the repose angles of the small, intermediate, and large grains, respectively. The phase diagram suggests regions in which these patterns appear: (1) A stratification pattern appears in the region where θ11 /θ33 1; (2) a segregation pattern appears in the region where θ11 /θ33 1, θ11 /θ22 1, and θ22 /θ33 1; (3) an upper stra.-lower seg. pattern appears in the region where θ11 /θ33 1 and θ11 /θ22 1; (4) an upper seg.-lower stra. pattern appears in the region where θ11 /θ33 1 and θ22 /θ33 1; and (5) ambiguous patterns emerge near the boundaries of the above fundamental patterns (1)–(4). This result reveals that the two parameters, θ11 /θ33 and θ22 /θ33 , are important for the pattern formation of a sandpile.
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IV. NUMERICAL RESULTS A. Phenomenological model with a roll-or-stay rule
To understand the pattern formation obtained in our experiments, we consider a phenomenological model based on a roll-or-stay rule [2,10,13]. In the model, three kinds of rectangular blocks 1, 2, and 3 with heights of h1 , h2 , and h3 are piled up one by one and a sandpile of blocks is formed. The height of the sandpile at the horizontal coordinate i is denoted by Hi , and the local slope is expressed as Si = Hi − Hi+1 . The update of the position of each block is determined by a roll-or-stay rule parametrized by the Sab of the critical slopes. That is, a block a at horizontal site i rolls down to the neighboring site i + 1, when block a is located on top of block b and the local slope Si is larger than Sab . On the other hand, block a stays at site i or block a is piled up on top of b, then the height of the sandpile increases by ha , when the local slope Si is smaller than Sab . On the other hand, block a stays at site i or block a is piled up on top of b, then the height of the sandpile increases by ha , when the local slope Si is smaller than Sab . We assume that blocks 1, 2, and 3 correspond to small, intermediate, and large grains, respectively, and that the critical slope parameters Sab satisfy S31 < S21 < S32 < S23 < S12 < S13 from the physical consideration that the critical slope for small grains on top of large grains is larger than the critical slope for large grains on top of small grains. The critical slopes S11 , S22 , and S33 are located between S32 and S23 , but the order of S11 , S22 , and S33 cannot be determined from such a consideration. One block is randomly chosen from the three kinds of blocks and put on top of the left-edge site i = 1. The block rolls down according to the roll-or-stay rule until the stay condition is satisfied, and the block is piled up there. The above process is repeated and a sandpile develops. Because the volumes for the three kinds of grains were the same in our experiments, the probability choosing blocks a = 1,2, or 3 are set to be inversely proportional to the height ha of the blocks.
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FIG. 3. (Color online) Morphology diagrams on sandpile patterns. Solid circles, solid squares, solid triangles, solid diamonds, and crosses indicate regions with stratification (stra.), segregation (seg.), upper stra.-lower seg., upper seg.-lower stra., and ambiguous patterns, respectively. The physical parameters θ11 , θ22 , and θ33 used in the axis denote the repose angles for grains with the smallest size, a middle size, and the largest size, respectively.
We perform numerical simulations of the phenomenological model given in Sec. IV A. The parameters are set to be h1 = 1, h2 = 2, and h3 = 3, and S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12. We change the critical heights of S11 and S22 as control parameters. We show numerical results only for this parameter set, but we have checked that similar patterns appear for different parameter sets of Sab . Figure 4 shows sandpile patterns when 6000 blocks are piled up. We obtain similar patterns to those observed in our experiments. Figure 4(a) shows a stratification pattern at S11 = 7 and S22 = 11, Fig. 4(b) an upper stra.-lower seg. pattern at S11 = 15 and S22 = 18, Fig. 4(c) an upper seg.-lower stra. pattern at S11 = 18 and S22 = 6, and Fig. 4(d) a segregation pattern at S11 = 18 and S22 = 15. Several intermediate patterns between these typical four patterns appear, as shown in Figs. 4(e)–4(h). The stratification pattern has a ternary layer for each block. We observe kink structures climbing up the sandpile toward the left edge during a formation process of the stratification. The behavior is similar to that observed in experiments. In the 3-segregation pattern at S11 = 18 and
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mixture appears in the upper region because S11 < S22 . In the upper seg.-lower stra. pattern at S11 = 18 and S22 = 6, the stratification pattern in the lower region is composed of blocks 2 and 3, and the upper region is composed of block 1. This upper seg.-lower stra. pattern can also be interpreted as a 2-segregation pattern of a mixture of blocks 2–3 and block 1. That is, block 1 and blocks 2–3 are segregated because the average value of S22 and S33 is smaller than S11 , and a stratification pattern appears in the lower region because S22 < S33 . Figure 4(f) is an intermediate pattern between the segregation pattern and the upper stra.-lower seg. pattern. A segregation pattern composed of three regions is seen, but some layers composed of blocks 1 and 2 appear in the middle region. Figure 4(h) is an intermediate pattern between the segregation pattern and the upper seg.-lower stra. pattern. A segregation pattern composed of three regions is seen, but some layers composed of blocks 2 and 3 appear in the lower layer. Figures 4(e) is an intermediate pattern between the stratification pattern and the upper stra.-lower seg. pattern, and Fig. 4(g) is an intermediate pattern between the stratification pattern and the upper seg.-lower stra. pattern, C. Phase diagram of the sandpile pattern obtained from our simulations
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FIG. 4. (Color online) Patterns of a sandpile with ternary mixture grains, obtained from our simulations at critical heights S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12. Each grain is assigned a height of h1 = 1, h2 = 2, or h3 = 3. (a) Stratification pattern at S11 = 7 and S22 = 11. (b) Upper stra.lower seg. pattern at S11 = 15 and S22 = 18. (c) Upper stra.-lower seg. pattern at S11 = 18 and S22 = 6. (d) Segregation pattern at S11 = 18 and S22 = 15. (e) Intermediate pattern at S11 = 10 and S22 = 19. (f) Intermediate pattern at S11 = 16 and S22 = 17. (g) Intermediate pattern at S11 = 11 and S22 = 8. (h) Intermediate pattern at S11 = 18 and S22 = 12. Letters on the right-hand side are used as pattern labels in Fig. 5.
S22 = 15, the upper, middle, and lower regions consist of blocks 1, 2, and 3, respectively. The slopes Si of the sandpile in the upper and middle regions are nearly 19 and 16, which are close to S11 + 1 and S22 + 1. In the upper stra.-lower seg. pattern at S11 = 14 and S22 = 19, the upper region is composed of blocks 1 and 2, and the lower region is composed of blocks 3. This upper stra.-lower seg. pattern can be interpreted as a 2-segregation pattern composed of a mixture of blocks 1–2 and block 3. That is, a mixture of blocks 1–2 and block 3 is segregated because the average value of S11 and S22 , that is an effective critical slope for a mixture of two smaller grains, is larger than S33 . A stratification pattern of the binary granular
In Sec. III B, a phase diagram of patterns in our experiments is shown in a θ11 /θ33 − θ22 /θ33 parameter space. Since the block width in the horizontal direction is an irrelevant quantity in the phenomenological model, the slope in our model is not directly related to the slope angle, but we can construct a similar phase diagram in an S11 /S33 − S22 /S33 parameter space. The values of the critical slopes are set to be the same values, S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12, as in Sec. IV B. The block heights are also set to be h1 = 1, h2 = 2, and h3 = 3. Figure 5 is a phase diagram obtained by a numerical simulation by changing S11 and S22 . Stratification patterns, upper stra.-lower seg. patterns, segregation patterns, and upper seg.-lower stra. patterns are, respectively, observed in regions A, C, E, and G of Fig. 5. These patterns are fundamental patterns in ternary granular mixtures. Intermediate patterns appear in regions B, D, F, and H, which are boundary regions between the fundamental patterns. We have checked that the phase diagrams for different parameter sets such as h1 = h2 = h3 = 3 or h1 = h2 = h3 = 0.01 are almost the same as the phase diagram in Fig. 5. It is natural that the segregation pattern appears in the region of S33 < S22 < S11 or S11 /S33 > 1, S22 /S33 > 1, and S22 /S33 > S11 /S33 > 1 by analogy with the phase diagram of binary mixtures. We can make a rough estimate of the boundary line for the upper stra.-lower seg. pattern by considering the pattern as a 2-segregation pattern of a mixture of blocks 1–2 and block 3. If an effective critical slope for a mixture of smaller two grains is assumed to be an average value (S11 + S22 )/2, the critical line can be evaluated at (S11 + S22 )/2 = S33 or S11 /S33 + S22 /S33 = 2. Similarly, we can make a rough estimate of the boundary line for the upper seg.-lower stra. pattern by considering the pattern as another 2-segregation pattern of block 1 and a mixture of blocks 2–3.
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FIG. 5. Morphology diagrams on sandpile patterns provided in our simulations at critical heights of S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12. Stratification, segregation, upper stra.-lower seg., and upper seg.-lower stra. patterns appear in regions A, E, C, and G, where letters A–H show the regions corresponding to the ones drawn on the right-hand side in Fig. 4. Crosses denote regions where intermediate patterns emerge in Fig. 4. The S11 , S22 , and S33 are axis mean critical heights of each block with the smallest size, middle size, and the largest size. The heights of the blocks are h1 = 1, h2 = 2, h3 = 3. The number of blocks in the pattern is 6000 in our simulations.
If an effective critical slope for a mixture of two larger grains is assumed to be (S22 + S33 )/2, the critical line is evaluated at (S22 + S33 )/2 = S11 or S22 /S33 = 2S11 /S33 − 1. The two lines are roughly consistent with the boundary regions denoted by B and H in Fig. 5. D. Pattern formation in quaternary granular mixtures
We have furthermore performed numerical simulations of quaternary granular mixtures to understand the pattern formation in sandpiles more generally. Four kinds of rectangular blocks 1, 2, 3, and 4 are piled up by the roll-or-stay rule. The critical slopes Sab for a = b are assumed to be rb /ra , where rb = 1 + (b − 1)/3 and ra = 1 + (a − 1)/3 for a = 1,2,3,4 and b = 1,2,3,4. Here, additional parameters ra and rb are introduced to generate Sab , satisfying the condition of the order of Sab , which could be interpreted as grain sizes. The critical slopes Sab for a = b are control parameters. The heights of the blocks are set to be hi = 0.01. More varieties of patterns appear in sandpiles of quaternary granular mixtures. Figure 6 shows four typical patterns at (S11 ,S22 ,S33 ,S44 ) = (1.15,1.05.0.95,0.85) [Fig. 6(a)], (1.15, 0.95, 1.05, 0.85) [Fig. 6(b)], (1.15, 0.85, 0.95, 1.05) [Fig. 6(c)], and (1.05, 1.15, 0.85, 0.95) [Fig. 6(d)]. Figure 6(a) shows a 4-segregation pattern of blocks 1, 2, 3, and 4 for S11 > S22 > S33 > S44 . Figure 6(b) shows a 3-segregation pattern of blocks 1, 2–3, and 4 for S11 > S33 > S22 > S44 . The 3-segregation pattern appears, since S11 is larger than S33 and S22 , and S33 and S22 are larger than S44 . A stratification pattern of blocks 2 and 3 appears in the middle region owing to
FIG. 6. (Color online) Patterns of a sandpile with quaternary mixture grains, obtained from our simulations with critical heights S12 = 4/3, S13 = 5/3, S14 = 6/3, S21 = 3/4, S23 = 5/4, S24 = 6/4, S31 = 3/5, S32 = 4/5, S34 = 6/5, S41 = 3/6, S42 = 4/6, and S43 = 5/6, respectively. Each grain is assigned a height of h1 = h2 = h3 = 0.01. (a) 4-segregation pattern at S11 = 1.15, S22 = 1.05, S33 = 0.95, and S44 = 0.85. (b) 3-segregation pattern with middle stratification at S11 = 1.15, S22 = 0.95, S33 = 1.05, and S44 = 0.85. (c) 2-segregation pattern with lower ternary stratification at S11 = 1.15, S22 = 0.85, S33 = 0.95, and S44 = 1.05. (d) 2-segregation pattern with two different stratification patterns at S11 = 1.05, S22 = 1.15, S33 = 0.85, and S44 = 0.95. The number of blocks is 10000.
a reversed order S33 > S22 . There are three kinds of 3-segregation patterns: an upper stratification pattern, a middle stratification pattern, and a lower stratification pattern. Figure 6(c) shows a 2-segeragation pattern of blocks 1 and 2–3–4 for S11 > S44 > S33 > S22 . The 2-segregation pattern appears, since S11 is larger than S22 , S33 , and S44 . A stratification pattern of a ternary mixture appears in the lower region owing to a reversed order S44 > S33 > S22 . There is another type of 2-segregation pattern of blocks 1–2–3 and 4, in which a stratification pattern of blocks 1, 2, and 3 appears in the upper region. Figure 6(d) is a 2-segeragation pattern of blocks 1–2 and 3–4 for S22 > S11 > S44 > S33 . The 2-segregation pattern appears, since S11 and S22 are larger than S33 and S44 . Two kinds of stratification patterns of blocks 1–2 and blocks 3–4, respectively, appear in the upper and lower regions owing to reversed orders S22 > S11 and S44 > S33 . A stratification pattern composed of blocks 1, 2, 3, and 4 is another typical pattern, although it is not shown in Fig. 6. Varieties of patterns might be classified by considering the patterns as segregation patterns of stratified granular mixtures and homogeneous granular phases. From this viewpoint, the four fundamental patterns in ternary granular mixtures—segregation, upper-seg.-lower stra., upper stra.-lower seg., and stratification patterns—can be expressed respectively as (1,2,3), (1,2–3), (1–2,3), and (1–2–3), where the comma denotes a boundary of segregated regions, and the dash denotes block pairs forming a stratification pattern. For example, the upper-seg.-lower stra. pattern is expressed as (1,2–3), because it is a
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2-segregation pattern of a homogeneous phase of block 1 and a mixture of blocks 2 and 3, and blocks 2 and 3 form a stratification pattern. Similarly, the eight typical patterns in a sandpile of quaternary granular mixtures can be expressed as (1,2,3,4), (1,2,3–4), (1,2–3,4), (1–2,3,4), (1,2–3–4), (1–2–3,4),(1–2,3–4), and (1–2–3–4). V. SUMMARY AND DISCUSSION
We have investigated pattern formation in a sandpile of ternary granular mixtures. Ternary granular mixtures were poured into a vertical narrow cell, and segregation by granular size occurred in the avalanche. We found four typical regular patterns in the sandpiles: (1) a stratification pattern, (2) a segregation pattern, (3) an upper stratification– lower segregation (upper stra.-lower seg.) pattern, and (4) an upper segregation–lower stratification (upper seg.-lower stra.) pattern. The upper stra.-lower seg. and the upper seg.-lower stra. patterns are characteristic patterns in experiments with ternary granular mixtures. We constructed a phase diagram in the θ11 /θ33 − θ22 /θ33 parameter space, where θ11 , θ22 , and θ33 are the repose angles of the smallest, intermediate, and largest grains, respectively. To understand the pattern formation, a model according to a roll-or-stay rule was proposed. The model
[1] J. C. Williams, Powder Technol. 2, 13 (1968). [2] P. Ball, The Self-Made Tapestry (Oxford University Press, Oxford, UK, 2001). [3] S. B. Savage and C. K. K. Lun, J. Fluid Mech. 189, 311 (1988). [4] A. Kudrolli, Rep. Prog. Phys. 67, 209 (2004). [5] M. Schr¨oter, S. Ulrich, J. Kreft, J. B. Swift, and H. L. Swinney, Phys. Rev. E 74, 011307 (2006). [6] J. M. N. T. Gray and C. Ancey, J. Fluid Mech. 678, 535 (2011). [7] J. M. N. T. Gray and C. Ancey, J. Fluid Mech. 629, 387 (2009). [8] J. M. N. T. Gray and B. P. Kokelaar, J. Fluid Mech. 652, 105 (2010). [9] J. M. N. T. Gray and V. A. Chugunov, J. Fluid Mech. 569, 365 (2011).
produced sandpile patterns and a phase diagram similar to those obtained from our experiments. These results suggest that the ratios of the repose angles for different kinds of grains are important for pattern formation in a sandpile. As shown above, our simulation provides an understanding of our experimental results on pattern formations for a sandpile of ternary mixture grains. Our simulation, however, could not explain dynamical features such as size segregation in an avalanche and a granular shock wave, since our simulation rule is too simple. Recently, continuum models have been proposed to consider dynamical features in an avalanche [6–9]. In order to know the detailed dynamics of the pattern formation, it would be interesting to extend a model which includes the dynamical properties as shown in Refs. [6–9]. The extended model might be further applicable to pattern formation in a sandpile of more general granular mixtures.
ACKNOWLEDGMENTS
We would like to thank Professor H. Honjo and Professor S. Ohta of Kyushu University, Professor I. Maruyama of the Fukuoka Institute of Technology, and Professor H. Kitahata of Chiba University for their fruitful discussions and suggestions.
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Pattern formation in a sandpile of ternary granular mixtures Michiko Shimokawa,1,* Yuki Suetsugu,1 Ryoma Hiroshige,1 Takeru Hirano,1 and Hidetsugu Sakaguchi2 1
Fukuoka Institute of Technology, Wajiro-higashi, Higashi-ku, Fukuoka 811-0295, Japan 2 Kyushu University, Kasuga-Koen, Kasuga, Fukuoka 816-8580, Japan (Received 10 February 2015; published 9 June 2015)
Pattern formation in a sandpile is investigated by pouring a ternary mixture of grains into a vertical narrow cell. Size segregation in avalanches causes the formation of patterns. Four kinds of patterns emerge: stratification, segregation, upper stratification–lower segregation, and upper segregation–lower stratification. A phase diagram is constructed in a parameter space of θ11 /θ33 and θ22 /θ33 , where θ11 , θ22 , and θ33 are the repose angles of small, intermediate, and large grains, respectively. To qualitatively understand pattern formation, a phenomenological model based on a roll-or-stay rule is proposed. A similar pattern formation is found in a numerical simulation of the phenomenological model. These results suggest that the ratios of the repose angles of three kinds of grains are important for pattern formation in a sandpile. DOI: 10.1103/PhysRevE.91.062205
PACS number(s): 45.70.−n, 05.65.+b, 05.10.−a, 64.60.av
I. INTRODUCTION
Size segregation of granular mixtures is a ubiquitous phenomenon studied in widespread fields, including agricultural science, geophysics, materials science, and engineering [1,2]. The fundamental property of size segregation, however, is not yet fully understood, and it has been studied as a topic of physics. It has been reported that a mechanism called void-filling percolation causes size segregation in granular avalanches [3–6]. The process is as follows: Gaps are opened between grains due to random fluctuation in the avalanche, and smaller grains drop down below larger grains under the action of gravity. This leads to the emergence of layered avalanches [6–9] in which the upper and lower layers consist of larger and smaller grains, respectively. Size segregation in avalanches causes pattern formation in a sandpile; when binary mixtures of grains are poured into a narrow vertical cell, a segregation or a stratification pattern is formed spontaneously [1,2,6–15]. The pattern is determined by the angles of repose and the sizes of the two grains. Experimental results show that the segregation pattern appears when θ11 θ22 , and the stratification pattern appears when θ11 θ22 , where θ11 and θ22 represent the repose angles of the smaller and larger grains, respectively [10]. To understand the experimental results, phenomenological models based on the Bak-Tang-Wiesenfeld model [16] were proposed [2,10,13]. These models could reproduce the transition between the segregation and stratification patterns at θ11 ∼ θ22 . However, the model is too simple to explain the dynamical process of pattern formation. Then, other types of continuum models were proposed based on the Navier-Stokes equation [6–9]. The models focused on a complex dynamics in the avalanche, such as granular convection and granular size segregation. In contrast to binary mixtures, pattern formation in a sandpile of ternary granular mixtures has not yet to be investigated in detail. Lecocq and Vandewalle focused only on the stratification pattern in their experiments [17]. Noguchi numerically studied the classification of various patterns in a sandpile of ternary granular mixture using a roll-or-stay rule [13]. In this
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article, we first show experimental results with ternary granular mixtures and then compare these results with the numerical results of a phenomenological model for such mixtures. II. EXPERIMENTAL PROCEDURE
We performed experiments using a vertical narrow cell (see Fig. 1). The vertical cell consists of an acrylic board and a wooden board of the same size. The height and width of the vertical cell are 600 and 700 mm, respectively. The two boards are mounted parallel to one another on a horizontal base plate. The space between the two parallel boards is 5.0 mm. Nineteen different types of grains (see Table I) are used to make various kinds of ternary granular mixtures. The diameters are distributed between 0.11 and 1.60 mm, and the repose angles between 21.8° and 42.5°. The values in Table I are obtained by the measurement of 100 samples of each grain type. The grains all have very similar densities, as shown in Table I, and it is realized that the reverse-Brazil-nut effect cannot occur [18]. In our experiments, 100 ml volumes of each of three kinds of grains are uniformly mixed. The granular mixture is poured through a funnel at the left edge of the vertical cell, as shown in Fig. 1. The width d of the funnel is either 2.0 or 2.5 mm. The experiment is performed at a temperature of 24 ± 2 ◦ C and a room humidity of 47.5 ± 2.5%, since special care has to be taken to prevent cohesion due to moisture. III. EXPERIMENTAL RESULTS A. Patterns in a sandpile of ternary granular mixtures
Figure 2 shows the patterns found in experiments on ternary granular mixtures, including a stratification pattern [Fig. 2(a)], a segregation pattern [Fig. 2(b)], an upper stratification–lower segregation (upper stra.-lower seg.) pattern [Fig. 2(c)], and an upper segregation–lower stratification (upper seg.-lower stra.) pattern [Fig. 2(d)]. The stratification pattern in Fig. 2(a) has three layers in which the ternary mixtures segregate vertically. The upper, middle, and lower layers consist of burnt sand, African sand, and glass beads of diameter 0.4 mm. These grains correspond to large, intermediate, and small grains, respectively. Figure 2(b) shows a segregation pattern. The upper, middle, and lower regions in a sandpile consist of
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FIG. 1. Schematic drawing of the experimental setup. The width of the funnel d could be 2.0 or 2.5 mm.
African sand, glass beads of diameter 0.4 mm, and glass beads of diameter 1.0 mm, which are respectively small, intermediate, and large grains. Figure 2(c) shows an upper stra.-lower seg. pattern. This pattern is a characteristic pattern in a sandpile of a ternary mixture and is not observed in experiments with binary mixtures. The stratification pattern at the upper region consists of glass beads of diameter 0.1 mm (small grains) and large brick sand (intermediate grains). The lower region includes glass beads of diameter 1.0 mm (large grains). The upper seg.-lower stra. pattern in Fig. 2(d) is another characteristic pattern in experiments on ternary granular mixtures. The pattern includes African sand (small grains) at the upper region, and zirconia balls (intermediate grains) and glass beads of diameter 1.0 mm (large grains) at the lower region. The patterns in Figs. 2(a)–2(d) are the fundamental patterns observed in our experiments with ternary granular mixtures. Size segregation in avalanches forms the structures of these fundamental patterns. A kink, which is a wave that moves toward the top of the sandpile along the slope,
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FIG. 2. (Color online) Patterns of a sandpile. (a) Stratification pattern formed by glass beads of diameter 0.4 mm, African sand, and burnt sand. (b) Segregation pattern formed by African sand and glass beads of diameters 0.4 and 1.0 mm. (c) Upper stra.-lower seg. pattern formed by glass beads of diameters 0.1 and 1.0 mm and brick sand. (d) Upper seg.-lower stra. pattern formed by African sand, zirconia balls, and glass beads of diameter 0.8 mm. (e), (f) Mixture patterns formed by (e) coral sand, brick sand, and glass beads of diameter 1.0 mm and (f) brick sand, Lakanto, and black sand.
TABLE I. Properties of the grains we used. Material Glass beads 0.l mm Glass beads 0.4 mm Glass beads 0.6 mm Glass beads 0.8 mm Glass beads l.0 mm Zirconia ball Black sand Brick sand Canary sand Silica sand Lakanto sugar Coral sand Australian sand Color sand Natural sand Burnt sand African sand Grain of millet Salt
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0.11 ± 0.01 0.40 ± 0.03 0.59 ± 0.06 0.85 ± 0.08 0.93 ± 0.06 0.66 ± 0.02 0.82 ± 0.15 0.43 ± 0.10 0.71 ± 0.15 0.52 ± 0.31 0.69 ± 0.10 0.35 ± 0.10 0.22 ± 0.06 0.93 ± 0.15 0.85 ± 0.18 0.84 ± 0.16 0.25 ± 0.08 1.60 ± 0.24 0.38 ± 0.09
29.6 ± 0.4 31.2 ± 0.4 36.3 ± 0.8 33.8 ± 0.4 26.9 ± 0.5 21.8 ± 0.3 39.8 ± 0.2 39.2 ± 0.5 41.1 ± 0.2 39.0 ± 0.7 39.2 ± 0.3 38.5 ± 0.3 29.2 ± 0.3 42.5 ± 1.9 40.6 ± 0.2 41.5 ± 0.5 36.5 ± 0.3 39.4 ± 0.3 37.9 ± 0.5
1.49 1.52 1.51 1.57 1.54 1.88 1.88 1.23 1.52 1.31 1.52 1.57 1.49 1.52 1.56 1.53 2.87 0.83 1.28
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develops in the formation of stripe structures as shown in Figs. 2(a), 2(c), and 2(d). The behavior is similar to that observed in ordinary experiments with binary granular mixtures [6–15]. In addition, other ambiguous patterns are observed as shown in Figs. 2(e) and 2(f), which are difficult to classify into the categories of Figs. 2(a)–2(d). Insufficient size segregation in avalanches might make these patterns, as the sizes and repose angles of three kinds of grains are close to one another. B. Phase diagram of the pattern of a sandpile
We perform experiments using 46 different kinds of ternary granular mixtures to construct a phase diagram. Patterns are independently observed in experiments with different funnels of width d = 2.0 and 2.5 mm, although the stripe wavelength in the stratification pattern and the segregation area in the segregation pattern vary. The experimental results of the sandpile patterns are summarized as a phase diagram in a θ11 /θ33 − θ22 /θ33 parameter space, as shown in Fig. 3, where θ11 , θ22 , and θ33 are the repose angles of the small, intermediate, and large grains, respectively. The phase diagram suggests regions in which these patterns appear: (1) A stratification pattern appears in the region where θ11 /θ33 1; (2) a segregation pattern appears in the region where θ11 /θ33 1, θ11 /θ22 1, and θ22 /θ33 1; (3) an upper stra.-lower seg. pattern appears in the region where θ11 /θ33 1 and θ11 /θ22 1; (4) an upper seg.-lower stra. pattern appears in the region where θ11 /θ33 1 and θ22 /θ33 1; and (5) ambiguous patterns emerge near the boundaries of the above fundamental patterns (1)–(4). This result reveals that the two parameters, θ11 /θ33 and θ22 /θ33 , are important for the pattern formation of a sandpile.
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IV. NUMERICAL RESULTS A. Phenomenological model with a roll-or-stay rule
To understand the pattern formation obtained in our experiments, we consider a phenomenological model based on a roll-or-stay rule [2,10,13]. In the model, three kinds of rectangular blocks 1, 2, and 3 with heights of h1 , h2 , and h3 are piled up one by one and a sandpile of blocks is formed. The height of the sandpile at the horizontal coordinate i is denoted by Hi , and the local slope is expressed as Si = Hi − Hi+1 . The update of the position of each block is determined by a roll-or-stay rule parametrized by the Sab of the critical slopes. That is, a block a at horizontal site i rolls down to the neighboring site i + 1, when block a is located on top of block b and the local slope Si is larger than Sab . On the other hand, block a stays at site i or block a is piled up on top of b, then the height of the sandpile increases by ha , when the local slope Si is smaller than Sab . On the other hand, block a stays at site i or block a is piled up on top of b, then the height of the sandpile increases by ha , when the local slope Si is smaller than Sab . We assume that blocks 1, 2, and 3 correspond to small, intermediate, and large grains, respectively, and that the critical slope parameters Sab satisfy S31 < S21 < S32 < S23 < S12 < S13 from the physical consideration that the critical slope for small grains on top of large grains is larger than the critical slope for large grains on top of small grains. The critical slopes S11 , S22 , and S33 are located between S32 and S23 , but the order of S11 , S22 , and S33 cannot be determined from such a consideration. One block is randomly chosen from the three kinds of blocks and put on top of the left-edge site i = 1. The block rolls down according to the roll-or-stay rule until the stay condition is satisfied, and the block is piled up there. The above process is repeated and a sandpile develops. Because the volumes for the three kinds of grains were the same in our experiments, the probability choosing blocks a = 1,2, or 3 are set to be inversely proportional to the height ha of the blocks.
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FIG. 3. (Color online) Morphology diagrams on sandpile patterns. Solid circles, solid squares, solid triangles, solid diamonds, and crosses indicate regions with stratification (stra.), segregation (seg.), upper stra.-lower seg., upper seg.-lower stra., and ambiguous patterns, respectively. The physical parameters θ11 , θ22 , and θ33 used in the axis denote the repose angles for grains with the smallest size, a middle size, and the largest size, respectively.
We perform numerical simulations of the phenomenological model given in Sec. IV A. The parameters are set to be h1 = 1, h2 = 2, and h3 = 3, and S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12. We change the critical heights of S11 and S22 as control parameters. We show numerical results only for this parameter set, but we have checked that similar patterns appear for different parameter sets of Sab . Figure 4 shows sandpile patterns when 6000 blocks are piled up. We obtain similar patterns to those observed in our experiments. Figure 4(a) shows a stratification pattern at S11 = 7 and S22 = 11, Fig. 4(b) an upper stra.-lower seg. pattern at S11 = 15 and S22 = 18, Fig. 4(c) an upper seg.-lower stra. pattern at S11 = 18 and S22 = 6, and Fig. 4(d) a segregation pattern at S11 = 18 and S22 = 15. Several intermediate patterns between these typical four patterns appear, as shown in Figs. 4(e)–4(h). The stratification pattern has a ternary layer for each block. We observe kink structures climbing up the sandpile toward the left edge during a formation process of the stratification. The behavior is similar to that observed in experiments. In the 3-segregation pattern at S11 = 18 and
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mixture appears in the upper region because S11 < S22 . In the upper seg.-lower stra. pattern at S11 = 18 and S22 = 6, the stratification pattern in the lower region is composed of blocks 2 and 3, and the upper region is composed of block 1. This upper seg.-lower stra. pattern can also be interpreted as a 2-segregation pattern of a mixture of blocks 2–3 and block 1. That is, block 1 and blocks 2–3 are segregated because the average value of S22 and S33 is smaller than S11 , and a stratification pattern appears in the lower region because S22 < S33 . Figure 4(f) is an intermediate pattern between the segregation pattern and the upper stra.-lower seg. pattern. A segregation pattern composed of three regions is seen, but some layers composed of blocks 1 and 2 appear in the middle region. Figure 4(h) is an intermediate pattern between the segregation pattern and the upper seg.-lower stra. pattern. A segregation pattern composed of three regions is seen, but some layers composed of blocks 2 and 3 appear in the lower layer. Figures 4(e) is an intermediate pattern between the stratification pattern and the upper stra.-lower seg. pattern, and Fig. 4(g) is an intermediate pattern between the stratification pattern and the upper seg.-lower stra. pattern, C. Phase diagram of the sandpile pattern obtained from our simulations
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FIG. 4. (Color online) Patterns of a sandpile with ternary mixture grains, obtained from our simulations at critical heights S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12. Each grain is assigned a height of h1 = 1, h2 = 2, or h3 = 3. (a) Stratification pattern at S11 = 7 and S22 = 11. (b) Upper stra.lower seg. pattern at S11 = 15 and S22 = 18. (c) Upper stra.-lower seg. pattern at S11 = 18 and S22 = 6. (d) Segregation pattern at S11 = 18 and S22 = 15. (e) Intermediate pattern at S11 = 10 and S22 = 19. (f) Intermediate pattern at S11 = 16 and S22 = 17. (g) Intermediate pattern at S11 = 11 and S22 = 8. (h) Intermediate pattern at S11 = 18 and S22 = 12. Letters on the right-hand side are used as pattern labels in Fig. 5.
S22 = 15, the upper, middle, and lower regions consist of blocks 1, 2, and 3, respectively. The slopes Si of the sandpile in the upper and middle regions are nearly 19 and 16, which are close to S11 + 1 and S22 + 1. In the upper stra.-lower seg. pattern at S11 = 14 and S22 = 19, the upper region is composed of blocks 1 and 2, and the lower region is composed of blocks 3. This upper stra.-lower seg. pattern can be interpreted as a 2-segregation pattern composed of a mixture of blocks 1–2 and block 3. That is, a mixture of blocks 1–2 and block 3 is segregated because the average value of S11 and S22 , that is an effective critical slope for a mixture of two smaller grains, is larger than S33 . A stratification pattern of the binary granular
In Sec. III B, a phase diagram of patterns in our experiments is shown in a θ11 /θ33 − θ22 /θ33 parameter space. Since the block width in the horizontal direction is an irrelevant quantity in the phenomenological model, the slope in our model is not directly related to the slope angle, but we can construct a similar phase diagram in an S11 /S33 − S22 /S33 parameter space. The values of the critical slopes are set to be the same values, S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12, as in Sec. IV B. The block heights are also set to be h1 = 1, h2 = 2, and h3 = 3. Figure 5 is a phase diagram obtained by a numerical simulation by changing S11 and S22 . Stratification patterns, upper stra.-lower seg. patterns, segregation patterns, and upper seg.-lower stra. patterns are, respectively, observed in regions A, C, E, and G of Fig. 5. These patterns are fundamental patterns in ternary granular mixtures. Intermediate patterns appear in regions B, D, F, and H, which are boundary regions between the fundamental patterns. We have checked that the phase diagrams for different parameter sets such as h1 = h2 = h3 = 3 or h1 = h2 = h3 = 0.01 are almost the same as the phase diagram in Fig. 5. It is natural that the segregation pattern appears in the region of S33 < S22 < S11 or S11 /S33 > 1, S22 /S33 > 1, and S22 /S33 > S11 /S33 > 1 by analogy with the phase diagram of binary mixtures. We can make a rough estimate of the boundary line for the upper stra.-lower seg. pattern by considering the pattern as a 2-segregation pattern of a mixture of blocks 1–2 and block 3. If an effective critical slope for a mixture of smaller two grains is assumed to be an average value (S11 + S22 )/2, the critical line can be evaluated at (S11 + S22 )/2 = S33 or S11 /S33 + S22 /S33 = 2. Similarly, we can make a rough estimate of the boundary line for the upper seg.-lower stra. pattern by considering the pattern as another 2-segregation pattern of block 1 and a mixture of blocks 2–3.
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FIG. 5. Morphology diagrams on sandpile patterns provided in our simulations at critical heights of S12 = 22, S13 = 23, S21 = 2, S23 = 21, S31 = 1, S32 = 3, and S33 = 12. Stratification, segregation, upper stra.-lower seg., and upper seg.-lower stra. patterns appear in regions A, E, C, and G, where letters A–H show the regions corresponding to the ones drawn on the right-hand side in Fig. 4. Crosses denote regions where intermediate patterns emerge in Fig. 4. The S11 , S22 , and S33 are axis mean critical heights of each block with the smallest size, middle size, and the largest size. The heights of the blocks are h1 = 1, h2 = 2, h3 = 3. The number of blocks in the pattern is 6000 in our simulations.
If an effective critical slope for a mixture of two larger grains is assumed to be (S22 + S33 )/2, the critical line is evaluated at (S22 + S33 )/2 = S11 or S22 /S33 = 2S11 /S33 − 1. The two lines are roughly consistent with the boundary regions denoted by B and H in Fig. 5. D. Pattern formation in quaternary granular mixtures
We have furthermore performed numerical simulations of quaternary granular mixtures to understand the pattern formation in sandpiles more generally. Four kinds of rectangular blocks 1, 2, 3, and 4 are piled up by the roll-or-stay rule. The critical slopes Sab for a = b are assumed to be rb /ra , where rb = 1 + (b − 1)/3 and ra = 1 + (a − 1)/3 for a = 1,2,3,4 and b = 1,2,3,4. Here, additional parameters ra and rb are introduced to generate Sab , satisfying the condition of the order of Sab , which could be interpreted as grain sizes. The critical slopes Sab for a = b are control parameters. The heights of the blocks are set to be hi = 0.01. More varieties of patterns appear in sandpiles of quaternary granular mixtures. Figure 6 shows four typical patterns at (S11 ,S22 ,S33 ,S44 ) = (1.15,1.05.0.95,0.85) [Fig. 6(a)], (1.15, 0.95, 1.05, 0.85) [Fig. 6(b)], (1.15, 0.85, 0.95, 1.05) [Fig. 6(c)], and (1.05, 1.15, 0.85, 0.95) [Fig. 6(d)]. Figure 6(a) shows a 4-segregation pattern of blocks 1, 2, 3, and 4 for S11 > S22 > S33 > S44 . Figure 6(b) shows a 3-segregation pattern of blocks 1, 2–3, and 4 for S11 > S33 > S22 > S44 . The 3-segregation pattern appears, since S11 is larger than S33 and S22 , and S33 and S22 are larger than S44 . A stratification pattern of blocks 2 and 3 appears in the middle region owing to
FIG. 6. (Color online) Patterns of a sandpile with quaternary mixture grains, obtained from our simulations with critical heights S12 = 4/3, S13 = 5/3, S14 = 6/3, S21 = 3/4, S23 = 5/4, S24 = 6/4, S31 = 3/5, S32 = 4/5, S34 = 6/5, S41 = 3/6, S42 = 4/6, and S43 = 5/6, respectively. Each grain is assigned a height of h1 = h2 = h3 = 0.01. (a) 4-segregation pattern at S11 = 1.15, S22 = 1.05, S33 = 0.95, and S44 = 0.85. (b) 3-segregation pattern with middle stratification at S11 = 1.15, S22 = 0.95, S33 = 1.05, and S44 = 0.85. (c) 2-segregation pattern with lower ternary stratification at S11 = 1.15, S22 = 0.85, S33 = 0.95, and S44 = 1.05. (d) 2-segregation pattern with two different stratification patterns at S11 = 1.05, S22 = 1.15, S33 = 0.85, and S44 = 0.95. The number of blocks is 10000.
a reversed order S33 > S22 . There are three kinds of 3-segregation patterns: an upper stratification pattern, a middle stratification pattern, and a lower stratification pattern. Figure 6(c) shows a 2-segeragation pattern of blocks 1 and 2–3–4 for S11 > S44 > S33 > S22 . The 2-segregation pattern appears, since S11 is larger than S22 , S33 , and S44 . A stratification pattern of a ternary mixture appears in the lower region owing to a reversed order S44 > S33 > S22 . There is another type of 2-segregation pattern of blocks 1–2–3 and 4, in which a stratification pattern of blocks 1, 2, and 3 appears in the upper region. Figure 6(d) is a 2-segeragation pattern of blocks 1–2 and 3–4 for S22 > S11 > S44 > S33 . The 2-segregation pattern appears, since S11 and S22 are larger than S33 and S44 . Two kinds of stratification patterns of blocks 1–2 and blocks 3–4, respectively, appear in the upper and lower regions owing to reversed orders S22 > S11 and S44 > S33 . A stratification pattern composed of blocks 1, 2, 3, and 4 is another typical pattern, although it is not shown in Fig. 6. Varieties of patterns might be classified by considering the patterns as segregation patterns of stratified granular mixtures and homogeneous granular phases. From this viewpoint, the four fundamental patterns in ternary granular mixtures—segregation, upper-seg.-lower stra., upper stra.-lower seg., and stratification patterns—can be expressed respectively as (1,2,3), (1,2–3), (1–2,3), and (1–2–3), where the comma denotes a boundary of segregated regions, and the dash denotes block pairs forming a stratification pattern. For example, the upper-seg.-lower stra. pattern is expressed as (1,2–3), because it is a
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2-segregation pattern of a homogeneous phase of block 1 and a mixture of blocks 2 and 3, and blocks 2 and 3 form a stratification pattern. Similarly, the eight typical patterns in a sandpile of quaternary granular mixtures can be expressed as (1,2,3,4), (1,2,3–4), (1,2–3,4), (1–2,3,4), (1,2–3–4), (1–2–3,4),(1–2,3–4), and (1–2–3–4). V. SUMMARY AND DISCUSSION
We have investigated pattern formation in a sandpile of ternary granular mixtures. Ternary granular mixtures were poured into a vertical narrow cell, and segregation by granular size occurred in the avalanche. We found four typical regular patterns in the sandpiles: (1) a stratification pattern, (2) a segregation pattern, (3) an upper stratification– lower segregation (upper stra.-lower seg.) pattern, and (4) an upper segregation–lower stratification (upper seg.-lower stra.) pattern. The upper stra.-lower seg. and the upper seg.-lower stra. patterns are characteristic patterns in experiments with ternary granular mixtures. We constructed a phase diagram in the θ11 /θ33 − θ22 /θ33 parameter space, where θ11 , θ22 , and θ33 are the repose angles of the smallest, intermediate, and largest grains, respectively. To understand the pattern formation, a model according to a roll-or-stay rule was proposed. The model
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produced sandpile patterns and a phase diagram similar to those obtained from our experiments. These results suggest that the ratios of the repose angles for different kinds of grains are important for pattern formation in a sandpile. As shown above, our simulation provides an understanding of our experimental results on pattern formations for a sandpile of ternary mixture grains. Our simulation, however, could not explain dynamical features such as size segregation in an avalanche and a granular shock wave, since our simulation rule is too simple. Recently, continuum models have been proposed to consider dynamical features in an avalanche [6–9]. In order to know the detailed dynamics of the pattern formation, it would be interesting to extend a model which includes the dynamical properties as shown in Refs. [6–9]. The extended model might be further applicable to pattern formation in a sandpile of more general granular mixtures.
ACKNOWLEDGMENTS
We would like to thank Professor H. Honjo and Professor S. Ohta of Kyushu University, Professor I. Maruyama of the Fukuoka Institute of Technology, and Professor H. Kitahata of Chiba University for their fruitful discussions and suggestions.
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