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PHYSICAL REVIEW E 85, 031309 (2012)

How relative humidity affects random packing experiments N. Vandewalle,1 G. Lumay,1,2 F. Ludewig,1 and J. E. Fiscina1,3 1

GRASP, Université de Liège, B-4000 Liège, Belgium 2 F.R.S.-FRNS, B-1000 Bruxelles, Belgium 3 Experimental Physics, Saarland University, D-66123, Saarbrücken, Germany (Received 24 December 2011; published 30 March 2012) The influence of relative humidity (RH) on the extremely slow compaction dynamics of a granular assembly has been experimentally investigated. Millimeter-sized glass beads are considered. Compaction curves are fitted by stretched exponentials with characteristic time τ and exponent δ, which are seen to be deeply affected by the moisture content. A kinetic model, taking into account both triboelectric and capillary effects, is in excellent agreement with our results. It confirms the existence of an optimal condition at a relative humidity ≈45% for minimizing cohesive interactions between glass beads. The exponent δ is seen to depend strongly on the diffusive character of grains and voids inside the packing: diffusion for cohesiveless particles and subdiffusion when cohesion plays a role. As a consequence, the RH represents a relevant parameter that should be reported for every experimental work on a slowly driven dense random packing. DOI: 10.1103/PhysRevE.85.031309

PACS number(s): 81.05.Rm, 81.20.Ev, 83.80.Fg

I. INTRODUCTION

How a large number of spherical objects fill a volume is one of the most persistent puzzles in mathematics and science [1]. The packing fraction η of a dense granular assembly is defined as the ratio between the volume of all grains and the volume of the container. When identical spherical grains are gently poured into a tube, a range of packing fractions between 0.59 and 0.64 is obtained. The largest value (0.64) is called the random close packing (RCP) fraction. Small mechanical disturbances of the random assembly induce local rearrangements of the grains, thus √ increasing the packing fraction. A crystal of grains at η = π/ 18 ≈ 0.74 is, however, difficult to obtain due to the extremely slow compaction dynamics. Indeed, numerous studies [2–5] have proved that the compaction dynamics of a “dry” packing submitted to a series of taps is extremely slow and similar to relaxation mechanisms in glassy systems. Steady state rheology and compaction behavior have been recently related [6] as a part of a more general theory of jamming [7,8]. Thousands of papers [9] have been published on the experimental physics of granular materials. Only 1% of them [9] include information about the hygrometry in the laboratory where the experiments were conducted. Earlier experiments [10] proved that humidity is a relevant parameter for the stability of static piles. Aging of avalanche angles has indeed been observed at high humidity and has been associated with capillary condensation [11], i.e., the formation of small liquid bridges between contacting grains. Although all physicists suspect a moderate effect of moisture content on the physical properties of a dense granular assembly, we discover a stronger and more complex effect than expected by analyzing relaxation mechanisms in slowly driven random packings. The results of the present study aim to change the way experiments should be conducted and reported in the scientific literature. II. EXPERIMENTAL SETUP

Following our earlier studies on compaction experiments [5,12,13] we modified our experimental setup in order to 1539-3755/2012/85(3)/031309(5)

control the relative humidity (RH, which we express in % in this paper) in the granular assembly. A sketch of the experimental setup is shown in Fig. 1. The protocol is as follows. The grains are monodisperse glass beads (ρ ≈ 2500 kg/m3 ) with a diameter d = 1 mm. Our deliberate choice of large beads is motivated by the fact that the physics of such a packing is generally supposed to be dominated by gravity instead of intergrain forces. Such beads are commonly used and characterized in the scientific literature. Indeed, one should compare the weight π6 ρgd 3 of a grain with the capillary force π γ ζ [14,15], where the surface tension is γ = 72 mN/m and ζ is the diameter of the liquid bridge between two neighboring grains. For capillary nucleation, ζ corresponds to the roughness of the bead surface. Since ζ d, gravity forces are much larger than capillary forces. A bottomless glass tube is inserted into the main one and is filled with a defined mass of granular materials. The diameter of the main tube is D = 35.3 mm. Thereafter, the inner tube is removed upward at a low and constant velocity fixed to v = 1 mm/s, leaving the grains to rearrange themselves into the larger tube. After this initialization stage, a reproducible and spatially homogeneous initial packing fraction is obtained [12]. To ensure compaction, the tube is fixed on a support which can perform some periodic free falls. The free fall height is fixed to z = 1 mm, meaning that the mechanical energy

injected into the system is kept constant for all experiments. This parameter is indeed a relevant parameter for such an experiment as demonstrated in Refs. [16,17]. Two successive taps are separated by at least 1 s, allowing the system to relax. An aluminum hollow cylinder is gently placed on the granular pile in order to keep the grain-air interface flat during the measurements. Moreover, this hollow cylinder is necessary to perform the measurement of the interface position with a distance sensor. The mass of the hollow cylinder (14 g) is lower than the mass of the same volume of granular material (22 g). From the vertical position of the interface between the granular bed and the hollow cylinder, the packing fraction η of the pile is determined after each tap. The resolution on the packing fraction η is around 0.001, thus providing accurate measurements.

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©2012 American Physical Society

N. VANDEWALLE, G. LUMAY, F. LUDEWIG, AND J. E. FISCINA


(a)

FIG. 1. (Color online) Sketch of the experimental setup, including the RH controller. See text for details.

Long compaction experiments up to 104 taps have been recorded. In order to control the RH, the tube containing the granular material is placed under a cover. An air flux of controlled humidity is injected continuously inside this cover during the initialization of the pile and during the compaction process. Moreover, between the initialization and the compaction process, we let the system stabilize for 1 h. The RH controller device works as follows. Some air flux is first dispersed into small bubbles in a bottle containing water. Therefore, at the output of the bottle, we obtain a saturated air flux. Afterward, the saturated air flux passes through a condenser, which is a block of copper into which a tortuous channel has been machined. The temperature of the condenser is controlled by a Peltier cell between 0.5 ◦ C (to avoid the formation of ice) and room temperature. By adjusting the temperature of the condenser, the RH of the air injected into the compaction device can be precisely tuned. III. RESULTS

Figure 2(a) presents typical compaction curves when the RH is changed from 20% to about 80%. Saturation is reached on all curves after t = 104 taps. The compaction curves η(t) look similar to the phenomenological Kohlrausch-WilliamsWatts (KWW) law [18], i.e., δ η − η0 t , (1) = 1 − exp − η∞ − η0 τ which is a stretched exponential where τ is a relaxation parameter related to reorganization mechanisms taking place in the random packing. In compaction experiments, the stretched exponential law is observed for monodisperse systems when some dense parts of the packing (like polytetrahedral structures [19]) become relevant [5], while an inverse logarithmic law is observed for polydisperse systems [16]. The stretching exponent δ in Eq. (1) is related to the physical mechanisms behind the growth of structures inside the packing. In Ref. [5], we found that δ is a signature for the diffusion of defects, i.e., voids, along grain boundaries in a two-dimensional (2D) packing. Figure 2(b) presents two compaction curves after rescaling of the data: x = (η − η0 )/(η∞ − η0 ) where η∞ has been estimated by averaging the last part of the compaction curve, i.e., the saturation. When the parameter x reaches 1 − 1/e ≈ 0.63, one should find t = τ . The observation of the two curves in Fig. 2(b) indicates different τ values, i.e., quite different compaction dynamics. Figure 2(c) presents

FIG. 2. (Color online) (a) Semilogarithmic plot of typical compaction curves η(t) for four different values of the air relative humidity: disks, 29%; squares, 40%; triangles, 61%; and diamonds, 75%. Fits using Eq. (1) are also shown. (b) Rescaled values x of the packing fraction for two typical curves (RH of 40% and 75%). Only the first parts of the compaction curves (and fits) are shown for clarity. This part evidences different relaxation times τ . (c) The same data after rescaling in a log-log plot. This plot evidences different stretching exponents δ. Straight lines illustrate typical values obtained for those data.

ln[1/(1 − x)] as a function of t in a log-log plot in order to show that the stretching exponent is the slope of the curves. Two different slopes are emphasized by specific lines in the figure. From Figs. 2(a)–2(c), one concludes that the relative humidity seems to have an effect on both the compaction dynamics (τ and δ) and the packing fraction. Nonlinear fits of Eq. (1) have been performed by using the Levenberg-Marquardt iterative algorithm. The parameter η0 is fixed to the first point of each curve. Three free fitting parameters remain: η∞ , τ , and δ. For fitting, the initial values of these free parameters are chosen to be those obtained in similar fits done for neighboring values of the RH. This initialization of the iterative fits allows us to obtain a fast convergence of the fits. We checked that the fits are robust since either linear or logarithmic weights associated with the data do not change the conclusion of the present paper. In particular, we performed various fits from t = 1 to t = T . For T > 1000, the fitted values remain unchanged with T within a few percent error because the compaction curves reach saturation. All fitted values given below correspond to T = 10 000. Figure 3 exhibits the evolution of four relevant parameters (η0 ,η∞ ,τ ,δ) as functions of RH. While the packing fraction η0 comes from measurements, the other three parameters are extracted from the fits of the compaction curves, using Eq. (1). Error bars are shown. The initial packing fraction η0

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minimum [21]. A remarkable result of our study is that a strong effect of RH is evidenced for a system of large beads. In a recent study [13], we showed that when a small liquid amount is added to a random assembly of smaller grains, corresponding to the high limit of RH and to large capillary bridges (ζ ≈ d), one obtains τ ≈ 100, in agreement with the values found herein when the RH is above 80%. Since the beads are the same in all experiments, the variation of τ with RH could only be explained by a large change of cohesion in the packing. As seen in the last plot of Fig. 3, the stretching exponent δ shows a large maximum around HR ≈ 45%. There, the maximum value is δ ≈ 1/2. However, for cohesive grains, δ is lower than 1/2. This behavior will be discussed below. IV. DISCUSSION

FIG. 3. (Color online) Top: The initial packing fraction η0 (disks) and the final packing fraction η∞ (squares) as functions of RH. The dashed curves are a guide for the eye and emphasize a slight decrease of the packing fraction with humidity. Middle: The relaxation time τ as a function of the relative humidity RH. A clear minimum is seen for relative humidity HR ≈ 45%. The continuous curve is a fit using our model explained in the text. Bottom: The stretching exponent δ as a function of RH. The dashed curve is a guide for the eye. Error bars are drawn for all data points. Horizontal lines at δ = 1/2 and 1/3.8 correspond to limiting values of anomalous diffusion in disordered (percolation) systems.

is seen to fluctuate around 0.59 whatever the RH value. The packing preparation protocol implies small error bars for η0 . A small slope is seen, however, and emphasized by a dashed curve, which is only a guide for the eye. Such a decrease is a signature of the intergrain cohesion induced by the RH. Indeed, interactions between grains allow larger voids between grains, as studied in magnetic systems [12,20]. One should remark that η0 is measured before numerous vibrations, such that electrical charges are not yet present in the system for the η0 measurement. However, after a complete series of taps at low RH values, some grains stick together due to the presence of electrical charges. It should be noted also that the η∞ values present a similar behavior, i.e., a slight negative slope as a function of RH. Therefore, the RH has a moderate effect on the packing fraction before and after tapping. In Fig. 3, the relaxation time τ shows large variations when the RH goes from dry to wet conditions. The relaxation time τ extracted from the compaction curves is seen to decrease greatly at moderate RH, while it increases at high humidity. A minimum of τ is found around a relative humidity HR ≈ 45%. There, the flowability of the granular assembly is supposed to present a maximum while cohesion is expected to exhibit a

Cohesion appears in both dry and wet conditions. When the RH is low, grain-grain friction due to tapping induces the appearance of electrical charges everywhere in the packing, as drawn in Fig. 4. This triboelectric effect has been characterized with similar glass beads in Ref. [22]. It has been found that the electrical charge q on beads decreases exponentially as a function of the RH with a characteristic rate 30%, i.e., q = q0 exp(−HR /30). This particular decrease should be associated with microscopic capillary condensation on the beads which neutralizes a fraction of the charges. The electric potential associated with this charge is proportional to q 2 and therefore decays as exp(−HR /15). The condensation at high humidity implies also the formation of liquid bridges between contacting grains. For high RH values, liquid bridges increase the cohesion as depicted in Fig. 4. Taking both effects (triboelectricity and capillarity) into account, the energy to separate two contacting grains defines a free energy barrier b −HR − , (2) B = a exp 15 ln(HR /100) which should be overcome by the grains in order to move in the packing. The first term is the empirical exponential law reported in Ref. [22], while the second term is based on the Kelvin relation and is derived from the capillary adhesion force proposed in Ref. [10] by neglect of the effect of waiting times because the system is perturbed at fixed time intervals (taps).

FIG. 4. (Color online) Sketch of the sphere packing at three different RH values. Cohesion is due to either electrical charges (red spots) or capillary nucleation (blue spots). At moderate RH, the presence of a liquid phase neutralizes the electrical charges, and low cohesion is observed. When cohesion plays an important role, the grains stick together and grain motion is stopped for a fraction of the assembly (grains colored in dark gray in the picture). Only a part of the packing (in light gray) participates in the compaction process. Motion of grains and voids takes place on percolationlike clusters.

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N. VANDEWALLE, G. LUMAY, F. LUDEWIG, AND J. E. FISCINA

F

B

configurations FIG. 5. (Color online) A naive representation of granular compaction within a free energy diagram F in the space of local configurations. In order to fill a small void with a single grain, a few grains should be separated from each other, thus implying overcoming a barrier B linked to cohesive forces.

The parameters a and b represent the strength for each effect. The local grain rearrangements can be described following the so-called energetic approach [13,16]. The model considers the mechanical energy per grain injected in the granular bed at each tap. We consider the normalized packing fraction x. As depicted in Fig. 5, compaction involves the motion of grains filling small voids in the packing. This microscopic motion implies some energy B needed in order to separate grains from each other, allowing grain rearrangements. One understands that the energy barrier is mainly related to the cohesion between neighboring grains. If an energy barrier B exists, xB becomes the energy fraction of the grains which participate in the compaction process while the amount of energy (1 − x)

contributes to the fluidized part of the granular bed. Assuming that at each tap the packing fraction evolves as a function of the free volume fraction and the energy balance within an Arrhenius-like form, the kinetic equation reads −x B dx = α(1 − x) exp , (3) dt 1−x

where α is a dissipation factor which should be related to

. Herein, the parameter α is considered as a constant since all experiments are conducted on the same apparatus. One should note that the ratio x/(1 − x) found in the Arrhenius (exponential) form was already assumed in the free volume kinetic model of Boutreux and de Gennes [23] and rationalized in Ref. [24]. By integration of Eq. (3) and by taking x = 1 − 1/e when t ≈ τ , one obtains B exp −

eB B τ≈ Ei − Ei , (4) α

y where Ei (y) = −∞ [exp(θ )/θ ]dθ is the exponential integral of the dimensionless variable y. The fit of Eq. (4) taking (2) into account is shown in Fig. 3. The agreement between this energy approach and real measurements is excellent. From the fit, one obtains a ratio a/b ≈ 115 ± 12, meaning that both tribolelectric and capillary effects play relevant roles in cohesion. These parameters could slightly change with the grain nature or the bead size. An interesting feature to


be mentioned is that sand granular materials tested in our equipment exhibit a minimum around 45%. The analysis of our data in Fig. 3 emphasizes a deep change of the compaction process with RH. The stretching exponent δ reaches quite different values when the level of cohesion varies in the packing. The energetic approach is able to capture time scales but cannot explain the probabilistic nature of the local grain rearrangements during compaction. As proposed in an earlier 2D experiment [5], the δ value is closely related to physical mechanisms governing the motion of grains and more precisely the voids (allowing grain motion) in the packing. The δ = 1/2 value obtained for cohesionless spheres evidences the diffusive character of the voids in the packing, allowing grain rearrangements of local loose configurations. A lower δ value therefore corresponds to anomalous diffusion mechanisms and more precisely subdiffusion. Indeed, when cohesion increases in the packing, more and more grain contacts become permanent. Cohesion rigidifies a fraction of the grains in the assembly and voids cannot freely diffuse in the bulk. An analogous system is diffusion on percolation clusters near the critical point, exhibiting anomalous diffusion [25,26]. Extreme values of subdiffusion exponents range from 1/2 to 1/3.8. This is consistent with our findings, as depicted by the horizontal lines in Fig. 3. Recent theoretical work [27,28] has demonstrated that a percolation model predicts stretched exponential relaxation in glassy materials like granular systems. The range of possible stretching exponents [28] corresponds to our findings. V. CONCLUSION

In summary, we have observed the compaction dynamics of a random sphere assembly. Although the bead size (1 mm) is large, the relative humidity of air is seen to play an important role. Indeed, due to friction between contacting beads, electrical charges appears on the beads when the air is dry such that some energy barrier appears for individual motions of grains in the packing. When the moisture content increases, the charges disappear exponentially and liquid bridges can form between grains by capillary condensation. Liquid bridges, like electrical charges, represent a barrier for local reorganizations. Cohesive forces could mask (or modify) the physical properties of granular materials [13]. The consequence of our findings is that the moisture content should be characterized and reported in studies about granular materials. The best condition for conducting granular experiments with glass beads corresponds to a relative humidity of about 45%, since cohesion is minimum for that moisture content. Future work could concern similar measurements on other hydrophobic or hydrophilic granular materials in order to characterize triboelectric and moisture effects. Since the dry and wet limits induce cohesion, an optimum is expected for all types of material.

ACKNOWLEDGMENTS

G.L. would like to thank FNRS for financial support. J.E.F. thanks the Alexander von Humboldt foundation. This work

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has been supported by INANOMAT Project No. IAP P6/17 of the Belgian Science Policy. The authors thanks O. Gerasimov,

F. Boschini, and E. Mersch for valuable discussions and J.-C. Remy for his technical support.

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