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9 Appendix: Basic concept of cascade theory

# 9 Appendix: Basic concept of cascade theory

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237

8.11 Procedure of polyfunctional polymerization. Marked unit denotes

a monomer unit chosen at random in the system.

3

2

1

0

2

1

2

(a)

(b)

8.12 Tree-like model regards a polymer as a genealogical tree.

When one unit is chosen at random from the system, the macromolecule

to which that unit belongs establishes the unit as a root and can be

considered to be a rooted tree as shown in Fig. 8.12(a). Furthermore, this

macromolecule regards this unit as a zeroth generation of a genealogical

treeandtheotherunitsbelongingtothismacromoleculecanbeclassified

as first-generation, second-generation, etc. (see Fig. 8.12(b)). The zeroth

generation unit is linked to a number of first-generation units as shown

in Fig. 8.13(a). here, ak represents the probability of having k amount of

offspring.

3

S ak = 1

k =0

[8.19]

Furthermore, the first-generation unit is linked to a number of secondgeneration units as shown in Fig. 8.13(b). here, bk represents the probability

of having k amount of offspring. Beyond second-generation units, the

distribution of the probability of having offspring is the same as it would

beforfirst-generationunitsaccordingtotheMarkovprocessassumption:

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Chemistry, Manufacture and Applications of Natural Rubber

1

1

1

1

1

1

a1

a0

a2

a3

(a)

2

1

1

2

1

2

b0

b1

b2

(b)

8.13 The number of offspring and this probability: (a) for zerogeneration unit and (b) for first-generation unit.

bk =

ak

a1 + a2 + a3

[8.20]

By implementing this kind of modeling, it is possible to calculate the average

value and distribution of the degree of polymerization of the polymer that

an arbitrary monomer unit in the system belongs to.

Formalism and calculation by probability generating function

The probability generating function (pgf) is an extremely useful tool for

studying the stochastic process. here, the pgf that denotes the distribution

of polymerization, w(q),isdefinedasfollows:

w(q ) = a1q 1 + a 2q 2 + . . . + a kq k + . . . = S a kq k

k=

=1

1

[8.21]

here, ak represents the probability of an arbitrary monomer unit in the

system belonging to a polymer of a certain degree of polymerization. k is the

expected value and ak is the probability. When the arbitrary unit is a zerogenerationunit,thepgfrelatingtothenumberoffirst-generationoffspring

can be described as:

F0(q) = a0 + a1q + a2q2 + a3q3

[8.22]

Next, the pgf relating to the number of offspring of first-generation units

can be described as:

F1(q) = b0 + b1q + b2q2

[8.23]

Furthermore, the pgf relating to the number of offspring of all units beyond

Computer simulation of network formation in NR

239

second-generationunitsisequaltothispgfoffirst-generationaccordingto

the Markov process assumption:

F1(q) = F2(q) = F3(q) = . . . = Fj(q) = . . .

[8.24]

Then, new pgf, U, which describes the total degree of polymerization

distribution of offspring is introduced (Fig. 8.14).

U (q ) = S b kq k

[8.25]

k =1

here, bk represents the probability of an arbitrary branch in the system having

k units. When the branch pgf U is applied, Eq. [8.21] from Eq. [8.24] is:

W(q) = qF0(U(q))

[8.26]

also, the branch pgf from Eq. [8.24] is:

U(q) = qF1(qF2(qF3 . . .) . . .)

= qF1(qF1(qF1 . . .) . . .)

[8.27]

Consequently, this can be described as the following recursive formula:

U(q) = qF1(U(q))

[8.28]

in other words, when applying the branch pgf, the tree pgf, w(q), can

be described in the simultaneous equations, [8.26] and [8.28], and each

average degree of polymerization and gelation point are derived from these

equations.

Thegelationpointisdefinedasaconversioninwhichtheweightaverage

degreeofpolymerizationdivergesinfinitelyandthereforecanbecalculated

as follows:

W(q) = qF0(U(q))

[8.29]

U(q) = qF1(U(q))

Ê ∂F0 (u )ˆ

ÁË ∂u ˜¯

u =1

Ê ∂W (q )ˆ

Mw = Á

=1+

Ë ∂q ˜¯ q –1

Ê Ê ∂F1 (u )ˆ

ÁË1 – ÁË ∂u ˜¯

q

ˆ

˜

u =1¯

[8.30]

U(q)

8.14 Branch pgf U(q) expresses the distribution of total number of

offspring.

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Chemistry, Manufacture and Applications of Natural Rubber

Ê ∂F (u )ˆ

1–Á 1 ˜ =0

Ë ∂u ¯ u–1

Ê

M n= Á

Ë

Ú

1

0

W (q ) ˆ

dq˜

q

¯

[8.31]

–1

[8.32]

if the pgf can be obtained, then not only can the average value for the degree

of polymerization be calculated, but also the variance:

ÏÔÊ ∂2W (q )ˆ ¸Ơ

Ê ∂2W (q )ˆ

Ê ∂W (q )ˆ

Var = Á

– ÌÁ

˝

˜

˜

Ë ∂q ¯ q =1 Ë ∂q ¯ q =1 ÓÔË ∂q ˜¯ q =1 ˛Ô

2

[8.33]

The distribution of degree of polymerization can be calculated from the

following equation using the Lagrange theorem:

z(qn)W(q) = z(qn–1)F0(u(q))

=

∂F1 (q ) ¸˘

1 È ∂n–1 ÏF 0 (q )(

n–1

)(F1 (q ))n–

1 – 1q

Í

Ì

˝˙

n

1

(n – 1)!Í∂q ĨƠ

F (q ) ∂q ˛Ơ˙˚

q =0

[8.34]

The cascade theory can be expanded naturally by applying vector pgf even

for complex systems with multiple types of units. in the case of an s-type

unit, the pgf and simultaneous equation are as follows:

W(q) = (W1(q), W2(q), . . . Ws(q))

(

= S p1 (X )q1x1q1x2 … q sxs, S p2 (X )q1x1q1x2 … q sxs, S ps (X )q1x1q1x2 …q sxs

X

X

X

)

[8.35]

W(q) = q Ÿ F0(U(q))

U(q) = q Ÿ F1(U(q))

[8.36]

here, the operator Ÿ represents the direct product,

(a) Ÿ (b) = (a1b1, a2b2, . . . asbs)

[8.37]

and F0i(U) and F1i(U) in

F0(U) = (F01(U), F02(U), . . . F0s,)

F1(U) = (F11(U), F12(U), . . . F1s,)

[8.38]

are the pgf relating to the i-typeunitoffspringforzero-generationandfirstgeneration units, respectively. The average degree of polymerization and

gelation point can be calculated following the same procedure as for Eqs

[8.30] and [8.31]. That is, if

Computer simulation of network formation in NR

241

Ê ∂Wq (q )ˆ

Ê ∂U q (q )ˆ

Ê ∂Fiq (U )ˆ

Wqp ∫ Á

U qp ∫ Á

Fiqp ∫ Á

[8.39]

˜

˜

Ë ∂q p ¯ q =1

Ë ∂q p ¯ q =1

Ë ∂U p ˜¯ u =1

then,

Wi = qi · F0i(U)

\

s

∂Wi ∂qi

Ê ∂F (U )ˆ Ê ∂uj (q )ˆ

=

· F0i (U ) + qi · S Á 0i

˜Á

˜

j Ë ∂uj ¯ Ë ∂q k ¯

∂q k ∂q k

[8.40]

where

Ê W1 W 2 ºˆ

1 ˆÊ

Ê1

ˆ Ê F01

U11

1

1

Á

˜

Á

˜Á

Á

˜

1

1

1

O

Á W2

˜

Á F02

˜ Á U 21

Á

˜ =ÁO  ˜ +Á

˜Á

˜

Á

˜ ÁÁ

Á  ˜Á 

˜

1

1

¯ Ë

ÁË Ws

˜¯ Ë

¯Ë

ˆ

˜

˜,

˜

˜

¯

[8.41]

and

Wik = d ik + F0kj [d ij – F11jij ]–1

[8.42]

is obtained. From here, the gelation point can be calculated from the following

equation:

d ij – F1ij = 0

[8.43]

The distribution of the degree of polymerization is obtained by multivariate

Lagrange series expansion:

F0i(U(q))

q1n1q 2n2 …q sns È ∂n1 + n2º++ns ÏÔ

n2

F (a )F

)F1111 (a )n1 F12

12 (a ) …F1s (a )

n1!n2!ºns! ÍỴ∂a1n1 ∂a2n2 …∂asns ÌĨƠ 0i

am

∂F1m (a ) ¸˘

d mg –

·

F1m (a )

∂ag ˝˛Ơ˙˙

˚ a =0

=S

[8.44]

Gordon et al. expanded cascade theory to apply the substitution effect of

reversible condensation and loop formation:

f

f –1

F0 (q ) = S S

i =0 j =0

f Ci g

i

(1 – g ) f –1f (i )qai qsj qwk

[8.45]

kajiwara introduced ‘path weighted function’, and succeeded in deriving

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Chemistry, Manufacture and Applications of Natural Rubber

scattering function and average hydrodynamic radius of polymer chain in

solution:

W(q) = qf0(1 – a + au1(q))f

um(q) = qfn(1 – a + aun+1(q))f–1

[8.46]

Here, we introduce a recently developed cascade theory.

Tree decomposition

The conventional theory was only applied to polymer with a tree-like structure

and without a closed circuit. By applying tree decomposition of graph

theory (Diestel, 2012), the cascade theory can be applied even for general

structures containing a closed circuit. An example of tree decomposition is

illustrated in Fig. 8.15.

With regards to the arbitrary vertex within the graph, the partial order

between parts can be determined using the partial order determined by

making the parts that contains this vertex as the root. When a Markov chain

is assumed on top of this partial order, the cascade theory can be applied.

That is, the cascade theory can also be applied to general graphs with closed

circuits using the tree decomposition.

Forest polynomial

The analysis object in conventional cascade theory was the degree of

polymerization or the polymer chain length. This was further expanded to

apply to conformation. First, the macromolecule configuration is expressed as

a rooted tree and a forest polynomial is defined that expands Eq. [8.21] using

the term T that corresponds one-to-one with the rooted tree structure:

W = Sa kT k

[8.47]

Here, ak represents the probability that the unit arbitrarily chosen in the

system belongs to the kth type macromolecule, Tk. Two examples are shown

in Table 8.2.

The variable is commutative for the terms in brackets:

Tl = q(qq (q(qq)) q) = q(qqq(q(qq)))

[8.48]

When the unit that is the root of the tree is regarded as zero-generation and

the other units are regarded as the offspring, this forest polynomial can be

calculated from the following simultaneous recurrence equations based on

the Markov branching process assumption:

Computer simulation of network formation in NR

243

(a)

t1

Vt1

Vt2

t2

(b)

8.15 Tree and tree decomposition. (a) A model tree of nine-mer.

The vertices and the edges, respectively, denote monomers and the

bonds between monomers. The symbols  and  indicate vertex

and root, respectively. (b) Tree decomposition of nine parts. The

representative vertices of the subgraphs are connected to the tree,

and t1 and t2 adduce representative vertices of the Vt1 and Vt2,

respectively.

Table 8.2 Trees and corresponding termsa

Tree

Term

Tk = (q(q(qq(q))q(q)))

Tl = (q(qq(q(qq))q))

a

Circle denotes a unit, straight line indicates connecting

link, and filled circle is a unit chosen randomly in each

system that is regarded as the root of each tree.

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Chemistry, Manufacture and Applications of Natural Rubber

W = qF0(U)

U = qF1(U)

[8.49]

U denotes the distribution of the tree branches and is the pgf for the

branches:

U = Sb k Bk

[8.50]

k

here, bk represents the probability that the branch to which the arbitrary

descendant belongs is the kth structural branch, Bk. Examples of sub-trees

and corresponding terms are shown in Table 8.3.

By classifying the terms in the recursive formula and forest polynomial

into an equivalence class according to purpose, a new recursive formula and

forestpolynomialisdefinedforthatclass.Furthermore,variousdistributions

canbecalculatedusinguniquelydefinedoperations:

W = S a k Tk

k

W = (q F0 (U ))

U = (q F1 (U ))

j

ỉỈ

Wj = S a j k Tj k

j

Wj = (qj Fj 0 (Uj )))

ỉỈ

k

Uj = (qj Fj 1 (Uj )))

[8.51]

The following is an example of sorting terms into equivalence class

according to purpose as shown in Table 8.4.

Case1:Allpartsareclassifiedthesame.

Case2:Classifiedaccordingtogeneration.

Table 8.3 Sub-trees and corresponding termsa

Sub-tree (branch)

Term

Bj = (q)

Bk = (q(q))

Bl = (q(q(qq)))

Bm = (q(qq(q)))

a

Circle with dot denotes the unit regarded as the ancestor of

the sub-tree.

Computer simulation of network formation in NR

245

Table 8.4 Corresponding terms transformed for each

classification

Case

Tk

Tl

1

2

3

4

q7

q0q12q23q3

q13q23q31

q02q12q13q221q22q31

q7

q0q13q2q32

q14q2q32

q03q 221q22q33q 241

Case3:Classifiedaccordingtotheconversionoffunctionalgroups.

Case4:Classifiedaccordingtogenerationandconversion.

InCase1,thedifferenceinthemacromoleculeconfigurationisnotdistinguished

and only the number of units is considered. Therefore, Case 1 is equivalent to

by generation. The corresponding forest polynomial is as follows:

W = S ck (P q sk (s ))

k

s

[8.52]

The fact that Case 2 is equivalent to kajiwara’s ‘path weighted function’ is

ascertained from the following relation. accordingly, the scattering function

can be derived by following the same procedure as kajiwara:

q s ~ q fs

[8.53]

Case 3 can be applied to analysis of the substitution effect of irreversible

polycondensationpolymerizationbecausetheunitsareclassifiedaccording

to conversion. Case 4 can, for example, be applied to the analysis of the

length distribution of an elastically active network chain because the units

areclassifiedaccordingtogenerationandconversion.

Formalism by rational generating function

By applying a rational function, various distributions can be expressed

and it is also possible to perform the calculation in the same way as for a

polynomial. That is, the pgf and simultaneous recursive equations:

W = a kq k + … +

P0 (U )

Q0 (U )

P (U )

U =q Ÿ 1

Q1 (U )

P(q )

P(q )

+ … + qm

Q(q )

Q(q )

[8.54]

W =q Ÿ

[8.55]

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Chemistry, Manufacture and Applications of Natural Rubber

and the gelation point and weight average degree of polymerization,

Ê Ê P (U )ˆ

ˆ

∂Wi ∂qi P0 (U )

=

·

+ q S Á ∂Á 0

˜ ∂U j ˜

j Ë Ë Q0 (U )¯

∂q k ∂q k Q0 (U )

¯

Ê ∂U j ˆ

·Á

Ë ∂q k ˜¯

Ê Ê P (U )ˆ

ˆ Ê ∂U j ˆ

∂Ui ∂qi P1 (U )

=

·

+ q S Á ∂Á 1

˜ ∂U j ˜ · Á ∂q ˜

j Ë Ë Q1 (U )¯

∂q k ∂q k Q1 (U )

¯ Ë k¯

[8.56]

are obtained by solving,

Ê ∂U j ˆ

ÁË ∂q k ˜¯

[8.57]

from equations and substituting this into the following equation:

Ê ∂W ˆ

Wik = Á i ˜

Ë ∂q k ¯ q =1

(MW)W = (m(Wik), M)

[8.58]

The distribution of the degree of polymerization,

U1 · Q1(U1, U2, . . . Un) = q1P1(U1, U2, . . . Un)

U2 · Q2(U1, U2, . . . Un) = q2P2(U1, U2, . . . Un)

[8.59]

Un · Qn(U1, U2, . . . Un) = qnPn(U1, U2, . . . Un)

can be calculated by applying Groebner base (Cox et al., 1997) and rearranging

the equations as follows:

ϕ1(q1, q2, . . ., qn, U1) = 0

ϕ2(q1, q2, . . ., qn, U1, U2) = 0

ϕn(q1, q2, . . ., qn, U1, U2, . . . Un) = 0

[8.60]

Part II

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9 Appendix: Basic concept of cascade theory

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