Table Of ContentParticulate Morphology
Particulate Morphology
Mathematics Applied to Particle
Assemblies
Keishi Gotoh
Emeritus Professor
Toyohashi University of Technology
Toyohashi, Japan
AMSTERDAM(cid:1)BOSTON(cid:1)HEIDELBERG(cid:1)LONDON(cid:1)NEWYORK(cid:1)OXFORD
PARIS(cid:1)SANDIEGO(cid:1)SANFRANCISCO(cid:1)SINGAPORE(cid:1)SYDNEY(cid:1)TOKYO
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Preface
In the field of science and technology, mathematics can be considered as a general
tool for expressing a state and/or phenomena of various substances so that it is
applicabletoubiquitoussubjects.Accordingly,itisrecommendedtostudyasmuch
as possible before we engage in any research subject. However, there are too many
tools to learn so that we usually start our research without sufficient knowledge to
deal with, exceptthe personwho hasinterestinmathematics itself. There are many
regrettable cases in which more effective results might be derived by use of suffi-
cientmathematicalknowledge.
The author has been involved for a long time in the research of “Flow and
Measurement of Particulate Materials,” especially in the statistical geometry of
“SpatialStructureofRandomParticleAssemblies.”Thistextisacollectionofmathe-
maticaltoolsusedintheseriesofpapers.Inotherwords,thecontentisappliedmath-
ematics concerning personal research subjects. The author will be grateful if it helps
manystudentsandyoungresearchersmakefurtheradvancement.
Mathematical derivations are shown in as much detail as possible without omis-
sionsothatreaderscanfollowthemeasily.Referencescitedarelistedattheendof
thechapters.
Theauthoracknowledges sincerelyallstaffsofElsevier Insightsfortheir labori-
ouseditingwork.
1
Spatial Structure of Random
Dispersion of Equal Spheres
in One Dimension
Although the one-dimensional structure of random dispersion of equal spheres may
not exist in so many in practical applications, it is useful such that we can derive
the exact analytical solution and examine the accuracy of computer experiments.
Also itishelpfulfor studying the comparisonbetweena discrete system anda con-
tinuousone.
1.1 Discrete System
Consider a line of equal spheres placed at unit interval as depicted in Figure 1.1.
Two neighboring spheres are connected one at a time by placing a stick on the
intervalundertheconditionthatoverlappingconnectionisforbidden.
We call the connected two spheres the cluster of size 2, the connected three
spheres the cluster of size 3, and so on. On the other hand, we call n series of iso-
latedspherestheisolatedgroupofsizen.
Figure 1.1 shows an isolated group of size 4 where three different cases are
inherently possible to exist for the isolated group of size 2, and two different cases
for the isolated group of size 3. In general, the isolated group of size n inherently
contains two possible cases for the isolated group of size (n21), three possible
casesfortheisolatedgroupofsize(n22),andsoon.
Consider M lines of N series of isolated equal spheres. Two neighboring spheres
are connected by placing a stick on the interval repeatedly under the condition that
duplicate connection is forbidden. The probability of producing a cluster of size 2
during the time interval t2(t1dt) is expressed by k(t)dt, and the number of the
isolatedgroupsofsizenattimetinthejthlinebyC(j)(n,t).Accordingly,weobtain
(CohenandReiss,1963)
2dCðjÞðn;tÞ=dt5kðtÞ½ðn21ÞCðjÞðn;tÞ12CðjÞðn11;tÞ(cid:2) ð1:1Þ
ParticulateMorphology.DOI:10.1016/B978-0-12-396974-3.00001-1
©2012ElsevierInc.Allrightsreserved.
2 ParticulateMorphology
Size4 Figure1.1 Isolatedgroupofsize4.
Size2 Size2 Size2
Size3 Size3
Equation(1.1)expressesthedecreasingrateofthenumberoftheisolatedgroups
of size n at time t. There are (n21) possible cases for connecting two neighboring
spheres in the isolated group of size n. Therefore, the first term of the right-hand
side of Eq. (1.1) expresses the total number of the possible cases that the isolated
group of size n is internally destroyed by placing the stick. When either end-sphere
of the isolated group of size n is connected to its outside neighbor, the group is
destroyedtobecomethesize(n21).Theconnectiontotheoutsideneighbormeans
the existence of C(j)(n11,t). The other side of the isolated group is also taken into
account,leadingtothesecondtermoftheright-handsideofEq.(1.1).
For M lines of N series of spheres, we define the average number of the isolated
groupsofsizenasfollows:
XM
hCðn;tÞi5M21 CðjÞðn;tÞ ð1:2Þ
j51
Hence,Eq.(1.1)becomes
2dhCðn;tÞi=dt5kðtÞ½ðn21ÞhCðn;tÞi12hCðn11;tÞi(cid:2) ð1:3Þ
Because there are (N2n11) possible cases for making the isolated group of
sizenfromNseriesofspheres,theinitialconditionbecomesasfollows:
hCðn;0Þi5N2n11 ð1:4Þ
SpatialStructureofRandomDispersionofEqualSpheresinOneDimension 3
Ð
Byuseofz5 tkðtÞdt ordz5k(t)dt,Eq.(1.3)issimplifiedtobecome
0
2dhCðn;zÞi=dz5ðn21ÞhCðn;zÞi12hCðn11;zÞi ð1:5Þ
ThesolutionofEq.(1.5)withtheinitialcondition(1.4)becomes
XN2n
hCðn;zÞi5exp½2ðn21Þz(cid:2) ðN2n2s11Þð2e2z22Þs=s! ð1:6Þ
s50
or
NX2n21
hCðn11;zÞi5exp½2nz(cid:2) ðN2n2sÞð2e2z22Þs=s!
s50
Substitution of Eq. (1.6) into Eq. (1.5) can verify the solution, where replacing
(s21)byk,thesummationforthederivativetermshouldbefromk50to(N2n21).
Accordingly, the survival rate of the isolated groups of size n becomes as
follows:
Pðn;zÞ5hCðn;zÞi=ðN2n11Þ
XN2n
5e2ðn21Þz ½12s=ðN2n11Þ(cid:2)ð2e2z22Þs=s!
s50 ð1:7Þ
XN2n
!e2ðn21Þz ð2e2z22Þs=s!: N !N
s50
5e2ðn21Þz exp½22ð12e2zÞ(cid:2) ð1:8Þ
The survival probability of a single isolated sphere after a long time is obtain-
ablebysettingn51andz5NinEq.(1.8).
Pð1;NÞ5e22 ð1:9Þ
a. Caseofplacingtheclusterofsizeksmallerthantheisolatedgroup(ksn)
In the previous discussion, we considered the case of placing a single stick to
connect two neighboring spheres every time. Here we consider the case of placing
the cluster of size k every time under the condition that overlapping connection
is forbidden, where ksn. The survival probability of the isolated group of size
n is expressed by P(n,t) at time t. Then, similarly to Eqs. (1.1)(cid:3)(1.3), the
following relation is derived for the decreasing rate of the isolated cluster of size n
(Rodgers,1992).
4 ParticulateMorphology
Xk21
2dPðn;tÞ=dt5ðn2k11ÞPðn;tÞ12 Pðn1j;tÞ ð1:10Þ
j51
whereksn.
In the right-hand side of Eq. (1.10), (n2k11) expresses the possible number of
casesinwhichtheisolatedgroupofsizenisdestroyedbyplacingtheclusterofsizek
insidethegroup.Theisolatedgroupofsizenisalsodestroyedbyplacingthecluster
ofsizek partially insidethe group:one sideoftheclusterlapsbytheinterval j from
theendpointnoftheisolatedgroup.Inordertomakesuchanoccasionpossible,the
isolatedgroupofsize(n1j)doesexistfirstandtheclusterofsizekisplaced,where
j51, 2, ..., (k21). The second term of the right-hand side of Eq. (1.10) is for this
typeofthepartialdestructionoftheisolatedgroupofsizen.Assuming
Pðn;tÞ5FðtÞexp½2ðn2k11Þ(cid:2) ð1:11aÞ
andsubstitutingintoEq.(1.10),
2dPðn;tÞ=dt52fdFðtÞ=dtgexp½2ðn2k11Þt(cid:2)
1FðtÞðn2k11Þexp½2ðn2k11Þt(cid:2)
5 ðn2k11ÞFðtÞexp½2ðn2k11Þt(cid:2)
Xk21
12 FðtÞexp½2ðn1j2k11Þt(cid:2)
j51
Xk21
dlnFðtÞ=dt522 exp½2jt(cid:2)
j51
UsingtheinitialconditionsP(n,0)51andF(0)51,
" #
Xk21
FðtÞ5exp 22 ð12e2jtÞ=j :k^2 ð1:12Þ
j51
Therefore,Eq.(1.11a)becomesasfollows:
" #
Xk21
Pðn;tÞ5exp½2ðn2k11Þt(cid:2)exp 22 ð12e2jtÞ=j :n^k^2 ð1:11bÞ
j51
SpatialStructureofRandomDispersionofEqualSpheresinOneDimension 5
b. Caseofplacingtheclusterofsizeklargerthantheisolatedgroup(k^n)
Inthecaseofk^n,Eq.(1.10)becomesasfollows:
Xn21
2dPðn;tÞ=dt5ðk2n11ÞPðk;tÞ12 Pðk1j;tÞ ð1:13aÞ
j51
In order to place the cluster larger than the isolated group of size n without any
overlap,theisolatedgroupofsizekmustexistfirst.
There are (k2n11) possible cases for the isolated group of size n to exist
in the isolated group of size k, leading to the first term of the right-hand side
of Eq. (1.13a). The second term expresses the partial destruction of the isolated
group ofsizenbyplacingtheclusterofsize k.Considerthattheclusterofsizekis
placed on the isolated group leaving j isolated spheres: j51, 2,..., (n21). The
isolated group of size (k1j) must exist first for this purpose. The partial destruc-
tion of the isolated group of size n is possible to occur from both ends, leading to
thesecondtermoftheright-handsideofEq.(1.13a).
In Eq. (1.11a) with Eq. (1.12), which is the solution of Eq. (1.10), the substitu-
tionsofkintonand(k1j)intongive
Pðk;tÞ5FðtÞexp½2ðk2k11Þt(cid:2)5FðtÞe2t
Pðk1j;tÞ5FðtÞexp½2ðk1j2k11Þt(cid:2)5FðtÞe2ðj11Þt
" #
Xk21
FðtÞ5exp 22 ð12e2jtÞ=j
j51
Hence,Eq.(1.13a)becomesasfollows:
Xn21
2dPðn;tÞ=dt5ðk2n11ÞPðk;tÞ12 Pðk1j;tÞ
j51
( ) " #
Xn21 Xk21
5 ðk2n11Þ12 e2jt exp 2t22 ð12e2jtÞ=j
j51 j51
ð1:13bÞ
6 ParticulateMorphology
Therefore,weobtain(Rodgers,1992)
( ) " #
ð
t Xn21 Xk21
Pðn;tÞ512 ðk2n11Þ12 e2jt exp 2t22 ð12e2jtÞ=j dt:k^n
0 j51 j51
ð1:14aÞ
wheretheinitialconditionisP(n,0)51.
Especiallyinthecaseofn5k,
( ) " #
ð
t Xk21 Xk21
Pðk;tÞ512 112 e2jt exp 2t22 ð12e2jtÞ=j dt
0 ( j51 ) " j51 #
ð
t Xk21 Xk21
511 2122 e2jt exp 2t22 ð12e2jtÞ=j dt
0 ( "j51 #j5)1
ð
t d Xk21
511 exp 2t22 ð12e2jtÞ=j dt
dt
"0 j51#
Xk21
5exp 2t22 ð12e2jtÞ=j
j51
FromEq.(1.14a),
" #
ð
t Xk21
Pð1;tÞ512k exp 2t22 ð12e2jtÞ=j dt ð1:14bÞ
0 j51
FromEq.(1.14b),fork51:
ð
t
Pð1;tÞ512 e2t dt
0
ð
t
yðtÞ512Pð1;tÞ5 e2t dt512e2t ð1:15Þ
0
andfork52:
ð
t
Pð1;tÞ5122 exp½2t22ð12e2tÞ(cid:2)dt
0
