Abstract
Granular media take on great importance in industry and geophysics, posing a severe challenge to materials science. Their response properties elude known soft rheological models, even when the yieldstress discontinuity is blurred by vibrofluidization. Here we propose a broad rheological scenario where average stress sums up a frictional contribution, generalizing conventional μ(I)rheology, and a kinetic collisional term dominating at fast fluidization. Our conjecture fairly describes a wide series of experiments in a vibrofluidized vane setup, whose phenomenology includes velocity weakening, shear thinning, a discontinuous thinning transition, and gaseous shear thickening. The employed setup gives access to dynamic fluctuations, which exhibit a broad range of timescales. In the slow dense regime the frequency of cageopening increases with stress and enhances, with respect to μ(I)rheology, the decrease of viscosity. Diffusivity is exponential in the shear stress in both thinning and thickening regimes, with a huge growth near the transition.
Introduction
Dry granular materials are collections of macroscopic particles, interacting through frictional contact forces^{1,2,3}. The resistance of a granular aggregate to an applied shearing force is sensitive to many aspects of the experimental setup and may present analogies with macroscopic frictional laws, plasticity, soft glassy rheology and the shear thinning or thickening phenomena of suspensions^{4,5,6,7,8,9,10,11}. Recently, consensus has been achieved on a certain class of steady slow flows which obey the socalled μ(I)rheology^{6,12,13,14}. In such a framework the shear stress σ is proportional to normal pressure p through a friction coefficient μ(I) = σ/p, which slightly depends on the shear rate itself through the adimensional “inertial number” I, according to the following formula:
where μ_{1}, μ_{2} and I_{0} are constants. The above formula (see red curve in Fig. 1a) expresses (at constant p) a monotonic growth of σ from a minimum yield stress σ_{1} = μ_{1}p to a saturation (frictional) stress σ_{2} = μ_{2}p. The inertial number is the ratio between the shear rate and the microscopic frequency (d the diameter of a grain, ρ its material density, m its mass, D_{s} the space dimension). Basically f_{m} is the inverse of the time needed by a grain to move by d under the acceleration given by the pressure, if starting at rest. The validity of the μ(I) scenario has been probed in different setups and is typically associated with a dilatancy effect in the form of a Idependent packing fraction ϕ(I)^{12}. For this reason the scenario is better appreciated in experiments where the volume is not constrained. Note that Eq. (1) corresponds to a monotonic thinninglike reduction of effective viscosity which goes from ∞ to 0 as the shear rate is increased.
A more complex picture emerges in the presence of vibrofluidization, that is, under vertical vibration of the granular container^{15,16}. In applications, vibrofluidization is a renowned technique that enhances homogenization and surface of contact at the solidgas interface for combustion chambers and chemical reactors. A parameter that characterizes the intensity of vibration is Γ = a_{max}/g, that is the maximum vertical acceleration a_{max} (in the case of sinusoidal vibration) normalized by gravity acceleration g (in our experiment we have also used nonsinusoidal vibrations and therefore a more general definition of Γ, see Methods). Even at mild values of Γ (Γ < 1), an internal diffusion of kinetic energy cooperates with the applied stress and softens the discontinuities provided by enduring contacts^{15}. The result is the introduction of a thermallike energy scale (absent in nonfluidized granular media), an evident reduction of the yield stress and a faster fluidization of the material under increasing rates of deformation. Rheological studies in a splitbottom cell under vertical vibrofluidization demonstrated the existence of a thinning transition^{15}, whose exact nature is under scrutiny^{17,18}, recently ascribed to an internal distribution of microscopic stresses and a local HerschelBulkley ratestress relation^{19}.
A parallel line of investigation has approached the problem of dry granular rheology by introducing the concept of partial fluidization^{20,21}. In this context there is agreement about the hybrid nature of granular internal stress, modelled as a superposition of a frictional contribution, sustained by enduring contacts stabilized by normal pressure, and a kinetic contribution, where momentum is transferred through instantaneous collisions of the fluidized particles. The kinetic contribution is expected to be negligible in the densest and slowest regimes, while it emerges in liquidlike flows and finally becomes dominant in gaslike configurations. Notwithstanding the immediacy of the concept of partial fluidization, very different recipes and analyses have been suggested in the literature, focusing on different aspects and setups. A relevant role in this framework is played by models of nonlocal rheology^{7,21,22}.
Results
A unified rheological model
Our aim, here, is to put under scrutiny a conjecture of ours for a minimal rheological model, based upon superposition between frictional and collisional contributions to internal stresses, that can embrace the full spectrum of rotationally forced granular flows under vibrofluidization, specifically a large range of values of I ∈ [10^{−5}, 10] and Γ ∈ [0, 40]. In general, normal stress (pressure) p depends upon the degree of fluidization, i.e. upon both I and Γ. For this reason we take as a pressure scale p_{00} which is the pressure at total rest (I = 0 and Γ = 0): the inertial number I takes the same definition as above, by replacing p with p_{00}. Our proposal, illustrated in Fig. 1, takes the following form for a rheological curve at constant Γ:
where the modified friction coefficient (blue and cyan curves in Fig. 1a) has the form
the Bernoulli pressure correction function α(I) (see dotdashed purple curve in Fig. 1b) is defined as
and finally the Bagnold rheology function (dashed purple curve in Fig. 1b) is simply
with μ_{1}, μ_{2}, c, I_{0}, I_{1}, I_{2} and I_{3} model parameters. Our proposal is not only supported by a wide agreement with experimental data, discussed below, but is substantiated through the following physical arguments.
First, in contrast with the original μ(I) function, a ~ additional contribution appears at the denominator of : it represents “activated fluidization”, that is, the enhancement of the breakage rate of enduring contacts due to the applied stress. We note that the Idependence of the friction coefficient μ can be ascribed to the variation of the fraction P_{s}(I) of enduring “solid”like contacts, namely μ(I) ∝ P_{s}(I). A minimal model for P_{s}(I) consists in neglecting memory effects (expected to be important only at very slow shear rates) and writing down a balance equation^{23} , whose stationary state reads with W(f → s) and W(s → f ) the transition rates from fluid to solid state and viceversa, respectively. Comparison with the usual μ(I) rheology, Eq. (1), suggests that W(f → s) and W(s → f ) are linear in I. On the contrary, the correction in the , Eq. (3), implies that W(s → f ) is enhanced by an additional contribution ~. In our experiment detailed below, the analysis of fluctuations provides a transparent interpretation of such an additional term as a cageexit frequency. The correction can also reproduce rheological behaviors of the kind shown as the cyan curve in Fig. 1a, i.e. cases of velocityweakening (an initial reduction of from the μ_{1} value) which appear in certain experiments in the absence of vibrofluidization. Weakening cannot be explained by the usual μ(I) function, which is necessarily monotonic. The α(I) correction to pressure dictates the drop in pressure in the presence of finite fluid velocity, in analogy with classical Bernoulli’s principle. Finally, the Bagnold rheology function B(I) provides us with the inertial contribution of instantaneous collisions, dominating at large I, where one expects a viscous contribution σ ~ γ(I)I and the “thermal” fluctuations underlying effective viscosity are ruled by the shear rate itself, that is γ ~ I. The Bagnold relation is usually indicated as a case of shear thickening, even if there is no universal consensus on whether the word “thickening” should be reserved for dense suspensions, or it also applies to inertial effects arising in diluted fluids.
Gathering all the pieces together, a general rheological curve is obtained, an instance of which is shown as solid purple curve in Fig. 1b. At the transition between the soliddominated and the kineticdominated regions it is possible to observe a nonmonotonic van der Waalslike behavior of σ which, in stresscontrolled experiments, appears as a discontinuous thinning^{15,18}. It is straightforward to verify that a continuous change of parameters appearing in Eq. (2) transforms the nonmonotonic crossover in a monotonic one, as seen in the experiments. We underline that the nonmonotonic crossover between and B(I) is clearly distinct from the velocityweakening effect discussed above, which belongs to the behavior of alone.
The unified rheological formula, Eq. (2), contains a series of parameters which depend, among other physical aspects of the setup, upon the intensity of vibrofluidization Γ. We remark that in the frictional contribution the dependence on Γ is expected to have a behavior opposite to that in the kinetic contribution B(I). Indeed, vibrofluidization reduces the steady fraction of enduring contacts, while increasing the thermal agitation of flying/colliding particles. Such contrasting dependencies neatly reflect our experimental observations, as described below.
Comparison with experiments
The theoretical picture of Eq. (2) fairly describes the broad phenomenology observed in the experiments we carried out. These are inspired by vanetest tools for the insitu rheology of soils^{24,25}, while the granular medium undergoes mechanical vibrofluidization in the vertical direction. Experiments are detailed in the Methods section. The observed rheological curves σ vs I explore ranges of I which depend upon p_{00}. The six frames in Fig. 2 show several representative cases together with their best fits through Eq. (2).
Frame (a) illustrates a case at high p_{00} which provides us with a high resolution at low , i.e. zooming in the first part of Eq. (2), where the kinetic contribution is negligible and α(I) ~ c. The ~ behavior is evident, as well as a small but nonnegligible yield stress μ_{1} > 0. At intermediate values of p_{00} (frames (b), (c), (d) and (e)) the flow curve σ vs I exhibits the crossover from the soliddominated regime to the collisionaldominated regime, which at low Γ is nonmonotonic. Increasing Γ the parameters change continuously, leading to a point where the curve becomes monotonic. The pressure at rest in cases (c), (d) and (e) is low enough to allow a series of data at Γ = 0 (see black circles) where a large yield stress can be measured. Finally, frame (f) reports a low pressure situation, where the collisional part of Eq. (2) dominates, leading to thickeninglike behavior, that is an increasing differential effective viscosity ∂σ/∂I. The six frames confirm what we argued in the above theoretical discussion: when the stress is dominated by the solid contribution, an increase of Γ leads to a reduction of stress, while the opposite occurs when the kinetic contribution dominates. At a given p_{00}, the value of I corresponding to the crossover between the two regimes does not depend upon Γ: indeed the nonmonotonic curves (cases (b), (c), (d) and (e) at low Γ) cross, roughly, at a single point.
In Supplementary Section S1 we discuss the systematic behavior of the fit parameters (which are reported in Supplementary Table S2). The physical consistency of the model is corroborated by a general smoothness of parameters’ behavior when Γ is increased. It is more difficult to find a clear connection between the values of the parameters and p_{00} (controlled by material density and by N), an aspect which certainly deserves further investigation.
Microrheological properties
Further support for our picture comes from the study of fluctuations, made feasible by our vanetest experiment where the rotating blade behaves also as a microrheological probe^{26,27,28}. In particular we have measured diffusivity (where θ(t) is the angular position of the blade), the frequency of relaxation of the angular velocity ω(t) defined as f_{ω} = 〈(ω − 〈ω〉)^{2}〉/D, and the frequency of typical cage exit f_{cage} (which is well defined only in the slow dense cases at high p_{00}^{26}). The precise definition of those quantities is given in the Methods section.
In order to identify the relevant physical quantities responsible for the different regimes observed in our system, we define the Péclet number Pe = 〈ω〉/D, the Reynolds number Re = 〈ω〉/f_{ω} and the Mach number , which are shown in Fig. 3, frame (a), for a set of experiments. For not too high values of the inertial number I, we find that, in both cases at high and intermediate p_{00}, (green and red triangles) and (green and red circles): this behavior can be therefore interpreted as a regime where damping dominates over diffusion.
In the opposite limit, at high values of I, for the case p_{00} = 540, we find both Pe and Re > 1, with Pe > Re. This regime corresponds to the case where inertia dominates, leading to an increase of friction with a consequent thickening behavior, in agreement with the Bagnold contribution appearing in our phenomenological model, and with the general picture presented in ref. 29. Interestingly, in the intermediate regime, an inversion occurs  with Pe becoming smaller than 1 and Re larger than 1  at a value of I corresponding to Ma crossing 1, comparable to that where the unstable branch of σ(I) begins.
For the low pressure data, p_{00} = 78, we again find (blue triangles) and (blue circles), i.e. a regime dominated by damping. The measure of the Mach number allows us to distinguish between the cases at high and low pressure: indeed we find the crucial difference that Ma < 1 in the high p_{00} case (green line) whereas Ma > 1 in the low p_{00} case (blue line). This result reflects the observation that in the latter case the vane can drag the surrounding granular medium.
Green diamonds in Fig. 3b indicate f_{cage} ~ exp(σ) which at low values of stress is well approximated by f_{cage} ~ 1 + σ. This observation, together with the behavior σ ~ seen in Fig. 2a, is compatible  at low rates I  with our interpretation of the denominator of : the main responsible factor for the loosening of solidlike contacts is the activated escape from trapping cages^{30}. A further observation concerns the dependence of D on σ, again displayed in Fig. 3b: in all regimes, excluding the densedilute crossover region, we observe a striking exponential behavior D ~ exp(σ). This law seems universal and denotes a wide variability of D when σ is varied keeping Γ constant. For instance in cases near the transition a variation of more than three decades appears. Those findings reveal an extreme sensitivity of microdynamics to external disturbances which is critical in designing industrial processes or predicting geophysical hazards.
Discussion
We have introduced a novel rheological model, Eq. (2), which merges a corrected μ(I)like frictional contribution dominating at small I and a Bagnoldlike term which gives high velocity thickening effects (Eq. (5)). The frictional contribution is modified to take into account the fact that the cageopening rate is Idependent (Eq. (3)), and that (particularly in vibrofluidized experiments) the pressure due to enduring contacts is reduced with increasing I (Eq. (4)). The unifield model is able to describe the softening of yield stress with vibrofluidization, velocity weakening, shear thinning, the often observed discontinuos thinning transition, and shear thickening at large I. We have employed a vibrofluidized vane setup to reproduce most of the mentioned phenomena and compare the flow curves with Eq. (2), confirming its wide applicability. A microrheological study of experimental fluctuations has offered further insight, giving solid arguments to the cageopening interpretation of the ~ correction to the standard μ(I) formula. We have also observed a striking sensitivity of diffusivity to the shear stress.
Methods
Details of the experiment
The granular medium was made of a number N ∈ [300, 2600] of spheres of diameter d = 4 mm made of nonmagnetic steel (mass of each sphere: 0.267 g), glass (mass 0.0854 g), or delrin^{®} (mass 0.0462 g). They were housed in a plexiglas^{®} cylinder with a conicalshaped floor (diameter 90 mm, minimum height 28.5 mm, maximum height 47.5 mm) in which a plexiglas vane (height 15 mm, width 6 mm, length 35 mm) was suspended in order to be in contact with the granular medium and not with the container^{31,32,33}. The container was vertically vibrated by an electrodynamic shaker (LDS V450) fed with an acceleration signal a(t). In most of the experiments a(t) is a white noise with a bandpass filter between 200 Hz and 400 Hz, while in the lowest p_{00} case (p_{00} = 78 Pa) we used a sinusoidal signal at frequency 53 Hz. This choice is motivated by two empirical observations: 1) a lower number of particles (as in the case of low p_{00}) requires a larger energy input to be homogeneously fluidized and to reach the blade, and this can be obtained by supplying energy through a sinusoidal signal at low frequency; 2) in dense cases a sinusoidal signal induces spurious resonances, while in diluted cases such resonances are never observed. We have checked that performing the same experiments with noise signal for a(t) (pushing the shaker to its working limits) gives flow curves with the same shape. An accelerometer placed on the container side measured a(t), allowing us to define . The vane, mounted through its rotation axis to a rotary encoder, was also connected to a dc motor (typical working voltage 12 V) as the source of the driving torque. The motor was directly fed by a dc voltage supply in the range 0 to 7 V. No limit was set for the maximum current absorbed by the motor that, averaged on the duration of the experiment, was never higher than 450 mA. A data acquisition system collected data for the angular position/velocity of the vane, the effective motor voltage, the current circulating in the motor and the rootmeansquare vertical acceleration of the container. A procedure of calibration allowed us to translate average values of current into average values of applied torque. The same procedure helped determining the moment of inertia of the rotating block, 3.2 × 10^{2} g mm^{2} (the blade with its axis and the gears linking it to the motor). The typical experiment, at a given Γ and applied motor voltage, was 3600 s long, with the granular always “reset” at the beginning of each run for 30 s at high shaking intensity (Γ = 42) and motor off. This procedure  together with periodic replacement of used spheres  guaranteed reproducible results at a distance of several weeks. Packing fractions was nonhomogeneous (it was larger in regions far from the borders of the container): its value at rest was estimated to be in the range 55–70%, while it decreased when vibration was switched on. In the analysis we have identified the shear rate with the average of the angular velocity ω(t) of the rotating blade, i.e. , while the shear stress σ is proportional to the average of the applied torque T(t) through the shear stress constant κ, i.e. σ = κ^{−1}〈T(t)〉 with κ = 2πR^{2}H (with R and H the blade halflength and height, respectively)^{24}.
Details of data analysis
Velocity power density spectra (VPDS) are defined as with t_{TOT} the timelength of an experiment (=3600 seconds). Some examples of S(f ) curves are shown in the Supplementary Fig. S4. In ref. 26 VPDS in a similar vibrofluidized experimental setup, without applied torque (σ = 0), has been investigated. In the dilute or gaslike limit, e.g. low number of spheres at high shaking, the VPDS takes a simple Lorentzian shape S(f ) = D/[1 + (2πf/f_{visc})^{2}], with D the asymptotic (long time) diffusivity and f_{visc} the effective viscosity due to granular gasvane collisions. When the number of particles (density) is increased and/or the intensity of shaking (Γ) is reduced, the system approaches a slow liquid regime and the VPDS develops a wide bump (or smooth peak) with a maximum near f ~ 20 Hz, which is associated to oscillations of the velocity autocorrelation induced by liquid cages. At much smaller frequencies the VPDS reaches a plateau whose height, lim_{f→0}S(f), corresponds to diffusivity D: indeed the blade is not trapped in a cage forever, eventually it manages to explore a much larger phase space and reaches normal diffusion. From the low frequencies plateau of VPDS we have extracted values of D for Fig. 3. We have defined the cageexit frequency f_{cage} as the xposition, in the VPDS plot, of the minimum separating the cage bump from the lowfrequency diffusive plateau (see filled circles in Supplementary Fig. S4).
Data availability
The data that support the plots within this paper and all results reported in this study are available from the corresponding author upon request.
Additional Information
How to cite this article: Gnoli, A. et al. Unified rheology of vibrofluidized dry granular media: From slow dense flows to fast gaslike regimes. Sci. Rep. 6, 38604; doi: 10.1038/srep38604 (2016).
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Acknowledgements
We thank Giorgio Pontuale for useful discussions and suggestions. We also thank Rachele De Felice and Francisco Vega Reyes for reading the manuscript. AL acknowledges the financial support of the Ministerio de Economía y Competitividad (Spain) through Grant No. FIS201342840P and the Junta de Extremadura (Spain) through Grant No. GR15104 (partially financed by FEDER funds).
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A.G., A.S. and A.P. conceived the experiment. A.G. realized the setup and acquired the data. All authors analyzed and discussed the experimental results. A.P. conceived the theory and applied it to the data. All authors contributed to the writing of the manuscript.
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Gnoli, A., Lasanta, A., Sarracino, A. et al. Unified rheology of vibrofluidized dry granular media: From slow dense flows to fast gaslike regimes. Sci Rep 6, 38604 (2016). https://doi.org/10.1038/srep38604
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