(Submitted to Physical Review Letters) Phase Diagram for Avalanche Stratification of Granular Media. J. P. Koeppe, M. Enz and J.

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1 1 (Submitted to Physical Review Letters) Phase Diagram for Avalanche Stratification of Granular Media J. P. Koeppe, M. Enz and J. Kakalios School of Physics and Astronomy The University of Minnesota Minneapolis, MN 4 When a binary mixture of granular materials is poured into a quasi-two dimensional Hele-Shaw cell, alternating stratified layers of large and small particles are formed along the top surface. This effect is studied as the plate separation of the cell and the flow rate at which the granular mixture is poured are systematically varied. A non-trivial phase diagram is found, with pairing of the stratification layers occuring for a finite range of plate separation and flow rate values. These results suggest that velocity gradients transverse to the flowing surface, near the walls of the Hele-Shaw cell, play an important role in this effect. PACS no z, Te, g Segregation of granular materials is a significant problem for particle processing industries such as pharmaceutical and agricultural firms. Binary mixtures of powders or grains can separate by particle property when vertically [1-3] or horizontally [4] shaken, when rotated in a horizontal cylinder like a drum mixer [-7], or when simply poured into a vertical Hele- Shaw cell [8-1]. This last phenomenon, termed avalanche stratification, occurs when an initially homogeneous mixture of two different granular materials, such as sand and sugar, are poured between two vertical plates held a narrow distance apart. Two types of demixing can be observed, depending on the properties of the granular media. Segregation can occur, whereby the larger particles reside near the base of the quasi-two-dimensional sandpile, while the smaller particles are on top. In addition, certain granular mixtures also exhibit stratification, where the large and small materials separate into alternating layers. An elucidation of the mechanisms responsible for avalanche stratification is important for understanding many geological formations [8,11-14], such as striation patterns in sandstone, as well as industrial processes which require that mixtures poured from a hopper remain homogeneous. Models by Makse, Cizeau and Stanley [1], and Boutreaux and degennes [16] can successfully account for the observed avalanche segregation and stratification patterns for a given Hele-Shaw [17] cell. However, these models are strictly twodimensional, while experiments find that the

2 2 stratification pattern is sensitive to the paper we report experimental studies of avalanche stratification as the plate separation and flow rate of the poured mixture are systematically varied. These studies yield a non-trivial phase diagram of the stratification effect and indicate that transverse interactions with the sidewalls play an important role. The experimental set-up (fig. 1a) consists of two vertical Plexiglas sheets 6 mm thick of area 1. inches by 8 inches, mounted parallel to each other on a horizontal base plate. One sheet is fixed to the base plate, while the second sheet is attached at one end to the fixed plate. By employing spacers which extend the full height of the vertical sheets (so that one end remains closed) the separation between the plates can be varied from 3 to 24 mm. A / mixture by mass of black sand (roughly spherical with an average diameter.4 mm, density 2.6 mg/mm 3 ) and white sugar (roughly cylindrical, with an average long axis of.8 mm, density 1. mg/mm 3 ) is poured against the closed edge of this Hele-Shaw cell using a titrating bulb with a rotating stopcock. The granular mixture is premixed by stirring prior to pouring. The flow rates can be varied from.3 to 3.7 g/sec. The static angle of repose of the sand is 37.3 and of the sugar is 4, while the angle of repose of sand on sugar is 4.3 and of sugar on sand is Digital images of the stratification pattern (fig. 1b) are taken with a monochrome charge coupled device (CCD) camera (Cohu 491) in conjunction with a Scion LG-3 frame grabber and a Power Macintosh 71/8. Data image analysis is performed using the public domain program Image from the NIH. The stratification pattern in fig. 1b was obtained for a plate separation d of 4 mm and a flow rate f of.78 g/sec, when the separation of the vertical plates [1]. In this sand/sugar mixture is poured against the closed edge of the cell (similar patterns are found when the mixture is poured in the center of the cell [1]). There are several features of note in fig. 1b, as described by Makse and co-workers [9]. There is a dead zone of mixed material in the lower left hand corner of the pile. Second, the concentration of sand (sugar) is greater near the top (bottom) of the pile. In addition to this segregation, there is pronounced (a) (b) FIG. 1. Sketch of the avalanche stratification apparatus (a) wherein a mixture of two different granular materials is poured against the closed edge of a quasi-twodimensional Hele-Shaw cell. Fig. 1 b is a digital image of the resulting stratification pattern for a / mixture (by mass) of sand and sugar. The flow rate was.78 g/sec and the plate separation was 4 mm.

3 3 stratification, that is, alternating layers of sand and sugar in the center of this quasitwo-dimensional sandpile, which is the focus of the studies described below. In order to quantify the intensity and wavelength of the stratification effect, digital images of the banding pattern near the top of the sandpile are Fourier analyzed. The digital image in the white box in fig. 1b is converted into a plot of pixel intensity against position. Since the sand is dark and the sugar is white, a high pixel value corresponds to a large concentration of sand next to the transparent vertical plate in the Hele-Shaw cell, while a low pixel value corresponds to a high concentration of sugar and an intermediate pixel value reflects a mixture of the two. The resulting plot of varying pixel value against position (depth into the sandpile) is then Fourier transformed to yield the structure function of the stratification pattern. While the structure function is the square of the Fourier amplitude and is typically plotted against wavevector, for simplicity the FFT amplitude is plotted against wavelength in fig. 2. As shown in fig. 2, the characteristic wavelength and the degree of stratification, as reflected in the position and amplitude of the peak of the FFT amplitude, respectively, are sensitive to the plate separation d of the Hele-Shaw cell. For a flow rate of.78 g/sec, as d is increased from 3 to 1 mm, both the wavelength and FFT amplitude decrease. Moreover, for an intermediate plate separation of ~ 7 mm, clear pairing of adjacent stratification layers of sand is evident, confirmed by the second peak in the corresponding FFT amplitude in fig. 2b. By d = 1 mm there is little indication of stratification. Fig. 3 plots the FFT amplitude (fig. 3a) and the initial wavelength (fig. 3b) against plate separation (filled and open circles) for a flow rate of.78 g/sec. The error bars in fig. 3a indicate the range of FFT amplitudes observed when the avalanche stratification pattern is repeated for the same plate separation and flow rate. The decrease in FFT amplitude A and wavelength λ on plate separation d can be fit to a power law dependence, where A ~ d -α and λ ~ d -β with α = 1.6 and β =.8. However, equally good fits can be obtained to an exponential dependence on plate separation [1]. Further experiments which can extend the dynamic range over which d can be varied and still yield a measurable banding pattern are needed to determine the functional form of the plate separation dependence of the stratification effect. As indicated in fig. 3b, when the plate separation is ~ 6 mm, a second wavelength (represented by the open circles) develops, indicating the pairing of stratified bands discussed above. The error bar in fig. 3b reflects the width of the second peak in the FFT amplitude (see for example the FFT amplitude in fig. 2b for a plate separation of 7 mm). In general the broadness of this second peak associated with the pairing of the bands obscures any variation of this second wavelength with plate separation. The sensitivity of the stratification pattern to varying the flow rate f at which the granular mixture is poured into the Heleshaw cell, for a fixed plate separation, has also been investigated. The resulting banding patterns are Fourier analyzed as in fig. 2, and the results are summarized in fig. 3 (open squares) for a plate separation of 3 mm. Fig. 3a demonstrates that the degree of stratification initally increases with flow rate, and then exhibits a sharp decrease at a critical flow rate of ~ 1. g/sec. For flows

4 4 d 3 mm 4 mm 2 2 FFT 1 Amp. (a.u) 1 d = 3mm d = 4mm Wavelength (cm) 6 mm 7 mm 2 2 FFT 1 Amp. 1 (a.u) d = 6mm d = 7mm Wavelength (cm) 8 mm 1 mm FIG. 2. Digitial images (a) of the stratification banding pattern near the top of the pile (indicated by the white box in fig. 1b) and plots of the corresponding Fourier amplitude against wavelength (b) for the 2 2 FFT 1 Amp. 1 (a.u) d = 8mm d = 1mm Wavelength (cm) sand/sugar mixture of fig. 1 as the plate separation of the Hele-Shaw cell is varied from 3 to 1 mm. The arrows in fig. 2a indicate the layers of sand (dyed dark). The flow rate for these data is.78 g/sec.

5 greater than this critical rate (which depends on the plate separation) the stratification pattern abruptly disappears. In contrast to the plate separation dependence, the wavelength of the stratification pattern is roughly insensitive to the flow rate (fig. 3b), (a) FFT Amplitude (a.u.) (b) Wavelength (cm) Flow Rat e ( g/ s) Plate Width (cm) Flo w Rat e ( g/ s) Plate Width (cm) FIG. 3. Plots of the FFT amplitude (fig. 3a) and wavelength (fig. 3b) against plate separation (solid circles) for a flow rate of.78 g/sec, and as a function of flow rate (open squares) for a plate separation of 3 mm. Fig. 3b also shows the wavelength of the paired bands (open circles) for plate separations greater than 6 mm. The dashed and solid lines are guides to the eye. though the rapid loss of stratification above the critical flow rate limits the range for which this parameter can be investigated. These experiments have been repeated for a series of plate separations and flow rates. The results are summarized in fig. 4, which is a phase diagram, indicating whether or not stratification is observed for a given flow rate and plate separation. The open squares indicate that stratification occured (defined as a Fourier amplitude greater than 6 as in fig. 3a), while the solid diamonds represent those flow rates and plate separations for which no layering was found. The crosses denote those f and d values for which pairing of the stratification layers was observed. The data in fig. 4 have been reproduced for many separate measurement runs over a period of several months. Qualitatively similar results are found for avalanche stratification of a mixture of poppy seeds and sugar. The results of fig. 4 demonstrate that the stratification effect occurs only for a finite range of flow rates and plate separations. Moreover the asymmteric distribution of FFT amplitudes in the stratification band, indicated in fig. 3, indicate that this effect does not follow a simple reciprocity relationship, that is, doubling the plate separation and doubling the flow rate does not yield the same stratification pattern (and, in fact, depending on the d and f values chosen, the stratification pattern may not persist). We now address possible mechanisms underlying the phase diagram for avalanche stratification. As mentioned above, previous models for stratification in a Hele-Shaw cell have been explicitly two dimensional. In these models the stratification results when the larger and more faceted granular material has a larger angle of repose compared to the smaller and smoother material. This

6 6 4 Stratification Pairing Mixed Flow Rate (g/s) Plate Width (mm) FIG. 4. Phase diagram for the avalanche stratification effect for a / mixture of sand and sugar. The open squares indicate those plate separations and flow rates for which stratification is observed, while the solid diamonds denote that no layering was found. The cross symbols represent the plate separation and flow rates for which pairing of the stratified bands occured. difference between angles of repose of the two materials leads to separation of the two species as they roll down the free surface, with the smaller material becoming fixed in irregularities of the granular surface [9, 1, 16]. Grains either become fixed at locations on the free surface determined by the value of the local angle of repose, or continue to roll down the pile. All motion is only down the free surface of the pile, and interactions with the sidewalls are neglected. Computer simulations based upon these models [1] are able to faithfully reproduce the stratification pattern observed, as in fig. 1b. However, the results presented here indicate that stratification is sensitive both to the separation of the sidewalls, and the rate at which material is deposited on the top of the pile. As granular material is poured into the Hele-Shaw cell, some of the grains immeadiately begin rolling down the top surface of the pile, as in the models of ref. [1] and [16], while some of the granular material can remain at the top of the pile, forming a metastable heap. This heap will grow until the maximum angle of stability is reached, at which point the material in the heap avalanches down the top surface, returning the top of the pile to the critical angle of repose. When this excess mass flows down the top surface, there is a large shear on the granular material due to the strong velocity gradient normal to the surface of the pile. This shear leads to dilation [18] which in turn enables separation or sieving of the smaller or denser granular material to fall into small gaps or voids in the flowing surface beneath the larger or less dense material in the flowing layer [19-21]. As the sand/sugar mixture avalanches down the top surface of the pile, it experiences

7 7 essentially the same dynamics as granular flow through a narrow inclined chute. However, it is well known that the velocity profile of the flowing material is significantly depressed in the region adjacent to each sidewall in a chute, with the relative decrease in velocity increasing for wider chutes [22]. Consequently, as the plate separation d is increased, there is an associated increase in the flow transverse to the down slope motion, as the particles follow the velocity gradients. The onset of this transverse velocity occurs when the plate separation is approximately twice the diameter of the orifice in the titrating bulb from which the granular mixture is poured into the Heleshaw cell. The transverse flow tends to oppose the stratification tendency within the flowing layer, as reflected in the decrease in FFT amplitude for increasing d. This increase with d of the velocity gradient, both normal and transverse to the free surface, may also be responsible for the decrease in wavelength of the stratification layers. The pairing of the layers (fig. 2a and 2b for d = 6 and 7 mm) can then be thought of as an Eckhaus instability [23], where the changes in the control parameter (plate separation d) which sets the wavelength of the pattern formation in the driven dynamical system can lead to period doubling of the spatial wavelength [24], though more work on this point is needed. The sensitivity of the stratification effect to variations in the flow rate, for a fixed plate separation, are harder to understand. The stratification pattern is fairly stable until a critical flow rate f c is reached. For f < f c, these results are therefore compatible with the models of refs. [1] and [16], for which the rate at which material is deposited at the top of the pile is not a significant factor. However there is no obvious reason in these models for the stratification pattern to disappear for f > f c. In the alternative scenario sketched above, stratification occurs when the metastable heap at the top of the pile avalanches down the free surface. When f > f c, the grainfall along the top surface is so fast that there is no time for the development of metastable heaps at the top of the pile, which would then avalanche down the top surface and produce stratified layers. Hence the banding pattern should disappear for too fast a pouring rate, and the wider the plate separation, the larger an f c is needed to inhibit heap formation, consistent with our observations. However this explanation would also predict that the wavelength of the banding pattern will depend on the timing of the avalanches, and consequently, should be sensitive to the rate at which the granular mixture is added to the top of the pile. The weak dependence of the wavelength of the banding pattern on flow rates less than f c (fig. 3b) is therefore in conflict with this explanation, and further work is need to elucidate the role that flow rate plays in the stratification effect. In summary, we have observed a nontrivial phase diagram for avalanche stratification as a function of plate separation and flow rate for a binary mixture of granular media poured into a quasi-two-dimensional Hele-Shaw cell. In addition, pairing of stratified layers is found for a finite range of d and f values. These results indicate that two-dimensional models for avalanche stratification are incomplete, and must possibly include the effects of velocity gradients induced by grain/sidewall interactions. This research was supported by the Minnesota Graduate School Grant-in-Aid program, the Undergraduate Research Opportunity Program and Grant no. NSF- CTS We gratefully acknowledge experimental assistance from K. M. Hill and W. Weisman and helpful comments from J.

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