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Branched polyesters: recent advances in synthesis and performance
Matthew G. McKeeb, Serkan Unala, Garth L. Wilkesb, Timothy E. Longa,*
aDepartment of Chemistry, Virginia Polytechnic Institute and State University, 124A Davidson Hall, Blacksburg, VA 24061, USA
bDepartment of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
Received 28 September 2004; revised 12 January 2005; accepted 12 January 2005
Abstract
The synthesis, characterization, physical properties, and applications of branched polyesters are discussed. This review
describes recent efforts in the synthesis of statistically and tailored branched systems, and performance advantages compared to
linear counterparts. In particular, an emphasis is placed on long-chain branching, where the branches are sufficiently long
enough to form entanglements. Step-growth polymerization methodologies that employ various combinations of multi and
mono-functional groups to achieve different levels of branching are reviewed in detail. The performance of branched
polyesters, including behavior in dilute and semi-dilute solutions, and melt and solid-state properties are discussed. The
implications of topological parameters including branch length, number of branches, and branching architecture on rheological
performance are also reviewed. Although the majority of this review focuses on the synthesis and rheological behavior of
branched polyesters, some discussion is devoted to the influence of branching on solid-state properties, sub-micron fiber
formation, and controlled biodegradation for drug-delivery applications. Finally, a perspective of future directions in highperformance applications for branched polyesters is provided.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Branching; Polyesters; Rheology; Entanglements; Crystallization; Step-growth polymerization
Contents
1. Scientific rationale and perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
2. Synthesis of long-chain branched polyesters via step-growth polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
2.2. Synthesis of branched polyesters via A2 and B2 monomers in the presence of An or Bn (nO2) monomers 511
2.3. Synthesis of branched polyesters via A2 and B2 monomers in the presence of An or Bn (nO2) Monomers
and a monofunctional endcapping reagent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
2.4. Synthesis of branched polyesters via AB monomers in the presence of A2B monomers . . . . . . . . . . . . . . 514
Prog. Polym. Sci. 30 (2005) 507539
www.elsevier.com/locate/ppolysci
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doi:10.1016/j.progpolymsci.2005.01.009
* Corresponding author. Tel.: C1 540 231 2480; fax: C1 540 231 8517.
E-mail address: [email protected] (T.E. Long).
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3. Characterization of branched polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
3.2. Contraction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
3.3. Endgroup analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
4. Influence of branching on melt rheological properties: model systems and long-chain branched polyesters . . . . . 518
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
4.2. Number of branches per chain and branch length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
4.2.1. Randomly branched polyesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
4.2.2. Star-branched polyesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
4.2.3. H-shaped and comb-branched polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
4.3. Flow activation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
5. The influence of branching on solution rheology properties in the semidilute regime . . . . . . . . . . . . . . . . . . . . . 526
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
5.2. Effect of branching on the entanglement concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
5.3. Recent advances in electrospinning of long-chain branched polyesters . . . . . . . . . . . . . . . . . . . . . . . . . . 527
6. Influence of branching on thermal properties of polyesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
6.2. Glass transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
6.3. Melting behavior and quiescent crystallization growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
6.4. Controlling the biodegradation of aliphatic polyesters through branching . . . . . . . . . . . . . . . . . . . . . . . . . 532
7. Conclusions and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
1. Scientific rationale and perspective
Branched polymers are characterized by the
presence of branch points or the presence of more
than two end groups and comprise a class of polymers
between linear polymers and polymer networks [1].
Although undesirable branching can occur in many
polymerization reactions, controlled branching is
readily achieved. In fact, numerous studies on polymer
structure-property relationships have shown that
branched polymers display enhanced properties and
performance for certain applications [2]. Long-chainbranched polymers offer significantly different physi-
cal properties than linear polymers and polymer
networks. For example, a low concentration of long
chain branching in the polymer backbone influences
melt rheology, mechanical behavior, and solution
properties, while large degrees of branching readily
affects crystallinity [3,4]. The strong influence of only
one long chain branch per chain can be visualized by
looking at Fig. 1. The slip-links along the polymer
backbone represent entanglements with other chains.
The linear polymer is free to diffuse along a tubeimposed by other chains, while it is obvious from
Fig. 1b that the mobility of the long-chain branched
polymer is restricted, and must diffuse through
some other mechanism. Thus, it is not surprising that
long-chain branched polymers exhibit very different
properties where chain entanglements play a role.
(a) (b)
Fig. 1. Cartoon representing entangled linear chains (a), and long
chain branched chains (b). The slip links represent entanglements
due to other polymers.
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It is widely documented that a high degree of
branching in a polymer backbone provides enhanced
solubility, lower viscosity and lower crystallinity, for
the case of symmetric chains that readily crystallize,than a linear polymer of equal molecular weight [5].
Therefore, a fundamental understanding of branching
and how it influences polymer properties is essential
for tailoring a polymeric material for high perform-
ance applications. Numerous types of branched
polymers can be prepared using different polymeriz-
ation techniques. In a living polymerization, multi-
functional initiators or multifunctional linking agents
yield well-defined star-branched polymers. Alkyl-
lithium initiators are particularly efficient types of
multifunctional initiators, and polyfunctional silyl
halides are highly efficient multifunctional linking
agents [2]. Comb polymers, which contain extensive
branching along the polymer backbone, are syn-
thesized in the presence of a polyfunctional coupling
agent. Polyfunctional or multifunctional monomers of
a functionality greater than two result in randomly
branched polymers. Randomly branched polymers are
often prepared by step-growth or chain polymeriz-
ation in the presence of a multifunctional comonomer
[1]. Highly branched (hyperbranched) polymers are
prepared without gelation via the self condensation of
an ABx monomer containing one A functionalgroup, and two or more B functional groups that
are capable of co-reacting. Unlike dendrimers, which
exhibit a regular, tree-like branch structure from a
central core, hyperbranched polymers contain linear
segments (defects) due to a more randomly branched
architecture. Hyperbranched polymers are generally
produced more easily than dendrimers and exhibit
several similar properties [6,7].
The effect of branching on polymers prepared by
chain-growth polymerization and single site catalyzed
polymerizations has received significant attention.However, structure/property relationships for
branched polyesters are limited and further studies
are needed [8]. Polyesters offer good mechanical and
thermal properties and high chemical resistance at
relatively low cost. Many polyesters, such as poly
(ethylene terephthalate)s (PET), polycarbonates, bio-
degradable aliphatic polyesters, and liquid crystalline
polyesters are commercially available [9]. PET is
utilized for a wide range of applications including
injection-molding and blow-molding [10]. Processing
of some polyesters, however, is limited due to
insufficient melt strength and melt viscosity. For
example, while aliphatic polyesters such as poly
(butylene adipate) (PBA) and poly(butylene succi-nate) (PBS) decompose rapidly under natural environ-
mental conditions and are replacing some commodity
polymers due to environmental concerns, processing
these resins is often difficult due to low melt strength
and melt viscosity [1113]. Thus, many researchers
have focused on modifying polyesters for enhanced
melt strength and melt viscosity by introducing long
chain branches into the polyester backbone. In this
review, the synthetic methods for preparing various
long-chain branched polyesters are reported. More-
over, the influence of branching on polyester proper-
ties for new high performance applications is
discussed.
2. Synthesis of long-chain branched polyesters
via step-growth polymerization
2.1. Introduction
Multifunctional comonomer branching agents are
introduced into polycondensation reactions to obtainlong-chain branched polyesters. Unlike short chain
branches (SCB), a long chain branch (LCB) is long
enough to entangle with other chains in the melt and
concentrated solutions thereby drastically altering the
flow properties. The critical molecular weight (Mc) is
the minimum molecular weight at which a polymer
chain entangles, as often measured by the molecular
weight dependence of viscosity [14]. The value Mcseparates two regimes in the dependence of zero shear
rate viscosity (h0) on weight average molecular
weight (Mw
) for linear chains. Below Mc
the value
ofh0 scales directly with Mw and above Mc h0 scales
with M3:4w . The value of Mc for a given polymer is
directly related to the entanglement molecular weight
(Me), which is typically determined from the plateau
modulus G0N as shown in Eq (1),
MeZrRT
G0N(1)
where r is the polymer melt density, R is the gas
constant, and T is the absolute temperature. Fetters
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et al. relatedMc andMe through the packing length (p),
which is proportional to the cross-sectional area of a
polymer chain [15]. Values ofMc were reported in the
literature for several linear polyesters, including PET(3300 g/mol), poly(decamethylene succinate) (4600 g/
mol), poly(decamethylene adipate) (4400 g/mol), and
poly(decamethylene sebacate) (4500 g/mol) [16]. The
parametersMc andMe are discussed further in Sections
4.1 and 4.2 of this review.
Hudson et al. showed that long-chain branching in
the polymer backbone permits control over the
rheology of the polymer [17]. More recent studies
on the modification of polyesters with long-chain
branching have involved the use of PET, an
engineering thermoplastic, with good thermal and
mechanical stability, high chemical resistance, and
ease of processing [10,18]. Early work by Manaresi et
al. describes the preparation of long-chain branched
PET using low levels of trimesic acid [19]. Intrinsic
viscosity measurements and the extent of reaction
were reported along with the degree of branching.
It is well known that polycondensation reactions
with multifunctional comonomers may form an
infinite molecular weight polymer network, or gel,
above a certain multifunctional comonomer concen-
tration or at high conversions. The onset of gelation
occurs at a critical point of conversion during thepolymerization, and is dependent on the degree of
functionality and the concentration of the multi-
functional (fO2) branching agent. For example,
polyester networks are prepared using dicarboxylic
acids and tri- or tetrafunctional monomers [20]. The
critical extent of reaction (ac) at which a polymer is
predicted to form a gel is shown in Eq. (2).
acZ1
rCrpfK21=2(2)
This equation is valid for polymerization mixtures
with bifunctional A and B monomers and a multi-
functional A monomer. In Eq. (2), r is the ratio of A
functional groups to B functional groups and p is the
ratio of A functional groups with fO2 to the total
number of A groups. Low concentrations of multi-
functional comonomers are used at low conversions to
obtain long chain branching, and this method has
yielded low molecular weight polymers. Neff et al.
suggested the use of a monofunctional comonomer
together with bifunctional and multifunctional mono-
mers to overcome the gelation problem in high
multifunctional comonomer concentrations or athigh conversions [21]. Manaresi et al. were first to
report the preparation and characterization of
PETs synthesized in the presence of a high content
(O1 mol%) of trifunctional comonomer (trimethyl
trimesate), as well as monofunctional comonomers
(methyl 2-benzoylbenzoate) [22]. Rosu et al. reported
branched PETs using multifunctional and monofunc-
tional comonomers and subsequent solid-state pol-
ymerization was employed to increase the molecular
weight of the final product [23].
Jayakannan and Ramakrishna synthesized high
molecular weight branched PETs through the copo-
lymerization of an A2 monomer with small amounts
of an AB2 monomer [24]. However, as discussed in
detail later in this review, insoluble crosslinked
polymers were obtained at higher conversions.
Hudson et al. synthesized and characterized branched
PETs to study the balance between the branching
reagents and endcapping reagents [17]. The objective
of their study was to examine various branching
agents used for PETs and other polyesters and their
influence on polymer properties, both with and
without an endcapping reagent. Molecular weightwas controlled via endcapping reagents on branched
polyesters using a variety of branching agents.
More recently, Yoon et al. studied the effects of
multifunctional comonomers such as trimethylo-
lethane (TME) and pentaerythritol on the properties
of PET copolymers [18]. Molecular weights increased
with increasing comonomer content while the mole-
cular weight distribution broadened. Although solid-
state mechanical properties did not differ significantly
from linear analogues, the branched copolymers
exhibited earlier shear-thinning onset in the meltcompared to linear PET. Moreover, the crystallization
rates of the copolymers decreased with increasing
comonomer content as would be expected. Similar to
branched PETs, branched poly(butylene isophthalate)
(PBI) and poly(butylene terephthalate) (PBT) were
synthesized and characterized to investigate their melt
and crystallization properties. Linear and branched
PBIs were synthesized from dimethylisophthalate
(DMI) and 1,4-butanediol (BD) in the presence of
trifunctional comonomers [25]. Branched PBTs were
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synthesized by incorporating the trifunctional como-
nomer, 1,3,5-tricarboxymethylbenzene [26].
High molecular weight branched aliphatic polye-
sters such as poly(ethylene succinate), poly(butylenesuccinate) (PBS), and poly(butylene adipate) (PBA),
known as Bionollee polymers, were also prepared
[27]. Bionollee polymers are used in a variety of
applications, including film blowing, blow molding,
extrusion coating and extrusion foaming [28]. Han
et al. described synthetic conditions and thermal and
mechanical properties for high molecular weight
branched PBAs [11]. Ramakrishnan et al. reported
the synthesis of a series of branched thermotropic
liquid crystalline polyesters and their structural
features [29]. Other novel branched polyesters
such as branched poly(3-hydroxy-benzoates) and
poly(4-ethyleneoxy benzoate) were synthesized by
Kricheldorf et al. and Ramakrishnan et al., respecti-
vely, [30,31].
2.2. Synthesis of branched polyesters via A2 and B2monomers in the presence of An or Bn (nO2)
monomers
The most common method for synthesizing
branched polyesters is via the addition of small
amounts of tri- or tetrafunctional comonomers to thepolymerization. Manaresi et al. first reported the
synthesis of branched PETs from dimethyl terephthal-
ate (DMT) and ethylene glycol (EG) using a trifunc-
tional branching agent, trimethyl trimesate (Fig. 2)
[19]. To prevent gelation, only small amounts (!2%)
of the trifunctional branching agent were used and the
influence of long chain branching on PET properties
was also reported. Although Manaresi et al. did not
report the absolute molecular weights of the polymers,
intrinsic viscosities in o-chlorophenol at 25 8C and the
extents of reaction by end-group analysis were
reported. After ester-interchange, only one type of
functional group remained, i.e. the hydroxyl group of
the hydroxy ethyl ester (Fig. 3). In the subsequent
polycondensation step, high molecular weight PET is
formed via the evolution of ethylene glycol. (Fig. 4)Weisskopf used different trifunctional agents to
synthesize high molecular weight branched PETs
[32]. Trimethylolpropane (TMP) was used as a
trifunctional branching agent and pentaerythritol
was a suitable tetrafunctional branching agent.
Trimethylolethane (TME) and trimesic acid were
also used as multifunctional comonomers (Fig. 5).
Hess et al. recently described the syntheses of both
linear and branched PETs [33]. Branched PETs were
obtained via the ester-interchange route starting from
DMT and a 2.5 M excess of EG. The reactions were
performed in a stainless-steel reactor with different
amounts (0.07 to 0.43 mol% with respect to DMT) of
trimethylolpropane (TMP) (branching agent, Fig. 5a)
present during the transesterification step. Transester-
ification was catalyzed with the addition of manga-
nese acetate at a maximum temperature of 230 8C.
Following transesterification, polycondensation was
catalyzed by antimony acetate at a maximum
temperature of 290 8C under vacuum.
Yoon et al. synthesized branched PETs in a similar
manner with TME as a branching agent at concen-
trations from 0.04 to 0.15 mol% [18]. Titanium
isopropoxide was used as the catalyst for the
polycondensation reaction. High molecular weight
PET copolymers were obtained with broad molecular
weight distributions. The thermal properties of the
copolymers were not significantly influenced by the
comonomers due to the low concentrations, however
the branched PET displayed enhanced zero shear rate
viscosity (h0) and shear thinning behavior. In a similar
fashion, branched PBI and PBT samples were prepared
O
O O
OHO
OH
OO
O
OO
O
(a) (b) (c)
Fig. 2. Bifunctional and trifunctional monomers used in the
synthesis of branched PET. (a) dimethyl terephthalate (DMT),
(b) ethylene glycol (EG), and (c) trimethyl trimesate (trimethyl
1,3,5-benzenetricarboxylate) (TMT).
COOCH2CH
2OH
COOCH2CH
2OH
COOCH2CH
2OHHOCH
2CH
2OOC
COOCH2CH
2OH
Fig. 3. Hydroxy ethyl esters formed after the ester-interchange step
during the polycondensation of PET.
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using A2, B2, and A3/B3 type monomers. Branched
PBIs were synthesized with the trifunctional branching
agent tris(hydroxyethyl) isocyanurate (THEIC) during
the polymerization reaction of dimethyl isophthalate
(DMI) with 1,4-butanediol (BD) in the presence of aTi(OBu)4 catalyst (Fig. 6) [25]. The branched PBIs
were prepared via a two step polycondensation
reaction. In the first step, the reaction temperature
was raised from 140 to 200 8C and held at 200 8C until
about 90% of the theoretical amount of methanol was
collected. In the second step, the pressure was reduced
and the temperature was maintained in the range of
200230 8C. The temperature was maintained lower
than normally employed for polyesters, such as PBT, to
prevent side reactions. Compositional and structural
characterization included elemental analysis, mass
spectroscopy, 1H NMR spectroscopy, HPLC, and end
group analysis. Linear and branched PBTs were also
recently synthesized using DMT and BD as bifunc-
tional monomers and trimethyl trimesate (TMT) as a
trifunctional comonomer [26]. Using titanium tetra-
butoxide at 250 8C in the second step yielded randomly
branched PBTs [28].
Han et al. synthesized high molecular weight
branched PBAs from aliphatic dicarboxylic acids and
glycols in the presence of glycerol or pentaerythritol
[11]. The influence of reaction parameters such as
catalyst concentration, reaction time, temperature,
and concentration of branching agent on molecular
weight was examined. These branched PBAs were
prepared via the synthesis of linear PBA from adipicacid and BD in the presence of a titanium(IV)
isopropoxide (TIP) catalyst and a triethylamine
(TEA) cocatalyst. The resulting linear polymer was
reacted with adipic acid in the presence of TIP to
obtain prepolymers with carboxylic acid end groups.
In a second step, the carboxylic acid terminated PBA
prepolymers were condensed with the branching
agent (glycerol or pentaerythritol) in the presence of
TIP to obtain branched PBS. Han et al. studied
molecular weight with respect to multifunctional
comonomer concentration and showed that both the
molecular weight and the molecular weight distri-
bution of branched PBAs increased with increasing
concentration of glycerol up to 0.6 wt% relative to the
PBA prepolymer. The gel content of the branched
PBAs also increased with increasing glycerol con-
centration up to 0.6 wt%. Surprisingly, 0.9 wt% or
more glycerol resulted in lower gel content and lower
molecular weights. The authors did not offer an
explanation for this dependence of gel content and
molecular weight on branching content.
C C
C
O O
OCH2CH2OOCH2CH2O CC
O O
CC
O O
O
x y
z
OCH2CH2O C
O
C
O
Fig. 4. Structure of randomly branched PET.
(c) (d)(b)(a)
OOH
O OH
OHO
HO OH
OH
OH
HO
HO
OH
OH
HO OH
Fig. 5. Trifunctional and tetrafunctional branching agents used in the synthesis of branched PET. (a) trimethylolpropane (TMP),
(b) pentaerythritol, (c) trimethylolethane (TME), and (d) trimesic acid (TMA).
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When branched PBAs were prepared with either
glycerol or pentaerythritol, it was found that the
molecular weight of branched PBAs with pentaery-
thritol was higher due to the higher degree of
functionality. The introduction of a branching agent,
TMP, to the polycondensation system of succinic acidand BD resulted in high molecular weight randomly
branched poly(butylene succinate) (PBS) [34]. The
esterification was conducted using a 1.0 to 1.1 ratio of
succinic acid to BD under nitrogen in the presence of
a titanium isopropoxide catalyst. The temperature was
raised from 140 to 200 8C as water was removed. The
ensuing polycondensation step was performed by
introducing 0.10.5 wt% of TMP to the reaction
mixture at 140 8C. The reaction temperature was
raised to 240 8C and the reaction was completed at a
pressure less than 1 Torr. Absolute molecular weightsand molecular weight distributions were determined
using SEC (size exclusion chromatography) with a
multi-angle laser light-scattering (MALLS) detector.
The molecular weight distribution and the weight
average molecular weight increased with increasing
amounts of TMP, while the number-average molecu-
lar weight decreased.
It is possible to synthesize numerous types of
branched polyesters by introducing A3/B3 or A4/B4monomers into a polymerization of A2 and B2
monomers that involve transesterification and poly-condensation. Long chain branched polymers gener-
ally have higher weight average molecular weights
and broader molecular weight distributions compared
with linear polymers synthesized at equivalent reac-
tion conditions. In fact, a high level of multifunctional
comonomer results in insoluble crosslinked systems
or high gel content if the conversion proceeds too far.
Long chain branching strongly influences thermal,
mechanical, and rheological behaviors of polymers as
discussed in subsequent sections.
2.3. Synthesis of branched polyesters via A2 and B2monomers in the presence of An or Bn (nO2)
monomers and a monofunctional endcapping reagent
The introduction of long chain branches in
polyesters is accomplished using low levels of a
multifunctional comonomer and low conversions
since gelation occurs at high levels of multifunctional
comonomer and at higher conversions. However, the
use of monofunctional comonomers in the presence of
bifunctional and multifunctional monomers prevents
or decreases gel content, and high molecular weight
polymers with long chain branches are attainable at
higher conversions. Neff et al. incorporated a mono-
functional reagent as a chain terminator to prevent or
decrease gel formation [21]. Branched PETs were
synthesized from DMT, EG, and diethylene glycol
(DEG) as bifunctional monomers, with trimellitic
anhydride as the branching agent and stearic acid as a
monofunctional reagent (Fig. 7). Manaresi et al.
synthesized highly branched PETs using a two step
polycondensation reaction in the presence of a
monofunctional comonomer, methyl 2-benzoylbenzo-
ate, with bifunctional monomers, DMT and EG, and
the trifunctional monomer, trimethyl trimesate
(Fig. 2) [22]. Munari et al. used a monofunctional
comonomer to shift the gel point to higher percentconversions, and no gelation occurred when the ratio
of monofunctional monomer to trifunctional mono-
mer was greater than 3. When the ratio was less than
3, the gel point was reached at lower conversions.
Methyl 2-benzoylbenzoate was used as the mono-
functional comonomer, and polycondensation tem-
peratures caused an approximately 30 wt% loss of
monofunctional comonomer.
Rosu et al. recently reported the synthesis and
characterization of high molecular weight branched
PETs that were prepared using a two step polycon-densation reaction in the presence of monofunctional
NC N
CNC
O
O
O
OH
OH
HOO
O
O
O
HOOH
(a) (b) (c)
Fig. 6. Bifunctional and trifunctional monomers used in the
synthesis of branched PBI. (a) dimethyl isophthalate (DMI),
(b) 1,4-butanediol (BD), and (c) tris(hydroxyethyl) isocyanurate
(THEIC).
O
O
OO
HOO
HO
(a) (b)
Fig. 7. Trifunctional and monofunctional comonomers used by Neff
et al.29 in the synthesis of branched PETs. (a) trimellitic anhydride,
and (b) stearic acid.
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dodecanol or benzyl alcohol, DMT and EG, and
multifunctional monomers, glycerol or pentaerythritol
[23]. The polymers with glycerol and pentaerythritol
displayed different degrees of branching as pentaer-ythritol has four primary alcohol groups, while
glycerol has two primary and one secondary alcohol
group. Therefore, the degree of branching was
expected to be higher with pentaerythritol. In addition
to two-step polycondensation reactions, solid-state
polymerization was used to obtain high molecular
weights. Solid-state polymerization increases polye-
ster molecular weight, while avoiding thermal degra-
dation [35,36]. This method enables the preparation of
linear and branched ultra-high-molecular-weight
PETs with intrinsic viscosities of more than 2 dl/g
(which corresponds to number-average molecular
weights of 110,000 g/mole approximately) [37].
Solid-state polymerization of PET is typically con-
ducted 1535 8C below the melting point of the
polymer for various times [35]. It is also possible to
perform these reactions at various temperatures
(220, 230, 235 8C) under vacuum [36]. Rosu et al.
reported molecular weight control for branched PETs
when the reaction time in the solid-state polymeriz-
ation step was controlled and specific compositions of
reagents were used [23]. The branched PETs were
characterized by solution viscometry, thermal anal-ysis, and melt rheology.
Recently, Hudson et al. studied branched PETs
based on various branching agents and different
endcapping reagent compositions [17]. Branched
PETs were prepared using conventional polyconden-
sation reactions with a monofunctional monomer and
various multifunctional monomers with DMT and EG
or bis(2-hydroxyethyl) terephthalate. In addition to
branching agents such as glycerol, pentaerythritol,
and benzene-1,3,5-tricarboxylic acid (trimesic acid),
Hudson et al. used benzene-1,2,4,5-tetracarboxylicacid, dipentaerythritol, and tripentaerythritol as
branching agents with the endcapping reagent benzyl
alcohol (Fig. 8). A wide range of branched PETs (with
various branching agents) as well as their compo-
sitions with and without the presence of an end-capping reagent were subsequently reported. The
polymers were characterized using FTIR and 1H
NMR spectroscopy, light scattering, dilute solution
viscometry, and melt rheology to investigate the
influence of branching on solution and melt
properties.
Although end-group modification of linear PETs
for enhanced solubility and blend compatibility was
previously reported, Kim and Oh investigated the
effect of functional end groups on the physical
properties of PETs by synthesizing hydroxyl and
carboxylic acid end-capped linear and branched PETs
[38]. The end-capped polymers were characterized
using NMR spectroscopy, viscosity measurements,
SEC, and thermal analysis. The high molecular
weight branched PETs (MwO100,000) had broad
molecular weight distributions, and diethylene glycol
(DEG) units were present in the polymer backbone,
which was attributed to side reactions of ethylene
glycol during polycondensation.
2.4. Synthesis of branched polyesters via AB
monomers in the presence of A2B monomers
An alternate method for synthesizing branched
polyesters involves the copolymerization of A2/B2 or
AB monomers with AB2/A2B monomers Ramak-
rishnan et al. reported the synthesis and character-
ization of branched and kinked PETs through the
copolymerization of an A2 monomer with small
amounts of an AB2 monomer [24]. The term
kinked describes linear disruption in the PET
backbone due to meta substitution of the aromatic
group. Therefore, in order to understand theinfluence of kinks, linear and branched polymers
O
O
OH
OH
O
O
HO
HO
OH
OH
OHO OH
OH
OH
OH
OH
OH
OH
OH
OHO O
OH
OH
(a) (b) (c)
Fig. 8. Branching agents used by Hudson et al. (a) benzene-1,2,4,5-tetracarboxylic acid, (b) dipentaerythritol, and (c) tripentaerythritol.
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were also prepared. Branched and kinked PETs were
synthesized using melt polymerization of bis-(2-
hydroxyethyl) terephthalate (BHET) as A2 monomer
and ethyl-3,5-(2-hydroxyethoxy)benzoate (EBHEB)as the AB2 monomer (Fig. 9). Linear and kinked
PETs were synthesized via the polycondensation of a
BHET monomer with a 3-(2-hydroxyethoxy) benzo-
ate (E3HEB) monomer that has a 1,3 connectivity
rather than a 1,4 connectivity, and the reaction was
terminated early to prevent gel formation. Early gel
formation was attributed to the fact that the EBHEB
monomer behaved similarly to an A3 type instead of
AB2, primarily due to the polycondensation reaction.
Therefore, BHET was able to react with all three
sites of EBHEB during polycondensation, whichresulted in gel formation during the early stages
of the polymerization. Unfortunately, stopping
the reaction early to avoid gelation yielded low
molecular weight polymers.
In addition to PETs, poly(4-ethyleneoxy benzoate)
was synthesized using ethyl 4-(2-hyroxyethoxy)benzoate (E4HEB) as AB monomer and EBHEB as
AB2 monomer [31]. Crosslinked polymers were
formed at branching agent levels higher than
50 mol%. When compared to branched PETs, the
branching content in these materials was higher and
therefore, a wider range of branched polymers were
prepared to study the effect of branching on the
thermal properties. Kricheldorf et al. prepared linear,
long chain branched, and hyperbranched poly
(3-hydroxy-benzoates) via condensation of acid
chlorides, 3-(trimethylsiloxyl) benzoyl chloride asan AB type monomer, and 3,5-(bistrimethylsiloxyl)
benzoyl chloride as AB2 type monomer [30].
Fig. 9. Reaction schemes for the synthesis of branched and kinked PETs.
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3. Characterization of branched polymers
3.1. Introduction
Since branching has such a dramatic influence on
polymer properties, it is important to characterize
polymer architecture on a molecular level. Short chain
branches are recognized with spectroscopic methods,
while sparsely long chain branched polymers are
much more difficult to characterize. In practice, most
branched polyesters are random in nature, with
heterogeneous distributions of molecular weight,
number of branch points, and length of the branches.
Since random branching influences polymer molecu-
lar weight and molecular weight distribution, it is
important to deconvolute the effects of chain archi-tecture and molecular weight. The determination of
molecular weight of a branched polymer using size
exclusion chromatography (SEC) and a calibration
curve based on linear polystyrene results in large
errors since separation is based on hydrodynamic
volume and linear and branched chains can possess
the same hydrodynamic size, but different molecular
weights [39]. Consequently, light scattering, a method
that does not depend on any standards or shapes, is
critical for measuring the absolute Mw of branched
polymer chains [40].
3.2. Contraction factors
When compared in the same environment (tempe-
rature and solvent), a branched polymer has a higher
segment density and a lower hydrodynamic volume
than that of a linear polymer of equal molecular
weight. Solution or melt viscosity measurements,
coupled with SEC and light scattering experiments
yield information regarding polymer size [41]. The
mean square radius of gyration, hR2gi, is a measure of a
polymers hydrodynamic volume as measured using
static light scattering. Consequently, the ratio of the
hR2gi of a branched polymer to that of a linear polymer
of the same molecular weight in the environment
expresses the degree of branching, a quantity, g,
referred to as the index of branching or contraction
factor, shown in Eq. (3).
gZhR2gibranched
hR2gilinear(3)
Similarly, the ratio of the intrinsic viscosity ([h]) of
a branched chain to a linear chain, conventionally
denoted as g 0, is employed as shown in Eq. (4).
g0Z
hbranchedhlinear
(4)
The value of g 0 is easily determined using the
procedure of Hudson et al., where a multi-angle laser
light scattering (MALLS) detector and a viscosity
detector are coupled with SEC [17]. The value of
[h]branched is measured directly using the viscosity
detector and [h]linear is calculated using the Mark
Houwink relationship shown in Eq. (5).
hlinearZKMaw (5)
The parameters, K and a, are the MarkHouwink
constants for a linear polymer. For a linear chain, g
and g 0 are equal to 1.0 and decrease as the level of
branching increases. Fig. 10 shows the decrease ofg,
denoted gM in the figure, for branched poly(vinyl
acetate) as a function of molecular weight [42]. The
contraction factor decreases from about 0.95 at low
molecular weight to 0.45 at higher molecular weights,
indicating the high molecular chains have a larger
degree of branching.
Since the value of [h] of a branched polymer is
lower than that of its linear analog, the MarkHouwink exponent, a, for a branched polymer is
generally smaller than that of a linear chain.
Comparison of the intrinsic viscosity dependence on
Mw for a series of linear and branched polystyrenes
showed a systematic decrease in the MarkHouwink
Fig. 10. Contraction factor vs. molecular weight for randomly
branched poly(vinyl acetate). The contraction factor decreases from
about 0.95 for low molecular weight species to 0.45 for the higher
molecular weight chains, indicating the high molecular chains have
a larger degree of branching.
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exponent from 0.73 to 0.68 to 0.39 for linear, star-
branched, and hyperbranched topologies, respectively
[43,44]. It should be noted that other researchers have
showed that, in the limit of high molecular weights,linear polymer and star polymers that possessed
a large range of arm numbers exhibited equal
MarkHouwink exponents for polybutadienes and
polyisoprenes [45,46]. This discrepancy between the
dependence of [h] on molecular weight for star-
shaped polymers may possibly be attributed to the
lower polystyrene molecular weights that were
investigated in Ref. [43]. Lusignan, Mourey, Wilson,
and Colby utilized the disparity in the MarkHouwink
exponent for linear and branched chains to estimate
the average distance between branch points for a
randomly branched polyester [47]. Fig. 11 shows the
intrinsic viscosity as a function of Mw for SEC
fractions of the branched polyester. The two slopes
correspond to a values of 0.80 and 0.43, typical for
linear and branched chains, respectively, and the two
lines intersect at 66,000 g/mol. The authors concluded
that this crossover from linear behavior to branched
behavior marks the average linear chain length in the
polymer system and the weight average molecular
weight between branch points.
The contraction factor has been theoretically
correlated to the branching parameters of a polymer
chain. Zimm and Stockmayer related g values to the
average functionality and the number of branchingunits for randomly branched chains [48]. For star
polymers with monodisperse arms under theta con-
ditions, g can be calculated using Eq. (6),
gZ3fK2
f2(6)
where fis the number of arms. Unfortunately, it is not
possible to measure the mean square radius of
gyration for low molecular weight chains due to the
limitation of MALLS. In fact, hR2gi1=2 measurements
are unreliable for values less than 10 nm, which isroughly a value ofMw on the order of 104 g/mol [49].
Intrinsic viscosity and g 0 are more reliable measure-
ments, however a theoretical basis is not developed
that relates g 0 to molecular parameters since the
dependence ofg 0 and g is not understood. Much work
has focused on understanding this relationship, and
empirical correlations suggest the form
g0Zg
3 (7)
where 3 is between 0.5 and 1.5, and is dependent on
the type of branching [50]. Generally, 3 is equal to 0.5
for low levels of branching or for star polymers, while
3 is closer to 1.5 for comb-shaped polymers [51].
Jackson, Chen and Mays determined values of 3
between 0.8 and 1.0 for randomly branched poly
(methyl methacrylate), and concluded that the visco-
metric radius (Rv) is more sensitive to the radius of
gyration due to the higher segment density of the
branched chains [52]. Instead of trying to convert
intrinsic viscosity contraction factors to radius of
gyration contraction factors, Balke et al. developed an
empirical relationship between g 0 and number of
arms, ranging for star-branched poly(methyl metha-crylate) with 3270 arms [53]. The authors showed
the number of arms, f, was more accurately predicted
through the empirical fits than by estimating values of
3 and using Eqs. (6) and (7).
3.3. Endgroup analysis
The average number of branches per chain for a
step-growth polymer is determined from the basic
Fig. 11. Intrinsic viscosity as a function of Mw for a randomly
branched polyester. The intersection of the two slopes correspond-
ing to respective a values of 0.80 and 0.43 mark the average
molecular weight between branch points.
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theoretical concepts of Flory and Stockmayer. The
models are dependent upon the number of polymer
chain ends, molecular weights, and initial concen-
tration of the mono-, bi-, and tri-functional repeatunits. End-group analysis on the branched polymer
reveals the number of total end-groups. Researchers
calculated the number of end groups for PET
branched with a trifunctional compound from the
concentration of hydroxyl and carboxylic acid end-
groups [54]. Utilizing the number of end-groups,
Manaresi et al. and others employed the number
average branching density or the average number of
branches per chain, Bn,
BnZ
2r
3KrK3p(8)
where r is a parameter that represents the initial
polymer composition,
rZ3ntrif
3ntrifC2nbif(9)
and ntrifis the number of initial trifunctional molecules,
nbif is the initial number of bifunctional molecules,
and p is the extent of reaction given by Eq. (10),
pZ
1K
E
Mb
2K
r
Mb
2K
Mt
3
(10)
where Eis the sum of all end-groups (equiv/g) and Mband Mt are the molecular weights of the bi- and tri-
substituted repeat units, respectively.
A mono-functional agent is often added to a step-
growth reaction mixture in tandem with the multi-
functional branching agent for facile control of
molecular weight. Theory developed by Flory and
Stockmayer predict that addition of a mono-functional
agent shifts the conversion at which gelation occurs to
higher values. Moreover, when the molar ratio of
mono- to tri- substituted agents is greater than or equal
to three, the gel point cannot be reached [22]. The
addition of a mono-functional compound is easily
accounted for in the above analysis by introducing the
parameters,
r0Z
nmono
nmonoC2nbifC3ntrif(11)
where nmono is the number of moles of mono-
functional agent. The extent of reaction, p, is
redefined as,
pZ1KEMb
2C
Mt
3K
Mb
2
rC MmK
Mb
2
r
0
(12)
where Mm is the molecular weight of the mono-
functional unit. Finally, the average number of
branches, Bn, is given by Eq. (13).
BnZ2r
3KrC3r0K3p(13)
It should be noted that long chain branches are often
below the detection limit of end-group analysis and
dilute solution measurements for mixtures of linear
and branched chains. Consequently, the aforemen-
tioned methods are often insensitive to sparsely
branched chains [55,56]. Typically spectroscopic
techniques and SEC methods are limited to detection
of branching levels of 1 branch point per 10,000
carbons [57]. Moreover, in polymer systems that
contain both short chain and long chain branches,
like polyolefins, the above methods cannot discrimi-
nate between the two thereby making LCB detection
difficult. More recently 13C NMR measurements
detected branching levels of 0.35 branches per
10,000 carbons in polyethylenes [58]. Since the flow
behavior of polymers is sensitive to long chainbranches at concentrations far below the detection
limit of the above methods, rheology becomes the only
feasible way to identify low levels of this type of
branching.
4. Influence of branching on melt rheological
properties: model systems and long-chain
branched polyesters
4.1. Introduction
The dependence of viscosity on shear rate for
branched chains is very different from that of linear
chains, aand varies with the chain architecture
(random, star-branched, comb-branched, H-shaped,
etc). Typically, long-chain branched polymers exhibit
shear and extensional viscosities that are unobtainable
with linear chains. For example at low shear rates,
branched chains can exhibit a viscosity greater than
100 times that of linear polymer of equal molecular
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weight, while at high shear rates, the branched
polymer may exhibit a lower viscosity than the linear
polymer due to enhanced shear thinning [59].
The presence of less than one long-chain branchalong a polymer backbone on average is known to
significantly alter the flow properties [60]. Only long-
chain branches, branches with MwOMc, can greatly
alter rheological properties, while short-chain
branches (SCB) only do not affect the rheological
behavior [61]. Several branching parameters influence
rheological properties including branch length, degree
of branching, and chain architecture (random, star-
branched, comb, H-branched, etc.). Typically,
branches are introduced in a random fashion during
polymerization, thereby leading to a broad distri-
bution of branch lengths and branch density. Conse-
quently it becomes difficult to separate the effects of
branching distribution, molecular weight distribution,
and chain architecture.
As stated earlier, PET and other linear polyesters of
relatively low molecular weight and narrow molecular
weight distribution display poor melt strength and
shear sensitivity at typical processing conditions
[62,63]. Additives, molecular weight and molecular
weight distribution changes by chain extension during
reaction or post reactor processing, branching, chain
end functionalization, and controlled cross-linking areoften used to modify the melt rheology of polyesters
[64,65]. In addition, long chain branched polymers
display superior melt strength and extensional
viscosity compared to their linear analogs, which
aids in blow molding and other processing appli-
cations [62]. This section describes the influence of
branching on the melt and concentrated solution
rheological properties of polyesters, with particular
focus on the influence of the branching parameters
including number of branches, branch length, and
branch type on melt viscoelasticity.
4.2. Number of branches per chain and branch length
4.2.1. Randomly branched polyesters
For a randomly branched polymer chain, the
average number of branches per chain (degree of
branching) and the branch length are coupled for a
given molecular weigh t. For example, a
100,000 g/mol polymer with an average of one branch
per chain has an a ver age bra nc h l ength of
33,300 g/mol from the relationship,
MbZMw
2BnC1
(14)
where Mb is the molecular weight between branch
points, Mw is the total polymer molecular weight, and
Bn is the average number of branches per chain. Thus,
if a higher concentration of multi-functional agent
was added to a step-growth polymerization to yield a
polymer chain with the same overall molecular
weight, but with three branches per chain, the average
branch length becomes 14,300 g/mol. Since effects of
branch number and branch length cannot be separated
for equivalent molecular weights, this section
describes the influences of both parameters on
rheological properties.
The dependence ofh0 on Mw is well established for
linear, flexible chains. Two regimes are separated by a
critical molecular weight (Mc), below which h0 scales
directly with Mw and above which h0 generally scales
with M3:4w . Chains with molecular weights below Mcare too small to entangle, while the high molecular
weight chains are topologically constrained due to
entanglement couplings. Researchers have shown a
significant departure from the h0KMw relationship
exists for branched chains due to the reducedhydrodynamic volume of the branched chains at low
molecular weights, and increased entanglement coup-
lings at higher molecular weights [66].
Hess, Hirt, and Opperman varied the level of
random branching in PET by adding different levels of
trimethylolpropane (TMP) to the melt polymerization,
and the branched PET possessed a lower h0 compared
to linear chains of equal Mw (approximately
50,000 g/mol) [33]. Fig. 12 shows the systematic
decrease in the zero shear rate viscosity as the average
number of branches per chain was increased from 0.1to 0.5. The parameter g*, which is the ratio of the zero
shear rate viscosity of a branched and linear chain at
equal molecular weight, also decreased with the
average number of branches per chain. For a series
of linear and randomly branched poly(butylene
isophthalate) polymers, Munari et al. also showed a
decrease in h0 with w0.5 branches per chain
compared to a linear chain of equivalent Mw(55,000 g/mol) [67]. Moreover, when these same
authors employed the correlation that relates the ratio
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ofh0 of a branched and linear chain of equal Mw as
developed by Ajroldi et al., the branching index
increased from 0 to 1.0 and the zero shear rate
viscosity systematically decreased by four orders of
magnitude. Similar behavior was exhibited for
branched PET and branched poly(butylene tereph-
thalate) [54,68].
The influence of the level of random branching on
the rheological properties of aliphatic polyesters hasalso been investigated. Kim et al. observed a
systematic decrease in shear thinning onset for
branched poly(butylene succinate) (PBS) as the level
of trifunctional branching agent was increased from 0
to 0.5 wt% [34]. The onset of shear thinning behavior
was attributed to a higher entanglement density of the
branched chains [69]. Moreover, the authors observed
an increase in h0 for branched PBS over linear PBS.
However the authors of this review believe the
enhancement in both shear thinning and h0 was more
likely due to the significant increase in Mw and
polydispersity (Mw/Mn) for the branched chains
relative to the linear chains. Short-chain, ethyl and
n-octyl branches were also introduced into poly
(ethylene adipate) (PEA) and PBS. In contrast to
long-chain branched systems, a decrease in melt
viscosity and increase in shear thinning onset were
observed when compared to linear PEA and PBS of
approximately equal weight average molecular weight
[70]. The ethyl and n-octyl branches were not long
enough to form entanglements, and consequently shear
thinning and h0 enhancement were not as pronounced.
The melt viscosity of the branched polyester was lower
than that of the linear polyester due to the reduced
hydrodynamic volume of the branched chains. Leher-meier and Dorgan studied the influence of blending
linear polylactide and polylactides that were randomly
branched through a peroxide cross-linking reaction on
the melt rheological properties [71]. The authors
ensured that the polylactides did not undergo degra-
dation at the rheological conditions by adding the
stabilizer, tris(nonylphenyl) phosphite. They observed
excellent control over the rheological performance
with the blend composition. In particular, the authors
reported an increase in h0 and decrease in the
frequency at which shear thinning occurred with
increasing blend compositions of the branched chain.
Unfortunately, molecular weight information was not
reported, and the branching structures of the polylac-
tides were not characterized. Thus, it was difficult to
assess the relationships between branch structure and
rheological behavior.
Lusignan, Mourey, Wilson, and Colby studied the
linear viscoelastic properties of randomly branched
polyesters with varying branch lengths [47]. The
authors showed for low branch length of NZ2
monomeric units, the chains were unentangled and
accurately described by the Rouse model withouthydrodynamic or topological interactions [72]. More-
over, further studies showed that entanglements
between the randomly branched chains did not form
for N!20, since the Rouse model was adequate for
branched polyesters with up to 20 repeat units
between branch points [73]. Branched polyesters
with NZ900 were synthesized to demonstrate that
topological constraints dominate the viscoelastic
response of chains with branch lengths long enough
to entangle [47]. The average molecular weight
between branch points, Mb, was determined byanalyzing the intrinsic viscosity dependence of Mwas shown in Fig. 11. The Mb (66,000 g/mol) was
defined as the crossover from linear to branched
behavior, and was measured by the reduction of the
[h] vs. Mw slope. Below Mb, h0 scaled with M3:6w
which was consistent with experimental results for
entangled linear chains, and above Mb, h0wM6:0w due
to the increased entanglement constraints imposed by
the branched chains. Fig. 13 shows that the Rouse
model breaks down for N/NeO2, where N/Ne is
Fig. 12. Systematic decrease in (h0,b/h0,l) as a function of average
number of branches per chain for randomly branched PET withMwZ50,000 g/mol.
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the number of entanglements per branch, due to
entangled dynamics as the viscosity exponent, s,
varies significantly from the Rouse prediction of 1.33.
Consequently, the authors concluded that Mbw2Mefor entanglements to dominate the flow behavior.
Earlier experiments by Long, Berry, and Hobbs
observed that the rheological behavior of randomly
branched polymers is dependent on the branch length
[74]. They observed a larger h0 for branched
poly(vinyl acetate) compared to a linear chain of
equal molecular weight when the average molecular
weight between branches (Mb) was greater than
28,000 g/mol. Using MeZ9,100 g/mol, which was
reported by Fetters et al. for poly(vinyl acetate) [75],
Mbw3Me for entanglements between branches to
control the melt rheological performance. Several
other researchers have also investigated the rheologi-
cal response of randomly branched chains. Valles and
Macosko showed that Mc is higher for randomly
branched poly(dimethylsiloxane) (PDMS) chains
compared to linear chains as measured by the Mwdependence of h0 [76]. Moreover, the number of
branches per chain also influenced Mc, as Mc
increased from 33,000 for the linear chain to 98,000
and 110,000 g/mol for the trifunctional and tetrafunc-
tional polymers, respectively. Unlike the randomly
branched polyesters, the researchers observed aweaker h0KMw relationship for branched PDMS
above Mc. However, when h0 was plotted against the
product gMw (where g is the contraction factor) in Fig.
14, a viscosity enhancement was observed for the
randomly branched PDMS as seen previously with
star-branched polyisoprene. Finally, Masuda et al.
reported a dependence of viscoelastic properties on
Mb for randomly branched polystyrene in 50 wt%
solutions [77]. A clear plateau region was not
Fig. 13. For N/NeO2 deviation from the Rouse models is evident
due to entangled dynamics as the viscosity exponent, s, significantly
varies from the Rouse prediction of 1.33. The solid line is the Rouse
prediction for N/Ne!2, and a phenomenological equation that
describes entanglements for N/NeO2.
Fig. 14. Dependence of h0 on the product gMw for randomly
branched PDMS. The triangles and squares correspond to tri and
tetra functionally branched PDMS, respectively.
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observed for Mb/Me!2, due to relaxation of the short
backbone segments via Rouse-like motions, while
branched chains with Mb/MeZ6 showed h0
enhancement.
4.2.2. Star-branched polyesters
This review has focused on randomly branched
polyesters that contain a distribution of branch
lengths. Unlike randomly branched polymers, the
synthesis of star polymers allows for a high degree of
control over the molecular structure. Due to the well-
defined chain architectures that result from more
controlled synthetic strategies, star polymers have
received much attention in the area of structure/
property relationships. Fundamental investigations of
the dynamics of star polymers will provide useful
information for understanding the behavior of com-
mercially produced, randomly branched materials
[78]. At relatively low molecular weight, the viscosity
of a star polymer is lower than its linear analog,
however, the viscosity of the a star polymer increases
faster with molecular weight and exceeds that of the
linear analog at some specific molecular weight [79].
This molecular weight dependence occurs because the
star polymer exhibits a reduced hydrodynamic
volume compared to a linear polymer due to the
higher segment density, however, a competing effectarises since the star polymer possesses restricted chain
motion due to the constraint that one end of the arm is
anchored to the star core. Consequently, the branch
point hinders reptation, and relaxation only occurs
when the arm retracts back along the confining tube
and seeks a new direction [80]. McLeish and Milner
suggested two modes of relaxation of a star polymer.
Short relaxations occur at the chain end, where the
branch point does not restrict the arm, and long-scale
relaxations occur near the star core [80]. Experimental
results by Ye and Sridhar corroborated this theory andrelaxation times for a concentrated solution of
polystyrene stars were 20 to 150 times greater than
the relaxation times predicted for linear chains by
reptation theory [81]. In addition, star polymer
polymer blend miscibility was highly influenced due
to the impenetrable core of the star from which the
arms diffuse outward [82].
As mentioned previously, the zero shear rate
viscosity for linear polymers follows a power law
dependence above the critical molecular weight for
entanglements, Mc. However, for star-shaped poly-
mers with Mw greater than Mc, the zero shear rate
viscosity increases exponentially with the weight
average arm molecular weight [83]. Fetters et al.observed that the value h0 of a star-branched chain
does not depend on the total Mw, but only on the arm
molecular weight, Ma. Thus, h0 is independent of the
number of arms. Later, Fetters et al. showed that the
h0 of a 3-arm polyisoprene star was approximately
20% lower than that of a 4-arm star of equivalent Ma,
while for fO4, the degree of functionality is saturated
and the viscosity is only dependent on Ma [84]. These
experimental results were consistent with previously
developed theories that suggested 3-arm stars have an
additional mechanism of stress relaxation that stars of
higher functionality do not exhibit [85]. They
proposed for 3-arm stars, an arm could relax if the
branch point diffuses down one of the tubes. The
rheology of stars with higher degrees of functionality
was also studied. Pakula et al. performed linear
viscoelastic studies on polybutadiene stars with a
significantly larger number of arms than previously
studied [86]. The authors observed a high frequency
and low frequency relaxation corresponding to chain
segmental motion and terminal response, respecti-
vely. Fig. 15 shows stars with fZ64 and fZ128 arms
display an additional transition in the terminal flowrange that is not present for stars with fZ4 arms. This
additional transition for stars with high degrees of
functionality is attributed to cooperative rearrange-
ment of the colloidal or liquid like structures in the
melt [87]. It should be noted this additional relaxation
would not be applicable for randomly branched
polymer chains.
Since arm length controls the viscoelastic response
for stars with a relatively few number of arms, it is
important to understand the role of branch length in
order to provide viscosity enhancement. Kraus andGruver observed for equal overall molecular weight, a
3 arm star polymer with MaZ10Me showed viscosity
enhancement over the linear analog [83]. However,
rheological studies performed on a series of asym-
metric poly(ethylene-alt-propylene) stars showed that
the critical Ma for viscosity enhancement was less
than 10Me. Gell et al. studied a series of 3-arm
asymmetric stars where two branch lengths were kept
constant and one was varied, thereby providing a
constant molecular weight backbone [88]. The branch
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length ranged from Mb/MeZ0 to Mb/MeZ18, and was
shorter than the backbone length (Mbb/MeZ38).
Deviation from linear chain behavior and h0 enhance-
ment were observed for Mb/Mew23, with consider-
ably fewer entanglements per branch needed than for
symmetric stars due to the nature of the very long
backbone compared to the branch length. Dorgan et
al. showed that viscosity enhancement for 4 and 6-arm
star poly(lactic acid) occurred at approximately Mb/
MeZ4 [89]. The authors observed that the viscosity
enhancement factor, G,
GZh0;bM
h0:lM(15)
increased more rapidly for the 6-arm versus the 4-arm
star as shown in Fig. 16. In Eq. (15) h0,b and h0,l are
the respective zero shear viscosities of branched
and linear chains of equal molecular weight. This
result is in disagreement with theoretical treatments
and experimental results, which show viscosity to be
only dependent on arm length, but independent of the
number of arms [90]. The discrepancy was attributed
to a combination of polydispersity, hydrogen bonding
effects between ester groups, and the relatively short
arm lengths of the star polymers.
Claesson et al. prepared star-shaped polyesters
composed of poly(3-caprolactone) (PCL) initiated
from hydroxy-functional hyperbranched cores end-
capped with methacrylate units [91]. The end groups
served as a cross-linking agent for utilization in
powder coating applications. Due to the narrow range
of molecular weights studied, it was difficult to
determine if the star polymer exhibited exponential
or power law behavior, however, the h0 was an order
of magnitude lower compared to a linear polyester of
equal Mw. Since the Mw of the hyperbranched cores
cannot be neglected in the total Mw, there was a
dependence of arm number on the zero shear rate
melt viscosity for the PCL stars. The PCL star
polymers with methacrylate end groups were cured
with ultraviolet (UV) light [92]. Gelation occurred
within seconds of UV exposure based on the
crossover point of the storage modulus (G 0) and the
loss modulus (G 0). The time to reach gelation
increased linearly with the molecular weight of the
star polymer since the concentration of methacrylate
end groups decreased.
Fig. 15. Frequency dependence of polybutadiene stars with (a) fZ4,
(b) fZ64 and (c) fZ128 arms. The vertical dashed lines correspond
the frequencies associated with relaxation of the chain segment (us)
and arm (uR), respectively.
Fig. 16. Viscosity enhancement vs. branching length for 4 and 6 arm
star polylactide.
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4.2.3. H-shaped and comb-branched polymers
H-shaped architectures are considered the simplest
form of a comb polymer, where branching occurs only
at the two ends of the backbone. Although, little workhas focused on the synthesis and rheological analysis
of H-shaped polyesters, many structure/property
studies have been performed on model H-shaped
polymers. This section of the present review serves to
summarize work performed on model polymer
systems, as the results are applicable to polyester
architectures. Roovers studied the melt rheology of
H-shaped polystyrene and observed a decrease in h0for the branched polymers compared to linear analogs
at low molecular weight, and an increase in h0 at
high molecular weights similar to star polymers [93].
Fig. 17 shows that the viscosity enhancement factor,
G, increased faster for the H-shaped polymer than that
of either 3 or 4 arm stars as a function of chain
entanglement per branch (Mb/Me). This was attributed
to an additional mode of relaxation for the H cross-bar
which is not present for star polymers. Archer and
Varshney corroborated this extra relaxation mode for
H-shaped or multi-arm polybutadienes with three
branches per chain end [94]. The authors observed a
broader and lower frequency transition to the terminal
region for the multi-arm polybutadiene compared to
its linear counterpart, which was attributed to anincreased relaxation time of the cross-bar. Moreover,
they showed the terminal relaxation time and h0enhancement were primarily controlled by the branch
length when MbOMe and were relatively independent
of the cross-bar molecular weight. Houli et al. also
studied the rheological behavior of pom-pom type
polymers with a much greater number of arms (fZ
32)[95]. They also observed the dominant mechanism of
terminal relaxation was arm relaxation. However,
multi-arm polymers with Mb!Mc did not form
entanglements, which was marked by power law
behavior from the glass to the terminal region, typical
of Rouse-like motions. This is similar to the
unentangled behavior of high molecular weight
hyperbranched polymers that relax via segments that
are smaller than Me [96].
Although model star and pom-pom polymers were
extensively studied to determine the influence of
branching on rheological properties, most commer-
cially produced polymers are randomly branched.
Interest in the rheological characterization of comb-
branched chains may bridge the gap between the
behavior of model star polymers and randomly
branched commercial polymer. Noda et al. performed
viscoelastic measurements with polystyrene combs
and showed a lower h0 compared to linear polystyrene
of equal Mw, however when compared at equivalent
Rg, the combs displayed viscosity enhancement [97].
Roovers and Graessley also performed rheological
analyses on comb polystyrenes with backbonemolecular weights of 275,000 and 860,000 g/mol
with approximately 30 branches per chain varying in
Mw from 6,500 to 98,000 g/mol [98]. The comb
polystyrenes showed a reduction in h0 when com-
pared to linear chains of the same Mw and showed h0enhancement when compared at equivalent intrinsic
viscosity. Surprisingly, this enhancement was not
restricted to branch lengths above Me, and the h0enhancement was different for the combs with
different molecular weight backbones. However, the
authors found good agreement with the log GK
Mb/Merelationship for stars when the comb molecular weight
was normalized by the average end-to-end comb
molecular weight (MEE/Me). Daniels et al. studied the
linear rheological response of comb-branched poly-
butadiene and varied the molecular weight of the
polymer backbone, the molecular weight of the arms,
and the number of arms [99]. The researchers reported
the viscoelastic response was dependent on the
number of arms for low frequencies. At short time
scales, the comb polymers displayed Rouse-like
Fig. 17. Viscosity enhancement factor, G, as a function of the
number of entanglements per branch. Triangles and diamonds
denote H-polymers, the solid line denotes a 4-arm star, and the
dashed line denotes a 3-arm star.
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behavior similar to star polymers due to relaxation of
the dangling chain ends. For a relatively small number
of arms, the comb polymers behaved similar to H-
shaped polymers, marked by relaxation of the arms atintermediate frequencies and reptative motion of the
backbone at low frequencies [100]. For a larger
number of arms, the terminal region showed a
distinctly different response for the comb compared
to the H-shaped polymer, as a larger number of
relaxation modes were available to the comb-
branched polybutadiene. Roovers and Toporowski
attributed the broader low frequency relaxation to
additional couplings between a branch and a back-
bone that are unavailable to star polymers [101].
Namba et al. showed that branch spacing in comb
polymers is important in their viscoelastic response
[102]. They observed that highly branched comb
poly(methyl methacrylate), with approximately one
polystyrene branch (MbZ3450) per repeat unit did not
entangle, as a plateau region was not observed for the
storage modulus. Although Mb!Mc for the poly-
styrene branches, the backbone molecular weight was
well above Mc, so the lack of entanglements was
attributed to the high branch density of the comb
polymer. When the polystyrene macromonomer was
copolymerized with methyl methacrylate to yield a
branch structure of approximately 3 branches per 100repeat units, a clear plateau region was observed in the
dynamic shear modulus. However, the authors did not
address the incompatibility issues between poly-
styrene and poly(methyl methacrylate) chains.
Tsukarhara et al. also observed fewer entanglement
couplings for highly branched combs with a backbone
molecular weight greater than Mc. The researchers
calculated Me of a highly branched poly(methyl
methacrylate) comb from the plateau region, where
Mb!Mc, and discovered Me was approximately 3
orders of magnitude larger for the highly branchedPMMA compared to linear PMMA [103]. The authors
attributed this to the increased cross-sectional area of
the highly branched PMMA comb, which excluded
other chains from a unit volume and thereby hindered
entanglement couplings.
4.3. Flow activation energy
The activation energy explains the temperature
dependence of viscosity for as shown in Eq (16),
h0TZA0expEa
RT
(16)
where h0 is the zero shear rate viscosity, A0 is a pre-exponential factor, Ea is the activation energy, R is the
gas constant, and Tis the temperature in K. Generally,
Eq. (14) is only valid for temperatures ca. 80 8C above
the polymer glass transition (Tg) due to the exponential
rise in viscosity at temperatures near the Tg. The value
of Ea is independent of molecular weight and is only
dependent on the local segmental nature of the chain
[104]. Typical values for the Ea are in the range of 5 to
30 kcal/mol. In general, Ea increases with either chain
stiffness or bumpiness [105]. Consequently, the
temperature dependence of viscosity for branched
polymers differs significantly from the corresponding
linear analogs. In particular, the rheological behavior
of the former shows a greater temperature dependence
and thus Ea is enhanced. One of the most outstanding
examples is that Ea depends on the degree of
branching and branch length in polyethylene, as
several researchers reported a larger Ea for low-
density polyethylene compared to high-density poly-
ethylene. However, limited work has focused on the
influence of branching on Ea in polyesters.
Munari et al. investigated the influence of long-
chain branching on the flow activation energy for aseries of partially aromatic polyesters, and found
inconsistent results for the different polyesters. They
reported a 35 to 100% increase in Ea for a series of
branched poly(butylene terephthalate)s (PBT) com-
pared to their linear analogs, however, only a slight
increase in the Ea for branched PET compared to
linear PET was observed [54,68]. Moreover, no
enhancement in Ea was observed for branched
poly(butylene isophthalate) (PBI) compared to linear
PBI [106]. The authors attributed these discrepancies
to differences in Mb between the branched polyestersand different temperature coefficients of the repeat
units. Graessley related this inconsistent behavior to
differences in the temperature coefficients of linear
and branched polymer melts [107]. This discrepancy
arises when considering the mode by which entangled
chains relax. In particular, linear chains undergo
reptation, while long chain branches relax by retrac-
tion or short time-scale fluctuations along the tube
contour length of an arm. As an arm relaxes, it must
pass through a higher energy barrier due to the more
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compact conformational states that are dependent on
the temperature coefficient of the chain. In the cases
where differences in activation energy for linear and
branched polymers are observed, the quantity, DEZ
(Ea)BK(Ea)L, is often used to quantify the degree of
Ea enhancement. The quantity DE was shown to
exponentially increase with the number of entangle-
ments per branch (Mb/Me), and decrease to zero with
decreasing branch length [108].
5. The influence of branching on solution rheology
properties in the semidilute regime
5.1. Introduction
The dilute solution properties of branched chains
are consistent with their reduced hydrodynamic
dimensions compared to linear polymers of equivalent
molecular weight. The properties of branched poly-
mers in dilute solution were discussed in some detail
in Section 3.2. In dilute solution, the polymer chains
are widely separated from each other, and only the
interactions between two chains need to be con-
sidered. At a critical concentration, C*, the polymer
chains begin to crowd each other and overlap in
solution, and COC* is termed the semidilute regime.
As the polymer chains overlap, intrachain interactionsare screened at length scales longer than the
correlation length, where the correlation length is
defined as the average distance between neighboring
contacts points [109]. Polymer concentrations above
C* do not indicate that entanglement couplings
between chains have formed [110]. Consequently,
in the semidilute unentangled regime, C*!C!Ce(where Ce is the entanglement concentration), chain
overlap is not sufficient to topologically constrain the
polymer chain motion. Above Ce, the semidilute
entangled regime, chain crowding and interpenetra-tion is sufficient to constrain the chain motion, and
topological interactions dominate at distances longer
than the tube diameter [111]. Limited rheological
studies have shown that typically Ce/C* is in the range
of 510 for neutral, linear polymers [111]. Finally, as
polymer concentration is increased further into the
concentrated regime, which is defined as the point
where chain dimensions become independent of
concentration, the polymer coils are highly entangled
and behave similar to a melt [112].
5.2. Effect of branching on the entanglement
concentration
The entanglement concentration is experimentallymeasured by analyzing the concentration dependence
of specific viscosity (hsp),
hspZh0Khs
hs(17)
where h0 is the zero shear rate viscosity of the
polymer solution and hs is the solvent viscosity. For
neutral, linear polymers in a good solvent, hspwC1.0
in the dilute regime, hspwC1.25 in the semidilute
unentangled regime, hspwC3.8 in the semidilute
entangled regime as predicted by the reptation theory.
Finally, the value of hsp generally shows a weakerdependence in the concentrated regime compared to
the semi-dilute entangled regime [111]. For example,
Colby et al. measured the onset of the semidilute
unentangled and semidilute entangled regime for an
aqueous solution of sodium hyaluranote [113]. Fig. 18
shows the concentration dependence of viscosity and
the determination of C* and Ce from the change in
slope. Takahashi et al. measured h0 for linear poly
(a-methylstyrene) in good, poor, and q solvents and
observed the transition to the semidilute regime
decreased with molecular weight as expected sincelarger chains begin to overlap at lower concentrations
compared to smaller chains [114]. Moreover, the
authors reported that the transition was dependent on
Fig. 18. Concentration dependence of specific viscosity for a
biopolymer, sodium hyaluranote. C* and Ce were determined as
0.59 and 2.4 mg/mL, respectively.
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solvent quality. Other researchers investigated the
hspKC relationship for linear poly(a-methylstyrene)
in solvents of variable quality [115]. The investigators
discovered that in dilute solutions hsp was lower inpoor solvents compared to good solvents, while the
opposite was true in entangled solutions. It was
proposed that the viscosity increase in the poor
solvent was attributed to enhanced entanglement
couplings due to relatively weak interactions between
the chain segments and the solvent. Moreover, other
researchers have observed a weaker concentration
dependence for viscosity in the semidilute entangled
regime for a polymer in a good solvent compared to a
polymer in a theta solvent [116].
Our discussion of concentration dependence ofhspand the determination of C* and Ce has focused on
linear chains to this point. Not surprisingly, at equal
molecular weights, branched chains show different
behavior in solution compared to linear chains.
Generally, as shown in Fig. 19, a branched polymer
exhibits a higher overlap concentration compared to a
linear polymer of equal Mw since the branch points
act as obstacles to chain interpenetration [117].
Sendijarevic et al. studied the effect of branching of
AB/AB2 etherimide copolymer solutions on solution
rheology properties [118]. The authors observed a
weaker concentration dependence of h0 in thesemidilute entangled regime for more highly branched
structures. As the copolymer composition was varied
from 0 to 100 mol% AB2, corresponding to linear and
hyperbranched architectures, respectively, the expo-
nent decreased from 12.5 to 4.7, attributed to a largernumber of entanglements per chain in the linear
copolymers. Moreover, the overlap concentration
increased with higher levels of branching. Similarly,
solution rheology studies with linear and randomly
branched poly(ethylene terephthalate-co-ethylene iso-
phthalate) (PET-co-PEI) showed a weaker hsp vs. C
dependence for branched copolyesters compared to
linear copolyesters of similar molecular weight [119].
Furthermore, branching dramatically influenced the
onset of the entanglement regime, as Ce increased
from 4.5 to 10 wt% as g 0 for PET-co-PEI decreased
from 1.0 to 0.43. In a related study, Juliani and Archer
studied the rheology of unentangled and entangled
A3KAKA3 multi-arm polybutadiene solutions with
equivalent branch molecular weights and variable
backbone molecular weights [120]. The investigators
reported Ce decreased from 22 to 8 vol% as the
molecular weight of the cross-bar was increased by
w30% allowing a larger number of entanglements per
chain for the higher molecular weight crossbar.
5.3. Recent advances in electrospinning of long-chain
branched polyesters
Traditional melt processing of polyester fibers has
received significant commercial attention, however,
electrospinning has recently emerged as a specialized
processing technique for the formation of sub-micron,
high surface area fibers [121]. The utility of branched
polyesters in the electrospinning process has received
only limited attention. Typically, conventional poly-
mer fibers are melt spun using pressure-driven flow
through an extruder, yielding fibers on the order of
10100 m