Statistical characterization of microstructure of packings ...sharif.edu/~rahimitabar/pdfs/105.pdf · This paper is devoted to microstructural characterization of polydisperse packings

PHYSICAL REVIEW E 95 052902 (2017)

Statistical characterization of microstructure of packings of polydisperse hard cubes

Hessam Malmir and Muhammad Sahimi

Mork Family Department of Chemical Engineering and Materials Science University of Southern California Los Angeles California90089-1211 USA

M Reza Rahimi TabarDepartment of Physics Sharif University of Technology Tehran 11365-9161 Iran

(Received 8 January 2017 published 22 May 2017)

Polydisperse packings of cubic particles arise in several important problems Examples include zeolitemicrocubes that represent catalytic materials fluidization of such microcubes in catalytic reactors fabricationof new classes of porous materials with precise control of their morphology and several others We present theresults of detailed and extensive simulation and microstructural characterization of packings of nonoverlappingpolydisperse cubic particles The packings are generated via a modified random sequential-addition algorithmTwo probability density functions (PDFs) for the particle-size distribution the Schulz and log-normal PDFsare used The packings are analyzed and their random close-packing density is computed as a function of theparameters of the two PDFs The maximum packing fraction for the highest degree of polydispersivity is estimatedto be about 081 much higher than 057 for the monodisperse packings In addition a variety of microstructuraldescriptors have been calculated and analyzed In particular we show that (i) an approximate analytical expressionfor the structure factor of Percus-Yevick fluids of polydisperse hard spheres with the Schulz PDF also predictsall the qualitative features of the structure factor of the packings that we study (ii) as the packings becomemore polydisperse their behavior resembles increasingly that of an ideal systemmdashldquoideal gasrdquomdashwith little or nocorrelations and (iii) the mean survival time and mean relaxation time of a diffusing species in the packingsincrease with increasing degrees of polydispersivity

DOI 101103PhysRevE95052902

I INTRODUCTION

This paper is devoted to microstructural characterizationof polydisperse packings of hard cubic particles The studyhas been motivated by our recent work on the developmentof a new method of fabrication of porous materials [1] Inthis method a salt such as NaCl which consists of cubiccrystals and is suspended in a nonsoluble medium such as analcohol or ketone is used to coat a nonporous surface suchas a plastic film metal foil or a glass A polymeric materialis then hot-pressed over the salt layer that fills the void spacebetween the salt crystals and solidifies upon cooling The saltcrystals are then washed off with water creating voids andexposing the pore space As the melting temperature of mostsalts is very high they preserve the shape of their crystalsThus their packing retains its microstructure even at the hightemperature at which the polymeric material is hot-pressedover the salt layer The method has the advantage that the sizedistribution of the salt crystals before being hard-pressed bythe polymeric material is the same as the pore-size distributionof the porous medium after the salt crystals are washed offThus one has significant control over the microstructure of theporous medium fabricated by the method

In addition packing of cubic particles is encountered inbiological materials [2] colloids [3] and other systems ofscientific importance As an important practical exampleconsider evaporation of saline water in soil [4] As evaporationproceeds salt crystallizes and precipitates on the surface of thepores in which the saline water flows giving rise to a packing

moeuscedu

of cubic salt crystals that damages the surface and reduces thepermeability and porosity of the soil

In a previous paper [5] hereafter referred to as part I wedefined the essential part of the problem and we reportedsome preliminary results In a subsequent paper [6] whichwe refer to as part II we presented the details of thecomputational algorithm for generating a large packing ofcubic particles the solution of which belongs to a classof problems that are referred to as computationally hardproblems and we presented the results for various two-point correlation functions that characterize the packingsrsquomicrostructure The computational procedure that we havedeveloped is a modification of the random sequential-addition(RSA) algorithm which to our knowledge had not beendeveloped before Our study indicated that the maximumpacking fraction is asymp057 representing the boundary betweena liquid-crystal phase and a crystalline structure In additionwe studied the effect of the porosity and finite size ofthe packings on their characteristics demonstrating that thepackings possess both spatial and orientational long-rangeorder at high packing fractions

In both parts I and II we studied packings of cubicparticles with equal sizes In practice however the particlesor crystals are most likely polydisperse with their sizesdistributed according to a probability density function (PDF)Examples include ceramic powders paint pigments andcolloidal suspensions Using polydisperse powders leads toconsiderable advantages such as higher packing densities[7] and increased fluidity of concentrated suspensions [8]While the properties of packings of polydisperse sphericalparticles have been studied by many researchers [9ndash26] thesame is not true about packings of polydisperse nonspherical

2470-0045201795(5)052902(14) 052902-1 copy2017 American Physical Society

MALMIR SAHIMI AND RAHIMI TABAR PHYSICAL REVIEW E 95 052902 (2017)

FIG 1 The SEM images of (a) cubic aluminosilicate zeolite particles (Advera Rcopy 401) suspended in a Newtonian fluid of polyethyleneglycol (after Ref [33]) and (b) self-synthesized LTA zeolite crystals (after Ref [34])

particles Desmond and Weeks [19] studied the packingfraction as a function of the polydispersity and skewnessof various particle size distributions Baranau and Tallarek[27] investigated random close packings of polydispersehard spheres Furthermore the effective flow and transportproperties of packed beds of polydisperse hard spheres such asthe conductivity and permeability have been reported in a fewstudies [28ndash30] However random packings of polydispersenonspherical particles have rarely been explored

In the present paper we report a detailed simulation ofrandom packings of polydisperse hard cubes as well as ananalysis of a variety of microstructural descriptors for suchporous materials The present study in addition to sheddingfurther light on the properties of packings of cubic particlesalso contributes to the more general problem of understandingthe properties of polydisperse packings of nonspherical parti-cles In addition to the aforementioned problem of fabricationof a new class of porous materials the present study ismotivated by other important applications One example ispackings of zeolites a highly important catalytic materialthat has been used in the chemical industry for a long timeMore recently zeolites have been proposed [31] for separationof fluid mixtures They consist of cubic particles [32] animage of which obtained by a scanning electron microscopeis shown in Fig 1(a) showing polydisperse aluminosilicatezeolites (Advera Rcopy401) suspended in polyethylene glycol [33]Figure 1(b) shows self-synthesized Linde type A (LTA) zeolitecrystals with a variety of sizes [34]

The rest of the paper is organized as follows In Sec II thecomputational approach for generating random packings ofpolydisperse cubic particles and the PDFs of the particlesrsquo sizesare described We then present and discuss in Sec III the resultsfor a variety of the important microstructural descriptors ofthe packings Section IV summarizes the paper and discussespossible further research

II THE MODEL AND ALGORITHM

The most efficient numerical algorithms for generatingpackings of hard spheres are a Monte Carlo (MC) methoddeveloped by He et al [13] and a modification of theLubachevsky-Stillinger algorithm [35] developed by Kansalet al [14] Clusel et al [36] introduced the ldquogranocentricrdquomodel for random packing of jammed emulsions by which

they simulated random packing of polydisperse frictionlessspheres [18] and they investigated the structure of its jamming[20] However such methods cannot be used for generatingrandom packings of mono- or polydisperse hard cubes We alsonote that various molecular dynamics and MC methods thathave been used for generating hard-particle packings are notapplicable to packing of cubes because the overlap potentialfunctions cannot be constructed for particles with nonsmoothshapes including all the Platonic and Archimedean solidsFor such particles Torquato and Jiao [3738] developed anoptimization algorithm that they referred to as the adaptiveshrinking cell (ASC) method which is based on an MCmethod with the Metropolis acceptance rule Except fortetrahedra packings of other Platonic solids (cube octahedradodecahedra and icosahedra) generated by the ASC algorithmare their lattice packings Since a cube is the only Platonicobject that tiles the space its lattice packing generated bythe ASC algorithm is highly ordered with densities close tounity [63738] Delaney and Cleary [39] proposed anotheralgorithm for generating packings of particles the dynamicparticle expansion technique which was originally developedfor the so-called superellipsoids defined by(x

a

)p

+(y

b

)p

+(z

c

)p

= 1 (1)

in which p is a shape parameter and a b and c are thesemimajor axes lengths For large values of p the particlestake on shapes that approach cubes This method also generatesordered packings of hard cubic particles

In parts I and II we presented an algorithm that was basedon a modification of the RSA process which is also used in thepresent paper The simulation begins with a large empty regionof volume V in R3 in which cubic particles are inserted withrandom edge lengths D drawn from a given distribution for theparticlesrsquo sizes The cubes are then placed sequentially in thesimulation cell at randomly selected positions and orientationstaking into account the nonoverlapping constraint Adding theparticles is continued until the desired packing fraction φ =ρ〈v1(D)〉 is reached where ρ = NV is the total numberdensity and 〈v1(D)〉 = 〈D3〉 is the mean volume of the cubicparticles

A The computational algorithm

The details of the computational algorithm are as follows

052902-2

STATISTICAL CHARACTERIZATION OF PHYSICAL REVIEW E 95 052902 (2017)

FIG 2 The size distributions of the particles (a) The Schulz distribution and (b) the log-normal distribution

Step 1 The total number N of cubic particles and the sizeof the simulation cell Lx times Ly times Lz are specified

Step 2 The cubersquos edge length D is selected randomly froma normalized PDF f (D) for the size of the particles

Step 3 Three random numbers xc isin (0Lx) yc isin (0Ly)and zc isin (0Lz) are generated for the center of a new cubicparticle

Step 4 Two random numbers u isin [minus11] and φ isin [02π )are generated for the normal vector n of the upper face of thecubes given by

n =radic

1 minus u2 cos φi +radic

1 minus u2 sin φj + uk (2)

where i j and k are the three unit vectors in Cartesiancoordinates (xyz)

Step 5 The matrix R that rotates the unit vector k into theunit vector n through

R = I + A + A2 (1 minus k) middot nv2

(3)

is constructed where I is the identity matrix and the unitvector v = (v1v2v3) is defined by k times n Furthermore

A =⎡⎣ 0 minusv3 v2

v3 0 minusv1

minusv2 v1 0

⎤⎦ (4)

Note that if n = k then R = I and if n = minusk we have R =minusI

Step 6 The coordinates of the cubersquos eight vertices Vi i = [12 8] are computed by

Vi = Vin + Vc (5)

where Vc is the coordinate vector of thecubersquos center and Vin = RWi in which W1 =(minusd02minusd02minusd02) W8 = (d02d02d02)

Step 7 All the cubersquos vertices are examined to see whetherthey are outside the previously inserted particles If so the newparticle is accepted and one increases n rarr n + 1 where n isthe current number of accepted particles If n N return toStep 3 or if n = N the simulation is terminated

B Particle-size distributions

We utilize two PDFs as the particle-size distribution Oneis the Schulz distribution sometimes referred to as the Schulz-Zimm distribution which arises in a variety of problems

[40ndash43] and is given by

f (Dm) = 1

(m + 1)

(m + 1

〈D〉)m+1

Dm exp

[minus(m + 1)D

〈D〉]

(6)

where m is an integer in [0infin) (x) is the Gamma functionand 〈D〉 is the mean edge length of the cubic particles Thenth moment of the distribution is given by

〈Dn〉 = (m + n)

m

1

(m + 1)n〈D〉n (7)

implying that as m increases the variance 〈D2〉 minus 〈D〉2

decreases Hence m rarr infin represents a monodisperse packingwith limmrarrinfin f (Dm) = δ(D minus 〈D〉) Thus the maximumpacking polydispersivity is obtained with m = 0

A log-normal particle-size distribution is also encounteredin several problems [44ndash49] and is given by the well-knownequation

f (Dσ ) = 1

Dσradic

2πexp

minus[ln (D〈D〉)]2

2σ 2

(8)

in which ln D is distributed according to a GaussianPDF with the mean 〈D〉 and the standard deviation σ =radic

〈(ln D)2〉 minus 〈ln D〉2 The nth moment of the distribution isgiven by

〈Dn〉 = exp

(n2σ 2

2

)〈D〉n (9)

implying that σ rarr 0 represents the monodisperse packingswith limσrarr0 f (Dσ ) = δ(D minus 〈D〉) Figure 2 presents sam-ples of the two distributions for three values of the parametersm and σ

III RESULTS

We assume that the mean edge length of the particles is〈D〉 = 005L where L is the linear size of the simulationcell For simplicity we refer to the pore space and the solidparticles as phases 1 and 2 respectively The porosity andpacking density of the system are therefore respectively φ1

and φ2 Figure 3 presents the top view of six packings whosesizes are distributed according to the Schulz and log-normaldistributions with the packing fraction φ2 = 02 Most of theresults presented in this paper are for the packings shown in

052902-3


FIG 3 Examples of the packings Parts (a) (b) and (c) show packings with the Schulz distribution with respectively m = 0 2 and 10while parts (d) (e) and (f) present those generated with the log-normal PDF with standard deviations that are respectively σ = 1 06 and 02

Fig 3 The results for other packing fractions are qualitativelysimilar to those that we present below

We have computed the dependence of the particle densityat the random close packing (RCP) on the parameter of m

of the Schulz distribution and the standard deviation σ of thelog-normal PDF Moreover we have also computed several ofthe most important microstructural descriptors of the packingsfor the same ranges of m and σ In what follows all theresults were obtained with packings that contained at least2 000 particles while many of them were generated with up to18 000 cubes In addition the results represent averages overat least 10 realizations of each packing

A Random close-packing fraction

Before we present and discuss the results for the randomclose-packing (RCP) fraction we should point out that thevalue of the RCP fraction depends on the method or protocolby which a packing is generated Thus the RCP fraction is notunique On the other hand the jammed state of a packing isunique [5051] and it has to do with the mechanical stabilityof the packing In fact the jammed state of a packing of softcubic particles has been recently studied [52]

It is well known [71113ndash17192124252740] that in-creasing the polydispersivity increases the packing efficiencybecause smaller particles pack more compactly by fillingthe voids between adjacent large particles Some expressionsfor the packing fraction of polydisperse hard spheres thatdepend on the parameters of the particle-size distributions used[1920] as well as some limits for their RCP [27] fractionshave previously been reported The highest achievable packingfraction of polydisperse hard spheres is still unknown becauseas pointed out earlier the RCP fraction depends on themethod by which one generates the packings [19] Indeedthe maximum packing fraction of the structures generated by

the RSA algorithm is different from those generated by thegrowth [14] and the shrinking cell [21] algorithms

Using extensive simulations we computed the packingdensity at the RCP point as a function of the Schulzdistributionrsquos polydispersivity parameter m isin [010] and thestandard deviation of the log-normal PDF σ isin [01] Theresults are shown in Fig 4 While the maximum packingfraction for the most polydisperse packing generated by theSchulz distributionmdashthe limit m = 0mdashis around 081 thecorresponding estimate for the log-normal distribution is 071obtained with σ = 1 Moreover the packing fraction is aslarge as φRCP0 + (1 minus φRCP0)φRCP0 asymp 082 for a bidispersepacking with the size ratio D1D2 rarr 0 In the monodisperselimit ie m rarr infin or σ rarr 0 the close-packing density forboth distributions approaches the maximum packing fractionof the monodisperse packing asymp057 which we calculated inparts I and II (see also Refs [53ndash56] as well as the discussionsbelow for monosized nonspherical particles) For examplethe maximum packing fractions for σ = 01 and m = 100 areboth close to 058 Moreover while the close-packing densityappears to vary linearly with σ isin [01] it has a very sharpnegative slope for m isin [01] and varies slightly for m 3

Note that the maximum packing fraction for packings ofpolydisperse hard spheres is asymp075 if their size distribution islog-normal with σ = 06 [57] which should be compared with064 [58] for random monodisperse packing of hard spheresThe approximate formula given by Desmond and Weeks [19]is also applicable only to close-packing densities with σ isin[006] Note that the trends in Fig 4(b) indicating that themaximum random close-packing fraction increases with thestandard deviation of the log-normal distribution is consistentwith the reported experimental data reported by Brouwers [59](see also Ref [60])

Yu et al [50] studied packings of nonspherical particles andthey presented an approximate formula based on the Westman

052902-4


FIG 4 Random close-packing density vs (a) the polydispersivity parameter m of the Schulz distribution and (b) the standard deviation σ

of the log-normal PDF

equation [61] for the porosity of a binary mixture of suchparticles with intermediate size ratio D1D2 For this purposethey approximated nonspherical particles with spherical onesby introducing an ldquoequivalent packing diameterrdquo and theystudied the porosity as a function of the volume fraction andsize ratio of small and large particles Good agreement wasfound between the empirical equation and experimental data

B Radial distribution function and the structure factor

One of the most important microstructural descriptors forpackings of polydisperse particles (and for any material forthat matter) is the radial distribution function g(r) definedas the probability of finding a cubersquos centroid at a distance r

from a given reference cubersquos centroid at the origin Thus g(r)describes how the density of the packing varies as a functionof r

g(r) = 〈n(r)〉ρvs

(10)

where ρ is the total number density of the particles and 〈n(r)〉is the average number of particles in a spherical shell of volumevs at a radial distance r from a reference cubersquos centroid Theimportance of the radial distribution function is due to the factthat it can be determined experimentally by x-ray or neutronscattering via the structure factor S(k) which for isotropic

materials is given by [62ndash64]

S(k) = 1 + 4πρ

k

int infin

0sin(kr)[g(r) minus 1]r dr (11)

where k is the magnitude of the scattering wave vector If thereis no long-range order in the system g(r) and S(k) decay tounity very rapidly as r rarr infin

Figure 5 presents the average radial distribution function forseveral values of m the Schulz distribution polydispersivityparameter and σ the standard deviation of the log-normalPDF Also shown are the corresponding results for themonodisperse packings The average g(r) is a measure ofthe packingrsquos structure regardless of the cubesrsquo diametersA similar function was introduced by Stapleton et al [65] forpolydisperse Lennard-Jones fluids represented by sphericalparticles For narrow symmetric size distributions eg form = 10 and σ = 02 g(r) is very similar to that of monodis-perse packings with the same packing fraction [56] As the sizedistribution of the particles is broadened with higher skewnessthe first peak of g(r) and its oscillations beyond the peak aresuppressed Moreover for higher degrees of polydispersivitythe first peak of g(r) occurs at smaller distances r Unlike themonodisperse packings [56] for which g(r) has its first peak atr 13D and essentially vanishes for radial distances less than13D the polydisperse packings are characterized by radialdistribution functions that vary continuously starting fromzero until they reach their first peak For example for the most

FIG 5 Dependence of the radial distribution function g(r) on (a) the polydispersivity parameter m of the Schulz distribution and (b) thestandard deviation σ of the log-normal distribution m = infin and σ = 0 represent the monodisperse packings

052902-5


FIG 6 Dependence of the structure factor S(k) on (a) the polydispersivity parameter m of the Schulz distribution and (b) the standarddeviation σ of the log-normal distribution m = infin and σ = 0 represent the monodisperse packings

polydisperse packing generated by the Schulz distribution (thelimit m = 0) g(r) is nonzero at r 06〈D〉 and it achievesits first peak at r 〈D〉

Figure 6 presents the structure factor S(k) for the sixpackings of Fig 3 calculated via Fourier transforming ofg(r) along with those for the monodisperse packings Anotherway of computing S(k) is by direct Fourier transforming ofthe particlesrsquo positions rj leading to S(k) = |sumN

j=1 exp(ikj middotrj )|2N where N is the total number of particles That is S(k)is computed by averaging it over k = |k| As Fig 6 indicatessimilar to g(r) the structure factor S(k) of the polydispersepackings exhibits behavior similar to that of the monodispersepackings if the particle-size distribution is narrow (such as forexample m = 10 and σ = 02 for respectively the Schulz andlog-normal distributions) Stronger polydispersivity resultshowever in the suppression of the first peak followed byoscillations around unity and loss of the structure since thepackings contain particles of all sizes The first peak ofS(k) occurs at larger wave numbers k as the polydispersivityincreases (decreasing m or increasing σ ) All the radialdistribution functions and the structure factors are of coursefinite and approach unity as r rarr infin The peaks beyond the firstone indicate the structured behavior of the packings (disorderto order) As is well known sharper and stronger oscillationsaround unity indicate that the packings exhibit behavior closerto long-range order or quasicrystalline behavior

C Analytical approximation for the structure factor

An important unsolved problem is the derivation of aclosed-form analytical formula for the structure factor ofdisordered materials In the absence of the solution of theproblem analytical approximations have been proposed forS(k) In particular for the Percus-Yevick fluids representedby polydisperse hard spheres with a Schulz distribution of theparticlesrsquo size the following accurate approximation has beenproposed [436667]

S(k) = [ρP (k)]minus1I (k) (12)

with I (k) being the scattering intensity and P (k) representingthe scattering amplitude given by

P (k) = 16π2int infin

0f (D)y2(kD)dD (13)

where as before ρ is the total number density of the particlesand y(t) = sin t minus t cos t with t = kD2 I (k) the scatteringintensity is given by [4368]

I (k) = ρ

[ int infin

0f (Dα)F 2

α (kDα)dDα +int infin

0dDα

timesint infin

0dDβf (Dα)f (Dβ)Fα(kDα)Fβ(kDβ)Hαβ(k)

]

(14)

Here Fα(kDα) is the single-particle amplitude form andHαβ(k) = (ραρβ)12hαβ(k) with ρα being the number densityof particles of size fraction α The functions hαβ are the Fouriertransforms of the total correlation functions and they havea simple mathematical form for the Percus-Yevick solutionsof hard spheres [436667] To derive an expression for thesingle-particle form amplitude it is assumed [43] that thedistribution of the scattering contrast in the spherical particlescan be partitioned in a core of radius a1 = p1aM followed bya sequence of M minus 1 concentric shells with each extending toan outer radius of ai = piaM with pM = 1 Then [69]

Fα(kDα) = 4π

k3

Msumi=1

nii+1y(05kpiDαp) (15)

where nii+1 = ni minus ni+1 with ni being the electron densityin the shell i p = 05DαaM and the function y(t) was givenearlier More details are given elsewhere [436667]

Although the packings that we study are quite different fromthe Percus-Yevick fluids we used Eqs (12)ndash(15) to predictS(k) for the packings with a Schulz particle-size distributionThe results are presented in Fig 7 where they are comparedwith the predictions of Eq (12) All the qualitative trendsindicated by our computed S(k) are similar to those predictedby Eq (12) Note that one reason for the difference betweenthe computed and predicted S(k) for small k is due to Eq (12)not being able to precisely predict the scattering intensity atzero angle at high packing fractions and for broad particle-sizedistributions [67]

D Loss of structure

To quantitatively investigate the ldquoloss of structurerdquo orldquoloss of orderrdquo in the polydisperse packings relative to the

052902-6


FIG 7 Comparison of the structure factor S(k) computed bythe numerical simulation and the predictions of the analyticalapproximation Eq (12) The sizes of the particles follow the Schulzdistribution with the polydispersivity parameter m The monodispersepackings correspond to m = infin

monodisperse ones we use an order metric τ introduced byTorquato et al [70] and defined by

τ equiv 1

3

int infin

0[g(r) minus 1]2dr = 1

(2π )3ρ2

int infin

0[S(k) minus 1]2dk

(16)where is some characteristic length scale usually taken tobe unity in order to rescale the configuration of the systemand to make the number density ρ equal to unity [70] Thesecond equality in Eq (12) is based on the use of Parsevalrsquostheorem For spatially uncorrelated systems of particlesmdashtheldquoideal gasrdquomdashτ = 0 and therefore the deviation of τ from zerois a measure of order in the system For perfectly crystallinestructures τ = infin

Figure 8 presents the computed τ with both the Schulzand log-normal distributions The corresponding value of τ

for the monodisperse packings at the same packing fraction ofφ2 = 02 (see Fig 3) is asymp13 Consider the results for eg theSchulz distribution As pointed out earlier m = 0 correspondsto the highest degree of polydispersivity Figure 8 indicatesthat τ is a monotonically increasing function of m implying

that the more polydisperse the packings are the weaker are thecorrelations in the spatial distribution of the packings That isthe more polydisperse the packings are the closer they areto an ldquoideal gasrdquo one for which τ = 0 A similar trend isalso indicated by Fig 8 with a log-normal distribution of theparticlesrsquo sizes

E Two-point probability function

Another important microstructural descriptor of a givenphase i of a packing of particles is the two-point probabilityfunction S

(i)2 (x1x2) defined by

S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)

which represents the probability of finding two randomlyselected points x1 and x2 separated by a distance r belongingto phase i of a multiphase material The indicator functionis I (i)(x) = 1 if x isin phase i and it is zero otherwise Forstatistically homogeneous and isotropic media S

(i)2 (x1x2)

depends only on the distances ie

S(i)2 (x1x2) = S

(i)2 (r) (18)

One also has S(i)2 (0) = φi where φi is the volume fraction of

phase i In addition S(i)2 must satisfy limrrarrinfin S

(i)2 (r) rarr φ2

i There are certain relations between S

(i)2 and other microstruc-

tural descriptors [6364] so that any knowledge about S(i)2

leads directly to information about such characteristicsThe computed S

(1)2 (r) for the six packings of Fig 3 are

presented in Fig 9 along with those for the monodispersepackings Since the porosity φ1 of all the packings equals 08S

(1)2 (0) 08 Furthermore S

(1)2 (r) approaches 064 for large

r since theoretically limrrarrinfin S(1)2 (r) = φ2

1 The approachingrate to φ2

1 is however different for various degrees ofpolydispersivity The lower the degree of polydispersivitythe faster is the approaching rate For the highest degreeof polydispersivity m = 0 in the Schulz distribution S

(1)2 (r)

approaches φ21 at radial distances r gt 5〈D〉 Thus the prob-

ability of finding two points separated by a distance r in thepore space of a polydisperse packing is greater than that ina monodisperse packing with the same porosity or packingfraction

In addition S(1)2 (r) has a minimum at r = 〈D〉 for the

narrowest polydispersivity that we studied namely for m = 10

FIG 8 Loss of order in the polydisperse packings as characterized by the order metric τ defined by Eq (12)

052902-7


FIG 9 Dependence of the two-point probability function S(1)2 (r) for the pore space of the packings on (a) the polydispersivity parameter

m of the Schulz distribution and (b) the standard deviation σ of the log-normal PDF m = infin and σ = 0 represent the monodisperse packings

and σ = 02 in the two particle-size distributions This impliesthat in the monodisperse packings the probability of findingin the pore space two end points of a line segment r which isequal to the linear size of the cubes is lower than any otherdistance r Thus we may deduce that compared with themonodisperse packings the polydisperse ones contain moreconnected void space This feature which is more distinctivefor higher degrees of polydispersivity affects the flow andtransport properties of such packings

F Specific surface

The specific surface s defined as the interfacial area perunit volume contains information about the internal surfaceof porous media For the packings under study s is computedby

s = 6η〈D2〉〈D3〉 (19)

where η = ρ〈D3〉 is the reduced density of the systemequivalent to φ2 The surface area ratio A is the ratio ofthe specific surface of a polydisperse system and that of amonodisperse one with the same packing fraction φ2 Figure 10presents the quantity versus m the polydispersivity parameterof the Schulz distribution and σ the standard deviation of thelog-normal PDF In both cases A decreases with increasingpolydispersivity implying that the more dispersed the packingsare the smaller is the specific surface they possess which is

due to filling up the small holes between the large particleswith smaller ones

G Lineal-path function

A very useful statistical characteristic of random packingsof solid objects is their lineal-path function L(i)(z) theprobability of finding a randomly thrown line segment oflength z entirely in phase i Since L(i)(z) quantifies connect-edness along a lineal path it is also known as the coarse-scaleconnectedness function Its limiting values are L(i)(0) = φi

and L(i)(infin) = 0 In the limit z = 0 one has L(1)(0) = φ1the porosity of the packing whereas the tail of the functionas z rarr infin yields information about the largest possible linesegment in the packingrsquos pore space

Figure 11 presents the lineal function for the six packingsthree each with the Schulz and log-normal distributions Forboth distributions the higher the degree of polydispersivitythe more likely it is to find a line segment of length z entirelyin the pore phase The longest possible line segment entirelyin the pore space belongs to the packing with the highestdegree of polydispersivity the limit m = 0 in the Schulzdistribution The maximum possible length of a line segment inthe pore space is a bit larger than 08〈D〉 For the monodispersepackings represented by the limits m rarr infin and σ = 0 inthe two particle-size distributions the maximum length of aline segment entirely in the void space is around z asymp 04〈D〉almost half of that for the highest degree of polydispersivity

FIG 10 The surface area ratio A vs (a) the polydispersivity parameter m of the Schulz distribution and (b) the standard deviation σ of thelog-normal PDF

052902-8


FIG 11 Dependence of the lineal-path function L(1)(z) of the pore space of the packings on (a) the polydispersivity parameter m of theSchulz distribution and (b) the standard deviation σ of the log-normal distribution m = infin and σ = 0 represent the monodisperse packings

This provides further evidence that the polydisperse packingscontain more connected pore space than the monodisperseones

H Pore-size distribution

The pore-size distribution function P (δ) of a porousmedium is defined such that P (δ)dδ is the probability thata randomly selected point in the pore phase is at a distancebetween δ and δ + dδ from the nearest point at the pore-solidinterface Thus P (δ) is related to the probability of inserting asphere of radius δ into the system Since P (δ) is a probabilitydensity function it normalizes to unity ie

int infin0 P (δ)dδ = 1

Note that P (δ) is not the same as the classical pore-sizedistribution that is used to quantify the statistical distribution ofthe effective sizes of the pore bodies and pore throats in a porespace which is measured by a variety of techniques [71] Theassociated cumulative distribution function F (δ) represents thefraction of the pore space that has a pore radius larger than δ

F (δ) =int infin

δ

P (r)dr (20)

Hence the limiting values of F (δ) are F (0) = 1 and F (infin) =0 The mean pore size 〈δ〉 is then given by

〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)

Figure 12 illustrates the cumulative pore-size distributionsF (δ) computed for the six packings of Fig 3 For both particle-size distributions the area under the F (δ) curves is greater forhigher degrees of polydispersivity implying that the meanpore size 〈δ〉 in the polydisperse packings is greater than thatof the monodisperse one While the largest pore size is aroundasymp175〈δ〉 for the highest degree of polydispersivity with theSchulz distribution it is around asymp075〈δ〉 for σ = 02 in thelog-normal distribution This is due to the fact that the longtail of the log-normal distribution generates some very largeparticles that can be better filled up by very small particleshence reducing the pore sizes

I Mean survival time and principal relaxation time

Two other important physical quantities are closely relatedto the cumulative distribution F (δ) One is the mean survivaltime Tm of a diffusing species that undergoes a first-orderreaction among partially absorbing traps in the porous mediumthat a packing represents A lower bound to Tm is given by[7273]

Tm 1

(int infin

0F (δ)dδ

)2

(22)

The second property is the principal relaxation time T1 inthe same diffusion problem which also arises in the problemof the relation between nuclear magnetic resonance and theeffective permeability of a porous medium [6874ndash76] with

FIG 12 Dependence of the cumulative pore-size distribution function F (δ) on (a) the polydispersivity parameter m of the Schulzdistribution and (b) the standard deviation σ of the log-normal distribution m = infin and σ = 0 represent the monodisperse packings

052902-9


FIG 13 Lower bounds for the mean survival time Tm and the principal relaxation time T1 of a diffusing species in the two types of packings

the permeability K given by K prop Tf

1 with f asymp 2 [71] Thefollowing rigorous lower bound has been derived for T1

[7273]

T1 2

int infin

0δF (δ)dδ (23)

where as previously mentioned is usually taken to be unityThe computed lower bounds for Tm and T1 are presented

in Fig 13 According to Fig 13 the monodisperse packingshave the shortest survival and relaxation times but the timesincrease as the polydispersivity of the packings increasesThe reason is due to the nature of the reaction-diffusionphenomenon Absorbing traps in a packing with equal-sizeparticles in which a species diffuses cannot be avoided for along time since the diffusion paths are not too tortuous Onthe other hand as the packings become more polydispersesmaller particles fill up the pores between the large particleshence providing more tortuous diffusion paths for the speciesimplying that it takes longer to encounter the traps and beabsorbed by them

J Nearest-neighbor functions

The particle nearest-neighbor probability density functionHP (r) is defined such that HP (r)dr is the probability of findingthe nearest-neighbor particlersquos center at a distance between r

and r + dr from an arbitrary reference particlersquos center in thepacking A second nearest-neighbor function HV (r) referred

to as the void nearest-neighbor PDF is defined in a similarmanner ie HV (r)dr is the probability of finding the nearest-neighbor particlersquos center at a distance between r and r + dr

from an arbitrary point in the void region of the system Whilethe first moment of HV (r) is related to that of the pore-sizefunction P (δ) the first moment of HP (r) yields the meannearest-neighbor distance lP between the particles

lP =int infin

0rHP (r)dr (24)

To compare the results with those for the monodispersepackings we define LP as the ratio of the mean nearest-neighbor distance of a polydisperse system and that of amonodisperse packing with the same packing fraction φ2 (orporosity φ1) Figure 14 presents the dependence of LP on thepolydispersivity parameter m of the Schulz distribution andon σ the standard deviation of the log-normal PDF In bothcases the scaled mean nearest-neighbor distance decreasesupon increasing the degree of polydispersivity because smallerparticles can fill the void space between the large ones hencedecreasing LP With the Schulz distribution LP varies from035 for the highest degree of polydispersivity (ie for m = 0)to around 08 for the narrowest distribution It approachesunity with a slight tilt as m increases With the log-normaldistribution on the other hand LP varies from unity for themonodisperse packing (ie for σ = 0) to around 048 for themost polydisperse system that we simulated The implicationsare clear (i) The distances between nearest-neighbor particles

FIG 14 The scaled mean nearest-neighbor distance LP vs (a) the polydispersivity parameter m of the Schulz distribution and (b) thestandard deviation σ of the log-normal PDF

052902-10


FIG 15 Dependence of the specific surface area s on the porosity for various values of (a) the polydispersivity parameter m of the Schulzdistribution and (b) the standard deviation σ of the log-normal PDF m = infin and σ = 0 represent the monodisperse packings

in the polydisperse packings are considerably less than thosein the monodisperse packings and (ii) due to the long tail ofthe log-normal distribution that gives rise to some very largecubes the distances between the nearest-neighbor particleswith a log-normal particle-size distribution are smaller thanthose in the packing with a Schulz distribution In other wordsthe presence of very large cubes also creates very large voidsin between the particles that are filled by multiple small par-ticles hence reducing the distance between nearest-neighborparticles

K Effect of porosity

All the results presented so far were for packings with afixed porosity of φ1 = 08 In this section we present theporosity dependence of two important properties namely thespecific surface area s and the mean nearest-neighbor distancelp between the particles Figure 15 presents the dependenceof s on the porosity φ1 of the packings with the two typesof particle-size distribution As the porosity decreases theinterfacial area between the particles increases since thedistance between them decreases Moreover upon increasingthe degree of polydispersivity at a fixed porosity the interfacialarea decreases since once again the small particles fill up thevoid between the large ones The trends are identical for bothtypes of particle-size distribution In addition for any m and σ which are the parameters of the two distributions the specific

surface area depends on the porosity more or less linearlywhich is then useful for estimating s for any other porosity

Figure 16 presents the results for the nearest-neighbordistance lp Clearly as the porosity increases so also shouldlp as the void space expands at higher φ1 The expectation isconfirmed by the results shown in Fig 16 for both particle-size distributions Similar to the specific surface area s thenearest-neighbor distance lp varies linearly with the porosityThe linear dependence of s and lp on the porosity is thedirect result of the nonoverlapping constraint that we haveimposed on the particles If the particles are allowed a degreeof overlapmdashthe so-called cherry-pit model [6364]mdashthe lineardependence should disappear and be replaced by a morecomplex dependence

IV SUMMARY

This paper reports on the statistical characterization of themicrostructure of packings of nonoverlapping polydisperse cu-bic particles Two types of particle-size distributions namelythe Schulz and log-normal distributions were utilized andthe random close packing density of the porous media wascomputed for various degrees of polydispersivity In additionsome of the most important microstructural descriptors in-cluding the radial distribution function and the correspondingstructure factor the two-point probability function the specificsurface and lineal-path function as well as the pore-size

FIG 16 Dependence of the mean nearest-neighbor distance lp on the porosity of the packings for several values of (a) the polydispersivityparameter m of the Schulz distribution and (b) the standard deviations σ of the log-normal PDF m = infin and σ = 0 represent the monodispersepackings

052902-11


and nearest-neighbor functions were calculated and analyzedfor numerous degrees of polydispersivity With the Schulzdistribution with the highest degree of polydispersivity themaximum packing fraction is computed to be about 081whereas the corresponding value computed with the log-normal particle-size distribution with a standard deviationσ isin [010] is about 071 computed with σ = 1

Furthermore it was demonstrated that the probability offinding two points separated by a distance r in the pore space ofa polydisperse packing is greater than that in a monodisperseone with the same porosity The maximum length of a linesegment entirely in the void space of a packing with the largestdegree of polydispersivity as well as the maximum pore size inthe same packing are twice those of a monodisperse packingHence polydisperse packings contain better connected voidspace Their specific surface areas are also smaller than thecorresponding value in monodisperse packings

More importantly we showed that (i) an approximateanalytical expression for the structure factor of Percus-Yevickfluids of polydisperse hard spheres in which the particlesrsquo sizesfollow a Schulz distribution also predicts all the qualitativefeatures of the structure factor of the packings that havebeen studied in this paper (ii) as the packings become morepolydisperse their behavior resembles increasingly that of anideal systemmdashldquoideal gasrdquomdashwith little or no correlations and(iii) the mean survival time and mean relaxation time of adiffusing species in the packings increase with higher degreesof polydispersivity

ACKNOWLEDGMENTS

Work at the University of Southern California was sup-ported in part by the Petroleum Research Fund administeredby the American Chemical Society

[1] A R Mehrabi L Vaziri R Mehrabi J M de Santos and MSahimi Novel method for fabrication of porous materials withcontrolled morphology Adv Mater (unpublished)

[2] Z A Almsherqi S D Kohlwein and Y Deng Cubicmembranes A legend beyond the Flatland of cell membraneorganization J Cell Biol 173 839 (2006)

[3] L Rossi S Sacanna W T M Irvine P M Chaikin D JPine and A P Philipse Cubic crystals from cubic colloids SoftMatter 7 4139 (2011)

[4] M Norouzi Rad N Shokri and M Sahimi Pore-scale dynamicsof salt precipitation in drying porous media Phys Rev E 88032404 (2013)

[5] H Malmir M Sahimi and M R Rahimi Tabar Microstructuralcharacterization of random packings of cubic particles Sci Rep6 35024 (2016)

[6] H Malmir M Sahimi and M R Rahimi Tabar Packing ofnonoverlapping cubic particles Computational algorithms andmicrostructural characteristics Phys Rev E 94 062901 (2016)

[7] E Dickinson Polydisperse suspensions of spherical colloidalparticles Analogies with multicomponent molecular liquidmixtures Ind Eng Chem Prod Res Dev 25 82 (1986)

[8] A P Shapiro and R F Probstein Random Packings ofSpheres and Fluidity Limits of Monodisperse and BidisperseSuspensions Phys Rev Lett 68 1422 (1992)

[9] B Lu and S Torquato General formalism to characterize themicrostructure of polydispersed random media Phys Rev A43 2078 (1991)

[10] B Lu and S Torquato Nearest-surface distribution functions forpolydispersed particle systems Phys Rev A 45 5530 (1992)

[11] Z Adamczyk B Siwek M Zembala and P Weronski Influenceof polydispersity on random sequential adsorption of sphericalparticles J Colloid Interface Sci 185 236 (1997)

[12] A Daneyko A Houmlltzel S Khirevich and Ul Tallarek Influenceof the particle size distribution on hydraulic permeability andeddy dispersion in bulk packings Anal Chem 83 3903 (2011)

[13] D He N N Ekere and L Cai Computer simulation of randompacking of unequal particles Phys Rev E 60 7098 (1999)

[14] A R Kansal S Torquato and F H Stillinger Computergeneration of dense polydisperse sphere packings J ChemPhys 117 8212 (2002)

[15] H J H Brouwers Particle-size distribution and packing fractionof geometric random packings Phys Rev E 74 031309 (2006)

[16] Y Shi and Y Zhang Simulation of random packing of sphericalparticles with different size distributions Appl Phys A 92 621(2008)

[17] R S Farr and R D Groot Close packing density of polydispersehard spheres J Chem Phys 131 244104 (2009)

[18] E I Corwin M Clusel A O N Siemens and J Brujic Modelfor random packing of polydisperse frictionless spheres SoftMatter 6 2949 (2010)

[19] K W Desmond and E R Weeks Influence of particle sizedistribution on random close packing of spheres Phys Rev E90 022204 (2014)

[20] C Zhang C B OrsquoDonovan E I Corwin F Cardinaux T GMason M E Mobius and F Scheffold Structure of marginallyjammed polydisperse packings of frictionless spheres PhysRev E 91 032302 (2015)

[21] D Chen and S Torquato Confined disordered strictly jammedbinary sphere packings Phys Rev E 92 062207 (2015)

[22] D Hlushkou H Liasneuski U Tallarek and S TorquatoEffective diffusion coefficients in random packings of polydis-perse hard spheres from two-point and three-point correlationfunctions J Appl Phys 118 124901 (2015)

[23] W Schaertl and H Sillescu Brownian dynamics of polydispersecolloidal hard spheres Equilibrium structures and random closepackings J Stat Phys 77 1007 (1994)

[24] S I Henderson T C Mortensen S M Underwood and W vanMegen Effect of particle size distribution on crystallisation andthe glass transition of hard sphere colloids Physica A 233 102(1996)

[25] S-E Phan W B Russel J Zhu and P M Chaikin Effects ofpolydispersity on hard sphere crystals J Chem Phys 108 9789(1998)

[26] C E Zachary Y Jiao and S Torquato Hyperuniformityquasi-long-range correlations and void-space constraints inmaximally random jammed particle packings I Polydispersespheres Phys Rev E 83 051308 (2011)

[27] V Baranau and U Tallarek Random-close packing limits formonodisperse and polydisperse hard spheres Soft Matter 103826 (2014)

052902-12


[28] J F Thovert I C Kim S Torquato and A Acrivos Bounds onthe effective properties of polydispersed suspensions of spheresAn evaluation of two relevant morphological parameters JAppl Phys 67 6088 (1990)

[29] Y Li and C-W Park Permeability of packed beds filled withpolydisperse spherical particles Ind Eng Chem Res 37 2005(1998)

[30] S Deridder and G Desmet Calculation of the geometricalthree-point parameter constant appearing in the second orderaccurate effective medium theory expression for the B-termdiffusion coefficient in fully porous and porous-shell randomsphere packings J Chromatogr A 1223 35 (2012)

[31] N Rangnekar N Mittal B Elyassi J Caro and M TsapatsisZeolite membranes A review and comparison with MOFsChem Soc Rev 44 7128 (2015)

[32] S M Auerbach K A Carrado and P K Dutta Handbook ofZeolite Science and Technology (CRC Baton Rouge LA 2003)

[33] C D Cwalina K J Harrison and N J Wagner Rheology ofcubic particles suspended in a Newtonian fluid Soft Matter 124654 (2016)

[34] P Sharma J-S Song M H Han and C-H Cho GIS-NaP1 zeolite microspheres as potential water adsorption ma-terial Influence of initial silica concentration on adsorp-tive and physicaltopological properties Sci Rep 6 P22734(2016)

[35] B D Lubachevsky and F H Stillinger Geometric properties ofrandom disk packings J Stat Phys 60 561 (1990)

[36] M Clusel E I Corwin A O N Siemens and J Brujic Aldquogranocentricrdquo model for random packing of jammed emulsionsNature (London) 460 611 (2009)

[37] S Torquato and Y Jiao Dense packings of polyhedra Platonicand Archimedean solids Phys Rev E 80 041104 (2009)

[38] Y Jiao and S Torquato Maximally random jammed packingsof Platonic solids Hyperuniform long-range correlations andisostaticity Phys Rev E 84 041309 (2011)

[39] G W Delaney and P W Cleary The packing properties ofsuperellipsoids Europhys Lett 89 34002 (2010)

[40] P J Flory Molecular size distribution in linear condensationpolymers J Am Chem Soc 58 1877 (1936)

[41] J Welsh and V A Bloomfield Fitting polymer distribution datato a Schulz-Zimm function J Polym Sci Polym Phys 111855 (1973)

[42] R K Pandey and D N Tripathi Schulz distribution functionand the polydispersity of the binary suspension of chargedmacroions Coll Surf A 190 217 (2001)

[43] M Nayeri M Zackrisson and J Bergenholtz Scatteringfunctions of core-shell-structured hard spheres with Schulz-distributed radii J Phys Chem B 113 8296 (2009)

[44] E Limpert W A Stahel and M Abbt Log-normal distributionsacross the sciences Keys and clues BioScience 51 341 (2001)

[45] J Soumlderlund L B Kiss G A Niklasson and C G GranqvistLognormal Size Distributions in Particle Growth Processeswithout Coagulation Phys Rev Lett 80 2386 (1998)

[46] R Chantrell J Popplewell and S Charles Measurements ofparticle size distribution parameters in ferrofluids IEEE TransMag 14 975 (1978)

[47] G M Kondolf and A Adhikari Weibull vs lognormal distri-butions for fluvial gravels J Sedimentary Res 70 456 (2000)

[48] C Reimann and P Filzmoser Normal and lognormal datadistribution in geochemistry Death of a myth Consequences

for the statistical treatment of geochemical and environmentaldata Environ Geol 39 1001 (2000)

[49] M Madadi and M Sahimi Lattice Boltzmann simulation offluid flow in fracture networks with rough self-affine surfacesPhys Rev E 67 026309 (2003)

[50] S Torquato T M Truskett and P G Debenedetti Is RandomClose Packing of Spheres Well Defined Phys Rev Lett 842064 (2000)

[51] M van Hecke Jamming of soft particles Geometry mechanicsscaling and isostaticity J Phys Condens Matter 22 033101(2010)

[52] K C Smith I Srivastava T S Fisher and M AlamVariable-cell method for stress-controlled jamming of athermalfrictionless grains Phys Rev E 89 042203 (2014)

[53] A B Yu N Standish and A McLean Porosity calculation ofbinary mixtures of nonspherical particles J Am Ceram Soc76 2813 (1993)

[54] J Baker and A Kudrolli Maximum and minimum stablerandom packings of platonic solids Phys Rev E 82 061304(2010)

[55] U Agarwal and F A Escobedo Mesophase behavior ofpolyhedral particles Nat Mater 10 230 (2011)

[56] R P Zou and A B Yu Evaluation of the packing characteristicsof mono-sized non-spherical particles Powder Technol 88 71(1996)

[57] H Y Sohn and C Moreland The effect of particle sizedistribution on packing density Can J Chem Eng 46 162(1968)

[58] G D Scott and D M Kilgour The density of random closepacking of spheres J Phys D Appl Phys 2 863 (1969)

[59] H J H Brouwers Packing fraction of particles with lognormalsize distribution Phys Rev E 89 052211 (2014)

[60] A Yang C T Miller and L D Turcoliver Simulation ofcorrelated and uncorrelated packing of random size spheresPhys Rev E 53 1516 (1996)

[61] A E R Westman The packing of particles Empirical equationfor intermediate diameter ratios J Am Ceram Soc 19 127(1936)

[62] J-P Hansen and I R McDonald Theory of Simple Liquids(Elsevier New York 1990)

[63] S Torquato Random Heterogeneous Materials (Springer NewYork 2002)

[64] M Sahimi Heterogeneous Materials I (Springer New York2003)

[65] M R Stapleton D J Tildesley T J Sluckin and N QuirkeComputer simulation of polydisperse liquids with density- andtemperature-dependent distributions J Phys Chem 92 4788(1988)

[66] P van Beurten and A Vrij Polydispersity effects in the small-angle scattering of concentrated solutions of colloidal spheresJ Chem Phys 74 2744 (1981)

[67] W L Griffith R Triolo and A L Compere Analyticalscattering function of a polydisperse Percus-Yevick fluid withSchulz- (-) distributed diameters Phys Rev A 35 2200(1987)

[68] L Blum and G Stell Polydisperse systems I Scatteringfunction for polydisperse fluids of hard or permeable spheres JChem Phys 71 42 (1979) Polydisperse systems I Scatteringfunction for polydisperse fluids of hard or permeable spheres72 2212 (1980)

052902-13


[69] J B Hayter in Physics of Amphiphiles Micelles Vesicles andMicroemulsions edited by V Degiorgio and M Corti (North-Holland Amsterdam 1985) p 59

[70] S Torquato G Zhang and F H Stillinger Ensemble Theory forStealthy Hyperuniform Disordered Ground States Phys Rev X5 021020 (2015)

[71] M Sahimi Flow and Transport in Porous Media and FracturedRock 2nd ed (Wiley-VCH Weinheim 2011)

[72] S Torquato and M Avellaneda Diffusion and reaction inheterogeneous media Pore size distribution relaxation timesand mean survival time J Chem Phys 95 6477 (1991)

[73] G Zhang F H Stillinger and S Torquato Transportgeometrical and topological properties of stealthy disordered

hyperuniform two-phase systems J Chem Phys 145 244109(2016)

[74] J R Banavar and L M Schwartz Magnetic Resonance as aProbe of Permeability in Porous Media Phys Rev Lett 581411 (1987)

[75] W E Kenyon P I Day C Straley and J F WillemsenThree-part study of NMR longitudinal relaxation proper-ties of water-saturated sandstones SPE Form Eval 3 622(1988)

[76] A H Thompson S W Sinton S L Huff A J KatzR A Raschke and G A Gist Deuterium magnetic resonanceand permeability in porous media J Appl Phys 65 3259(1989)

052902-14


FIG 1 The SEM images of (a) cubic aluminosilicate zeolite particles (Advera Rcopy 401) suspended in a Newtonian fluid of polyethyleneglycol (after Ref [33]) and (b) self-synthesized LTA zeolite crystals (after Ref [34])

particles Desmond and Weeks [19] studied the packingfraction as a function of the polydispersity and skewnessof various particle size distributions Baranau and Tallarek[27] investigated random close packings of polydispersehard spheres Furthermore the effective flow and transportproperties of packed beds of polydisperse hard spheres such asthe conductivity and permeability have been reported in a fewstudies [28ndash30] However random packings of polydispersenonspherical particles have rarely been explored

In the present paper we report a detailed simulation ofrandom packings of polydisperse hard cubes as well as ananalysis of a variety of microstructural descriptors for suchporous materials The present study in addition to sheddingfurther light on the properties of packings of cubic particlesalso contributes to the more general problem of understandingthe properties of polydisperse packings of nonspherical parti-cles In addition to the aforementioned problem of fabricationof a new class of porous materials the present study ismotivated by other important applications One example ispackings of zeolites a highly important catalytic materialthat has been used in the chemical industry for a long timeMore recently zeolites have been proposed [31] for separationof fluid mixtures They consist of cubic particles [32] animage of which obtained by a scanning electron microscopeis shown in Fig 1(a) showing polydisperse aluminosilicatezeolites (Advera Rcopy401) suspended in polyethylene glycol [33]Figure 1(b) shows self-synthesized Linde type A (LTA) zeolitecrystals with a variety of sizes [34]

The rest of the paper is organized as follows In Sec II thecomputational approach for generating random packings ofpolydisperse cubic particles and the PDFs of the particlesrsquo sizesare described We then present and discuss in Sec III the resultsfor a variety of the important microstructural descriptors ofthe packings Section IV summarizes the paper and discussespossible further research

II THE MODEL AND ALGORITHM

The most efficient numerical algorithms for generatingpackings of hard spheres are a Monte Carlo (MC) methoddeveloped by He et al [13] and a modification of theLubachevsky-Stillinger algorithm [35] developed by Kansalet al [14] Clusel et al [36] introduced the ldquogranocentricrdquomodel for random packing of jammed emulsions by which

they simulated random packing of polydisperse frictionlessspheres [18] and they investigated the structure of its jamming[20] However such methods cannot be used for generatingrandom packings of mono- or polydisperse hard cubes We alsonote that various molecular dynamics and MC methods thathave been used for generating hard-particle packings are notapplicable to packing of cubes because the overlap potentialfunctions cannot be constructed for particles with nonsmoothshapes including all the Platonic and Archimedean solidsFor such particles Torquato and Jiao [3738] developed anoptimization algorithm that they referred to as the adaptiveshrinking cell (ASC) method which is based on an MCmethod with the Metropolis acceptance rule Except fortetrahedra packings of other Platonic solids (cube octahedradodecahedra and icosahedra) generated by the ASC algorithmare their lattice packings Since a cube is the only Platonicobject that tiles the space its lattice packing generated bythe ASC algorithm is highly ordered with densities close tounity [63738] Delaney and Cleary [39] proposed anotheralgorithm for generating packings of particles the dynamicparticle expansion technique which was originally developedfor the so-called superellipsoids defined by(x

a

)p

+(y

b

)p

+(z

c

)p

= 1 (1)

in which p is a shape parameter and a b and c are thesemimajor axes lengths For large values of p the particlestake on shapes that approach cubes This method also generatesordered packings of hard cubic particles

In parts I and II we presented an algorithm that was basedon a modification of the RSA process which is also used in thepresent paper The simulation begins with a large empty regionof volume V in R3 in which cubic particles are inserted withrandom edge lengths D drawn from a given distribution for theparticlesrsquo sizes The cubes are then placed sequentially in thesimulation cell at randomly selected positions and orientationstaking into account the nonoverlapping constraint Adding theparticles is continued until the desired packing fraction φ =ρ〈v1(D)〉 is reached where ρ = NV is the total numberdensity and 〈v1(D)〉 = 〈D3〉 is the mean volume of the cubicparticles

A The computational algorithm

The details of the computational algorithm are as follows

052902-2







n =radic






(3)



v3 0 minusv1

minusv2 v1 0

⎤⎦ (4)



Vi = Vin + Vc (5)






f (Dm) = 1

(m + 1)

(m + 1

〈D〉)m+1

Dm exp

[minus(m + 1)D

〈D〉]

(6)


〈Dn〉 = (m + n)

m

1

(m + 1)n〈D〉n (7)




f (Dσ ) = 1

Dσradic

2πexp


2σ 2

(8)



〈Dn〉 = exp

(n2σ 2

2

)〈D〉n (9)


III RESULTS



052902-3













052902-4









(10)



S(k) = 1 + 4πρ

k

int infin





052902-5








S(k) = [ρP (k)]minus1I (k) (12)



0f (D)y2(kD)dD (13)


I (k) = ρ

[ int infin

0f (Dα)F 2


0dDα

timesint infin


]

(14)


Fα(kDα) = 4π

k3

Msumi=1




D Loss of structure


052902-6




τ equiv 1

3

int infin


(2π )3ρ2

int infin

0[S(k) minus 1]2dk








S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)


(i)2 (x1x2)


S(i)2 (x1x2) = S

(i)2 (r) (18)



(i)2 (r) rarr φ2












(1)2 (r)






052902-7





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14







n =radic






(3)



v3 0 minusv1

minusv2 v1 0

⎤⎦ (4)



Vi = Vin + Vc (5)






f (Dm) = 1

(m + 1)

(m + 1

〈D〉)m+1

Dm exp

[minus(m + 1)D

〈D〉]

(6)


〈Dn〉 = (m + n)

m

1

(m + 1)n〈D〉n (7)




f (Dσ ) = 1

Dσradic

2πexp


2σ 2

(8)



〈Dn〉 = exp

(n2σ 2

2

)〈D〉n (9)


III RESULTS



052902-3













052902-4









(10)



S(k) = 1 + 4πρ

k

int infin





052902-5








S(k) = [ρP (k)]minus1I (k) (12)



0f (D)y2(kD)dD (13)


I (k) = ρ

[ int infin

0f (Dα)F 2


0dDα

timesint infin


]

(14)


Fα(kDα) = 4π

k3

Msumi=1




D Loss of structure


052902-6




τ equiv 1

3

int infin


(2π )3ρ2

int infin

0[S(k) minus 1]2dk








S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)


(i)2 (x1x2)


S(i)2 (x1x2) = S

(i)2 (r) (18)



(i)2 (r) rarr φ2












(1)2 (r)






052902-7





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14













052902-4









(10)



S(k) = 1 + 4πρ

k

int infin





052902-5








S(k) = [ρP (k)]minus1I (k) (12)



0f (D)y2(kD)dD (13)


I (k) = ρ

[ int infin

0f (Dα)F 2


0dDα

timesint infin


]

(14)


Fα(kDα) = 4π

k3

Msumi=1




D Loss of structure


052902-6




τ equiv 1

3

int infin


(2π )3ρ2

int infin

0[S(k) minus 1]2dk








S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)


(i)2 (x1x2)


S(i)2 (x1x2) = S

(i)2 (r) (18)



(i)2 (r) rarr φ2












(1)2 (r)






052902-7





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14









(10)



S(k) = 1 + 4πρ

k

int infin





052902-5








S(k) = [ρP (k)]minus1I (k) (12)



0f (D)y2(kD)dD (13)


I (k) = ρ

[ int infin

0f (Dα)F 2


0dDα

timesint infin


]

(14)


Fα(kDα) = 4π

k3

Msumi=1




D Loss of structure


052902-6




τ equiv 1

3

int infin


(2π )3ρ2

int infin

0[S(k) minus 1]2dk








S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)


(i)2 (x1x2)


S(i)2 (x1x2) = S

(i)2 (r) (18)



(i)2 (r) rarr φ2












(1)2 (r)






052902-7





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14








S(k) = [ρP (k)]minus1I (k) (12)



0f (D)y2(kD)dD (13)


I (k) = ρ

[ int infin

0f (Dα)F 2


0dDα

timesint infin


]

(14)


Fα(kDα) = 4π

k3

Msumi=1




D Loss of structure


052902-6




τ equiv 1

3

int infin


(2π )3ρ2

int infin

0[S(k) minus 1]2dk








S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)


(i)2 (x1x2)


S(i)2 (x1x2) = S

(i)2 (r) (18)



(i)2 (r) rarr φ2












(1)2 (r)






052902-7





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14




τ equiv 1

3

int infin


(2π )3ρ2

int infin

0[S(k) minus 1]2dk








S(i)2 (x1x2) = 〈I (i)(x1)I (i)(x2)〉 (17)


(i)2 (x1x2)


S(i)2 (x1x2) = S

(i)2 (r) (18)



(i)2 (r) rarr φ2












(1)2 (r)






052902-7





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14





F Specific surface


s = 6η〈D2〉〈D3〉 (19)








052902-8








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14








F (δ) =int infin

δ

P (r)dr (20)


〈δ〉 =int infin

0δP (δ)dδ =

int infin

0F (δ)dδ (21)




Tm 1

(int infin

0F (δ)dδ

)2

(22)



052902-9





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14





[7273]

T1 2

int infin

0δF (δ)dδ (23)








lP =int infin

0rHP (r)dr (24)



052902-10








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14








IV SUMMARY



052902-11





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14





ACKNOWLEDGMENTS





























052902-12












































052902-13











052902-14












































052902-13











052902-14











052902-14

Documents

Statistical characterization of microstructure of packings ...sharif.edu/~rahimitabar/pdfs/105.pdf · This paper is devoted to microstructural characterization of polydisperse packings