DTIC ADA509885: Dynamic Length-Scale Characterization and Nonequilibrium Statistical Mechanics of Transport in Open-Cell Foams

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PRL 103, 218001 (2009) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2009
DynamicLength-ScaleCharacterizationandNonequilibriumStatisticalMechanicsofTransport
in Open-Cell Foams
Tyler R. Brosten,1 Sarah L. Codd,1Robert S. Maier,2 and JosephD. Seymour3,*
1Department of Mechanical and Industrial Engineering, Montana State University, Bozeman, Montana 59717-3920, USA
2U.S. Army Engineer Researchand Development Center, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199,USA
3Department of Chemical and Biological Engineering, Montana State University, Bozeman, Montana 59717-3920, USA
(Received29 September 2009; published 20 November 2009)
Nuclear magnetic resonance measurements of scale dependent dynamics in a random solid open-cell
foam reveal a characteristic length scale for transport processes in this novel type of porous medium.
These measurements and lattice Boltzmann simulations for a model foam structure indicate dynamical
behavioranalogoustolowerporosityconsolidatedgranularporousmedia,despiteextremelyhighporosity
in solid cellular foams. Scaling by the measured characteristic length collapses data for different foam
structures as well as consolidated granular media. The nonequilibrium statistical mechanics theory of
preasymptoticdispersion,developedforhierarchicalporousmedia,isshowntomodelthehydrodynamic
dispersive transport in a foam structure.
DOI: 10.1103/PhysRevLett.103.218001 PACS numbers: 81.05.Rm, 05.40.(cid:1)a, 47.56.+r, 87.64.kj
Solid open-cell random foams occur throughout the measurements and lattice Boltzmann (LB) simulation of
natural and technological worlds, ranging from natural the hydrodynamic dispersion molecular dynamics during
sponges and cancellous bone to manufactured metal, ce- pressure driven fluidflow.
ramic, and polymer foams [1]. In these systems, fluid NMR imaging (MRI) has been used to spatially image
transportthroughthefoamstructuremayrelatetobiologi- foam structures [13] and the data used to calculate geo-
cal function [2] as well as the control of energy and mass metricproperties[8].Figure1showsanopticalimageand
transport [3] in heat exchanger applications [4,5], filters, MRI of the polyurethane 110 ppi (pores per inch, an
catalysts, and monolithic absorbents [1,6,7]. Character- industry standard) open cell random polymer foam
ization of the structure of foam systems is an old and (Foamex, Inc.) of volume fraction ’¼0:97 used in this
important problem predating the classic work of Kelvin study. Alternately, dynamics measured by pulsed gradient
and Plateau [1,6]. Detailed geometric analysis of foam spinecho(PGSE)NMRprovidelength-scalecharacteriza-
structureisbasedonstrutlength,cellvolume,andnumber tionduetoporestructuresamplingbymoleculardiffusion
of polygonal faces and subsequent vertex connections, [14,15] and advection [16,17]. PGSE NMR measurement
with ordered foams generating Kelvin’s tetrakaidecahe- of scale dependent hydrodynamic dispersion has been ex-
dron and both wet and solid random foams having distri- tensivelyappliedtocompactedgranularmedia[17–23]but
butionsofstrutlengthsandgeometricstructures[1,6,8,9]. not to characterize transport in open cell foams. NMR
Consideration of random foams as porous media leads to measurements [21,23] and LB simulations [24] of hydro-
theconceptofatransportlengthscale[3,10]andcompari- dynamic dispersion in random spherical bead packs dem-
sontononequilibriumstatisticalmechanicstransportmod-
onstrate a correlated motion in the transverse direction
els [11].
perpendiculartotheappliedflow.Thisresultsinanegative
In the study of consolidated granular porous media the
length scale is taken to be a characteristic pore size l(cid:2)
’=ð1(cid:1)’ÞðV=SÞ based on the surface to volume ratio
(S=V) and volume fraction ’, which derives from the
concept of hydraulic radius in fluid mechanics. In highly
poroussolidfoams,complicationsinstructuralsimulation
and modeling are posed by the solid phase forming a
sample spanning cluster, which is not the case in lower
porosity granular porous systems where phases become
isolated as in percolation theory when p<pc [3,12].
This renders definition of a characteristic length scale
governingtransportinfoamsanopenandimportantques-
tion.ThisLetterpresentsthefirstdirectcharacterizationof
the transport length scale and nonequilibrium statistical
mechanics modeling of transport in an open cell solid FIG. 1 (color). Images of 110 ppi foam sample (a) optical
random foam, using nuclear magnetic resonance (NMR) (b) MRI.
0031-9007=09=103(21)=218001(4) 218001-1 (cid:1) 2009 The American Physical Society
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Dynamic Length-Scale Characterization and Nonequilibrium Statistical
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Mechanics of Transport in Open-Cell Foams
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PRL 103, 218001 (2009) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2009
FIG. 2 (color). Velocity maps for the
longitudinal, i.e., flow, direction by
NMR for 110 ppi foam (sample 1)
(a) NMR—transverse plane,
(b)NMR—axialplane,and(c)LBsimu-
lationfor50ppifoamofMontminyetal.
[3]—axial plane.
lobe in the transverse velocity autocorrelation function force autocorrelation, !2v¼Kð0Þ¼ð3mÞ(cid:1)1r2UðrÞ with m
(VACF), and consequent preasymptotic maxima in the the fluid particle mass and UðrÞ the molecular energy po-
time dependent hydrodynamic dispersion coefficient, tential, and (cid:1)o, the correlation time of the exponential re-
analogous to that observed in molecular dynamic simula- laxation to asymptotic diffusion at equilibrium [25,26].
tions of dense fluids [25,26]. The solution to Eq. (1) for the exponential memory func-
Nonequilibrium statistical mechanics has been used to tionis
rigorouslyderiveatheoryofpreasymptotichydrodynamic (cid:1) (cid:2) (cid:3) (cid:4)
1
dispersion for porous media with hierarchical structure cðtÞ¼expð(cid:1)t=2(cid:1)oÞ cos$tþ 2$(cid:1) sin$t ; (2)
[11]inthecontextoftimecorrelationfunctionsofdynami- o
cal variables [25,26]. The dynamic variable of interest in where $ ¼ð!2v(cid:1)1=4(cid:1)2oÞ1=2 and from which the time
hydrodynamic dispersion is the fluctuation in Lagrangian dRependent hydrodynamic dispersion DðtÞ¼h½uð0Þ(cid:4)2i(cid:5)
particle velocity uðtÞ arising from a Reynold’s decompo- t cðt0Þdt0 is calculated from the nonequilibrium Green-
0
sitionofthedynamicsintoameanandfluctuationcompo-
Kubo relation.
nent vðtÞ¼hviþuðtÞ. The evolution of the normalized
Figure 2(a) shows the spatially resolved longitudinal
VACF cðtÞ¼huð0Þ(cid:3)uðtÞi=h½uð0Þ(cid:4)2i is governed by the
velocity in the 110 ppi foam measured by NMR imaging.
memory function equation [11,25,26]
The fluid follows a clear streamline path dictated by the
dcðtÞ Zt structureofthefoam,withstrongspatialcorrelationinthe
¼(cid:1) Kðt(cid:1)t0Þcðt0Þdt0: (1) flowdirection.Figure2(b)showsthelongitudinalvelocity
dt
0
in a plane transverse to the flow direction. The velocity
A simple exponential memory function KðtÞ¼ field exhibits little spatial correlation in the transverse
!2vexpð(cid:1)t=(cid:1)oÞ in Eq. (1) generates the negative lobe of plane. Figure 2(c) shows longitudinal velocities obtained
the velocity autocorrelation function [25,26]. The time by LB simulation of flow [28] through the digitized poly-
evolution of the VACF reflects the nonequilibrium varia- urethanefoamstructureofMontminyetal.[8]with50ppi
tionofsystemtransportduetocorrelationsandinteractions and’¼0:93(AirProducts,Inc.).Althoughthelargercell
ofdynamics.Theexponentialmemoryfunctionrepresents structure of the 50 ppi foam is obvious, the similarity
aPoissonprocessapproximationtocagediffusionwithina across length scales between the two foam structures in
fluctuating cage in the case of dense hard sphere fluids Figs. 2(b) and 2(c) is quite evident.
[25], and is a model for the position evolution of the NMRmeasurementofthescaledependentdisplacement
Brownianmotionofaharmonicoscillator[27].Theexpo- dynamics transverse to the direction of flow are shown in
nential memoryfunction is derived fromthe second order Fig. 3 as a function of the average longitudinal displace-
approximationofacontinuedfractionrepresentationofthe ment length h(cid:2)ki¼hvki(cid:1), obtained by varying the PGSE
formalLaplacetransformedsolutionofEq.(1)[26].Inthe NMR displacement observation time (cid:1). Data were mea-
hard sphere fluid model, the memory function parameters suredfor3flowratesinthe110ppifoamandbasedonthe
are the second frequency moment of the VACF, i.e., the known linear scaling of transverse dispersion with mean
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PRL 103, 218001 (2009) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2009
for the 80 ppi foam (sample 2) and l(cid:2)475 (cid:3)m for the
50ppifoam[8].Thenonequilibriumstatisticalmechanics
model scales identically, in excellent agreement with the
NMR data and LB simulation. Also plotted is LB simula-
tionofarandomspherepackof’¼0:44[24]inwhichthe
transport scaling length used is l(cid:2)’=ð1(cid:1)’Þd¼0:8d
forspheres ofdiameter d.Remarkably,despite significant
structuralandporositydifferences,thespheredatacollap-
ses withthe foam data for l(cid:7)h(cid:2)ki,and shows only slight
variation at longer displacements. The consistency be-
tween these results and NMR data and LB simulations
for consolidated spherical grain porous media of much
smaller porosity [21,23,24], indicates the universality of
dynamics over a broader range of porous structures than
previously recognized. Converting the hydrodynamic dis-
FIG. 3 (color online). NMR measured hydrodynamic disper- persionintoavarianceandtaking thesecondtimederiva-
sion coefficient normalized by the molecular diffusion of the tive of the NMR data for each flow rate of the 110 ppi
fluid and the mean longitudinal velocity hvki as a function of foam, then averaging the amplitude scaled values yields
displacementobservationtime(cid:1)intermsofmeandisplacement
thevelocityautocorrelationdatainFig.4(b).Again,when
length h(cid:2)ki¼hvki(cid:1). For the 110 ppi foam sample 1 (open cir- plotted versus longitudinal displacement scaled by the
cles),3flows(hvki¼7:1,12.1,and15:0 mm=s)weremeasured transportlength,theagreementbetweenexperiment,simu-
and due to linear scaling of D? with mean velocity have the
lation and analytical theory is excellent.
same amplitude. Overlaid is the fit to the diffusion calculated
The results of Figs. 3 and 4 indicate the potential for
from the velocity autocorrelation for the exponential memory
further development of memory function approaches to
function model Eq. (2) (blue or dark gray line). Data from the
80ppifoamsample2at2flowrates(hvki¼6:0and8:3 mm=s) modeltransportinporousmedia.Theabilitytoincorporate
(opensquares)anddatafromtheLBsimulation(redorgrayline) pore structure into model parameters for the scale depen-
for the 50 ppi foam structure of Montminy et al. [8] at hvki¼ dence of dynamics provides a means to design porous
4:73 mm=s. structureswhichcontroltransport.Intheexponentialmem-
ory function model for dispersion, the !2v represents a
velocity[21,23,24]normalizationby1=hvkiyieldsasingle potentialwhichdeterminesthetransversevelocityoscilla-
tionsduetotransportaroundthefoamstructure,analogous
curve. All flow rates exhibit increasing dispersion with a
to the strength of the restoring force in the harmonic
maximum at the characteristic dynamic transport length
oscillator analogy [27] and consistent with an oscillatory
scale l(cid:6)h(cid:2)kijðdD?=dh(cid:2)kiÞ¼0(cid:2)250 (cid:3)m induced by the flowtoymodelemployedtoelucidatethephysicsofsimi-
structure of the foam [3,10]. The data are compared to
lar dynamics in consolidated granular porous media [21].
the hydrodynamic dispersion coefficient calculated using
FollowingtheargumentsofBerneetal.,thenegativelobe
Eq. (2) in the nonequilibrium Green-Kubo relation. The
oftheVACFinFig.4(b)indicatesreversaloffluidmotion
agreementbetweenthetheoryandexperimentisexcellent.
due to the foam structure [25]. The mean displacement a
The NMR measurement directly provides the character- fluid particle with initial vRelocity uð0Þ trRavels before ve-
istictransportlengthscaleforthefoam,thuscomplement- locity reversal h(cid:1)XðtÞi¼ thuðt0Þidt0 ¼ uð0Þcðt0Þdt0 is
o
ing traditional complex geometric analysis and
givenbythefirstzerocrossingoftheVACF[25].Inapure
measurements. Since the characterization and design of
oscillatory flow such as the toy model of Callaghan and
open cell foams is limited in part by effective structural Codd [21] this length is the order of a quarter wavelength
characterization methods, this result is significant in pro- of the oscillation. The transport length scale l defined by
viding a new and unique method[8]. themaximainthetransversedispersioncorrespondstothe
The transverse dispersion for NMR experiments at 2 length scale at the minima of the VACF and is half the
flowratesinan80ppifoam(Foamex,Inc.)alsoofvolume wavelength of a pure oscillatory flow. In the foam, this
fraction ’¼0:97 and the LB simulation for the 50 ppi transportlengthrepresentsthemeanintercelldistanceover
’¼0:93foamofMontminyetal.[8]areshowninFig.3. which a fluid particle experiences the entire transverse
Thedispersionbehaviorisqualitativelysimilarinallthree velocity distribution. The heterogeneity of the velocity
casesbuttheexactlocationofthemaximum,thetransport field induces a decay of this correlation with time scale
length, depends on the porous structure. (cid:1)o. Development of a potential function based on the
Figure 4(a) demonstrates data collapse with the dis- porous structure that generates particular fluid path lines
placement length h(cid:2)ki scaled by the transport length l and wouldrepresentamajoradvancetowardconnectingstruc-
the dispersion coefficient normalized by the maximum in ture and transport in poroussystems [3,12].
D?ðh(cid:2)jjiÞ for the foam systems. The transport lengths are Hydrodynamicdispersionoffluidflowthroughopencell
l(cid:2)250 (cid:3)mforthe110ppifoam(sample1),l(cid:2)300 (cid:3)m foam has been demonstrated to provide a dynamic trans-
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PRL 103, 218001 (2009) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2009
FIG. 4 (color online). (a) The transverse dispersion coefficient normalized by its maximum amplitude as a function of the axial
displacementlengthscaledbythetransportlengthscalelforLBsimulation(redorgrayline)ofthefoamstructureofMontminyetal.
[8],theNMRdataforthe110ppifoamsample1(opencircles)athvki¼12:1,the80ppisample2(opensquares)athvki¼8:3 mm=s,
thescaledmemoryfunctionmodel(blueordarkgrayline),andLBsimulationforaconsolidatedrandomspherepacking24(black
line). (b) The velocity autocorrelation for the average of all the NMR experiments on the 110 ppi foam sample 1 and the
nonequilibrium statistical mechanics model (blue or dark gray line) and L-B simulation (red orgray line).
port length scale based on the correlated dynamics of the [7] U. Tallarek, F.C. Leinweber, and A. Seidel-Morgenstern,
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