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
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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
+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˙˚
Ỵ
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
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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]
8.9.2 Generalized cascade theory
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:
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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.
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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
conventionalcascadetheory.InCase2,themacromoleculeunitsareclassified
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
Applications of natural rubber