review article geometric models for isotropic random...

Review ArticleGeometric Models for Isotropic RandomPorous Media A Review

Helmut Hermann1 and Antje Elsner2

1 Institute for Solid State and Materials Research IFW Dresden PO Box 270116 01171 Dresden Germany2 Institute for Informatics TU Dresden 01062 Dresden Germany

Correspondence should be addressed to Helmut Hermann hhermannifw-dresdende

Received 9 October 2013 Accepted 26 February 2014 Published 30 April 2014

Academic Editor Thomas Hipke

Copyright copy 2014 H Hermann and A Elsner This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Models for random porous media are consideredThe models are isotropic both from the local and the macroscopic point of viewthat is the pores have spherical shape or their surface shows piecewise spherical curvature and there is no macroscopic gradient ofany geometrical feature Both closed-pore and open-pore systems are discussedThePoisson grainmodel themodel of hard spherespacking and the penetrable sphere model are used variable size distribution of the pores is included A parameter is introducedwhich controls the degree of open-porosity Besides systems built up by a single solid phase models for porous media with theinternal surface coated by a second phase are treated Volume fraction surface area and correlation functions are given explicitlywhere applicable otherwise numerical methods for determination are described Effective medium theory is applied to calculatephysical properties for themodels such as isotropic elasticmoduli thermal and electrical conductivity and static dielectric constantThe methods presented are exemplified by applications small-angle scattering of systems showing fractal-like behavior in limitedranges of linear dimension optimization of nanoporous insulating materials and improvement of properties of open-pore systemsby atomic layer deposition of a second phase on the internal surface

1 Introduction

Thedevelopment ofmodels for randomporousmedia and thesetting-up of structure-property relationships have benefittedsubstantially from the rich body of theoretical work ondisordered matter Beginning with Zimanrsquos famous bookon models of disorder [1] treatises like that on the physicsof structurally disordered matter [2] the physics of foam[3] mechanical properties of heterogeneous materials [4]effective medium theory for disordered microstructures [5]structure-property relations of random heterogeneous mate-rials [6] transport and flow in porous media [7] followedParallel to this progress in physics mathematical methodsfor the description of random systems have been improvedas documented in textbooks on stochastic geometry and itsapplications [8ndash10] Activities of bridging ideas developedin physics and materials science on the one hand and ofmethods established by mathematicians on the other handcan be found for example in [10ndash14]

In this review we consider models for random isotropicporous media of both closed-pore and open-pore type Thepore surfaces have at least piecewise spherical curvaturewitharbitrary distribution of radii of curvature (for simplicitycalled pore size distribution below) There is no restrictionregarding the volume fraction of pores The consideredmodels are based on rules for arranging spheres in the three-dimensional space Three types of rules are used randomarrangement of noninteracting spheres (Boolean model thismodel is not restricted to spherical shape of pores) dense-random packing of hard spheres (DRP) and random packingof partially penetrable spheres each consisting of a hard coreand a soft shell (cherry-pit model)The latter one appears as anumerical interpolation scheme between the Boolean modeland the DRPmethod A model for a porous medium is madeby arranging spheres according to one of the above rules andthereafter allotting solid material to the space of the modelwhich is not covered by any sphere Such a model can berefined by coating the pore surfaces with a second solid phase

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014 Article ID 562874 16 pageshttpdxdoiorg1011552014562874

2 Advances in Materials Science and Engineering

The models are characterized by their basic geometricalfeaturesmdashvolume fraction of pores 119881

119881 specific surface area

119878119881 pore size distribution 119891(119903) and correlation function

119862(119903) On one hand these characteristics are required tointerpret experimental data for example small-angle scat-tering curves On the other hand they are important forunderstanding and control of the interplay between structureand physical properties of random porous media

Physical properties of the models are discussed in termsof an effective medium approach Explicit expressions formacroscopic elastic moduli thermal and electrical conduc-tivity and static dielectric constant are given

The definitions of the geometrical characteristics areexplained in Section 2 the models are described in Section 3physical properties are discussed in Section 4 and applica-tions to experimental situations are presented in Section 5

2 Geometrical Characteristics

The pore space is described as a random set 119860 We considervolume fraction 119881

119881(119860) specific surface area 119878

119881(119860) pore

size distribution 119891(119903) and correlation function 119862119860(119903) as

geometrical characteristics of 119860 It is useful to introduce thecomplement of 119860 consisting of the part of the space whichdoes not belong to 119860 The complement of 119860 is indicated by119860119888 This notation is standard in stochastic geometry [10] For

example if 119860 denotes the pore space of the porous medium119888119901= 119881119881(119860) and 119888

119887= 119881119881(119860119888) = 1 minus 119888

119901are the volume fraction

of the pore space and the solid bulk phase respectively119878119881(119860) = 119878

119881(119860119888) is the same for 119860 and 119860119888

The pore space 119860 is described by the shape function

119904119901(r) =

1 r isin 1198600 otherwise

(1)

The volume fraction 119888119901of pores is defined by the probability

of finding a random test point r1in 119860

119888119901= 119881119881 (119860) = 119875 (r

1isin 119860) (2)

If 119904119901(r) is known 119888119901can be calculated by

119888119901=1

119881int119881

119904119901(r) 119889119881 (3)

The correlation function

119862119901 (r) = 119875 (r

1isin 119860 r

2isin 119860) r = r

1minus r2 (4)

denotes the probability of finding two random test points(r1 r2) inside the pore space It is calculated from the shape

function by

119862119901 (r) =

1

119881int119881

119904119901(u + r) 119904119901 (u) 119889119881u (5)

Because of isotropy it depends only on the distance 119903 = |r1minus

r2| between the test points

119862119901 (119903) =

1

4120587intΩ

119862119901 (r) 119889Ω (6)

whereΩ is the solid angle Replacing index 119901 by 119887 and setting119904119887(r) = 1 minus 119904

119901(r) one obtains from ((1) (5) and (6)) for the

correlation function of the solid phase

119862119887 (119903) = 1 minus 2119888

119901+ 119862119901 (119903) (7)

The specific surface area of 119860 is defined by

119904119901= 119878119881 (119860) = lim

120598rarr0

119875 (r1isin 119878120598)

2120598 (8)

where 119875(r1isin 119878120598) is the probability of finding a random test

point r1 in 119878120598and 119878

120598is defined by all points of the three-

dimensional space the distance of which from the surface119878(119860) is less than 120598 If the correlation function of the systemis known 119904

119901is related to 119862

119901(119903) by [10]

119904119901= minus4 lim119903rarr0

119889119862119901 (119903)

119889119903 (9)

For illustration we consider the most simple case of a sin-gle porewith radius119877 in a sample of volume119881 as amodel for adilute system of equal pores that is 119888

119901≪ 1 The volume frac-

tion and the specific surface area are given by 119888119901= 43120587119877

3119881

and 119904119901= 4120587119877

2119881 respectively The integral in formula (5)

is evaluated as the convolution integral of a spherical shapefunction [16] Neglecting terms of the order 1198882

119901one obtains

119862119901 (119903) =

119888119901(1 minus

3119903

4119877+

1199033

161198773) 0 le 119903 le 2119877

0 119903 gt 2119877

(10)

Polydisperse systems are additionally characterized by asize distribution function 119891(119903) with mean value

119898 = int119909119891 (119909) 119889119909 (11)

and variance

1205902= int (119909 minus 119898)

2119891 (119909) 119889119909 (12)

3 Models

31 Poisson Grain Model The Poisson grain model (alsoknown as Boolean model see [10]) is an important andversatile model for generating random two-phase structuresIt is created in two steps In the first step a random (Poisson)point field is generated (here in the three-dimensional space)with number density ] of points This point field has thefollowing property Considering an arbitrary finite region 119861

with volume119881(119861) the number119873(119861) of points situatedwithin119861 is a random variable The probability that 119873(119861) takes thevalue 119899 is

119875 (119873 (119861) = 119899) =[]119881 (119861)]

119899

119899exp (minus]119881 (119861)) 119899 = 0 1

(13)

Advances in Materials Science and Engineering 3

The numbers 119873(1198611)119873(119861

2) are independent random

variables if the regions 1198611 1198612 do not overlap Practically

the point field that is the coordinates of the points can beproduced using a random number generator for numbersequally distributed in finite coordinate intervals

In the second step respectively one sphere is placed oneach point of the point field where the size distribution of thespheres is arbitrary The set-theoretical union of all spheresrepresents a random set 119860 with volume fraction

119881119881 (119860) = 1 minus exp (minus]119881) (14)

where 119881 = (41205873) intinfin

01199093119891(119909)119889119909 is the mean volume of the

spheresConsider definition (2) of volume fraction for the special

case that119860 is generated using equal spheres with radius119877 andchoose a sphere of the same radius for 119861 in (13) Then theprobability that 119861 does not contain any point of the Poissonpoint field that is 119899 = 0 in (13) is equal to the probabilitythat the center of the sphere 119861 has always a distance 119903 gt 119877

to any point of the point field This is exactly definition (2)for the volume fraction of 119860119888 119881

119881(119860119888) = exp(minus](41205873)1199033) =

1minus119881119881(119860)which gives (14) for the considered special caseThe

general result (14) is obtained using the same ideaThe specific surface area of the model is

119878119881= ] [1 minus 119881

119881 (119860)] 119878 (15)

where 119878 = 4120587intinfin

01199092119891(119909)119889119909 is the mean surface area of the

spheresThe correlation function as defined in (5) and (6) is

119862119860 (119903) = 2119888

119860minus 1 + (1 minus 119888

119860)2 exp (]120574 (119903)) (16)

where 119888119860

= 119881119881(119860) and 120574(119903) is the correlation function

of a single sphere averaged over the size distribution 119891(119903)

according to

120574 (119903) =4120587

3int

infin

1199032

1199093(1 minus

3119903

4119909+

1199033

161199093)119891 (119909) 119889119909 (17)

Thefirst formulation of themodel was given by Porod [17]for equal spheres A fundamental and general description canbe found in [10]

Identifying the space covered by spheres in the Booleanmodel with the pore space of a random porous medium oneobtains

119888119901= 119881119881 (119860)

119904119901= 119878119881

(18)

Figure 1 shows a cutout of a Boolean model with constantdistribution of radii in the interval from 1 to 2 Note that someof the small spheres may be not visible because they can becovered completely by large spheres

32 Random Packing of Hard Spheres The model of hardspheres is easily defined by the condition that two particles

Figure 1 Cutout of a Boolean model with sphere radii equallydistributed in the interval (1 2) box size = 5 and volume fraction119881119881= 081

cannot overlap Despite the simplicity of the model it appearsvery complex especially if the packing fraction of spheres ishigh For equal spheres packing fractions of 074 (regularhexagonal and face-centered cubic structure) 068 (regularbody-centered cubic) and 0636 (dense-random packing)are obtained While regular packing of spheres has beenknown as a model for the arrangement of atoms in simplecrystal structures [18] Bernal [19] proposed the model ofdense-randompacked spheres as an approach to the structureof simple liquids Early reviews on construction algorithmsand applications of randomly packed sphere models canbe found in [20ndash22] More recently Lowen [23] reviewedmethods for analyzing hard sphere systems such as thePercus-Yevick theory the density-functional approach andmethods of computer simulations from the physical point ofview Mathematical theories are described in [10]

In this work we use the force-biased algorithm forthe generation of computer models of hard spheres Thisalgorithm includes the simulation of systems with arbitrarysize distribution of spheres and works very efficiently [24ndash26] The algorithm starts with a dilute system of randomlydistributed spheres in a parallelepipedal container with peri-odic boundary conditions applied Initially each sphere 119894 hasan inner hard core diameter 2119903ℎ0

119894and an outer diameter 21199031199040

119894

During iteration step 119899 distances 119903119894119895lt 119903ℎ119899

119894+ 119903ℎ119899

119895between

two spheres are not allowed No interaction occurs for 119903119894119895ge

119903119904119899

119894+119903119904119899

119895 If 119903ℎ119899119894+119903ℎ119899

119895lt 119903119894119895lt 119903119904119899

119894+119903119904119899

119895 sphere 119894 is shifted from

position x119899119894to position x119899+1

119894according to

x119899+1119894

= x119899119894+

119911119894

sum

119895=1

119875119894119895

x119899119894minus x119899119895

10038161003816100381610038161003816x119899119894minus x119899119895

10038161003816100381610038161003816

(19)

where 119875119894119895depends on the diameters of spheres 119894 and 119895 and on

control parameters and 119911119894is the number spheres which have

overlappedwith sphere 119894 Quantity119875119894119895is only used to optimize

the convergence of the algorithm It does not represent anyphysical interaction After each step 119899 the packing fraction is


0204

0608

068

067

066

065

0908

0706

05

flarge

rsmallrlarge

Figure 2 Volume fraction of systems of dense-random packedsphereswith bimodal distribution of radii versus size ratio 119903small119903largeof spheres and number fraction 119891large of large spheres

enhanced by adjusting the sphere diameters at constant boxsize Denoting the maximum number of iteration steps by119899max 119903

119904119899

119894minus 119903ℎ119899

119894rarr 0 for all 119894 if 119899 rarr 119899max

The algorithm is terminated if either the desired packingfraction is attained or if no further progress can be achievedor if 119899 = 119899max For details see [24ndash27]

The volume fraction of spheres and the specific surfacearea of a system of packed hard spheres with number density] = 119873119881 are simply given by

119881119881= ]intinfin

0

4120587

31199033119891 (119903) 119889119903 (20)

119878119881= ]intinfin

0

41205871199033119891 (119903) 119889119903 (21)

respectivelyFigure 2 shows 119881

119881of systems of dense-random packed

hard spheres with bimodal size distribution For a size ratio of045 of small-to-large spheres the volume fraction has a max-imum of 0684 at a number fraction of 02 of large spheresThe maximum packing fraction decreases as the size ratioapproaches 1 and takes the value of 0636 for equal spheres

There are no exact formulas for the correlation functionof dense hard sphere systems Current approximations anddiscussions of this problem can be found for example in[10 11 23] Monte-Carlo simulations based on the definition(4) can be used to calculate the correlation function fornumerically simulated hard sphere systems

33 Penetrable Sphere Models Penetrable sphere modelscan be used to describe open-pore structures Widom andRowlinson [28 29] introduced a penetrable sphere model forthe study of liquid-vapor phase transitions In this modelthe spheres are allowed to overlap completely The maindifference to the Boolean model described above consists inthe definition of a potential energy which is related to thevolume occupied by the overlapping spheres The Widom-Rowlinson model was first extended and applied to studieson phase behavior of complex fluids in [30 31] Similarmodels were described by a number of authors [32ndash40] Notethat the force-biased algorithm described in the previous

Figure 3 Construction scheme of the cherry-pit model Circles(solid lines) mark the pits gray regions are the spheres whichpartially overlap and the solid material is black

section shows features of a penetrable sphere model duringthe rearrangement process

Torquato [41ndash43] introduced the penetrable concentric-shell model which is called cherry-pit model for brevityIn this model each sphere of radius 119877 is composed of animpenetrable core of radius 120582119877 (the pit) surrounded by aperfectly penetrable shell of thickness (1minus120582)119877 For the case ofrandomly distributed sphere centers the limits of 120582 = 0 and120582 = 1 correspond to the Booleanmodel and the random hardsphere one respectively Approximate analytical expressionsfor volume fraction and specific surface area of this modelwere presented in [6 39 40 44] Results of numericalsimulations with equally sized spheres [45] agreed reasonablywith the predictions for the volume fractions by [39 40]Formulas with improved accuracy for volume fraction andspecific surface area of the cherry-pit model with equallysized spheres have been given in [46 47] The cherry-pitmodel is a variable model which appears as an interpolationscheme communicating between the Boolean model and thehard sphere packing paradigm Also it allows to vary thedegree of open-porosity by changing the value of 120582 Whileprevious studies of the cherry-pit model concentrated onusing equally sized spheres actual progress has been achievedby introducing arbitrary size distributions of the spheres [48]

Samples of the cherry-pit model can be generated bymeans of the force-biased procedure for packing of hardspheres After choosing the number 119873 of the spheres thedesired size distribution and the intended packing fractionthe procedure is started The result consists in the set ofthe coordinates (119909

119894 119910119894 119911119894) 119894 = 1 2 3 119873 the radii 120582119903

119894

of the spheres and the achieved packing fraction 120601ℎgiven

by (20) The corresponding cherry-pit model is obtainedby identifying the spheres of the simulated system withthe pits of the cherry-pit model The union of all sphereswith radii 119903

119894represents the pore space of the model The

procedure is illustrated in Figure 3 Figure 4 shows cherry-pit models generated from a system of randomly packed pitswith Gamma distributed radii

119891 (119903) =119887119901

Γ (119901)119903119901minus1 exp (minus119887119903) 119903 ge 0 (22)

and 119903 = 119901119887 1205902= 119901119887

2 Γ(119901) = int

infin

0119905119901minus1 exp(minus119905)119889119905


(a) (b)

(c) (d)

Figure 4 Models for porous media generated using the cherry-pit model Gamma distributed pore size mean pore size 119903 = 1 and variance1205902= 05 packing fraction of the pit system 120601

ℎ= 060 120582 = 075 085 095 100 for ((a) (b) (c) and (d))

The samples shown in Figure 4 differ by the values of 120582chosen for the construction of the cherry-pit models Clearlythe degree of open-porosity rises with decreasing 120582-values

The volume fraction of a simulated model is determinedusing the point-count method where random test points arescattered in the simulation box The number of test pointssituated in pores divided by the number of all test pointsgives an estimate for the volume fraction of poresThe specificsurface area is calculated in a similar way Here the randomtest points are distributed on the surfaces of the spheres ofthe cherry-pit model The number of points on the surface ofa sphere not covered by any other sphere contributes to thesurface of the model The number of contributing points tothe number of all points gives an estimate for the surface areaof the model Figure 5 illustrates the statistical reliability ofthe simulations and its dependence on the size of the modelmeasured by the number of spheres of the simulated cherry-pit model The standard deviation of both 119881

119881and 119878

119881 each

divided by the mean value of the corresponding quantitydecreases with increasing box size The statistical accuracy ofthe volume fraction is of the order of 025 for119873 ge 10

4 The

specific surface area is determinedwith an statistical accuracyof less than 2 for119873 = 10

4 and 05 for119873 = 105

Statistical analysis of large numbers of simulatedmonodisperse [47] and polydisperse [48] cherry-pit modelsrevealed that the volume fraction 119881

119881 of spheres and the

specific surface area 119878119881 are given by the approximate

formulas

119881119881= 120601ℎ+ (1 minus 120601

ℎ)Φ(

3radic2120587

2

(1 minus 120582) 120601ℎ

120582 (1 minus 120601ℎ)) (23)

Φ (119911) =2

radic120587int

119911

0

exp (minus1199052) 119889119905 (24)

119878119881=

3

120582119903ref120601ℎexp (minus9120587

4

(1 minus 120582)21206012

ℎ

1205822(1 minus 120601ℎ)2) (25)

Again relations (18) apply if the spheres describe the porespace of a medium Expressions (23) and (25) depend on thequantities 120601

ℎand 120582 which characterize the basic system of

randomly packed hard spheres and the construction scheme


2 3 4 5001

01

1

10

100

120590S NS

V(

)

120590VNVV

120590SNSV

Log10 N

Figure 5 Normalized standard deviation 120590119881

119873119881119881and 120590

119878

119873119878119881of

respectively volume fraction 119881119881 and specific surface area 119878

119881 of

cherry-pit models with Gamma distributed radii and 120590119903 = 0707

versus number 119873 of spheres used for the construction of a modelThe standard deviation is calculated from 10 samples for eachmodel119878119881 rhombuses and solid line 120582 = 075 triangles and dashed line

120582 = 095 119881119881 circles and dotted line 120582 = 075 squares and dashed

line 120582 = 095

of the cherry-pit model respectively Parameter 119903ref definesthe unit of length of the system For monodisperse systemsit is equal to the sphere radius In the polydisperse case therelation

119903ref119903

= 119886 + 119887(120590

119903)

2

(26)

with 119886 = 1014 plusmn 0002 119887 = 0454 plusmn 0006 follows from fitting(25) to the values of the specific surface areas of simulatedcherry-pit models [48] Parameters 119903 and 120590 are the meanvalue and the standard deviation of the radii distributionrespectively defined by ((11) (12)) The error appearing for120590 rarr 0 in (26) is below 2

Expression (23) does not depend on any parameterrelated to the size distribution of spheres So it appears asquite a general relation Detailed analyses in [48] showed that(23) is most reliable for systems with 040 le 120601

ℎle 071 119888

119901le

08 and for size distributions with 120590119903 up to about 35 (119903 is themean radius of spheres) Figure 6 compares (23) to simulateddata obtained from systems with 10000 spheres generated atperiodic boundary conditions where the radii are distributedaccording to a Gamma distribution [48]

Formula (25) is accurate in the limits of few percent for120590 le 025 The impact of the skewness of the size distributionson the accuracy of (25) is negligible Figure 7 illustrates thevalidity of (25)

34 Interphase Models

341 Limit of Very Thin Interphase Layer We start with oneof the models for a porous medium with a single bulk phaseThe pore space is characterized by its volume fraction 119881

119881

and the specific surface area 119878119881 Then the internal surface

070 075 080 085 090 095 100

06

07

08

09

10

074 075 076 077095

096

097

098

099

Impenetrability parameter 120582

Volu

me f

ract

ionVV

Figure 6 Volume fraction of cherry-pit models for Gamma dis-tributed pore size with 120590119903 = 025 (rhombs) 071 (squares) and 100(circles)

070 075 080 085 090 095 10000

05

10

15

20

25

30

35


Spec

ific s

urfa

ce ar

eaS V

Figure 7 Specific surface area of cherry-pit models for Gammadistributed pore size with 120590119903 = 025 (rhombs) 071 (squares) and100 (circles)

of the model is coated by a layer of a second solid phaseof thickness 120575119905 This layer is called interphase Denoting thesmallest radius of curvature of a pore of themodel by 119903min for120575119905 ≪ 119903min the volume fraction of the layer is approximately

119888119894= 120575119905119878119881 (27)

the volume fraction of the pores reduces to

119888119901= 119881119881minus 120575119905119878119881 (28)


Figure 8 Planar intersection of a Boolean interphase model pores-light gray bulk phase-black and interphase layer-red

and the volume fraction of the bulk phase remains

119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)

342 Boolean Interphase Model Another way to calculatethe volume fraction of the interphase is to compare twomodels differing only by the size of the spheres used for theconstruction Consider a Boolean model 1 generated usingspheres with size distributions 119891

1(119903) The space not covered

by any sphere defines the bulk phase 1 Reducing the radiusof each of the spheres of exactly this model by 120575119905 a secondmodel 2 is obtained with size distribution 119891

2(119903) = 119891

1(119903 minus 120575119905)

where 1198911(119903) = 0 for 119903 le 120575119905 is required This model defines

the pore space Then the interphase of thickness 120575119905 is givenby the space neither belonging to the bulk phase 1 nor to thepore space and the volume fraction of the interphase followsas

119888119894= 119881119881 (1) minus 119881119881 (2) (30)

Figure 8 shows the intersection of a Boolean interphasemodel For the special case of Boolean models with spheresof equal radius 119877 one obtains

119888119894= 1 minus exp(minus]4120587

31198773) minus 1 minus exp(minus]4120587

3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)

follows which corresponds to ((15) (27))

343 Hard Sphere Interphase Models For interphase modelsbased one hard sphere systems the volume fraction ofthe interphase is calculated similarly Assuming a Gaussiandistribution of radii

119891 (119903) =1

radic2120587120590exp(minus(119903 minus 119903)

2

21205902) (33)

the volume fraction of the interphase is according to (20)

119888119894=4120587120588

3119903 [1199032+ 31205902] minus (119903 minus 120575119905) [(119903 minus 120575119905)

2+ 31205902] (34)

In the limit 120575119905 ≪ 119903 one obtains with ((21) (33)) analogouslyto (27)

119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)

344 Cherry-Pit Interphase Model The cherry-pit modelwith one bulk phase can be transformed to an interphasemodel in a similar way as the Boolean model and the hardspheres one However expression (14) for the volume fractionof pores for the cherry-pit model does not depend explicitlyon the pore size Remembering that the hard core radius 119903

ℎ

and the corresponding pore radius 119903119901 are related by

119903ℎ= 120582119903119901 (36)

and considering two monodisperse cherry-pit models 1 and2 characterized by the same values of 120601

ℎand 119903ℎbut different

pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=

119903ℎ1199031199012 the volume fraction of the interphase of the corre-

sponding cherry-pit interphase models is

119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)

Setting 1205821= 120582 120575119905 = 119903120575120582120582 replacing

119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)

and carrying out 120575120582 rarr 0 one obtains from ((23) (25))

119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)

which corresponds with (27)

35 Reconstruction It is often important to generate modelsnot from mathematical algorithms or rules but from experi-mentally obtained structural information Such informationis basically incomplete for random systems The generationof models from experimental data or reconstruction musttherefore focus on reproducing important structure parame-ters like volume fraction pore size distribution or correlationfunctions The problem of reconstruction is related to thewell-known field of stereology (see chapter 10 in [10]) Atypical problem of stereology is for example to estimate thesize distribution of particles from the length distribution ofchords inside particles where the chords are determined fromlinear sections of the considered sample Few problems areexactly solved for example the determination of the sizedistribution of pores from planar sections of the sample if


the pores have spherical shape The size distribution of thepores determines however not yet the spatial distribution ofthe spheres In this case small-angle scattering data would behelpful to complete information about spatial correlations ofthe 3-dimensional arrangement of the pores

Procedures for the reproduction of random structureshave been successfully developed in recent years Robertsgenerated 3-dimensional models from 2-dimensional images[49] Yeong and Torquato [50 51] proposed a procedureapplicable to very different types of random media Hilferand Manwart [52] used physical properties like conductivityand permeability in order to improve reconstruction Recon-struction from microtomographic images was carried out byOslashren and Bakke [53] Ohser and Schladitz [14] presentedgeneral methods of computational image analysis Methodsfor the reconstruction of randommedia of the type describedby the Poisson grain model have been developed by Arnset al [54 55] Problems occurring with reconstruction on thenanometer scale have been addressed by Prill et al [56]

4 Structure-Property Relationships

Properties of random heterogeneous materials includingporous media have been considered in many theoreticalstudies The effective medium theory see for example [4 5]and its combination with the maximum entropy approach[57] have been proven as a powerful tool to understandmacroscopic properties and various structure models havebeen considered for the study of structure-property relationsof heterogeneousmaterials [6 7 58ndash60] Among the differentstructure models the Poisson grain model appears as oneof the most successful approaches for establishing structure-property relationships of random media [58 61ndash66] Recentwork on the optimization of property combinations of ran-dom media by means of structure variation can be found in[67 68]

We consider the composite sphere assemblagemodel (seeeg [4]) and related effective medium theories (see eg [5])as most adequate for the models described in the previoussections Explicit expressions for dielectric constants thermalconductivity and isotropic elastic moduli of models consist-ing of amatrix and spherical inclusions have been derived anddiscussed in [4 5 67ndash69] Below the idea of this approach isexplained for the example of the thermal conductivity of aporous medium with coated pore surface

A corresponding composite sphere element (CSE) ismade up by a central sphere with radius 119903

119901filled with phase

119901 a concentric spherical shell of phase 119894with thickness 119903119894minus119903119901

and a second shell of phase 119887 with thickness 119903119887minus 119903119894(see

Figure 9) In our case phase 119901 represents the pore content(vacuum air or something else) phase 119894 is the pore coatinglayer and phase 119887 is the bulk material of the porous mediumThe CSE is embedded in the effective medium The volumefraction of the pores the coating and the bulk material aredenoted by 119888

119901 119888119894 and 119888

119887 respectively Writing

119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)

Effective porous medium

rb

ri

uΘ

urrp

Figure 9 Composite sphere element

the volume fractions of the phases are

119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)

The temperature distribution in the porous medium followsthe Laplace equation

Δ119879 = 0 (43)

And the corresponding intensity field is

f = minusnabla119879 (44)

Heat flux j and thermal conductivity 120590 are defined by

j = 120590f (45)

According to [70] the solution of the Laplace (43) for the CSEunit embedded in the effective medium is

f =

119862119901sinΘu

Θminus 119862119901cosΘu

119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)

The boundary conditions of the heat flux through the inter-faces pore to coating coating to bulk and bulk to effectivemedium require continuity of the radial component ju

119903of

the heat flux j and of the tangential component fuΘof the

temperature gradient minusf Denoting the values of the thermalconductivity of the pore phase the coating interlayer the bulk


Table 1 Mathematically equivalent properties (according to [5])

Problem Potential Field Flux PropertyHeat conduction Temperature 119879 Temperature gradient minusnabla119879 Heat flux f Heat conductivity 120590Electricalconductivity Electrical potential Electric field Current density Electrical

conductivityDiffusion Density Density gradient Current density Diffusion constantElectrostatics Electric potential Electric field Electric displacement Dielectric constantMagnetostatics Magnetic potential Magnetic field Magnetic induction Permeability

material and the effective medium by 120590119901 120590119894 120590119887 and 120590lowast one

obtains a system of linear equations

(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)

This systemhas a solution if the determinant of the coefficientmatrix vanishes Solving the resulting equation for 120590lowast oneobtains for the effective thermal conductivity

120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)

Quantity 120590 can be interpreted as the effective conductivity ofa subsystem consisting of spherical pores with conductivity120590119901embedded in a matrix with property 120590

119894and the corre-

sponding volume fractionsIt is known that a number of physical problems are

mathematically equivalent to the formulation of thermalconductivity [5]Therefore (43) to (49) apply also to electricalconductivity static dielectric property magnetostatics anddiffusion (see Table 1)

The elastic moduli of the considered model can becalculated using either the CSA model [4] or the extendedreplacement method [71] The results can be expressed ina similar form as expressions ((48) (49)) Denoting the

effective property of the system by 119886lowast one can write [48 6869]

119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)

The symbol 120590 stands for dielectric constant thermal con-ductivity and related properties119870119866 denote elastic bulk andshear modulus for isotropic systems and index 119898 = 119901 119894 119887

stands for pore interphase layer and bulk phaseExpressions ((50) to (53)) are rigorous results obtained

for geometrical situations as given in the structure modelsdescribed above The results for the properties are uniqueand no uncertainty due to unknown topology of phasedistribution occurs Nevertheless there are relations to theHashin-Sthrikman bounds for properties of heterogeneousmaterials [72] which we discuss here for the case of thedielectric constant119896 For simplicity we use amodel consistingof a matrix and spherical inclusions Choosing 119888

119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901

and changing the topology of the system by interchangingproperties and volume fractions ofmatrix and inclusions thatis 119896119887harr 119896119901 119888119887harr 119888119901 one obtains from ((50) (51)) theHashin-

Sthrikman bounds for the effective dielectric constant 119896lowast ofa two-component composite (see also [5] (23))

119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)

The same procedure can be carried out for the Hashin-Sthrikman bounds for the isotropic elastic moduli (see [4]


Sections 32 and 41) For a given distribution of phases onpores interphase layer and bulk there is no uncertainty ofthe topology and consequently expressions ((50) to (53)) forthe properties are unique

5 Applications

51 Fractal-Like Pore Structure Porous media may havefractal features (see eg [73ndash75]) We consider here a modelfor surface fractals which is based on the Boolean modeldescribed in Section 31 This model is tunable with respectto the fractal dimension of the internal surface and explicitexpression for the geometrical characteristics defined inSection 2 is known [68 76] The size distribution of spheresused for the construction of the model is defined by self-similarity rules for the radii 119903

119906and the partial number density

]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]

119906are radius and number density of spheres

belonging to the interval (119906minus119889119906 119906) Parameter 120581 = 11990301199031= 119906119904

0

describes the ratio of minimum and maximum sphere sizeThe normalized size distribution 119891(119903) is then given by

119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)

Number density of spheres volume fraction specificsurface area and correlation function are given by ((14) (15)and (16)) and

] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578

1 minus 1205813minus120578ln 1

1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578

Figure 10 Volume fraction (insert) and specific surface area of self-similar Booleanmodels versus self-similar parameter 120578 119903

1= 1 ]

1=

001 and 119904 = 1 120581 = 01 (dotted) 001 (dashed) and 0001 (solidline)

The self-similar model described by ((14) (15) and (16)) and((58) (59) (and 60)) exhibits the properties of a surfacefractal if the specific surface area tends to infinity while thevolume fraction remains finite for 120581 rarr 0 This is the case for

2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)

is the dimension of the fractal internal surface of the modelaccording to the Minkowski-Bouligand definition [77 78]Quantity 119878

120598is the same as in (8) Fractality of a porous

medium can be detected by means of small-angle scatteringmethods [73] In such cases the limit 119903119903

1≪ 1 is of interest

which determines the behavior of the scattering intensity atlarge scattering angles One obtains from ((7) (60))

119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)

which is in accordance with [73 79]The behavior of the self-similar system is illustrated by

Figures 10 and 11The volume fraction of pores increases withincreasing values of the self-similarity parameter 120578 and alsowith decreasing ratio of radii 120581 of smallest to largest pores(Figure 10) Of course 119881

119881keeps limited in the fractal limit

120581 997888rarr 0 where 120578 takes the meaning of the fractal dimensionof the internal surface (see expressions ((14) (58)) In theextreme case of 120581 = 0 and 119889

119891997888rarr 3 the porosity 119881

119881

tends to 1 The specific surface area given by ((15) (59)) iscontrolled by the term 120581

2minus120578 Clearly 119878119881tends to infinity for

120581 997888rarr 0 independent of the value 2 lt 119889119891lt 3 of the fractal

dimension of the internal surface The behavior of 119878119881in the

fractal limit appears also applying (9) to (63) Obviously 119878119881

tends to infinity for each fractal dimension 2 lt 119889119891lt 3 of the


00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]

Figure 11 Small-angle scattering intensity of self-similar Booleanmodels 119902

0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21

(dotted curve shifted on the ordinate by 2) 120578 = 21 (dashed shiftedby 1) 120581 = 005 and 120578 = 25 (solid line)

internal surface Figure 10 illustrates the dependence of 119878119881on

the self-similarity parameter 120578 for different values of 120581The scattering intensity of the self-similar system can be

calculated [11 68 78] using

119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)

where 119902 is the absolute value of the scattering vector definedin the usual way [74] If the size of pores is on the nanometerscale 119868(119902) is called small-angle scattering intensity Inserting((16) (60)) in (64) an explicit expression for the small-anglescattering intensity can be obtained The limit of small 119903-values given in (63) becomes apparent in 119868(119902) in the form

119868 (119902) prop 1199026minus119889119891 (65)

in the fractal limit for not to small 119902-valuesFigure 11 shows scattering intensities of a self-similar

system with fractal-like behavior for small 120581 The scatteringintensity is dominated in the region log(119902119902

0) lt 05 by the

largest pores of the system (1199031= 1 in suitable units eg

in nanometers) The curve for the system with small range120581 = 005 of self-similarity follows approximately expression(65) in the interval 05 lt log(119902119902

0) lt 1 and for log(119902) gt 1 the

scattering intensity 119868(119902) is determined by the smallest poresof size 119903

0= 1205811199031 Systems showing such scattering curves are

sometimes called fractal-like Seriously speaking the systemis self-similar over about one order of magnitude of lengthscale The curves for 120581 = 0001 represent systems which areself-similar over 3 orders of magnitude This is reflected bythe linear behavior in the log(119868(119902)) versus log(119902119902

0) plot for

05 lt log (1199021199020) lt 23 where the limit (65) applies Here the

term fractal-like would be justified

52 Optimization of Porous Dielectric Materials Theimprovement of silicon based microelectronic deviceslike microprocessors has been a continuing process overdecades [80] This includes the further development of manycomponents and also the replacement of components by newmaterials for example the replacement of aluminum wiresfor connecting billions of transistors in a microprocessorcircuit by copper wires This replacement was necessarybecause the performance of microprocessors is increasinglyaffected by the on-chip interconnect design and technologySignal delay caused by the resistor-capacitor (RC) interactionin the system of copper wires and insulating layers becomesmore and more important compared to the transistorsrsquosignal delay [81 82] Additionally cross-talking and powerdissipation worsen due to increasing RC coupling causedby continuous reduction of device dimensions [83 84]The introduction of the copper technology was one stepto reduce these unfavorable implications of continuousminiaturization Another one concerns the reduction ofthe dielectric constant of the insulating layers separatingthe copper wires Manufacturable materials with effectivedielectric constants of 290 to 260 are available at the presenttime however feasible solutions with 119896 lt 260 are difficult toobtain [85]

Research is in progress to define dielectric materialswith extremely low dielectric constant Besides dense low-119896 materials like organosilicate glass and the challengingairgap technology insulating materials with porosity on thenanometer scale are considered as a promising way to reducethe above problems [85] Among possible nanoporous insu-lating materials [86ndash88] nanoporous silica has the advantagethat it can be implemented comparably easily in the currentsilicon technology

We consider the following problem optimize the struc-ture of nanoporous silica in such a way that the dielectricconstant 119896 is as low as possible for example less than 2(vacuum 119896 = 1 SiO

2 119896 = 4) and the Youngrsquos modulus

119864 as high as possible for example higher than 5GPa (SiO2

119864 = 74GPa) which would match future technologicalrequirements Using an interphase model specified by thevolume fractions of pores 119888

119901 and interphase 119888

119894 and choosing

a favorable second material additionally to silica effectiveproperties can be calculated for the models with ((50) (51)and (52)) and (53) with the corresponding values for 120572119896

119886 the

Youngrsquos modulus is given by

119864 =9119870119866

3119870 + 119866(66)

for isotropic systems The properties of the components usedfor the simulations are quoted in Table 2 Figure 12 showscontour plots of effective property combinations in the planespanned by 119888

119894and 119888119901for amodel consisting of porous Parylene

coated with SiO2 Due to the porosity and the low 119896-value of

the bulk material very low values of the dielectric constantcan be achieved However the Youngrsquos modulus is too smallfor pure porous Parylene Coating with SiO

2improves the

effective modulus with the result that for 119888119894asymp 02 and pore

fraction of at least 065 excellent values for 119896lowast lt 15 can becombined with reasonable values 119864lowast ge 5GPa The required


Table 2 Properties of the structural components for models ofcoated porous ultralow 119896 dielectrics

Component 119870 (GPa) 119866 (Gpa) 119896

Parylene 47 10 2SiO2 367 312 4Vacuum 0 0 1

040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

Figure 12 Effective property map for porous Parylene coated withSiO2versus volume fractions 119888

119901and 119888

119888of pores and coating layer

respectively Properties of the components are given in Table 2Regions with 119896lowast lt 15 119864lowast gt 5GPa (yellow) 119896lowast lt 16 119864lowast gt 4GPa(light blue) 119896lowast lt 17 119864lowast gt 3GPa (blue) and 119896lowast ge 17 119864lowast le 3GPa(dark blue)

porosity of at least 065 is however very high and cannotbe achieved with random hard sphere packing models withequal spheres Instead the pores must have a polydispersesize distribution to obtain 119888

119901gt 065 The parameters of

the required size distribution can be determined using themethods described in Section 3 see also Figure 2

Starting with porous silica and coating it with Paryleneanother effective property map appears (Figure 13) Systemswith an effective dielectric constant of 119896lowast lt 18 can beachieved at moderate porosity of about 060 for valuesof the effective Youngrsquos modulus of not less than 12GPawhich appears as an excellent combination of dielectric andmechanical properties for the purpose under consideration

53 Simulation of Coating Processes The previous sectionshowed for a specific case that introducing an interphase layerinto a porous system may come out as a useful method toimprove effective properties of a porous system Coating ofporous media has been proven to be a successful method tomodify their properties in many fields of application Thisapplies not only to the properties discussed above but also

045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040

Figure 13 Effective property map for porous SiO2coated with

Parylene versus volume fractions 119888119901and 119888

119894of pores and coating

layer respectively Properties of the components are given in Table 2Regions with 119896lowast lt 18 119864lowast gt 12GPa (yellow) 119896lowast lt 20 119864lowast gt 10GPa(light blue) 119896lowast lt 22 119864lowast le 5GPa (blue) and 119896lowast ge 22 119864lowast le 5PGa(dark blue)

to flow and transport in porous media [7] percolation [89]enzyme immobilization [90 91] adsorption processes [92ndash94] and others Different methods of coating and surfacerefinement of open-pore materials are known for examplespray coating [95] and microarc oxidation coating [96] ofmetallic foams [97] and atomic layer deposition [15 98ndash101]The latter method is of special interest for media which arecharacterized by open-porosity on the nanometer scale

We study the growth of a layer on the internal surface ofopen-pore systems within the framework of the cherry-pitmodel Expression (37) for the volume fraction of a coatinglayer is written in the form

119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)

where 120591 denotes the coating time For simplicity we considera system with narrow pore size distribution In a continuouscoating process the radius 119903

120591of a given pore reduces accord-

ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)

where120583 is the growth rate of layer thicknessmeasured in unitsof the pit radius of the cherry-pit model For discontinuousprocesses like atomic layer deposition 120583120591 has to be replacedby Δ119905119899 where 119899 is the number of the deposition cycle and


0 20 40 60 8000

01

02

03

04

05

06

Number of ALD cycles

Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)

Figure 14 Volume fraction of pores (circles) and ALD layer(squares) of Al

2O3coated polypropylene Experimental data points

taken from [15] Curves calculated using formula (36)

Table 3 Properties of the structural components used for the atomiclayer deposition process

Component 119870 (GPa) 119866 (Gpa) 120590 (WmK)Polypropylene 6 06 012Al2O3 165 124 18

Δ119905 is the increment of layer thickness achieved by one cyclemeasured in units of the pit radius We apply the presentapproach to experimental results for atomic layer depositionof Al2O3on porous polypropylene [15] Figure 14 compares

the evolution of the porosity and the volume fraction of theAl2O3layer with the number of deposition cycles The cycle

numbers are corrected for the induction period [15] Themodel curves reproduce the trend of the experimental datawith an accuracy of about 20

The estimated volume fractions can now be used topredict effective properties of the coated material Usingknown data (see Table 3) for bulk 119870 shear 119866 modulusand the thermal conductivity 120590 of polypropylene and forAl2O3the effective elastic and thermophysical properties

of the coated material are calculated Figure 15 shows theresults for the effective thermal conductivity and the Youngrsquosmodulus Both for the Youngrsquos modulus and the thermalconductivity the essential increase is achieved within thefirst 20 deposition cycles These results supplement theexperimental data obtained in [15] by predicting the changeof physical properties during the deposition process

6 Conclusion

There are appropriate theoretical tools for a consistent de-scription of the structure and effective properties of isotropicrandom porous media These tools provide impr-oved facili-ties to develop suitable models for porous materials from ex-perimentally obtained structural parameters and to optimize

0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)

Figure 15 Effective thermal conductivity and Youngrsquos modulus ofAl2O3coated polypropylene versus ALD cycle number

effective properties by simulating the response of effectiveproperties on structural modification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J M Ziman Models of Disorder Cambridge University PressCambridge UK 1979

[2] N E Cusack The Physics of Structurally Disordered MatterAdam Hilger Bristol UK 1987

[3] D Weaire The Physics of Foam Clarendon Press Oxford UK1999

[4] W Kreher and W Pompe Internal Stresses in HeterogeneousSolids Akademie-Verlag Berlin Germany 1999

[5] T C Choy Effective Medium TheorymdashPrinciples and Applica-tions Clarendon Press Oxford UK 1999

[6] S Torquato Random Heterogeneous Materials MicrostructureandMacroscopic Properties SpringerNewYorkNYUSA 2002

[7] M Sahimi Flow and Transport in Porous Media and FracturedRock From Classical Methods to Modern Approaches Wiley-VCH Weinheim Germany 1995

[8] D Stoyan W S Kendall and J Mecke Stochastic Geometry andIts Applications Akademie-Verlag Berlin Germany 1987

[9] D Stoyan W S Kendall and J Mecke Stochastic Geometry andIts Applications Wiley Chichester UK 2nd edition 1995

[10] S N Chiu D Stoyan W S Kendall and J Mecke StochasticGeometry and Its Applications Wiley Weinheim Germany 3rdedition 2013

[11] H Hermann Stochastic Models of Heterogeneous MaterialsTrans Tech Zurich Switzerland 1991

[12] K R Mecke and D Stoyan Eds Statistical Physics and SpatialStatisticsmdashThe Art of Analyzing and Modeling Spatial Structuresand Pattern Formation Springer Berlin Germany 2000


[13] J Ohser and F Mucklich 3D Images of Materials StructuresProcessing and Analysis Wiley-VCH Weinheim Germany2000

[14] J Ohser andK Schladitz 3D Images ofMaterials and StructuresProcessing and Analysis Wiley-VCH Weinheim Germany2009

[15] S Y Jung A S Cavanagh L Gevilas et al ldquoImprovedfunctionality of lithium-ion batteries enabled by atomic layerdeposition on the porous microstructure of polymer separatorsand coating electrodesrdquo Advanced Energy Materials vol 2 no8 pp 1022ndash1027 2012

[16] R Hosemann and S N BagchiDirect Analysis of Diffraction byMatter North-Holland Amsterdam The Netherlands 1962

[17] G Porod ldquoDie Rontgenkleinwinkelstreuung von dichtgepack-ten kolloiden Systemenrdquo Kolloid-Zeitschrift vol 125 no 1 pp51ndash57 1952

[18] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1996

[19] J D Bernal and J Mason ldquoPacking of spheres co-ordination ofrandomly packed spheresrdquo Nature vol 188 no 4754 pp 910ndash911 1960

[20] G S Cargill III ldquoStructure of metallic alloy glassesrdquo Solid StatePhysics vol 30 pp 227ndash320 1975

[21] J L Finney ldquoModelling the structures of amorphousmetals andalloysrdquo Nature vol 266 no 5600 pp 309ndash314 1977

[22] M R Hoare ldquoPacking models and structural specificityrdquoJournal of Non-Crystalline Solids vol 31 no 1-2 pp 157ndash1791978

[23] H L Lowen ldquoFun with hard spheresrdquo in Statistical Physics andSpatial StatisticsmdashThe Art of Analyzing and Modeling SpatialStructures and Pattern Formation K R Mecke and D StoyanEds pp 295ndash331 Springer Berlin Germany 2000

[24] W S Jodrey and E M Tory ldquoComputer simulation of closerandom packing of equal spheresrdquo Physical Review A vol 32no 4 pp 2347ndash2351 1985

[25] J Moscinski and M Bargiel ldquoC-language program for theirregular close packing of hard spheresrdquo Computer PhysicsCommunications vol 64 no 1 pp 183ndash192 1991

[26] A Bezrukov M Bargiel and D Stoyan ldquoStatistical analysisof simulated random packings of spheresrdquo Particle amp ParticleSystems Characterization vol 19 no 2 pp 111ndash118 2002

[27] A Elsner Computer-aided simulation and analysis of randomdense packings of spheres [thesis] 2009 (German)

[28] B Widom and J S Rowlinson ldquoNew model for the study ofliquid-vapor phase transitionsrdquoThe Journal of Chemical Physicsvol 52 no 4 pp 1670ndash1684 1970

[29] B Widom ldquoGeometrical aspects of the penetrable-spheremodelrdquoThe Journal of Chemical Physics vol 54 no 9 pp 3950ndash3957 1971

[30] C N Likos K R Mecke and H Wagner ldquoStatistical morphol-ogy of random interfaces in microemulsionsrdquo The Journal ofChemical Physics vol 102 no 23 pp 9350ndash9361 1995

[31] K R Mecke ldquoA morphological model for complex fluidsrdquoJournal of Physics Condensed Matter vol 8 no 47 pp 9663ndash9667 1996

[32] F H Stillinger ldquoPhase transitions in the Gaussian core systemrdquoThe Journal of Chemical Physics vol 65 no 10 pp 3968ndash39741976

[33] C N Likos M Watzlawek and H Lowen ldquoFreezing andclustering transitions for penetrable spheresrdquo Physical ReviewE vol 58 no 3 pp 3135ndash3144 1998

[34] C N Likos A Lang M Watzlawek and H Lowen ldquoCriterionfor determining clustering versus reentrant melting behaviourfor bounded interaction potentialsrdquo Physical Review E vol 63no 3 Article ID 031206 2001

[35] C N Likos B M Mladek D Gottwald and G Kahl ldquoWhy doultrasoft repulsive particles cluster and crystallize Analyticalresults from density-functional theoryrdquoThe Journal of ChemicalPhysics vol 126 no 22 Article ID 224502 2007

[36] M-J Fernaud E Lomba and L L Lee ldquoA self-consistentintegral equation study of the structure and thermodynamicsof the penetrable sphere fluidrdquoThe Journal of Chemical Physicsvol 112 no 2 pp 810ndash816 2000

[37] A Santos and A Majewsky ldquoFluids of spherical moleculeswith dipolarlike nonuniform adhesion an analytically solvableanisotropic modelrdquo Physical Review E vol 78 no 2 Article ID021201 2008

[38] L Blum and G Stell ldquoPolydisperse systemsmdashI Scatteringfunction for polydisperse fluids of hard or permeable spheresrdquoThe Journal of Chemical Physics vol 71 no 1 pp 42ndash46 1979

[39] P A Rikvold and G Stell ldquoPorosity and specific surface forinterpenetrable-sphere models of two-phase random mediardquoThe Journal of Chemical Physics vol 82 no 2 pp 1014ndash10201985

[40] P A Rikvold and G Stell ldquoD-dimensional interpenetrable-sphere models of random two-phase media microstructureand an application to chromatographyrdquo Journal of Colloid andInterface Science vol 108 no 1 pp 158ndash173 1985

[41] S Torquato ldquoBulk properties of two-phase disordered mediamdashI Cluster expansion for the effective dielectric constant ofdispersions of penetrable spheresrdquo The Journal of ChemicalPhysics vol 81 no 11 pp 5079ndash5088 1984

[42] S Torquato ldquoBulk properties of two-phase disordered mediamdashII Effective conductivity of a dilute dispersion of penetrablespheresrdquo The Journal of Chemical Physics vol 83 no 9 pp4776ndash4785 1985

[43] S Torquato ldquoBulk properties of two-phase disordered mediamdashIII New bounds on the effective conductivity of dispersions ofpenetrable spheresrdquoThe Journal of Chemical Physics vol 84 no11 pp 6345ndash6359 1986

[44] K Gotoh M Nakagawa M Furuuchi and A Yoshigi ldquoPoresize distributions in random assemblies of equal spheresrdquo TheJournal of Chemical Physics vol 85 no 5 pp 3078ndash3080 1986

[45] S B Lee and S Torquato ldquoPorosity for the penetrable-concentric-shell model of two-phase disordered media com-puter simulation resultsrdquo The Journal of Chemical Physics vol89 no 5 pp 3258ndash3263 1988

[46] D Stoyan A Wagner H Hermann and A Elsner ldquoStatisticalcharacterization of the pore space of random systems of hardspheresrdquo Journal of Non-Crystalline Solids vol 357 no 6 pp1508ndash1515 2011

[47] A Elsner A Wagner T Aste H Hermann and D StoyanldquoSpecific surface area and volume fraction of the cherry-pitmodel with packed pitsrdquo The Journal of Physical Chemistry Bvol 113 no 22 pp 7780ndash7784 2009

[48] H Hermann A Elsner and D Stoyan ldquoSurface area andvolume fraction of random open-pore systemsrdquoModelling andSimulation in Materials Science and Engineering vol 21 no 8Article ID 085005 2013

[49] A P Roberts ldquoStatistical reconstruction of three-dimensionalporous media from two-dimensional imagesrdquo Physical ReviewE vol 56 no 3 pp 3203ndash3212 1997


[50] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E vol 57 no 1 pp 495ndash506 1998

[51] C L Y Yeong and S Torquato ldquoReconstructing randommediamdashII Three-dimensional media from two-dimensionalcutsrdquo Physical Review E vol 58 no 1 pp 224ndash233 1998

[52] R Hilfer and C Manwart ldquoPermeability and conductivity forreconstructionmodels of porousmediardquo Physical Review E vol64 no 2 Article ID 021304 4 pages 2001

[53] P-E Oslashren and S Bakke ldquoProcess based reconstruction ofsandstones and prediction of transport propertiesrdquo Transport inPorous Media vol 46 no 2-3 pp 311ndash343 2002

[54] CHArnsMAKnackstedt andK RMecke ldquoReconstructingcomplex materials via effective grain shapesrdquo Physical ReviewLetters vol 91 no 21 Article ID 215506 4 pages 2003

[55] CHArnsMA Knackstedt andK RMecke ldquoBoolean recon-structions of complex materials integral geometric approachrdquoPhysical Review E vol 80 no 5 Article ID 051303 17 pages2009

[56] T Prill K Schladitz D Jeulin M Faessel and CWieser ldquoMor-phological segmentation of FIB-SEM data of highly porousmediardquo Journal of Microscopy vol 250 no 2 pp 77ndash87 2013

[57] WKreher andW Pompe ldquoField fluctuations in a heterogeneouselastic material-an information theory approachrdquo Journal of theMechanics and Physics of Solids vol 33 no 5 pp 419ndash445 1985

[58] D JeulinMorphologie mathematique et proprietes physiques desagglomeres de miinerai de fer et de coke metullirrgique [PhDthesis] School of Mines Paris France 1979

[59] C H Arns M A Knackstedt W V Pinczewski and K RMecke ldquoEuler-Poincare characteristics of classes of disorderedmediardquo Physical Review E vol 63 no 3 Article ID 031112 13pages 2001

[60] C H Arns M A Knackstedt and K R Mecke ldquoCharacterisa-tion of irregular spatial structures by parallel sets and integralgeometric measuresrdquo Colloids and Surfaces A vol 241 no 1ndash3pp 351ndash372 2004

[61] H Hermann and W Kreher ldquoStructure elastic moduli andinternal stresses of iron-boron metallic glassesrdquo Journal ofPhysics F Metal Physics vol 18 no 4 pp 641ndash655 1988

[62] M D Rintoul and S Torquato ldquoPrecise determination of thecritical threshold and exponents in a three-dimensional contin-uumpercolationmodelrdquo Journal of Physics AMathematical andGeneral vol 30 no 16 pp L585ndashL592 1997

[63] A P Roberts and E J Garboczi ldquoComputation of the linearelastic properties of random porous materials with a widevariety of microstructurerdquo Proceedings of the Royal Society Avol 458 no 2021 pp 1033ndash1054 2002

[64] K Mecke and C H Arns ldquoFluids in porous media a morpho-metric approachrdquo Journal of Physics Condensed Matter vol 17no 9 pp S503ndashS534 2005

[65] F Willot and D Jeulin hal-00553376 version 1ndash7 2011[66] D Jeulin ldquoMulti scale random sets from morphology to

effective behaviourrdquo in Progress in Industrial Mathematics atECMI 2010 M Gunther A Bartel M Brunk S Schops andM Striebel Eds pp 381ndash393 Springer Berlin Germany 2012

[67] S Torquato ldquoOptimal design of heterogeneous materialsrdquoAnnual Review of Materials Research vol 40 pp 101ndash129 2010

[68] H Hermann ldquoEffective dielectric and elastic properties ofnanoporous low-k mediardquo Modelling and Simulation in Mate-rials Science and Engineering vol 18 no 5 Article ID 0550072010

[69] V Kokotin H Hermann and J Eckert ldquoTheoretical approachto local and effective properties of BMG basedmatrix-inclusionnanocompositesrdquo Intermetallics vol 30 pp 40ndash47 2012

[70] L D Landau and E M Lifschitz Elektrodynamik der KontinuaAkademie-Verlag Berlin Germany 1967

[71] I Shen and J Li ldquoEffective elastic moduli of composites rein-forced by particle or fiber with an inhomogeneous interphaserdquoInternational Journal of Solids and Structures vol 40 no 6 pp1393ndash1409 2003

[72] Z Hashin ldquoThe elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 pp 143ndash150 1962

[73] P W Schmidt A Hohr H B Neumann H Kaiser D Avnirand J S Lin ldquoSmall-angle X-ray scattering study of the fractalmorphology of porous silicasrdquoThe Journal of Chemical Physicsvol 90 no 9 pp 5016ndash5023 1989

[74] P W Schmidt ldquoSmall-angle scattering studies of disorderedporous and fractal systemsrdquo Journal of Applied Crystallographyvol 24 pp 414ndash435 1991

[75] B Yu ldquoAnalysis of flow in fractal porous mediardquo AppliedMechanics Reviews vol 61 no 5 Article ID 050801 19 pages2008

[76] H Hermann and J Ohser ldquoDetermination of microstructuralparameters of random spatial surface fractals by measuringchord length distributionsrdquo Journal of Microscopy vol 170 no1 pp 87ndash93 1993

[77] U Zahle ldquoRandom fractals generated by random cutoutsrdquoMathematische Nachrichten vol 116 no 1 pp 27ndash52 1984

[78] H Hermann ldquoA new random surface fractal for applications insolid state physicsrdquo Physica Status Solidi B vol 163 no 2 pp329ndash336 1991

[79] H D Bale and O W Schmidt ldquoSmall-angle X-ray-scatteringinvestigation of submicroscopic porosity with fractal proper-tiesrdquo Physical Review Letters vol 53 no 6 pp 596ndash599 1984

[80] ldquoInternational Technological Roadmap for SemiconductorsrdquoInterconnect Semiconductor Industry Association 2007httpwwwitrsnet

[81] S P Jeng R H Havemann and M C Chang ldquoProcessintegration and manufacturasility issues for high performancemultilevel interconnectrdquoMaterials Research Society SymposiumProceedings vol 337 article 25 1994

[82] S H Rhee M D Radwin J I Ng and D Erb ldquoCalculationof effective dielectric constants for advanced interconnectstructures with low-k dielectricsrdquo Applied Physics Letters vol83 no 13 pp 2644ndash2646 2003

[83] S Yu T K S Wong K Pita X Hu and V Ligatchev ldquoSurfacemodified silica mesoporous films as a low dielectric constantintermetal dielectricrdquo Journal of Applied Physics vol 92 no 6pp 3338ndash3344 2002

[84] M RWang R Rusli M B Yu N Babu C Y Li and K RakeshldquoLow dielectric constant films prepared by plasma-enhancedchemical vapor deposition from trimethylsilanerdquo Thin SolidFilms vol 462-463 pp 219ndash222 2004

[85] K ZagorodniyDChumakov C Taschner et al ldquoNovel carbon-cage-based ultralow-k materials modeling and first experi-mentsrdquo IEEE Transactions on Semiconductor Manufacturingvol 21 no 4 pp 646ndash660 2008

[86] J L Plawski W N Gill A Jain and S Rogojevic ldquoNanoporousdielectric films fundamental property relations and microelec-tronics applicationsrdquo in Interlayer Dielectrics for SemiconductorTechnology S P Murarka M Eizenberg and A K Sinha Edspp 261ndash325 Elsevier Amsterdam The Netherlands 2003


[87] K Zagorodniy H Hermann and M Taut ldquoStructure andproperties of computer-simulated fullerene-based ultralow-kdielectric materialsrdquo Physical Review B vol 75 no 24 ArticleID 245430 6 pages 2007

[88] K Zagorodniy G Seifert and H Hermann ldquoMetal-organicframeworks as promising candidates for future ultralow-kdielectricsrdquo Applied Physics Letters vol 97 no 25 Article ID251905 2010

[89] J Ohser C Ferrero O Wirjadi A Kuznetsova J Dull andA Rack ldquoEstimation of the probability of finite percolationin porous microstructures from tomographic imagesrdquo Interna-tional Journal of Materials Research vol 103 no 2 pp 184ndash1912012

[90] H Takahashi B Li T Sasaki C Miyazaki T Kajino and SInagaki ldquoCatalytic activity in organic solvents and stability ofimmobilized enzymes depend on the pore size and surfacecharacteristics of mesoporous silicardquo Chemistry of Materialsvol 12 no 11 pp 3301ndash3305 2000

[91] J-K Kim J-K Park and H-K Kim ldquoSynthesis and character-ization of nanoporous silica support for enzyme immobiliza-tionrdquo Colloids and Surfaces A Physicochemical and EngineeringAspects vol 241 no 1ndash3 pp 113ndash117 2004

[92] C Vega R D Kaminsky and P A Monson ldquoAdsorptionof fluids in disordered porous media from integral equationtheoryrdquoThe Journal of Chemical Physics vol 99 no 4 pp 3003ndash3013 1993

[93] G V Kapustin and J Ma ldquoModeling adsorption-desorptionprocesses in porous mediardquo Computing in Science amp Engineer-ing vol 1 no 1 pp 84ndash91 2002

[94] A Touzik and H Hermann ldquoTheoretical study of hydrogenadsorption on graphiticmaterialsrdquoChemical Physics Letters vol416 no 1ndash3 pp 137ndash141 2005

[95] M Maurer L Zhao and E Lugscheider ldquoSurface refinement ofmetal foamsrdquoAdvanced EngineeringMaterials vol 4 no 10 pp791ndash797 2002

[96] J Liu X Zhu Z Huang S Yu and X Yang ldquoCharacterizationand property of microarc oxidation coatings on open-cellaluminum foamsrdquo Journal of Coatings Technology and Researchvol 9 no 3 pp 357ndash363 2012

[97] J Banhart ldquoManufacture characterisation and application ofcellular metals and metal foamsrdquo Progress in Materials Sciencevol 46 no 6 pp 559ndash632 2001

[98] M Ritala M Kemell M Lautala A Niskanen M Leskelaand S Lindfors ldquoRapid coating of through-porous substratesby atomic layer depositionrdquo Chemical Vapor Deposition vol 12no 11 pp 655ndash658 2006

[99] J W Elam G Xiong C Y Han et al ldquoAtomic layer depositionfor the conformal coating of nanoporous materialsrdquo Journal ofNanomaterials vol 2006 Article ID 64501 5 pages 2006

[100] J W Elam J A Libera M J Pellin A V Zinovev J P Greeneand J A Nolen ldquoAtomic layer deposition of W on nanoporouscarbon aerogelsrdquo Applied Physics Letters vol 89 no 5 ArticleID 053124 3 pages 2006

[101] F Li Y Yang Y Fan W Xing and Y Ang ldquoModification ofceramic membranes for pore structure tailoring the atomiclayer deposition routerdquo Journal of Membrane Science vol 297-298 pp 17ndash23 2012

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


The models are characterized by their basic geometricalfeaturesmdashvolume fraction of pores 119881

119881 specific surface area

119878119881 pore size distribution 119891(119903) and correlation function

119862(119903) On one hand these characteristics are required tointerpret experimental data for example small-angle scat-tering curves On the other hand they are important forunderstanding and control of the interplay between structureand physical properties of random porous media

Physical properties of the models are discussed in termsof an effective medium approach Explicit expressions formacroscopic elastic moduli thermal and electrical conduc-tivity and static dielectric constant are given

The definitions of the geometrical characteristics areexplained in Section 2 the models are described in Section 3physical properties are discussed in Section 4 and applica-tions to experimental situations are presented in Section 5

2 Geometrical Characteristics

The pore space is described as a random set 119860 We considervolume fraction 119881

119881(119860) specific surface area 119878

119881(119860) pore

size distribution 119891(119903) and correlation function 119862119860(119903) as

geometrical characteristics of 119860 It is useful to introduce thecomplement of 119860 consisting of the part of the space whichdoes not belong to 119860 The complement of 119860 is indicated by119860119888 This notation is standard in stochastic geometry [10] For

example if 119860 denotes the pore space of the porous medium119888119901= 119881119881(119860) and 119888

119887= 119881119881(119860119888) = 1 minus 119888

119901are the volume fraction

of the pore space and the solid bulk phase respectively119878119881(119860) = 119878

119881(119860119888) is the same for 119860 and 119860119888

The pore space 119860 is described by the shape function

119904119901(r) =

1 r isin 1198600 otherwise

(1)

The volume fraction 119888119901of pores is defined by the probability

of finding a random test point r1in 119860

119888119901= 119881119881 (119860) = 119875 (r

1isin 119860) (2)

If 119904119901(r) is known 119888119901can be calculated by

119888119901=1

119881int119881

119904119901(r) 119889119881 (3)

The correlation function

119862119901 (r) = 119875 (r

1isin 119860 r

2isin 119860) r = r

1minus r2 (4)

denotes the probability of finding two random test points(r1 r2) inside the pore space It is calculated from the shape

function by

119862119901 (r) =

1

119881int119881

119904119901(u + r) 119904119901 (u) 119889119881u (5)

Because of isotropy it depends only on the distance 119903 = |r1minus

r2| between the test points

119862119901 (119903) =

1

4120587intΩ

119862119901 (r) 119889Ω (6)

whereΩ is the solid angle Replacing index 119901 by 119887 and setting119904119887(r) = 1 minus 119904

119901(r) one obtains from ((1) (5) and (6)) for the

correlation function of the solid phase

119862119887 (119903) = 1 minus 2119888

119901+ 119862119901 (119903) (7)

The specific surface area of 119860 is defined by

119904119901= 119878119881 (119860) = lim

120598rarr0

119875 (r1isin 119878120598)

2120598 (8)

where 119875(r1isin 119878120598) is the probability of finding a random test

point r1 in 119878120598and 119878

120598is defined by all points of the three-

dimensional space the distance of which from the surface119878(119860) is less than 120598 If the correlation function of the systemis known 119904

119901is related to 119862

119901(119903) by [10]

119904119901= minus4 lim119903rarr0

119889119862119901 (119903)

119889119903 (9)

For illustration we consider the most simple case of a sin-gle porewith radius119877 in a sample of volume119881 as amodel for adilute system of equal pores that is 119888

119901≪ 1 The volume frac-

tion and the specific surface area are given by 119888119901= 43120587119877

3119881

and 119904119901= 4120587119877

2119881 respectively The integral in formula (5)

is evaluated as the convolution integral of a spherical shapefunction [16] Neglecting terms of the order 1198882

119901one obtains

119862119901 (119903) =

119888119901(1 minus

3119903

4119877+

1199033

161198773) 0 le 119903 le 2119877

0 119903 gt 2119877

(10)

Polydisperse systems are additionally characterized by asize distribution function 119891(119903) with mean value

119898 = int119909119891 (119909) 119889119909 (11)

and variance

1205902= int (119909 minus 119898)

2119891 (119909) 119889119909 (12)

3 Models

31 Poisson Grain Model The Poisson grain model (alsoknown as Boolean model see [10]) is an important andversatile model for generating random two-phase structuresIt is created in two steps In the first step a random (Poisson)point field is generated (here in the three-dimensional space)with number density ] of points This point field has thefollowing property Considering an arbitrary finite region 119861

with volume119881(119861) the number119873(119861) of points situatedwithin119861 is a random variable The probability that 119873(119861) takes thevalue 119899 is

119875 (119873 (119861) = 119899) =[]119881 (119861)]

119899

119899exp (minus]119881 (119861)) 119899 = 0 1

(13)


The numbers 119873(1198611)119873(119861





119881119881 (119860) = 1 minus exp (minus]119881) (14)

where 119881 = (41205873) intinfin





119881(119860119888) = exp(minus](41205873)1199033) =



119878119881= ] [1 minus 119881

119881 (119860)] 119878 (15)




119862119860 (119903) = 2119888

119860minus 1 + (1 minus 119888

119860)2 exp (]120574 (119903)) (16)

where 119888119860



according to

120574 (119903) =4120587

3int

infin

1199032

1199093(1 minus

3119903

4119909+

1199033

161199093)119891 (119909) 119889119909 (17)



119888119901= 119881119881 (119860)

119904119901= 119878119881

(18)







119894


119894+ 119903ℎ119899

119895between


119903119904119899

119894+119903119904119899

119895 If 119903ℎ119899119894+119903ℎ119899

119895lt 119903119894119895lt 119903119904119899

119894+119903119904119899



119894according to

x119899+1119894

= x119899119894+

119911119894

sum

119895=1

119875119894119895

x119899119894minus x119899119895

10038161003816100381610038161003816x119899119894minus x119899119895

10038161003816100381610038161003816

(19)






0204

0608

068

067

066

065

0908

0706

05

flarge

rsmallrlarge



119904119899

119894minus 119903ℎ119899




119881119881= ]intinfin

0

4120587

31199033119891 (119903) 119889119903 (20)

119878119881= ]intinfin

0

41205871199033119891 (119903) 119889119903 (21)










119894 119910119894 119911119894) 119894 = 1 2 3 119873 the radii 120582119903

119894





119891 (119903) =119887119901


and 119903 = 119901119887 1205902= 119901119887

2 Γ(119901) = int

infin

0119905119901minus1 exp(minus119905)119889119905


(a) (b)

(c) (d)


ℎ= 060 120582 = 075 085 095 100 for ((a) (b) (c) and (d))



119881and 119878

119881 each


4 The


4 and 05 for119873 = 105




formulas

119881119881= 120601ℎ+ (1 minus 120601

ℎ)Φ(

3radic2120587

2

(1 minus 120582) 120601ℎ

120582 (1 minus 120601ℎ)) (23)

Φ (119911) =2

radic120587int

119911

0

exp (minus1199052) 119889119905 (24)

119878119881=

3

120582119903ref120601ℎexp (minus9120587

4

(1 minus 120582)21206012

ℎ

1205822(1 minus 120601ℎ)2) (25)





2 3 4 5001

01

1

10

100

120590S NS

V(

)

120590VNVV

120590SNSV

Log10 N


119873119881119881and 120590

119878

119873119878119881of


119881 of




line 120582 = 095


119903ref119903

= 119886 + 119887(120590

119903)

2

(26)



ℎle 071 119888

119901le





119881


070 075 080 085 090 095 100

06

07

08

09

10

074 075 076 077095

096

097

098

099


Volu

me f

ract

ionVV


070 075 080 085 090 095 10000

05

10

15

20

25

30

35


Spec

ific s

urfa

ce ar

eaS V



119888119894= 120575119905119878119881 (27)


119888119901= 119881119881minus 120575119905119878119881 (28)




119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)




2(119903) = 119891

1(119903 minus 120575119905)



119888119894= 119881119881 (1) minus 119881119881 (2) (30)


119888119894= 1 minus exp(minus]4120587


3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)



119891 (119903) =1


2

21205902) (33)


119888119894=4120587120588


2+ 31205902] (34)


119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)


ℎ


119903ℎ= 120582119903119901 (36)



pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=



119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)


119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)


119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




The numbers 119873(1198611)119873(119861





119881119881 (119860) = 1 minus exp (minus]119881) (14)

where 119881 = (41205873) intinfin





119881(119860119888) = exp(minus](41205873)1199033) =



119878119881= ] [1 minus 119881

119881 (119860)] 119878 (15)




119862119860 (119903) = 2119888

119860minus 1 + (1 minus 119888

119860)2 exp (]120574 (119903)) (16)

where 119888119860



according to

120574 (119903) =4120587

3int

infin

1199032

1199093(1 minus

3119903

4119909+

1199033

161199093)119891 (119909) 119889119909 (17)



119888119901= 119881119881 (119860)

119904119901= 119878119881

(18)







119894


119894+ 119903ℎ119899

119895between


119903119904119899

119894+119903119904119899

119895 If 119903ℎ119899119894+119903ℎ119899

119895lt 119903119894119895lt 119903119904119899

119894+119903119904119899



119894according to

x119899+1119894

= x119899119894+

119911119894

sum

119895=1

119875119894119895

x119899119894minus x119899119895

10038161003816100381610038161003816x119899119894minus x119899119895

10038161003816100381610038161003816

(19)






0204

0608

068

067

066

065

0908

0706

05

flarge

rsmallrlarge



119904119899

119894minus 119903ℎ119899




119881119881= ]intinfin

0

4120587

31199033119891 (119903) 119889119903 (20)

119878119881= ]intinfin

0

41205871199033119891 (119903) 119889119903 (21)










119894 119910119894 119911119894) 119894 = 1 2 3 119873 the radii 120582119903

119894





119891 (119903) =119887119901


and 119903 = 119901119887 1205902= 119901119887

2 Γ(119901) = int

infin

0119905119901minus1 exp(minus119905)119889119905


(a) (b)

(c) (d)


ℎ= 060 120582 = 075 085 095 100 for ((a) (b) (c) and (d))



119881and 119878

119881 each


4 The


4 and 05 for119873 = 105




formulas

119881119881= 120601ℎ+ (1 minus 120601

ℎ)Φ(

3radic2120587

2

(1 minus 120582) 120601ℎ

120582 (1 minus 120601ℎ)) (23)

Φ (119911) =2

radic120587int

119911

0

exp (minus1199052) 119889119905 (24)

119878119881=

3

120582119903ref120601ℎexp (minus9120587

4

(1 minus 120582)21206012

ℎ

1205822(1 minus 120601ℎ)2) (25)





2 3 4 5001

01

1

10

100

120590S NS

V(

)

120590VNVV

120590SNSV

Log10 N


119873119881119881and 120590

119878

119873119878119881of


119881 of




line 120582 = 095


119903ref119903

= 119886 + 119887(120590

119903)

2

(26)



ℎle 071 119888

119901le





119881


070 075 080 085 090 095 100

06

07

08

09

10

074 075 076 077095

096

097

098

099


Volu

me f

ract

ionVV


070 075 080 085 090 095 10000

05

10

15

20

25

30

35


Spec

ific s

urfa

ce ar

eaS V



119888119894= 120575119905119878119881 (27)


119888119901= 119881119881minus 120575119905119878119881 (28)




119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)




2(119903) = 119891

1(119903 minus 120575119905)



119888119894= 119881119881 (1) minus 119881119881 (2) (30)


119888119894= 1 minus exp(minus]4120587


3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)



119891 (119903) =1


2

21205902) (33)


119888119894=4120587120588


2+ 31205902] (34)


119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)


ℎ


119903ℎ= 120582119903119901 (36)



pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=



119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)


119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)


119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0204

0608

068

067

066

065

0908

0706

05

flarge

rsmallrlarge



119904119899

119894minus 119903ℎ119899




119881119881= ]intinfin

0

4120587

31199033119891 (119903) 119889119903 (20)

119878119881= ]intinfin

0

41205871199033119891 (119903) 119889119903 (21)










119894 119910119894 119911119894) 119894 = 1 2 3 119873 the radii 120582119903

119894





119891 (119903) =119887119901


and 119903 = 119901119887 1205902= 119901119887

2 Γ(119901) = int

infin

0119905119901minus1 exp(minus119905)119889119905


(a) (b)

(c) (d)


ℎ= 060 120582 = 075 085 095 100 for ((a) (b) (c) and (d))



119881and 119878

119881 each


4 The


4 and 05 for119873 = 105




formulas

119881119881= 120601ℎ+ (1 minus 120601

ℎ)Φ(

3radic2120587

2

(1 minus 120582) 120601ℎ

120582 (1 minus 120601ℎ)) (23)

Φ (119911) =2

radic120587int

119911

0

exp (minus1199052) 119889119905 (24)

119878119881=

3

120582119903ref120601ℎexp (minus9120587

4

(1 minus 120582)21206012

ℎ

1205822(1 minus 120601ℎ)2) (25)





2 3 4 5001

01

1

10

100

120590S NS

V(

)

120590VNVV

120590SNSV

Log10 N


119873119881119881and 120590

119878

119873119878119881of


119881 of




line 120582 = 095


119903ref119903

= 119886 + 119887(120590

119903)

2

(26)



ℎle 071 119888

119901le





119881


070 075 080 085 090 095 100

06

07

08

09

10

074 075 076 077095

096

097

098

099


Volu

me f

ract

ionVV


070 075 080 085 090 095 10000

05

10

15

20

25

30

35


Spec

ific s

urfa

ce ar

eaS V



119888119894= 120575119905119878119881 (27)


119888119901= 119881119881minus 120575119905119878119881 (28)




119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)




2(119903) = 119891

1(119903 minus 120575119905)



119888119894= 119881119881 (1) minus 119881119881 (2) (30)


119888119894= 1 minus exp(minus]4120587


3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)



119891 (119903) =1


2

21205902) (33)


119888119894=4120587120588


2+ 31205902] (34)


119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)


ℎ


119903ℎ= 120582119903119901 (36)



pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=



119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)


119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)


119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) (b)

(c) (d)


ℎ= 060 120582 = 075 085 095 100 for ((a) (b) (c) and (d))



119881and 119878

119881 each


4 The


4 and 05 for119873 = 105




formulas

119881119881= 120601ℎ+ (1 minus 120601

ℎ)Φ(

3radic2120587

2

(1 minus 120582) 120601ℎ

120582 (1 minus 120601ℎ)) (23)

Φ (119911) =2

radic120587int

119911

0

exp (minus1199052) 119889119905 (24)

119878119881=

3

120582119903ref120601ℎexp (minus9120587

4

(1 minus 120582)21206012

ℎ

1205822(1 minus 120601ℎ)2) (25)





2 3 4 5001

01

1

10

100

120590S NS

V(

)

120590VNVV

120590SNSV

Log10 N


119873119881119881and 120590

119878

119873119878119881of


119881 of




line 120582 = 095


119903ref119903

= 119886 + 119887(120590

119903)

2

(26)



ℎle 071 119888

119901le





119881


070 075 080 085 090 095 100

06

07

08

09

10

074 075 076 077095

096

097

098

099


Volu

me f

ract

ionVV


070 075 080 085 090 095 10000

05

10

15

20

25

30

35


Spec

ific s

urfa

ce ar

eaS V



119888119894= 120575119905119878119881 (27)


119888119901= 119881119881minus 120575119905119878119881 (28)




119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)




2(119903) = 119891

1(119903 minus 120575119905)



119888119894= 119881119881 (1) minus 119881119881 (2) (30)


119888119894= 1 minus exp(minus]4120587


3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)



119891 (119903) =1


2

21205902) (33)


119888119894=4120587120588


2+ 31205902] (34)


119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)


ℎ


119903ℎ= 120582119903119901 (36)



pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=



119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)


119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)


119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




2 3 4 5001

01

1

10

100

120590S NS

V(

)

120590VNVV

120590SNSV

Log10 N


119873119881119881and 120590

119878

119873119878119881of


119881 of




line 120582 = 095


119903ref119903

= 119886 + 119887(120590

119903)

2

(26)



ℎle 071 119888

119901le





119881


070 075 080 085 090 095 100

06

07

08

09

10

074 075 076 077095

096

097

098

099


Volu

me f

ract

ionVV


070 075 080 085 090 095 10000

05

10

15

20

25

30

35


Spec

ific s

urfa

ce ar

eaS V



119888119894= 120575119905119878119881 (27)


119888119901= 119881119881minus 120575119905119878119881 (28)




119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)




2(119903) = 119891

1(119903 minus 120575119905)



119888119894= 119881119881 (1) minus 119881119881 (2) (30)


119888119894= 1 minus exp(minus]4120587


3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)



119891 (119903) =1


2

21205902) (33)


119888119894=4120587120588


2+ 31205902] (34)


119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)


ℎ


119903ℎ= 120582119903119901 (36)



pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=



119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)


119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)


119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






119888119887= 1 minus 119881

119881= 1 minus 119888

119901minus 119888119894 (29)




2(119903) = 119891

1(119903 minus 120575119905)



119888119894= 119881119881 (1) minus 119881119881 (2) (30)


119888119894= 1 minus exp(minus]4120587


3(119877 minus 120575119905)

3)

(31)

For 120575119905 ≪ 119877

119888119894= exp(minus]4120587

31198773) 4120587]1198772120575119905 = ] (1 minus 119881

119881 (1)) 41205871198772120575119905 (32)



119891 (119903) =1


2

21205902) (33)


119888119894=4120587120588


2+ 31205902] (34)


119888119894= 4120587120588 (119903

2+ 1205902) 120575119905 = 119878

119881120575119905 (35)


ℎ


119903ℎ= 120582119903119901 (36)



pore size 1199031199011 1199031199012

with 1199031199012

= 1199031199011minus 120575119905 and 120582

1= 119903ℎ1199031199011 1205822=



119888119894= 119881119881(120601ℎ 1205821) minus 119881119881(120601ℎ 1205822) 120582

2=

1205821

1 minus 1205751199051199031199011

(37)


119881119881(120601ℎ 120582) minus 119881

119881(120601ℎ 120582 + 120575120582)

120575120582

(38)

by

120597119881119881(120601ℎ 120582)

120597120582

(39)


119888119894= 119903

120575120582

120582119878119881= 120575119905119878119881

(40)














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials














119901 119888119894 and 119888


119903119901119894= (

119903119901

119903119894

)

3

119903119894119887= (

119903119894

119903119887

)

3

119903119901119887= (

119903119901

119903119887

)

3

(41)


rb

ri

uΘ

urrp



119888119901= 119903119901119887 119888

119894= 119903119894119887minus 119903119901119887 119888

119887= 1 minus 119903

119894119887 (42)


Δ119879 = 0 (43)




j = 120590f (45)


f =

119862119901sinΘu


119903 0 le 119903 le 119903

119901

(119862119894+119863119894

1199033) sinΘu

Θ

+( minus 119862119894+ 2

119863119894

1199033) cosΘu

119903 119903119901le 119903 le 119903

119894

(119862119887+119863119887

1199033) sinΘu

Θ

+( minus 119862119887+2

119863119887

1199033) cosΘu

119903 119903119894le 119903 le 119903

119887

minus1198910sinΘu

Θ+ 1198910cosΘu

119903 119903

119887le 119903

(46)


119903of









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials









(((

(

1 minus119903119901119894

minus1 0 0 0

minus120590119901119903119901119894120590119894minus2120590119894

0 0 0

0 1 1 minus119903119894119887

minus1 0

0 minus120590119894

2120590119894

119903119894119887120590119887minus2120590119887

0

0 0 0 1 1 1

0 0 0 minus120590119887

2120590119887

minus120590lowast

)))

)

((((

(

119862119901

119862119894

119863119894

119862119887

119863119887

1198910

))))

)

= 0

(47)


120590lowast= 120590119887+

119903119894119887

(1 minus 119903119894119887) 3120590119887minus 1 (120590

119887minus 120590)

(48)

with

120590= 120590119894+

119903119901119894

(1 minus 119903119901119894) 3120590119894minus 1 (120590

119894minus 120590119901)

(49)







119886lowast

119886119887

= 1 +1 minus 119888119887

120572119886

119887119888119887minus 119886119887 (119886119887minus 119886)

(50)

119886

119886119894

= 1 +1 minus 119888

120572119886

119894119888 minus 119886

119894 (119886119894minus 119886119901)

(51)

119888=

119888119901

119888119901+ 119888119894

(52)

120572119886

119898=

1

3 119886 = 120590

3119870119898

3119870119898+ 4119866119898

119886 = 119870

6 (119870119898+ 2119866119898)

5 (3119870119898+ 4119866119898) 119886 = 119866

(53)




119894= 0 with

119888119901= 1 minus 119888

119887 119886 = 119896 and 120572

119896= 13 assuming 119896

119887lt 119896119901



119896119887+

119896119887119888119901

1198881199013 minus 119896

119901 (119896119901minus 119896119887)

le 119896lowastle 119896119901

+

119896119901119888119887

1198881198873 minus 119896

119887 (119896119887minus 119896119901)

(54)




5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





5 Applications



]119906of spheres 119903

119906

119903119906= 1199061199041199031 ]

119906= 119906minus119905]1 119906

0le 119906 le 1 (55)

where 119903119906and ]



0


119891 (119903) =

120578120581120578

(1 minus 120581120578) 1199031

(119903

1199031

)

minus120578minus1

1199030le 119903 le 119903

1

0 otherwise

120578 =119905 minus 1

119904

(56)


] = ]1

1 minus 120581120578

120581120578120578119904 (57)

]119881 = ]1

4120587

31199033

1

1 minus 1205813minus120578

(3 minus 120578) 119904 (58)

]119878 = ]141205871199032

1

1205812minus120578

minus 1

(120578 minus 2) 119904 (59)

]120574 (119903) =3 minus 120578


1 minus 119881119881

times 1 minus (120588119903

1)3minus120578

3 minus 120578+3

2(

119903

21199031

)1 minus (120588119903

1)2minus120578

120578 minus 2

minus1

2(

119903

21199031

)

31 minus (120588119903

1)minus120578

120578

0 le 119903 le 21199031

(60)

where

120588 =

1199030

119903

2lt 1199030

119903

2 1199030le119903

2le 1199031

(61)

20 22 24 26 28 300

20

40

60

80

100

20 22 24 26 28 30000

005

010

015

020

025

VV

120578

r 1S V

120578


1= 1 ]

1=



2 le 120578 lt 3

119889119891= lim120598rarr0

ln (119875 (r isin 119878120598) 1205983)

ln (1120598)= 120578

(62)






119862 (119903) minus 1198812

119881prop (

119903

21199031

)

3minus119889119891

(63)






119881








00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




00 05 10 15 20 25

0

2

minus2

minus4

minus6

Log[qq0]

Log[I(q)]

Log

[I(0)]


0= 1 119903

1= 1 ]

1= 005 119904 = 1 120581 = 0001 and 120578 = 21





119868 (119902) = 4120587int

infin

0

1199032[119862 (119903) minus 119881

2

119881]sin (119902119903)119902119903

119889119903 (64)


119868 (119902) prop 1199026minus119889119891 (65)



0) lt 05 by the







0) plot for










119894 and choosing


119886 the


119864 =9119870119866

3119870 + 119866(66)





2improves the







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







040 045 050 055 060 065 070

005

010

015

020

025

030

c i

cp


119901and 119888








045 050 055 060 065 070

005

010

015

020

025

030

c i

cp

040







119888119894= 119881119881(1205820) minus 119881119881(120582120591) (67)



ing to

119903 (120582120591) = 1199030minus 120583119903ℎ120591 =

119903ℎ

120582120591

1199030=119903ℎ

1205820

(68)

One obtains

120582120591=

1205820

1 minus 1205820120583120591

(69)



0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0 20 40 60 8000

01

02

03

04

05

06


Volu

me f

ract

ion

of p

ores

(∙) a

nd A

LD la

yer (◼

)











6 Conclusion


0 20 40 60 80

015

020

025

030

035

040


0 20 40 60 80

15

20

25

30


klowast

(Wm

middotK)

Elowast

(GPa

)





References
















































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






































































































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



review article geometric models for isotropic random...

Documents