福井大学 学術研究院工学系部門 研究報告 第 70 巻 2022 年 3 月
Mem. Fac. Eng., Univ. Fukui, Vol. 70（ March 2022 ）59

Branched Polymer Architecture
Formed in Conventional and Living Emulsion Polymerization
Hidetaka TOBITA*(Received January 14, 2022)
Emulsion polymerization proceeds in a unique locus of polymerization, having confinedsubmicron space with a higher polymer concentration from the beginning of polymerization.
Assuming the ideal polymerization kinetics during Interval II in emulsion polymerization, thebranched architecture formed by the chain transfer reaction to the polymer is investigated by usinga Monte Carlo simulation method, both for the conventional and the living free-radical
polymerization (FRP). The conventional FRP leads to form a broad molecular weight distribution(MWD), while the living FRP gives a rather narrow MWD. The expected contraction ratio g of themean-square radius of gyration Rg2 of the branched polymer to that of a linear polymer for a given
number of branch points k is essentially the same, at least approximately, both for the conventional
and the living FRP, irrespective of the reaction condition, and a universal equation is proposed forthe g-k relationship. The magnitude of Rg2 can be represented by using the maximum span length
LMS, Rg2 = 0.172 LMS, both for the conventional and the living FRP. Similar relationships, Rg2 = a
LMS with a = 0.17–0.18 apply to various types of branched polymers, and could be considered as auniversal relationship.
Key Words : Branched Polymers, Chain Transfer to Polymer, Emulsion Polymerization, Living

Weight fraction of polymer
in the polymerization locus

Radical Polymerization, Molecular Weight Distribution, Radius of Gyration

1. Introduction
Polymerization is a reaction process in whichmonomers are converted to polymers, and therefore, theweight fraction of polymer increases just linearly withthe monomer conversion x, as shown by the dotted line1 



0


Conversion; 1

in Fig. 1 for usual bulk polymerization. On the otherFig.1 Polymer weight fraction development in thehand, the locus of polymerization in emulsion

polymerization locus during emulsion (solid line) andpolymerization is the polymer particle, which isbulk polymerization.

schematically shown in Fig. 2. After a short nucleation
period, called Interval I, the weight fraction of polymer    

in the polymer particle is approximately kept constantuntil the depletion of monomer droplets, due to the    

monomer transfer from the monomer droplets to the   polymer particles. The constant polymer weight fraction
period, which leads to a constant polymerization rate
period in a typical emulsion polymerization, is called                    * Department of Materials Science and Engineering

Fig. 2 Illustration of the emulsion polymerization during
Interval II.

60

Interval II. Assuming the nucleation period is small

depending on when the unit is incorporated into theenough, the weight fraction of polymer in the polymer

polymer chain. The units that are incorporated in theparticle changes as shown by the solid line in Fig. 1. In

earlier stage of polymerization are subjected to thethis article, the branched polymer formation due to thepolymer transfer reaction for a longer period of time,

chain transfer to polymer is investigated theoreticallycompared with those bound in the later stage ofduring Interval II.

polymerization. The branching density of a unit in chain
is a function of the time when the monomer turns into 

     





kfp



Fig. 3 Schematic representation of the process of chaintransfer to polymer in free-radical polymerization (FRP).
Fig. 3 shows the process of chain transfer to polymer,
leading to the long-chain branching. The rate Rfp of chaintransfer to polymer is represented by:

Rfp = kfp [R • ][M]0 xc(1)where kfp is the rate constant for the chain transfer to•polymer, [R ] is the radical concentration, [M]0 is theinitial monomer concentration, and xc is the weightfraction of polymer in the polymer particle, which ispolymer.
The total number n of monomeric units incorporated
into polymer in a polymer particle increases with time.
Suppose n = np at the present time. The expectedbranching density ρunit of a unit bound into polymer at n
= nb is given by the following equation.[1]ρunit (nb ,np ) = Cfp⎛n ⎞xcln ⎜ p ⎟ for Interval II.
1− xc ⎝ nb ⎠(4)

Fig. 4 shows the calculated result of the branchingdensity distribution represented by Eq. (4) at np = 1x106.
A large variation in the branching density distribution is
an important characteristic of emulsion polymerization,although the average branching density (= ρ) is kept
constant throughout Interval II.

equal to the monomer conversion at which the monomer
droplets are depleted, as shown in Fig. 1. In usualpolymerization in a homogeneous medium, Rfp is zero atconversion x = 0, because polymer is not present. On theother hand, Rfp is large from the beginning in emulsion
polymerization. Emulsion polymerization promotes thebranching reaction.
The rate Rp of polymerization is given by:•(

Rp = kp [R ][M]0 1− xc)

Fig. 4 Branching density distribution ρunit during Interval(2)where kp is the propagation rate constant.
The branching density ρ is defined as the fraction ofunits that bear a tri-branch point. The instantaneous
branching density is given by the rate ratio of Rfp and Rp,
which is given by:

ρ=

Rfp
Rp

= Cfp

II at np = 1x106, calculated from Eq. (4).
When discussing the branched polymer architecture,
the primary polymer chain[2] is a useful concept. The
primary polymer chain is a linear chain when all branch
points are severed. In the conventional FRP, eachprimary chain is formed instantaneously, therefore, Fig.

xc
1− xc

(3)

where Cfp (= kfp/kp) is the chain transfer constant topolymer. Equation (3) shows that the branching density
is kept constant throughout Interval II, investigated inthis article.
Although the branching density of the whole polymer
is kept constant, the branching density or the probabilityof possessing a branch point in a unit in chain is different,

4 shows that the primary chains that are formed in theearlier stage of polymerization possess a larger
branching density than those formed in the later stage ofpolymerization. The variation in the branching density isamong the primary chains in the conventional FRP.
On the other hand, in the living FRP, or thereversible-deactivation radical polymerization in the
IUPAC terminology,[3] the variation in the branchingdensity is observed along the sequence of a primarypolymer chain. Note that when the growing primary

61

chain causes the polymer transfer reaction, this primary

backbone polymer chain is increased from 0.2857 (top:chain becomes a dead primary chain, while a new

C1, L1) to 0.8333 (bottom: C4, L4).

growing primary polymer chain is formed from a radical
on a backbone chain.

Table 1. Calculation conditions
Pnp 3)
Pn 1)
ρ2)

Pb4)

In this article, the statistical properties, such as theRunmolecular weight distribution and the radius of gyration,

C1, L1

200

0.002

142.9

0.2857

C2, L2

500

0.002

250

0.5

C3, L3

1000

0.002

333.3

0.6667

C4, L4

1000

0.005

166.7

0.8333of the branched polymer molecules formed in theconventional and the living emulsion polymerization areinvestigated, seeking for newer methods to design andcontrol the branched polymer architecture.
2. Method

1)

The number-average chain length of the product polymers.

2)

The average branching density of the product polymer. 3) The

number-average chain length of the primary polymer chains.
4)

The Monte Carlo simulation method to generate thebranched polymer architecture of each polymer molecule
is described in detail in ref.[4] As described therein, thenumber fraction distribution Np(r) of the primarypolymer chains in emulsion polymerization during
interval II is typically given by the following mostprobable distribution.

Fig. 5 shows the weight fraction distribution of the
primarythefigure,

PDI

means

the

weight- and number-average, representing the degree of

(5)

Pnp

is thenumber-average chain length of the primary polymer
molecules.
For the living FRP, the termination reaction isneglected, and the average number of monomeric units
added during a single growth period is set to be 2, as wasdone earlier.[4,5]The statistical analysis is conducted when the numberof monomeric units polymerized in a polymer particle

conventional FRP, PDI = 2 for all C1–C4, because the
PDI of the most probable distribution is 2.
For the living FRP, the smooth low molecular weight
curve

polymer transfer reaction to form a distribution close to
the most probable distribution whose PDI is nearly 2.
5

5

4
3
2

0
1.0

2.0

PDI = 1.25

1.5

2.0

W(log10r)

2.5

3.0

2

0
1.0

3.5

PDI = 21.5

2.0log10r

C3
L3

1.0

PDI = 2

0.5
1.5

2.0

2.5

2.5

3.0

3.5

log10r

PDI = 1.71

1.5

C2
L2

PDI = 1.48

3

1

PDI = 2

2.5

0.0
1.0

4

C1
L1

1

number of polymer particles are simulated to determine

3.0

3.5

log10r

1.2

C4
L4

PDI = 1.92

0.8
0.4
0.0
1.0

PDI = 2
1.5

2.0

2.5

3.0

3.5

log10rFig. 5 Primary chain length distribution.

3.1 Calculation Conditions
The systematic analyses are conducted by using the
Table

have

most of the primary chains in L4 are subjected to thepolymerization experiment during Interval II. A large

in

that

the sharp high molecular weight peak. On the other hand,

3.0

3. Results and Discussion

chainsthe polymer transfer reaction, which is represented by

1x106 is 68 nm, which conforms to a normal emulsion

statistically valid properties.

primary

reaction. In L1, most primary chains do not experience

W(log10r)g/cm3, the diameter of a dried polymer particle at n =theexperienced chain stoppage through the polymer transfer

reaches n = 1x106. Assuming that the molecular weightof the monomer is 100, and the density of polymer is 1

represents

W(log10r)⎛ r ⎞1
exp ⎜ − ⎟Pnp
⎝ Pnp ⎠

shownIn

breadth of the chain length distribution. For thewhere r is the chain length, andconditions

chains.

polydispersity index, which is the ratio between theW(log10r)

N p (r) =The probability that the chain end of a primary chain isconnected to the backbone chain, named branching probability.

1.

C1–C4

are

the

conventional FRP, and L1–L4 are the living FRP.
The branching probability Pb, which is the probability
for a primary polymer chain end to be connected to the

3.2 Molecular Weight Distribution (MWD)
3.2.1 Conventional FRP
Fig. 6 shows the weight fraction distribution of the
product branched polymers for the conventional FRP at n
= 1x106. As the branching probability Pb increases, the62

distribution becomes broader, and PDI is as large as 109

chain transfer constant experimentally.[7]for C4. The distribution is broader than the random
branching of the same primary polymer chains,[4] and it
is shown that the present type of nonrandom distributionof branch points, represented by Eq. (4) and Fig. 4, leads
to a broader molecular weight distribution in theconventional FRP.
When Pb is large, as in the cases of C3 and C4,
another high molecular weight (MW) peak appears. Thishigh MW peak is formed because of the limitation of thesmall particle size. In the present simulation, the total
number of monomeric units incorporated into polymer
molecules in a particle is n = 1x106, and therefore, it is
impossible to form a polymer molecule whose chain
length (degree of polymerization) is larger than 1x106. In
fact, the polymer molecules in the high MW peak in C4
are the largest polymer molecule in each polymer
particle. These polymer molecules want to grow further,
but they cannot because of the limitation of the particle
size. This type of confined space effect may become
important in nonlinear emulsion polymerization.
Fig. 7 Double logarithmic plot of the number fraction
distribution N(r) of the polymers formed in theconventional FRP, C1–C4.
The red bimodal MWD shown in Fig. 8 is theexperimental

resultreported[8]

polymerization of ethylene,

for

the

emulsion

which is known to have alarge frequency of branching through chain transfer toFig. 6 Weight fraction distribution of the productpolymers at n = 1x106, for the conventional FRP, C1–C4.

polymer. A bimodal MW distribution, as in the case of
C4, is observed. When the MWD, W(log10M) is
converted to W(M) and plotted in the doble logarithmic

Another interesting characteristic of the conventional
FRP

conducted

inaconstant

polymer/monomer

condition is that the MWD of the high MW tail follows
the

power

law,

represented

by

the

following[6,7]

form, one obtains the dotted curve, showing a power-law
relationship for the important MW region. Pb = 1/1.23 =0.813 leads to give the chain transfer constant Cfp thatagrees reasonably well with the reported value.[7]weight-based distribution function.

W (r) ∝ r

−1 Pb

,

(6a)

or equivalently, in the number-based functional form:

N (r) ∝ r ()− 1 Pb +1.

(6b)Fig. 7 shows the doble logarithmic plot of the number
fraction distribution N(r) of C1–C4. The power-law
distribution represented by Eq. (6b) is confirmed,

Fig. 8 MWD of the emulsion-polymerized polyethylene,

including the bimodal distributions observed for C3 andreported in ref.[8] In the figure, M is the molecular

C4 shown in Fig. 6. Because the slope is related with Pb,

weight, and is related with chain length (degree ofthe power-law distribution can be used to determine thepolymerization) r by M = 28 r.

63

g=

3.2.2 Living FRP
Fig. 9 shows the simulated weight fraction distribution

2
Rg Branched.
2
Rg Linear(7)for L1–L4. Although the branching density ρ is the same

As a standard branched polymer structure, theas for the corresponding conventional FRP, for example

Zimm-Stockmayer equation[9] that describes the random

L1 and C1, significant broadening of the MWD, as wasbranching of the primary chains with the most probableobserved for the conventional FRP, is not found. In fact,

distribution has been used widely. The expected g-value

the distribution is narrower than the random branching offor the polymer molecules having k branch points is[4]

the same primary chains.

The polymers with extremely

large MW are not formed in the living FRP.

given by:[Zimm-Stockmayer]⎡⎛ k ⎞ 0.5 4k ⎤⎥g = ⎢⎜ 1+ ⎟ +9π ⎥⎢⎣⎝ 7 ⎠
⎦

−0.5

(8)

Note that the branched architecture formed in thepresent emulsion polymerization system is not random,
as shown by the branching density distribution in Fig. 4.
For the primary chain length distribution, the mostprobable distribution applies for the conventional FRP,
Fig. 9 Weight fraction distribution of the product6

polymers at n = 1x10 , for the living FRP, L1–L4.

but not for the living FRP, as shown in Fig. 5.
Fig. 10 shows the g-k relationship for C4. Each red dot
represents a pair of values for each polymer molecule, g

Table 2 shows how the PDI changes with branching.

and k. The blue open circles show the average within theInterestingly, the PDI does not change significantly inintervals of Δk, representing the expected g-value for a

the living FRP, which is a notable difference in contrast

given k.

to the conventional FRP. Note that the primary chain
lengthdistribution

straightforward

cannotmannerbeindetermined
a

usual

in

a

emulsionpolymerization experiment, and all one can obtain is the
MWD of the product polymers. It may be difficult tofind the existence of branches, judging solely from theproduct polymer distribution because the PDI values areso small.
Fig. 10 Relationship between g and k for C4.
Table 2 Polydispersity index (PDI) of the primary chains
and the product branched polymers, for L1–L4.

Fig. 11 shows the expected g-k relationships for thePrimary chains

Product polymers

conventional FRPs. It is interesting to note that the

L1

1.25

1.27

relationships stay nicely on a single universal curve,

L2

1.48

1.51

irrespective of the calculation condition.

L3

1.71

1.78

The black solid line shows the Zimm-Stockmayer

L4

1.92

2.15

theory, represented by Eq. (8), and the conventional FRP
Because the primary chains in C1–C4 follow the most

3.3 Radius of Gyration
2

The mean-square radius of gyration Rg describes thespatial dimension of the polymer. To highlight the effect2of branching in polymer, the g-ratio of Rg of the
branched polymer to that of a linear polymer is a usefulmeasure.

shows smaller g-values, i.e., smaller radius of gyration.
probable distribution, which is the same as theassumption used in the Zimm-Stockmayer theory, thedifference must come from the nonrandom connection of
the primary chains. The compact architecture formed in
the conventional FRP, compared with the random
branching, could be rationalized, based on the branching

64

density distribution shown in Fig. 4. In the conventional
FRP, each primary chain is formed instantaneously, andthe primary chains formed in the earlier stage ofpolymerization are expected to possess larger values of[Random branch of the Schulz-Zimm chains]
0.5
⎡⎛
⎤
k ⎞
4k
⎥
g = ⎢⎜ 1++⎟(1+ 4PDI )π ⎥⎦⎢⎣⎝ 5 + PDI ⎠

−0.5

(9)

branching density. These primary chains are likely toNote that the Schulz-Zimm distribution, or the Gammabear a large number of branch chains, leading to form a

distribution in the mathematical term, is a generalizedcore region, resulting in an overall star-like architecture.

form of the most probable distribution, and Eq. (9)This would be the reason for showing smaller g-values.

reduces to Eq. (8) when PDI = 2.

-0.5

for the large k

The primary chain length distribution becomesregion, while the simulation results show g  k-0.8, as

broader in the order from L1 to L4, and therefore, it

Incidentally, Eq. (8) represents g k

represented by the dotted line in Fig. 11b. This finding

could be expected that the g-value increases from L1 towill be used to find the universal g-k relationship.

L4. However, the obtained g-k relationship is essentiallythe same, as shown in Fig. 12.

Fig. 11 Relationship between g and k for theconventional FRP, C1–C4. (a) Normal plot. (b) Double

Fig. 12 Relationship between g and k for the living FRP,

logarithmic plot.

L1–L4. (a) Normal plot. (b) Double logarithmic plot.

Fig. 12 shows the g-k relationships for the living FRPs,

To remove the difference of primary chain length

L1–L4. A slight increase by the branching probability Pb

distribution, the g-k relationships for the random

is observed, however, the difference is rather small, andbranched polymers of L1–L4 conditions shown in Fig. 5

it could be considered essentially the same g-k

are determined by using the MC simulation. Fig. 13

relationship for all conditions.

shows the comparison. When the branching probability

The solid black curve shows the Zimm-Stockmayer

Pb is small (L1), the g-value is larger than thetheory, and the g-k relationship appears to be close. On

corresponding random branch case, but the difference

the other hand, this seemingly close relationship is justbecomes smaller as Pb increases, and the g-value iscoincidental,

smaller than the random branch in L4.

becausetheprimary

chain

lengthdistribution is much different from the most probable

Incidentally, the black solid curves for the random
branched polymers in Fig. 13 are calculated from Eq. (9),distribution, as shown in Fig. 5.
It is already known that the narrower primary chain

and they agree with the MC simulation results well, evenlength distribution leads to a smaller g-value in thethough the primary chain length distribution assumed is[10]

For the primary chainsdifferent from the Schulz-Zimm distribution. Equationthat follow the Schulz-Zimm distribution, the following(9) could be used to estimate the g-values of the random

equation was proposed.

branched polymers in general.

random branched polymers.

65

As the branching probability Pb increases, the primarychain length distribution becomes broader, which would
make the g-value larger. On the other hand, a larger Pb
tends to reduce the above enlarging effect caused by thebranching density distribution along the chain. In thecase of L4 in which most of the primary chains
experience chain transfer reaction, the branching density
distribution is found among the primary chains, notalong the chain. The g-value is reduced by the branchingdensity distribution in the case, L4.
3.4 Universal g-k RelationshipAs shown in Fig. 11 and Fig. 12, it appears that while
the g-value for the conventional FRP is significantly
smaller than the Zimm-Stockmayer equation, the g-value
for

the

livingFRP

israther

close

to

the

Zimm-Stockmayer relationship. On the other hand, it isworth noting that the range of the k-values is muchdifferent. Even though the average branching density is
the same for the corresponding conventional and living
Fig. 13 Comparison of the g-k relationship. The primarychain length distribution is set to be the same for thecorresponding cases.

FRP, huge polymer molecules are not formed in theliving FRP, as shown in the weight fraction distribution,
Fig. 9. The branch points are distributed among polymer
molecules much more evenly in the living FRP.

For L4, most primary chains experience the chain
stoppage due to the chain transfer reaction, as can beguessed from the primary chain length distribution
shown in Fig. 5. In this case, the situation is similar tothe conventional FRP. The primary chains formed earlier
are expected to possess a larger number of branch chains,
and a core region could be formed. It is reasonable toconsider that more compact architecture is formed,
compared with random branching.
On the other hand, in L1, most primary chains do notexperience chain stoppage, and the branching density
distribution shown in Fig. 4 is expected to be observed
along the sequence of a primary chain. The probability ofpossessing a branch point is higher for the units locatedcloser to the starting point. If these earlier incorporated
units are subjected to the chain transfer reaction at theearlier stage of polymerization, the length of theconnected branch chains could be long, because they
have enough time for growth. This type of event would
introduce a long linear polymer part in the formed
branch polymer molecule, which makes the radius of
gyration larger. This would be the reason for obtaining alarger g-value, compared with the random branching.

Fig. 14 Universal g-k relationship found for both theconventional and living FRP. (a) Normal plot. (b)Double logarithmic plot.
Fig.14 shows the g-k relationship for C4 and L4,plotted on the same sheet. Both relationships agree well.
Note that the g-k relationships of C1–C4 lie on a single
curve (Fig. 11), and in addition, L1–L4 agree well (Fig.
12). Therefore, the g-k relationships for all cases can be
described by a single universal curve, i.e., the black66

dotted curve, which is given by the following empirical

0.172 LMS applies both for the conventional and living

equation.
[Emulsion polymerization during Interval II]

FRP.

0.5
⎡⎛
k ⎤
k⎞
g = ⎢⎜ 1+ ⎟ + ⎥5π ⎥⎢⎣⎝ 12 ⎠
⎦

−0.8

(10)

The above universal relationship for the emulsion
polymerization during Interval II, which is applicable
both for the conventional and living FRP, is proposedhere.
2

3.5 Relationship between Rg and the Maximum Span
Length, LMS

Fig. 16 Relationship between Rg2 and LMS for the casewith C4.

In a branched polymer molecule, there are many chain
ends, as shown in Fig. 15. Within various combination of
two chain ends, the longest one is named the maximum
span chain, whose length is described as LMS. It isreasonable to consider that LMS has a dominant effect on
the mean-square radius of gyration, Rg2.

Fig. 17 Relationship between Rg2 and LMS for the
Fig. 15 Schematic representation of the maximum span

conventional FRP, C1–C4.

chain (red), whose length is represented by LMS.
In this article, LMS is represented by the number ofmonomeric units, and therefore, Rg2 used here is the one
when each monomeric unit is considered as a freely
rotating segment. Obviously, a segment consisting of any
number of monomeric units can be used, as long as thenumber of segments is large enough. The above
definition (one monomeric unit forms a segment) is2

needed only for the absolute values of Rg and LMS
plotted on the figures.
Fig. 16 shows the Rg2-LMS relationship. Each red dot
represents a pair of values for each polymer molecule,Rg2 and LMS. The blue open circles show the average
within the intervals of ΔLMS, representing the expected

Fig. 18 Rg2-LMS relationship for the living FRP, L1–L4.
Essentially the same linear relationship is valid forvarious types of branched polymers, as summarized inTable 3. The relationship,

Rg 2 LMS = 0.17 – 0.18

(11)

Rg2-value for a given LMS. Nice linear relationship isapplies universally for various types of branchedobserved.

polymers that include randomness in their architecture.

Fig. 17 shows the expected Rg2 for the conventional
FRPs. All C1–C4 fall nicely on a single line, Rg2 = 0.172
LMS.
2

Fig. 18 shows the Rg -LMS relationships for the living
FRPs. Again, all the relationships for L1–L4 fall on thesame line, Rg2 = 0.172 LMS. The relationship, Rg2 =The value of a is a little larger than 1/6 = 0.167 for linear
polymers, which implies that the polymer chains otherthan the maximum span chain contribute slightly toincrease the a-value.
In order to synthesize branched polymers with large
radius of gyration, one should design a branched

67

architecture whose maximum span length is large.

relationship between the mean-square radius of gyration

Because LMS is a one-dimensional property, it is much

Rg2 and the maximum span length LMS is represented byeasier than directly imagining complex 3D architectures

Rg2 = 0.172 LMS. For various types of branched polymers,

for the molecular design.

from the polymers with long-chain branches to thehyperbranched polymers synthesized through various

Table 3 Linear relationship Rg2 = a LMS, found for

types

various types of polymers.

relationship applies. [Eq. (11)]aPolymer type
Emulsion polymerization during Interval II:
Conventional FRP with chain transfer to

0.172

reaction/reactor,

thefollowing

simpleRg 2 LMS = 0.17 – 0.18
For example, to produce branched polymers having alarge radius of gyration, one should think of a production

polymer
Emulsion polymerization during Interval II:

0.172

Living FRP with chain transfer to polymer
Random branching of polymer chains
Hyperbranchedpolymers

formed

[10]

0.178
in

in a batch reactor or in a CSTR
Hyperbranchedpolymers

length, LMS.
The present work would shed new light on the

0.18

polymers with controlled architecture.

[11]

formedconducted in a batch reactor or in a CSTR

Referencesin

self-condensing vinyl polymerization (SCVP)

0.18

[12]

Linear polymers[9]0.167[11,12]0.5

4. Conclusions
The branched architecture formed in the conventional
and living emulsion polymerization that involves chaintransfer to polymer was investigated theoretically. The
simulations were conducted for Interval II, where thepolymer/monomer ratio is kept constant.
The MWD formed in the conventional FRP can bevery broad, and a bimodal MWD could be formed when
the branching probability Pb is large enough. The highMW tail follows the power law, W(r)  −αα = 1/Pb.
Controllable power-law distribution can be obtained byusing the conventional FRP.

The living FRP leads to form rather narrow MWD.
One may overlook the formation of branches, judging
solely from the formed MWD.
Both for the conventional and living FRP, the g-k
relationship is represented by the following universal
relationship [Eq. (10)], which gives smaller g-value
compared with the Zimm-Stockmayer formula.
0.5
⎡⎛
k⎞
k ⎤
⎢
g = ⎜ 1+ ⎟ + ⎥5π ⎥⎢⎣⎝ 12 ⎠⎦process that leads to give larger values of maximum span

production technologies for various types of branchedsteppolymerization of AB2-type monomer conducted

Dendrimer

of

−0.8

Both for the conventional and living FRP, the[1] H. Tobita: Polym. React. Eng., 1, 357-378 (1993) .
[2] P.J. Flory: Principles of Polymer Chemistry,
Cornell University Press, Ithaca, NY, Chap. 9(1953).
[3] A.D. Jenkins, R.G. Jones and G. Moad: Pure Appl.
Chem., 82, 483-491 (2010).
[4] H. Tobita: Processes, 6, 14 (2018) [22 pages].
[5] H. Tobita: Macromol. Theory Simul. 28, 1900018(2019) [11 pages].
[6] H. Tobita: e-Polymers, 4, 878-887 (2004).
[7] H. Tobita: Macromol. Mater. Eng., 290, 363-371(2005).
[8] W.H. Starkweather, Jr. and M.C. Han: J. Polym.
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