Forced segregation in binary granular mixtures

Mixing granular particles of di erent sizes is a common way of increasing the packing fraction. Recently, a model predicting the packing fraction, taking into account the inhomogeneity of the mixed small and large particles, has been proposed by S. Pillitteri et al. Under certain conditions, this model can be simpli ed and analytical solutions can be found. We present here these solutions, compared to experimental data, and the physical interpretation they can bring.


Introduction
Granular binary mixtures are widely studied because of the large panel of properties observed in such systems [18]. The main quantity describing these mixtures is the packing fraction η which corresponds to the ratio between the true volume of the particles V and the apparent volume V a of the pile. Depending on the relative composition of both species, the packing fraction can be strongly dierent.
In such systems, the control parameters are the volume fraction of small particles [8] dened as where V s and V l are respectively the true volume of small and large particles in the mixture, and the size ratio whereR l and R s are the radii of large and small particles. Considering the denition of V s and V l , one consequently has V = V s + V l . However, depending on the history of the granular mixture, the packing fraction can be dierent. Indeed, during mixing or other manipulations, segregation often occurs and leads to inhomogeneous mixtures. This inhomogeneity is known to decrease the packing fraction [9,10].
In a previous article [11], we proposed a model depending on f and α that took into account the distributions of large and small particles. In this work we propose a specic view on the simplest case, when there is a segregated monodisperse phase of large or small particles in the mixture. We show how this model can be simplied in order to obtain analytical solutions. We present these solutions and describe their eect on the packing fraction curves. * e-mail: s.pillitteri@uliege.be

Method
We prepared several mixtures by varying f with a size ratio α ≈ 9 for which we forced the segregation. We dened p and b, two segregation parameters. The parameter p corresponds to the proportion of unmixed small particles which are placed at the bottom of the granular pile while b represents the proportion of unmixed large particles which are placed at the top of the pile. Sketches of such mixtures are presented in Figure 1.
Forcing a partial phase separation of particles ensures a control of the segregation. For each mixture with f , b and p xed, we measured the tap density curve with the GranuPack [12]. This device consists of a steel tube with a diameter D = 26 mm and a length L = 100 mm in which the powder is poured. A narrower tube inserted into the measurement tube is then removed in order to have a reproducible initial- and f = 0.5. The black curves correspond to ts using the logarithmic law eq. (3). Symbols correspond to the cases shown in Figure 1.
ization of the granular pile. The motion of the narrower tube is done at a low and constant velocity v = 1 mm/s. The measurement tube performs free falls or "taps" in order to densify the granular medium. The packing fraction η(t) is recorded after each free fall until t = 500 taps. The compaction curve is tted with the logarithmic law [13] where η i is the initial packing fraction and η ∞ and τ are free tting parameters corresponding respectively to the asymptotic packing fraction reached after the typical compaction time. We compare η ∞ for the different mixtures since it corresponds to the densest packing fraction reachable by each system.

Results
In Figure 2, three typical compaction curves are presented for α ≈ 9 and f = 0.5 for which dierent segregation patterns have been forced. One observes that the highest packing fraction is reached for the homogeneous case b = p = 0. Indeed, in this case, small particles can ll voids between large pores left by large particles. On the contrary, unmixing small and large particles decreases the packing fraction. Indeed, the monodisperse phase can only reach η ∞ ≈ 0.64, being the Random Close Packing fraction [14,15], while the mixed part exceeds this value. The global packing fraction is a weighted average of both packing fractions. It is therefore reduced due to this unmixed part since some small particles cannot ll voids between large ones when one of the species is segregated. Figure 3 presents the packing fraction as a function of f for mixtures with unmixed parts. One observes that higher packing fractions are obtained for homogeneous mixtures. When b = 0 and p = 0, one observes the decrease of the right part of the curve

Discussion
This can be understood by considering the model proposed in a previous work [11]. This model stems from Furnas' approach [16] which considers both extreme cases : (I) when the packing is dominated by large particles, and (II) when the packing is dominated by small ones. Our model diers from Furnas' approach because we consider that the packing fraction should be computed by integration of a local apparent volume V loc a to obtain the global apparent volume V a when the packing is inhomogeneous. One has to dene the distribution functions Ψ l (r) and Ψ s (r), respectively of the large and the small particles in the normalized unit volume V * of the granular pile. These functions follow the constraints In the rst case (I), when f → 0, the system is mainly made of large particles. One can write the local apparent volume as where η 0 l and η 0 s are the packing fractions of the monodisperse cases for respectively the large and the small beads. In the second case (II), when f → 1, the mixture is in a random packing conguration dominated by small beads. One can write the local apparent volume as and V (a) One remarks that only V  On the other hand, the case (b) gives and In this case, only V (b) II depends on b, which means that only the right part decreases when b increases.
One can observe the result of such modelling, knowing the xed proportions p and b of respectively segregated small and large particles, in Figure 3. The model is represented by curves of the same color as the data. One can see the good agreement between the model and the experimental data.
However one can point out that the smooth optimum observed in the data is not captured by the peak shape of the model. This dierence could come from the natural inhomogeneity of the mixtures. Indeed, it is highly probable that supplementary percolation occurs during manipulation. One can assume a gradient-like distribution of small particles depending on f to model this inhomogeneity [11]. Indeed, when there are few small particles with f → 0, one assumes that the gradient is strong. On the contrary, when f → 1, their distribution should be uniform since they cannot percolate when the system is dominated by small particles. One has the distribution The distributions Ψ l = D(z * , h * , 0) and Ψ s = G(z * , f ) are presented in Figure 4 (c). In this case however, the model cannot be simplied. In addition, h * must increase when f increases for a xed p what complicates the manipulation of the distributions. For more information about the gradient-like distribution of small particles we suggest to consult our previous article [11]. Nevertheless, we present by a dashed curve in Figure 3 the result of Eq. (7) for a mixture with Ψ s = G(z * , f ) and large particles homogeneously distributed with Ψ l = D(z * , 0, 0). Thereby, one can observe the apparition of the smooth optimum due to the gradient-like distribution of small particles.

Conclusion
We investigated the impact of partial segregation on packing fraction. We prepared mixtures with a proportion of small or large particles separated from the rest in a monodisperse phase. The decrease of the packing fraction was clearly observed as the proportion of segregated particles increases. We observed that the global packing fraction decreases in a dierent way if this monodisperse phase is made of large or small particles.
From these observations, we proposed here a discussion about the simplest case of our previous model [11] for inhomogeneous binary granular mixtures. Indeed, when segregation occurs by the existence of a monodisperse phase, the distributions Ψ l and Ψ s of large and small particles can be dened by step-like functions. According to the model and these distributions, one can remove the integration and simplify the expression of the apparent volume. In this way, one obtains a function η ψ which allows a direct interpretation of the decrease of the slope of the right or the left part of the curve. Depending on the nature of the monodisperse phase, made of large or small particles, the right or the left part of the curve will be aected.
We have shown that this model is adapted for the investigation of granular binary mixtures. Indeed, it takes into account the inhomogeneity of the system as well as common control parameters like the volume fraction of small particles f and the size ratio α. Since segregation often occurs in granular media, this model could be useful for granular mixtures designing and improvement. Moreover, new numerical studies about binary mixtures could be performed, with a forced gradient-like segregation, in order to test the quality of our model.