nmr experiments on a three-dimensional vibrofluidized granular

PHYSICAL REVIEW E 69, 041302 ~2004!

NMR experiments on a three-dimensional vibrofluidized granular medium

Chao Huan, Xiaoyu Yang,* and D. CandelaPhysics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA

R. W. Mair and R. L. WalsworthHarvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA

~Received 27 May 2003; revised manuscript received 16 October 2003; published 29 April 2004!

A three-dimensional granular system fluidized by vertical container vibrations was studied using pulsed fieldgradient NMR coupled with one-dimensional magnetic resonance imaging. The system consisted of mustardseeds vibrated vertically at 50 Hz, and the number of layersN,<4 was sufficiently low to achieve a nearlytime-independent granular fluid. Using NMR, the vertical profiles of density and granular temperature weredirectly measured, along with the distributions of vertical and horizontal grain velocities. The velocity distri-butions showed modest deviations from Maxwell-Boltzmann statistics, except for the vertical velocity distri-bution near the sample bottom, which was highly skewed and non-Gaussian. Data taken for three values ofN,

and two dimensionless accelerationsG515,18 were fitted to a hydrodynamic theory, which successfullymodels the density and temperature profiles away from the vibrating container bottom. A temperature inversionnear the free upper surface is observed, in agreement with predictions based on the hydrodynamic parametermwhich is nonzero only in inelastic systems.

DOI: 10.1103/PhysRevE.69.041302 PACS number~s!: 45.70.Mg, 76.60.Pc, 81.05.Rm

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I. INTRODUCTION

One of the basic granular flow phenomena is the creaof a fluidized state by vibration of the container that hothe granular medium. If the container is vibrated verticawith a sinusoidal wave formz(t)5z0 cos(vt), the dimen-sionless acceleration amplitude relative to gravity isG5v2z0 /g, andG*1 is required to set the granular systeinto motion. A second key dimensionless parameter isX5N,(12e), wheree is the velocity restitution coefficienfor grain-grain collisions@1# andN, is the number of grainlayers~the height of the granular bed at rest divided by tgrain diameter!. Experiments and simulations suggest thX,Xf with Xf;1 is required to reach auniformly fluidizedstate, in which the stochastic motions of the grains excmotions that are harmonically or subharmonically relatedthe container vibration@2–4#. For largerX.Xf inelastic col-lisions rapidly quench stochastic grain motion and the gralar bed moves as a coherent mass over much of the vibracycle @5,6#.

In this paper we report an experimental study of the uformly fluidized state in three dimensions, using NMR tecniques to probe the grain density profile and velocity disbution. Many authors have applied the methods of statistmechanics to the fluidized granular state@7–14#. Typically,one defines a ‘‘granular temperature’’ proportional to the rdom kinetic energy per degree of freedom of the grainshopes to apply hydrodynamic concepts such as thermalductivity and viscosity to the granular fluid. Using NMR ware able to directly measure the density and granular tperature profiles in a dense, three-dimensional samplecomparison with hydrodynamic theory.

*Present address: USA Instruments, Inc., 1515 Danner Drive,rora, OH 44202.

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Such comparisons are significant as the applicabilityhydrodynamics to granular systems is questionable. In abrofluidized state with energy injection from below, as stuied here, the granular temperature varies sharply with heover distances comparable to the mean free path. Henceseparation between microscopic and macroscopic~hydrody-namic! length scales is marginal, and hydrodynamics maymay not provide an accurate description of the system@15–17#.

Previous experiments have created~quasi-!two-dimensionalsystems by confining grains between verticplates or on a horizontal or tilted surface, enabling data clection by high-speed video@18–25#. Three-dimensionalgranular systems have been studied experimentally by aggate probes such as capacitance@26#, mutual inductance@27#,and diffusing-wave spectroscopy@28#. The distributions ofsingle-particle properties in three-dimensional granular flohave been probed noninvasively using NMR@29–34# andpositron-emission particle tracking~PEPT! @35#. The PEPTmethod has been used to measure the granular temperprofile of a vibrofluidized bed at volume filling fractions uto 0.15@35#. In the present study using NMR, we have mesured granular temperature profiles at much higher densiup to 0.45 volume filling fraction. Density is a key paramefor granular flows, as studies suggest the existence of lolived fluctuations and glassy, out-of-equilibrium behaviorthe density increases toward the random-close-packedume filling fraction'0.6, signaling a breakdown of conventional hydrodynamics@36–39#. Even in the absence oglassiness there are precollision velocity correlations whincrease with density@14,40#; these correlations are includein some hydrodynamic theories@11# but ignored in others~‘‘molecular chaos’’ assumption! @13#.

Other workers have used NMR to study the dense gralar flows that occur in a rotating drum@29,34#, for quasistatictapping@30#, in a period-doubled arching flow@31#, and in a

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HUAN et al. PHYSICAL REVIEW E 69, 041302 ~2004!

Couette shear apparatus@33#. In contrast to these earlier studies, we probe a fully fluidized system sufficiently simplebe described by first-principles granular hydrodynamics.addition, we use rapid gradient pulse sequences to pdeep within the ballistic regime, which has not been possfor these very dense systems.

In the experiments described here, small samples coning of 45–80 mustard seeds of mean diameter 1.84 mm wconfined to a cylindrical sample tube with 9 mm insideameter, and vibrated vertically atv/2p550 Hz, G515–18. The sample container was sufficiently tall to pvent grain collisions with the container top. The total numbof grains is small in our experiments~compared, for ex-ample, with the number of particles in recent moleculdynamics simulations@6#!. However, these physical experments include effects present in practical applications butusually addressed by simulations: irregular and variagrain size, grain rotation, realistic inelastic collisions, andeffect of interstitial gas.

The granular systems studied here were small both hzontally~5 grain diameters! and vertically (N,<4), and thusit may be surprising that they can be successfully describy a hydrodynamic theory. The small sample diameter wsimply a consequence of the inner diameter of the NMapparatus that was available, and could be increased in fustudies. Conversely, the sample depth is inherently limitea small number of layers by the fluidization conditionN,(12e)&1. Thereforephysical vibrofluidized systems~whichtypically havee<0.9) are limited toN,&10 and the appli-cability of granular hydrodynamics is a nontrivial questionbe tested by experiment.

A short report of some of this work was published in R@41#, using a single set ofN, ,G values. For the present report, both N, and G were varied. In addition, improvedmethods were developed for calibrating the NMR gradicoils, resulting in more accurate granular temperature pfiles. In particular, better agreement is found with hydrodnamic predictions for the ‘‘temperature inversion’’ that ocurs at the free upper surface.

II. EXPERIMENTAL METHODS

A. Experimental setup for NMR of vibrated media

The experiments were carried out in a vertical-bore supconducting magnet with a lab-built 40 MHz NMR probequipped with gradient coils for pulsed field gradient~PFG!and magnetic resonance imaging~MRI! studies. The lowstatic field value~0.94 T! was chosen to minimize magnetfield gradients in the sample due to the susceptibility diffence between grains and voids. This permitted the usesimple unipolar PFG sequence~Fig. 1! for the fastest pos-sible time resolution. The 9 mm inside diameter, 6.5 cm hsample chamber had a glass sidewall to minimize enetransfers between grains and the sidewall. The Teflon botwall was pierced by an array of 30 0.46 mm diameter homaking it possible to evacuate the sample container. Anternal vacuum gauge registered less than 200 mtorr duexperiments with evacuated samples.

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A loudspeaker connected to the chamber by a fibergsupport tube and driven by a function generator and auamplifier was used to vibrate the sample container verticwith a sinusoidal wave form. The amplitude and phase ofcontainer motion were precisely measured by fitting the dtized output of a micromachined accelerometer~Analog De-vices ADXL50! mounted to the support tube just below thNMR magnet. The gradient coils used for position~MRI!and displacement~PFG! measurements were calibrated bcombining data from measurements on phantoms of knodimensions, along with diffusion measurements on a wasample. In this way we obtained calibration maps for eagradient coil of both the gradient in the designed directand the total gradient magnitude including componentsthogonal to the designed direction. The accelerometervertical gradient calibrations were cross-checked by meaing the displacement wave form for a granular sample csolidated with epoxy. The dimensionless acceleration valG reported here are accurate to within60.1.

B. Granular samples

Previous NMR studies of granular media used plant seor liquid-filled capsules to take advantage of the long-livNMR signals produced by liquids@29–34#. For the presentstudy, we chose mustard seeds@42#, which showed no ten-dency to static charging. The seeds were ellipsoidal, wthree unequal major diameters and eccentricity<20%. Theaverage of all three diameters measured for a sample oseeds wasd51.84 mm with a standard deviation of 5%, anthe average mass wasm54.98 mg/seed. These average vues ofd andm are used below for comparison with a hydrdynamic theory which is formulated for samples of identicgrains.

FIG. 1. Pulse sequence used for PFG/MRI data acquisition. Tis a stimulated-echo PFG sequence followed by one-dimensifrequency-encoded MRI. The three solid vertical bars arep/2 rfpulses, the boxes with horizontal hatching are gradient pulses indisplacement-resolution~PFG! direction, and the open boxes argradient pulses in the position-resolution~MRI! direction. The gra-dient pulses are labeled with the gradient strengths in T/m, wnumbers on the time axis indicate durations of intervals in micseconds. The start of the sequence was triggered at a fixed phathe sample vibration, and the delay timeD1 was varied over therange 1–20 ms to select the time in the vibration cycle at whNMR data were acquired. The displacement measurement timeDtis equal to the time between PFG gradient pulses plus one-thirdduration of the gradient pulses; for the sequence shownDt5D21380 ms. An eight-sequence phase cycle was used to suppundesired coherences and to enable acquisition of both realimaginary parts of the complex PFG propagator.

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NMR EXPERIMENTS ON A THREE-DIMENSIONAL . . . PHYSICAL REVIEW E 69, 041302 ~2004!

To roughly measure the normal restitution coefficiente, avideo camera was used to record the bounce height whenseeds were dropped onto a stone surface. As some likinetic energy is converted to rotational energy upon impathis gives a lower bound one. Furthermore,e in real mate-rials varies with impact velocity@43,44#, and the smallesdrop velocity that could readily be used~44 cm/s! was largerthan the typical rms velocity in our vibrofluidized experments (<30 cm/s). Thee value deduced from the higherebound over 15 dropping trials on each of five seeds ranfrom 0.83 to 0.92, with an average of 0.86.

C. Data acquisition

Proton NMR signals from the mustard-seed sample wacquired using the PFG/MRI sequence shown in Fig. 1. AFourier transformation with respect to time, this sequeyields the position-dependent PFG signal~one-dimensionalimage!

h~qu ,v !5E p~Du,v !eiquDudDu, ~1!

wherev5x, y, or z is the direction of the MRI gradient anu5x, y, or z is the direction of the PFG gradient. In thequationqu is proportional to the area of the PFG gradiepulse, which is varied through an array of regularly spacvalues~Fig. 1!. Here the ‘‘propagator’’p(Du,v) is the jointprobability that a grain~more precisely, an oil molecule ingrain! is located at positionv and moves a distanceDu dur-ing the time intervalDt set by the pulse sequence@45#. FromEq. ~1!, the propagatorp(Du,v) can be obtained by Fourietransforming the NMR signalh(qu ,v) with respect toqu .With the gradient values used in these experimentspropagator was obtained with resolutions of 500mm in theposition variablev and 42mm in the displacement variablu.

This type of NMR experiment does not determine tpositions or displacements of individual grains. Rather,ensemble average, the propagator, is obtained. Providedthe propagator has a well defined time average at each pof the vibration cycle, the NMR acquisition can be repeaas many times as necessary to build up a good signanoise ratio. Allowing full longitudinal relaxation between aquisitions~the simplest but not the fastest way to take da!,the acquisition time for a time-resolved propagap(Du,v,t) was 21 h.

D. Measurement of rms displacement

For some purposes, the full propagatorp(Du,v) is ofinterest, while for other purposes~e.g., measuring the granular temperature! only the position-dependent rms displacment is required,

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Here n(v)5*p(Du,v)d(Du) is the position-dependen

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number density. Bothn(v) and Durms(v) can be obtaineddirectly from the NMR data~before it is Fourier transformedto yield the propagator! using

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For optimum signal-to-noise ratio we evaluated the righand side of these expressions by fitting the central~smalluquu) portion of h(qu ,v) to a function including constantquadratic, and higher terms.

III. EXPERIMENTAL RESULTS: DENSITY ANDDISPLACEMENT DISTRIBUTIONS

In the present experiments, all measured quantitiespended strongly on heightz, due to gravity and the localization of energy injection at the sample bottom. Therefomost data were taken with the MRI axisv5z. As both ver-tical and horizontal grain motion were of interest, for thPFG axis bothu5z andu5x were used. We assume densiand velocity distributions are the same in the two horizondirectionsx,y for the uniformly fluidized state studied heredue to the cylindrical symmetry of the experimental setabout the vertical axis. Thus, the primary experimental dshown in this paper are the height-dependent number denn(z), the vertical and horizontal displacement propagatp(Dz,z) and p(Dx,z), and the vertical and horizontal rmdisplacementsDzrms(z) andDxrms(z). In addition, some datawere taken using a two-dimensional MRI sequence to vethat the grain density was approximately uniform in the hozontal plane and to calibrate the number of layersN, ~Ap-pendix A!.

The granular state withN,53.0, G515, and sample celevacuated was selected as a reference state measurgreatest detail, and other states were selected by varyisingle parameter from the reference value. The followvariations were carried out:~a! increase of ambient air pressure up to 1 atm,~b! variation ofN, from 2.2 to 4.0, and~c!variation ofG up to 18. The upper limit onG was set by theapparatus, while the lower limit onG and the upper limit onN, were set by the requirement that the sample fully aconsistently fluidize, when observed visually outside of tNMR apparatus. For smallerG or largerN, we observe acomplex, hysteretic transition to a partially fluidized, patially jammed state which is beyond the scope of the presstudy.

A. Effects of interstitial air

Most of our data were taken with the sample containevacuated. Some of the measurements were repeated inmal atmospheric conditions, to determine how large theterstitial gas effects could be. Experimentally, the effectair is unobservably small~Fig. 2!. This agrees with the fol-lowing rough estimate of viscous energy losses due todrag. The mean free path varies widely with height in o

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system but a typical value isl'0.3d ~Appendix B!. Simi-larly, a typical grain velocity is on the order of 20 cm/Using these values and the drag force on a sphere inunbounded medium, we compute that the fraction of kineenergy lost to viscous drag over one mean free path931024 for p51 atm and 331024 for p5200 mtorr.These energy losses are negligible compared to the fracof kinetic energy lost in an intergrain collision, 12e2

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B. Time-resolved density and displacement profiles

In Sec. IV below the NMR data are fitted to a hydrodnamic theory which models the granular system as a flwith time-independent~but spatially varying! grain velocitydistribution. In this picture, the vibrating container bottoserves to inject energy~granular ‘‘heat’’! into the system. Forsuch a picture to make sense, the overall state of the gransystem must be nearly independent of time within the vibtion cycle.

We have taken some NMR data as a function oft withinthe 20 ms container vibration period~Figs. 3 and 4!. Thedensity profile~Fig. 3! is indeed largely independent oftover most of the sample, except for a noticeable disturbaat smallz for t near the top of the container’s motion, whethe container floor impacts the sample. In the bottom laye

FIG. 2. Effect of cell evacuation. Un-normalized number dens~top! and rms vertical displacement in time intervalDt51.38 ms~bottom! as functions of heightz for N,53.0 layers of mustardseeds vibrated atv/2p550 Hz and acceleration amplitude relativto gravity G515. The data labeled ‘‘Air’’ were taken in normaatmospheric conditions, while for the data labeled ‘‘Vacuum’’ tcell was evacuated so that an external pressure gauge registerethan 200 mtorr. The small differences between the Air and Vacudata are comparable to the typical experimental repeatabilitytherefore are not significant.

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the sample, this disturbance damps out~thermalizes! over atime span of less than 10 ms, and above the bottom layerhardly evident at all. Similarly, in the rms displacement prfiles ~Fig. 4! a large transient is visible in the vertical displacement due to grains impacted by the container bottwhile the horizontal displacement transient is much smarelative to the time-average value.

Over most of the cycle the sample ‘‘floats’’ well above thvibrating container floor~Fig. 3!, illustrating two importantpoints about the vibrofluidization process.~a! It is possible toachieve a nearly time-independent fluidlike granular steven though the driving vibration amplitude is much greathan typical stochastic motion within the sample. Thequired conditions are~i! v* 5v(d/g)1/2.1, which ensuresthat the sample itself falls only a grain-scale distance in ovibration period no matter how large is the container oslation amplitude, and~ii ! X5N,(12e)&1, allowing thesample to fully fluidize. For the experiments described hv* 54.3 and 0.29<X<0.52. ~b! For realistic multilayergranular systems, the primary contact between the grainsthe vibrating cell bottom occurs near the top of the vibratistroke, when the cell bottom velocity is much less thanrms value. Therefore, the energy transfer into the granmedium cannot be estimated, even roughly, from the rdriver velocity. This is because the energy transfer depeupon the mean cell bottom velocityduring grain impacts@46#.

Apart from Figs. 3 and 4 the data shown in this paphave all been averaged over the vibration cycle for improvsignal-to-noise ratio. It should be borne in mind that the stof the system is not, in fact time independent very nearsample bottom and that the complex process of energy trfer from the vibrating base is not captured by the time avages.

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FIG. 3. ~Color online! Un-normalized number density of grainn as a function of heightz and time in the vibration cyclet for thereference vibrofluidized stateN,53.0, G515. The thick blackcurve shows the container vibration wave form, obtained by fittthe digitized accelerometer output. For this figure the curvedrawn to show the motion of the container bottom, which impathe sample bottom att'12 ms. Apart from the disturbance causeby this impact, the density profile is largely independent oft.

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FIG. 4. ~Color online! Profiles of the vertical~left! and horizontal~right! rms grain displacementsDzrms and Dxrms in a time intervalDt51.38 ms, as functions of the heightz and the time in the vibration cyclet. The experimental conditions are the same as for Fig. 3

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C. Short time displacements: Crossover to the ballistic regime

Figure 5 shows the rms vertical and horizontal grain mtion in the reference stateN,53.0, G515 as functions of thedisplacement measurement timeDt, for slices at three differ-ent heightsz within the sample. At each height there iscrossover at shortDt to the power lawDzrms,Dxrms}Dt,which indicates ballistic behavior~displacements proportional to time!. For longerDt the data fall below the ballisticpower law, possibly crossing over to a diffusive power laDzrms,Dxrms}(Dt)1/2 @47#. The crossover time, which is aestimate of a typical grain collision time, ranges from 5 min the dense, lower portion of the sample to 10 ms in the ldense, upper portion of the sample. In Appendix B, itshown that the crossover time agrees well with the grcollision time computed from the mean free path andmeasured granular temperature.

A key feature of our experiments is the ability to take dafor displacement intervalsDt much less than the grain collision time, i.e., deep within the ballistic regime. The rms dplacement data shown in the remainder of this paper wtaken withDt51.38 ms, well within the ballistic regime foall values ofN, ,G used.

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D. Horizontal and vertical velocity distributions: Deviationsfrom Gaussian

Figures 6 and 7 show the distributions of vertical ahorizontal displacements, respectively, for the referencebrofluidized stateN,53.0, G515 and displacement obsevation timeDt51.38 ms within the ballistic regime. Thespropagators should be good approximations to the distrtions of vertical and horizontal velocitiesp(vz,x ,z) withvz,x5Dz,Dx/Dt.

In some previous experiments@20,21,23,24# and simula-tions @48# of granular fluids~mostly in reduced dimensions!significant deviations have been found between the measvelocity distribution, and the Gaussian~Maxwell-Boltzmann! distribution that describes an elastic fluidequilibrium, p(v i)}exp(2mvi

2/2T) ~setting Boltzmann’sconstant to unity!. For example, Rouyer and Menon@23#found that the distribution of horizontal velocities in a twdimensional vibrofluidized media robustly obeyed

p~vx!}exp~2uvx /saua! ~4!

with a51.5560.1. Using kinetic theory it has been show

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FIG. 5. Root-mean-square vertical~circles! and horizontal~squares! grain displacementsDzrms and Dxrms versus the displacemenmeasurement timeDt for a sample withN,53.0, G515. The data are shown for three different horizontal slices (z ranges!, which may becorrelated with the density profile in Fig. 3. In each log-log plot, the solid line has a slope of unity~rms displacement proportional to time!while the dashed line has a slope of 1/2.
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FIG. 6. ~Color online! Propagator, or distributionp(Dz,z) of vertical displacementsDz during time intervalDt51.38 ms, as a functionof height z for the reference vibrofluidized stateN,53.0,G515. As Dt is well within the ballistic regime,p(Dz,z) approximates thedistribution of vertical velocitiesp(vz ,z) with vz5Dz/Dt. The same data are shown from two different viewpoints in the left and right pNear the bottom of the sample~small z, most visible in the left plot! the velocity distribution is highly skewed, with a wide tail on thevz

.0 side due to particles traveling upward with large velocity after impacting the vibrating container bottom. Well away from the~right plot!, the velocity distribution is nearly symmetric.

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that thetails of the velocity distribution should follow Eq~4! with a51 for a freely cooling granular gas, anda53/2 for a granular gas with stochastic forcing appliedevery grain@49#. However, for the freely cooling case thcentralpart of the distribution is nearly Gaussian (a52) andthe crossover toa51 occurs very far out in the tails@50#.

In the present experiments and those of Ref.@23# a steadystate was achieved by supplying energy from the boundaof the system, which corresponds neither to the freely coing nor to the uniformly heated case. A few recent theoretand simulation studies have treated the two-dimensiogranular fluid with a strong thermal gradient, as occursvibrofluidization experiments, and have found features sas anisotropic granular temperature, asymmetric velocitytributions, and nonuniversal values of the velocity-stretchexponenta @48,51–53#.

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In previous work the measured velocity distribution typcally has been scaled by the rms velocitys ~note sÞsa).This is possible for simulations and for video experimentswhich every particle is tracked, allowings to be computedunambiguously. In NMR experiments there is a noise flovisible in Figs. 6 and 7, which could conceal tails of thdistribution contributing tos. One possibility would be touse direct measurement of the rms velocity fromq-spacedata, described in Sec. II D and used below to measuregranular temperature. Here we have taken the simplerproach of directly fitting the experimental velocity distributions to Eq.~4!, allowing the width parametersa to vary.

The results of these fits are shown in Fig. 8. The fittstretching exponents for vertical and horizontal velocity dtributionsaz ,ax generally lie between 1.5 and 2.0, with vaues closer to 2.0~Gaussian! near the top of the sample. Tw

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FIG. 7. ~Color online! Two views of the propagatorp(Dx,z) for horizontal displacementsDx in the same conditions as in Fig. 6. Thdistribution of horizontal velocities is symmetric at all heightsz, consistent with the symmetry of the experimental arrangement. A noiseof approximately 2% of the peak value is visible here and in Fig. 6: the apparent value ofp does not go precisely to zero for very largeDx.

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of the stretched-exponential fits to the horizontal displament data are shown in Fig. 9. As may be seen in this figthe deviations from a Gaussian distribution are smallsignificant near the bottom of the sample; near the top ofsample the distribution is indistinguishable from a Gaussto within the resolution of our measurements. Althoughazandax are not significantly different, the large error bars faz near the sample bottom reflect the fact that the expmental velocity distribution, Fig. 6, is asymmetric near tsample bottom and hence does not fit Eq.~4! well.

The present results forax in a three-dimensional systemappear marginally different from the two-dimensional resuof Ref. @23#. Our results agree more closely with the twdimensional simulations of Ref.@48#, which found that de-viations from Gaussian are most pronounced near the boof the system. In common with Ref.@48# our system has afree upper surface; in the experiments of Ref.@23# the systemwas confined and driven at much higherG by both lower andupper walls.

IV. FIT TO A HYDRODYNAMIC THEORY

A. Granular hydrodynamics for a stationaryvibrofluidized state

Granular hydrodynamic equations have been derivedmany groups, using increasingly sophisticated approximaschemes@7–13#. In this section we show how any of thestheories may be numerically integrated for comparison wexperiments. Section IV B shows this comparison betwthe theory of Ref.@13# and our experimental results.

We consider a system ofN identical spherical grains odiameterd and massm, for which the granular temperaturedefined byT5m^v2&/3. In Ref.@6# granular hydrodynamicswas successfully used to describe anonstationarysimulationwith large variations of the density profile over the coursea vibration cycle. Here, we specialize to the simpler situat

FIG. 8. Fitted stretching exponentsaz,x for the vertical and hori-zontal velocity distributions versus heightz, for N,53.0,G515.The Gaussian and exp(2v3/2) cases are shown by the dashed linThe large uncertainties foraz at smallz indicate a poor fit of thestretched-exponential function, which is symmetric, to the higasymmetric velocity distribution~Fig. 6!.

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in which the average velocity fieldv(r ,t) vanishes and thenumber density fieldn(r ,t) and temperature fieldT(r ,t) areindependent oft and depend uponr only via the height co-ordinatez. With these assumptions, the pressurep(z) andnumber densityn(z) satisfy a hydrostatic equilibrium equation

dp/dz52mgn, ~5!

and the upward heat fluxqz(z) obeys an energy flux balancequation

dqz /dz52Pc ~6!

wherePc is the rate of kinetic energy loss per unit volumdue to inelastic collisions. The heat flux is

qz52k~dT/dz!2m~dn/dz!, ~7!

wherek is the conventional thermal conductivity, andm is atransport coefficient that is nonzero only for inelastic sytems@9#. Due to the second term in Eq.~7!, it is possible tohave a flow of heat from colder regions to warmer onAlthough a number of authors have calculatedm @9,12,13#and discussed the consequences ofmÞ0 @54–56#, to ourknowledge no heuristic explanation has been proposed

FIG. 9. Horizontal displacement propagatorp(Dx,z) andstretched-Gaussian fits for two different heightsz. The data are asubset of the data plotted in Fig. 7 (N,53.0,G515), with fittedbackgrounds subtracted. The dotted lines show fits with the streing exponenta variable, while for the solid linesa was fixed at 2~Gaussian value!. The deviation from Gaussian is small but signicant near the sample bottom~bottom graph!, while the data fromhigher up in the sample~top graph! are indistinguishable from aGaussian distribution.

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the existence of a heat current driven by density gradieThis density-driven heat current has a marked effect inpresent experiments.

Equations~5!–~7! are simply the results of momentumand energy conservation and the definitions ofk and m. Ahydrodynamic theory closes this set of equations by proving expressions forp(n,T) ~the equation of state!, Pc , k,andm. It is convenient to nondimensionalize variables usthe grain diameterd, grain massm, and gravityg,

z* 5~z2zb!/d,

n* 5nd3,

T* 5T/mgd, ~8!

p* 5pd2/mg,

qz* 5qz~d/g!3/2/m.

Here zb is the height of the bottom of the sample, corrsponding toz* 50. Physically,zb is expected to be in thevicinity of the peak container floor position~Fig. 3!.

For a hard-sphere system with constant restitution coecient the equation of state takes the form

p/nT5 f ~n* !. ~9!

Using Eqs.~5!–~7! and ~9!, the following expression is derived for the density gradient:

dn

dz5

qz /k2mg/ f

~T/n!@11n* / f !~d f /dn* !] 2m/k. ~10!

Equations~5!, ~6!, and ~10! are coupled first-order ordinary differential equations for$p,qz ,n% that can be numeri-cally integrated from the sample bottom atzb to z51`. Theboundary conditions are as follows. For a system in gravwith no upper wall, the pressure and heat current satisfp(1`)50, qz(1`)50. Using Eq. ~5! this gives p(zb)5mgnA , where nA5*zb

1`n(z)dz is the total number of

grains in the sample per unit horizontal cross section. Theat current into the system at the bottomqz(zb) is an inputparameter determined by the vibration amplitude andsystem response. For given values ofqz(zb) and p(zb), thedensity at the bottomn(zb) must be adjusted to satisfy thupper boundary conditionqz(1`)50.

B. Fit of the experimental results to the Garzo-Duftyhydrodynamic theory

In this section, we show fits of our experimental data tcalculation of granular hydrodynamic coefficients by Gar´and Dufty @13#. This calculation is based on the ‘‘reviseEnskog theory,’’ which is accurate forelastic fluids over awide range of densities and is extended in Ref.@13# to arbi-trary values of the restitution coefficiente. Some other theo-ries of granular hydrodynamics have been derived as exsions in powers of 12e2 @12#, and thus may be accurate onfor e'1. On the other hand, the Garzo´-Dufty theory makes

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the molecular chaos assumption that the precollision gvelocities are uncorrelated. Precollision velocity correlatioare known to exist in inelastic hard-sphere fluids@14,40# andcan be treated theoretically using ‘‘ring kinetic theory’’@11#.

It is doubtful whether fits to the present data could distguish between these various hydrodynamic theories.chose to compare our data to Ref.@13# because this calculation is nominally valid for wide ranges of density and inelaticity, and because the theoretical results forf, Pc , k, andmare presented in an algebraic form convenient for numerintegration. In this theory the granular temperature issumed to be isotropic,Tx5Ty5Tz , whereTi5m^v i

2&, but itis not assumed that the velocity distribution is Gaussian.perimentally, we find modest differences~of order 20% orless! between the horizontal and vertical granular tempetures,Tx,yÞTz , and more generalized formulations of hdrodynamics may be able to treat this anisotropy@51,52,57#.For comparison with the isotropic-temperature theorymeasured vertical temperatureTz and horizontal temperaturTx were averaged usingT5 1

3 Tz123 Tx .

Using our convention, the dimensionless~starred! formsof Eqs.~5!–~10! and the boundary conditions are obtainedsettingm5g5d51. We have carried out the integrationsthis dimensionless form, using a simple first-order numerischeme and a bisection search onn* (0) to meet the upperboundary condition.

Figures 10–13 show a joint fit to the Garzo´-Dufty theoryof our data for three different values of the dimensionlebed heightN, and two different values ofG. For these fitsthe height of the sample bottomzb was allowed to be differ-ent for each data set, since the sample ‘‘floats’’ abovevibration center by an amount that depends upon thenamic interplay between energy input and gravity~Fig. 3!.Similarly, the energy inputqz* (0) was a separate fitting parameter for each data set. Conversely, a single value ofrestitution coefficiente was used to fit all four data setsThus the only fitting parameters arezb , qz* (0), ande. Allother quantities entering the fits shown in Figs. 10–13measured experimentally, including all of the axis variabin the figures.

The fitted value of the restitution coefficient ise50.87, inreasonable agreement with the approximate measuremenscribed in Sec. II B. The fitted values of the dimensionleheat flux at the sample bottomqz* (0) are listed in Table I.The total power input is computed asP5(pD2/4)qz , whereD58.7 mm is the measured effective sample diameter~Ap-pendix A!. By equating the rate of momentum transfer to ttotal sample weightNmg the input power is rigorously related to the average over grain impacts of the containertom velocity ^v i& by P52Nmg v i& @46#. Table I lists^v i&derived in this way from the fitted heat current, along wthe ratio of^v i& to the peak container velocityv05vz0.

The fitted power inputqz* (0) is an increasing function oG, as expected on physical grounds. The power input aincreases with bed heightN, , which is expected from themomentum-balance argument, but the increase is very sOne caveat is that theseqz* values are derived from the hy

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drodynamic fits, which tend to fail where the density at tsample bottom is much less than the peak density~Figs. 11and 13!. In these cases particularly, the fittedqz* representsaneffectivepower input reflecting the bottom boundary codition, which is greater than theactual power input becausethe hydrodynamic fit has larger granular temperature thanexperimental data and hence greater collision losses inlower part of the sample.

One robust conclusion from Table I is that the averacontainer-grain impact velocity is much less than the pcontainer velocity,^v i&!v0 ~substituting the actual inpupower for the effective power only strengthens this resu!.This conclusion corresponds to the experimental observathat the sample ‘‘floats’’ above the vibrating container btom ~Fig. 3!, so that most container-grain impacts occur nthe top of the vibration stroke where the container velocitymuch less than its peak value.

FIG. 10. Experimental density and granular temperature pro~symbols! and fit to the hydrodynamic theory of Ref.@13# ~curves!for N,53.0 layers of grains vibrated at dimensionless acceleraamplitudeG515. Dimensionless variables defined in Eq.~8! areused. In the top graph the symbols labelednz* show the densityprofile n* (z* ) derived from the data set used to measure vertdisplacements, while the symbols labelednx* show the same quantity derived from the horizontal-displacement data set; theseprofiles should be identical and differences reflect experimentalcuracy. In the bottom graphTz* andTx* are the granular temperaturprofiles for vertical and horizontal displacements, respectively, msured using the observation intervalDt51.38 ms. It is expectedthat Tz* .Tx* , at least near the bottom of the sample, as energinput only to the vertical degrees of freedom by the containerbration. The theory should be compared with the symbols labeT* , which are computed from the measured temperatures uT* 5Tz* /312Tx* /3.

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In both the data and the hydrodynamic fits in Figs. 10–there is a ‘‘temperature inversion’’—the temperaturecreases with height near the free upper surface of the samIn the hydrodynamic theory the heat flux is strictly upwaso the temperature inversion is made possible only byterm in m in Eq. ~7!. Furthermore, the heat current vanishat the sample top~as it must, as there is no energy sourcethe free upper surface! despite the roughly linear increasethe temperature with height. This also is made possiblethe m(dn/dz) term in Eq. ~7!, which cancels the conventional thermal current termk(dT/Tz). Even though there isa temperature inversion, the hydrodynamic theory predthat the density should fall off nearly exponentially wiheight near the free upper surface@56#. To show the expo-nential falloff clearly, the experimental and theoretical desity profiles are replotted on log-linear axes in Fig. 14.

The temperature inversion may be visible in some ofearliest experiments on quasi-two-dimensional syste@18,19# and has been extensively discussed theoretic@54–56#. The results presented here are perhaps thedemonstration that the temperature inversion in a physsystem can be quantitatively explained by granular hydronamics. As such, these experiments provide evidence tham term in the transport equation is necessary for an accudescription of real granular materials.

V. CONCLUSIONS

One conclusion of this work is that it is possible, undcertain conditions, to quantitatively model a physical vibr

s

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FIG. 11. Density and temperature profiles forN,52.2, withother conditions the same as in Fig. 10. Under these conditionsignificant discrepancy can be seen between the experimentatheoretical granular temperatures in the lower portion of the samz* ,3. We observe this discrepancy in conditions such thatdensity in the lower portion of the sample is well below the pedensity~upper graph!.

2-9

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fluidized granular system using granular hydrodynamThis is true despite the small sizes of the systems measin grain diameters: the systems were small horizontallyto apparatus constraints, and in any case were necesssmall vertically in order to meet the fluidization conditioX&1. The overall success of the hydrodynamic descriptidemonstrated by the fits in Figs. 10–13, can probablyattributed to the fact that hydrodynamics breaks down owhen the mean free path becomes much larger than orelevant length scales. It may appear paradoxical thatagreement with theory is best near the top of the samwhere the mean free path is longest. However, this simreflects the inability of the hydrodynamic theory to accrately describe the complex, time-dependent state ofsample near the vibrating container floor~Figs. 3 and 4!. Theactual velocity distribution is highly skewed and anisotropnear the sample bottom~Fig. 6!, and a discrepancy developbetween the theoretical and measured granular temperawhen the vibrational power input is large enough to deplthe particle density near the sample bottom~Figs. 11 and 13!.

For these reasons, present theory cannot adequatelyscribe the process of energy transfer from the container flto the sample. To calculate the power input, typically it hbeen assumed that the mean free path is much larger thavibration stroke, or that the vibration wave form is sawtooor triangular@46,55#. Such wave forms are unrealistic sincthey require infinite acceleration at the peak of the vibratstroke, which is precisely the point at which energy transto the grains occurs~Fig. 3!. Similarly, the mean free path imuch less than the vibration stroke in the current expe

FIG. 12. Density and temperature profiles forN,54.0, withother conditions the same as in Fig. 10. The peak densityn*50.85 observed in these conditions corresponds to volume-filfraction f5(p/6)n* 50.45, about 75% of the random-clospacked value.

04130

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ments~Appendix B!, which will be true in general for densuniformly fluidized systems unless very high vibration frquencies are used. One possibility for deriving the powinput theoretically might be to usetime-dependenthydrody-namics to model the formation and dissipation of shocksin Ref. @6#.

Thus it appears that current granular hydrodynamic thecan accurately describe the uniformly vibrofluidized stawell away from the vibrating energy source, but that derivtion of the time-averaged boundary condition for realisvibration wave forms remains an unsolved problem. If thproblem can be solved, a complete first-principles hydronamic description of the vibrofluidized state applicablereal granular systems will be in hand.

g

FIG. 13. Density and temperature profiles forG518, with otherconditions the same as in Fig. 10. As in Fig. 11 the density atbottom of the sample is much less than the peak density, andmeasured granular temperature does not agree well with the tretical granular temperature near the sample bottom. Note thatsituation is achieved here by increasingG relative to the referencestate, while in Fig. 11 it was achieved by decreasingN, .

TABLE I. Fitted dimensionless heat flux into the bottom of thsampleqz* (0) as a function of dimensionless vibration acceleratG and bed heightN, . Also listed are the average container impavelocity ^v i& derived fromqz* (0) and the ratio of v i& to the peakcontainer velocityv0.

G N, qz* (0) ^v i& ~cm/s! ^v i&/v0

15 2.2 3.3 8.7 0.18515 3.0 3.4 6.7 0.14315 4.0 3.5 5.3 0.11018 3.0 4.8 9.4 0.168

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ACKNOWLEDGMENTS

This work was supported by NSF Grant No. CT9980194. The authors thank N. Menon for useful discsions.

APPENDIX A: HORIZONTAL DENSITY PROFILES ANDCALIBRATION OF THE BED HEIGHT

The hydrodynamic model to which we have fitted odata assumes that all quantities depend only upon the hecoordinatez. This is not strictly true due to boundary effecat the vertical wall, as demonstrated by previous experime@35#. The boundary condition also affects the effective crosectional area of the sample tube, which is needed to cpute the number of grain layersN, for a given number ofgrainsN. To address these issues we have taken somedimensional MRI data~Fig. 15! to examine the horizontadensity profile of the vibrofluidized system,

p~x,z!5E n~x,y,z!dy. ~A1!

If the NMR signal from each grain comes precisely from t

FIG. 14. Density profiles for the four different experimentconditions plotted on log-linear axes. These graphs show the sdata as are shown in the top graphs of Figs. 10–13, using the ssymbols. The hydrodynamic theory~curves! predicts a nearly exponential falloff of density with heightz* near the free upper surfac~large z* ), resulting in straight lines on these semilog plots. Tdata~circles and triangles! generally agree with this prediction, bucurve away from the theory curves at very low values ofn* due tothe noise floor.

04130

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FIG. 15. Horizontal density profilesp(x,z) for z51.75 cm~top!and z51.55 cm~bottom!, for the N,53.0, G515 reference stateThe solid curves show fits to the profile for a uniform distributionpoint-source grains in a cylindrical volume. The dashed curshow fits assuming the grain centers are uniformly distributed,that the NMR signal source is a sphere with the full grain diameThe difference in the fitted cylinder diameter between the two fitnot significant.

FIG. 16. Fitted effective tube diameterD as a function of heightz for the N,53.0, G515 reference state. The error bars becolarge at the top and bottom of the sample where the density isThe dashed line shows the average valueD58.7 mm used to ana-lyze the data.

eme

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grain center and the grain centers are uniformly distribuhorizontally over a cylinder of diameterD, the profile has theform

p~x,z!5H f ~z!@12~2x/D !2#1/2 if uxu,D/2,

0 otherwise.~A2!

We also considered the possibility that the NMR signal froa grain comes from a sphere of diameterd<d, in which casethe profile of Eq.~A2! must be convoluted with the profilfor a single grain. We find that the fitted sample diametenot significantly changed by varyingd over the entire pos-sible range, Fig. 15. In this figure some systematic deviabetween the data and fit is visible, hinting that the densmay be slightly depressed in the sample center@35#. How-ever, these data are too noisy to be directly inverted to ythe radial density profile. The fitted diameter is consistwith a constant valueD58.7 mm~Fig. 16!. This is less thanthe actual inside diameter~9 mm! but by less than the graindiameter~1.84 mm!, suggesting a slight density enhancemeat the walls due to the dynamic boundary condition.

For the fits to the hydrodynamic theory, the height of tgranular bed is specified by the dimensionless number dsity nA* 5nAd25Nd2/(pD2/4). A second measure of the bedepth is the number of layersN, , defined as the height of thbed at rest divided by the grain diameter. The value ofN,

depends on the assumed volume filling fraction of the berest f0 via N,5(p/6f0)nA* . For theN, values quoted inthis paper, shown in Table II, we have usedf050.60 corre-sponding to a ‘‘loose random packing’’@58#. It should beemphasized that the hydrodynamic fits are independenthe assumed value forf0, as they are based onnA* and notN, .

APPENDIX B: THE MEAN FREE PATH

For hydrodynamics to be valid, the mean free pathl mustnot be too large relative to distances over which the hyd

TABLE II. Three measures of the sizes of the granular sampused in this study. The number of grainsN was counted preciselyThe dimensionless area densitynA* was computed fromN, the mea-sured average grain diameter, and the fitted effective samplediameter from MRI experiments on the reference vibrofluidizstate. The number of layersN, depends additionally upon the asumed volume filling fraction of the bed at rest. The quantityXwhich characterizes the possibility of fully fluidizing the samplesalso listed, using the valuee50.87 resulting from the hydrodynamic fits.

N nA* N, X5N,(12e)

45 2.56 2.2 0.2960 3.42 3.0 0.3980 4.56 4.0 0.52

04130

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dynamic fields vary. Therefore it is important to estimatelfor the experimental conditions, and this also provides a csistency check for the measured ballistic-to-diffusive croover time. Extending the estimate of Ref.@10# to three di-mensions gives

l* 5l/d5A~n0* 2n* !

n* ~B2n* !, ~B1!

whereA andB are constants chosen to give the exact dillimit l* 51/(pA2n* ) for n* →0 and the heuristic denslimit l* 5(12n* /n0* )/3 for n* →n0* . Heren0* 5(6/p)f0 isthe density at which the grains come into contact, whicharbitrarily assume occurs at the volume filling fraction floose random packingf050.60.

The mean free pathl* (n* ) computed from Eq.~B1! isshown in Fig. 17. At the peak densityn* 50.85 that occursin our experiments~Fig. 12!, Eq. ~B1! givesl* '0.1. Henceit is hardly surprising that the hydrodynamic theory can eplain the spatial dependence of the density and granular tperature in the high-density regions of the sample. Cversely, the data fit the hydrodynamic theory well fdensities extending belown* 50.1, which corresponds tol* .2. Thus, the theory continues to work for mean frpaths comparable to or even somewhat greater than theover which the temperature and density change.

The mean free path calculation can also be used to chthe consistency of the observed ballistic crossover time,5. For example, the center panel in this figure correspondan average dimensionless heightz* 55.2, and at this heighthe measured dimensionless density and temperature arn*50.41, T* 50.5 ~Fig. 10!. Using l5dl* (n* ) and v rms5(3T* gd)1/2 we calculate @59# the ballistic-to-diffusivecrossover timetcr5(32/3p)1/2l/v rms56.3 ms, in excellentagreement with the observed value~Fig. 5!.

s

be

FIG. 17. Mean free pathl as a function of number densityncomputed from Eq.~B1!. Both quantities are nondimensionalizeby the grain diameterd.

2-12

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