3 Formation and Structure of Polyesters and Alkyd Resins: 3.1 Reactions That Produce Polyesters

Reactions that produce polyesters
3 Formation and structure of

polyesters and alkyd resins
In the literature, polyesters and particularly alkyds are usually described in terms of their
building blocks. However, to facilitate an evaluation of the different factors that govern the
properties of polyesters and alkyd resins, this chapter deals with general structural influ-
ences, as opposed to those exerted by the choice of building blocks. Although there exist
overlapping interrelations, first the basic molecular structure of polyesters and alkyd resins
and the relevant influencing variables are discussed, which generally apply to the most
different material compositions. Building block selection is discussed in Chapter 4.
3.1 Reactions that produce polyesters

3.1.1 Fundamental reactions
3.1.1.1 Esterification of alcohols and carboxylic acids
Esterification of alcohols with carboxylic acids is a classic example of a condensation re-
action and useful for describing chemical equilibrium reactions. It is usually the reaction
chosen to explain the law of mass action. In the conventional view, which provides a good
model of how polyesters are prepared, esterification consists in addition of the electrophi-
lic H atom of the alcoholic OH group to the nucleophilic O atom of the carboxyl group to
Figure 3.1: Conventional model of the mechanisms behind esterification and saponification
Ulrich Poth: Polyester and Alkyd Resins

© Copyright 2020 by Vincentz Network, Hanover, Germany
17
Formation and structure of polyesters and alkyd resins
form an intermediate (see Figure 3.1). The steric effect of this intermediate structure with
its three oxygen atoms causes the adduct to decompose either into the starting substances
or into the ester and water. The ratio of reactants to products in the equilibrated reaction
mixture depends in part on the “R” groups of the reactants. Other factors are the concen-
trations of reactants and the temperature. The chemical equilibrium stems from the fact
that the ester on the product side can polarise itself to the extent that it can revert back to
the intermediate by reacting with the water. This reaction is defined as saponification.
Evidence for this has been provided by radioactively doping the oxygen atoms in the alco-
hol, which are subsequently found only in the ester molecule. When a carboxyl group
oxygen is radioactively doped, it is found in both the ester and the water molecules, be-
cause the two OH groups of the “ortho structure” of the adduct are wholly equivalent [19].
The Figure 3.2 shows the overall equilibrium equation for esterification and saponifi-
cation.
The reaction rates for esterification (vforward) and saponification (vback) at equilibrium
(vback = vforward) are governed by the law of mass action [20].
Equation 3.1
In other words, for quantitative ester preparation, the equilibrium must be shifted to the
product side. This is usually accomplished by removing the water of reaction through
distillation. The standard production processes for polyesters are based on this approach.
From the point of view of the distillation process, this is an example of “residue recovery”.
O O
R2 C OH + HO R1 R2 C O R1 + H OH
carboxylic acid alcohol ester water
Figure 3.2: Overall equilibrium equation for esterification and saponification
18
3.1.1.2 Transesterification
As the saponification reaction shows, the ester group can be polarised by water. But it can
also be polarised by alcohols. Thus, ester groups can react with mobile hydrogen atoms
from alcohols to form an intermediate structure. This intermediate structure can decom-
pose into its reactants, but it can also decompose into the ester formed with alcohol 2 and
into the free alcohol of ester 1 (see Figure 3.3 and overall equation in Figure 3.4).
The chemical equilibrium involved in transesterification is also subject to the law of
mass action.
Equation 3.2

Here, again, the equilibrium state is
influenced by the “R”-groups in the –
O O
carboxylic acid as well as the alco-
hol. Naturally, it is also influenced C C + ester 1
R O R1 R O R1
by the concentration of reactants
+
and by the temperature.
If the goal is to prepare one of –
H O R2 H O R2 alcohol 2
the esters in high yield, the reaction +
equilibrium needs to be shifted to

the correct side. This is usually ac-
O R1
complished by removing one of the
products through distillation. This C
OH
means that polyesters can be pro R O R2
duced from polycarboxylic esters of
lower alcohols. Full transesterifica-
O O –
tion with polyols is then achieved by
C C +
distilling off all the low boiling alco- + O R1
R O R2 R O R2 H
hol from the starting product.
Transesterification also plays a ester 2 alcohol 1
role in the preparation of alkyd re-

Figure 3.3: Mechanism of transesterification
sins. Starting products are then na-
tural oils or fats (triglycerides)
which are transesterified with poly-
ols. Because of the excess of hydro-
xyl groups, reaction equilibrium is
reached when the fatty acids are
distributed on all polyol molecules Figure 3.4: Overall equation for transesterification
19
(partial glycerol esters). These serve as intermediates for a second step in which dicarbo-
xylic acids or anhydrides (e.g. phthalic anhydride) are added to continue the polyconden-
sation and form the final alkyd resin.
Although transesterification in an industrially accepted production process has been
described in publications, it is usually ignored when it comes to theoretical descriptions
regarding the preparation of polyester molecules. The methods for determining molecular
weights and molecular weight distributions or defining gel points assume that only esteri-
fication reactions occur during the preparation of polyesters. However, transesterification
happens throughout the production process and is not restricted to the primary products.
Transesterification mainly affects the molecular weight distribution.
3.1.1.3 Reaction catalysis

Many authors believe that the order of the reaction for esterification, saponification and
transesterification (mostly 2nd order reactions) requires the use of catalysts. The addition
of such acid catalysts or Lewis acids causes polarisation of the carboxyl group on acids and
esters by protonation, and that subsequently the alcohol adds to an oxonium complex or
a carbonium ion (see Figure 3.5).
However, carboxylic acids themselves can liberate protons and so are able to polarise
carboxyl groups by themselves. Support for this is provided by the formation of dimers of
carboxylic acids (see Figure 3.6).
Thus, the reactions of carboxyl
+ groups or ester groups with alcohols
OH are quite complex, especially those
R C between carboxyl groups and tertia-
OH
O ry alcohols or phenols – which are
R C + H + completely different from those with
OH primary and secondary OH groups.
OH
For this reason, common polyesters
R C + are not usually prepared with the aid
OH of tertiary alcohols and phenols.
Figure 3.5: Acid catalysis at the start of esterification

3.1.1.4 Anhydride
addition
O H O
Due to ease of handling, more often
R C C R anhydrides (e.g. o-phthalic anhydride,
O H O tetrahydrophthalic anhydride, hexa-
hydrophthalic anhydride, trimellitic
Figure 3.6: Dimerisation of carboxylic
Figure 3.6: Carboxylic acids acids
form dimers anhydride, pyromellitic anhydride,
20
succinic anhydride, maleic anhydride) are used in industrial processes than their corre-
sponding diacids. Owing to the molecular stress in the anhydride ring and the exposed
position of the nucleophilic oxygen atom, anhydrides readily react with electrophilic active
hydrogen atoms, such as those of alcohols. The addition reaction yields an ester group and
a free carboxyl group (see Figure 3.7). The latter is then able to form a further ester group.
The reaction rate for the formation of the second ester group – especially in the case of
aromatic polycarboxylic acids – is lower than that for isolated carboxyl groups due to
steric hindrance. Higher temperatures can also cause reversion to the anhydride. This
mainly happens in the case of aromatic 1,2-carboxyl groups (e.g. phthalic, trimellitic, and
pyromellitic esters).
3.1.1.5 Epoxy addition O

anhydride
Formally, 1,2-epoxies are anhydri- C –
des of 1,2-diols. Ring stress in epo- + O – + H O R H O R
C +
xies makes them readily polarisable alcohol
to yield nucleophilic oxygen and O
electrophilic carbonium ions. The

O
electrophilic carbonium ion can add
C O R
nucleophilic oxygen from carboxyl -carboxy ester
groups to form ester groups and se- C OH
condary OH groups (see Figure 3.8). O
Under the influence of strong

Figure 3.7: Anhydride addition
acids, epoxies can react with them-
selves to form polyethers in a secon-
dary reaction. –
O O
R1 CH R1 CH
3.1.1.6 Other reactions CH2 CH2 +
Several other reactions are known in + epoxide

organic synthesis that can yield
–
polyesters. Two particular reactions O O
+
OH
R2 C H R2 C
are described to prepare a polyester R2 C
+ O O
OH
and a polyester-like product. –
carboxylic acid
Polycarbonates are prepared by
reaction of phosgene or other deri- O
vatives of carbonic acid with alkali- R2 C R1
O CH2 CH
ne alcoholates. Alkaline phenolates
ß-hydroxy ester
may also be used for this reaction. OH
The high enthalpy of formation of

alkaline halogenates ensures that Figure 3.8: Epoxy addition to form esters
21
_ _
O Na+ + Cl C Cl O C O + Na+ Cl
O O
sodium phenolate phosgene arylcarbonate sodium chloride
Figure 3.9:3.9:
Figure Preparation ofaryl
Preparation of aryl carbonates
carbonates
O O
C
R C O H
O + H R
O O
-caprolactone alcohol ester
Figure 3.10: Ring-opening reaction of ε-caprolactone
O O O O
HO R1 OH + HO C R2 C OH HO R1 O C R2 C OH
diol dicarboxylic acid – H 2O monoester
O O
HO R1 O C R2 C OH + HO R1 OH
– H 2O
monoester diol
O O
HO R1 O C R2 C O R1 OH
O O
HO C R2 C O R1 OH + HO C R2 C OH
– H 2O
O O
monoester dicarboxylic acid
O O
HO C R2 C O R1 O C R2 C OH
O O
O O O O
HO R1 O C R2 C OH + HO R1 O C R2 C OH
monoester – H 2O
monoester
O O O O
HO R1 O C R2 C O R1 O C R2 C OH
Figure 3.11: Initiation and formation of linear polyester molecules
22
there is a very high yield of carbonates, which are otherwise difficult to prepare. The car-
bonates formed (see Figure 3.9) are surprisingly stable.
Cyclic esters (lactones) can, to an extent depending on the number of atoms in the ring,
undergo ring-opening reactions with carboxylic acids or alcohols to form chain esters. While
the chemical equilibrium of cyclic esters of lactic acid, γ-butyrolactone, and δ-valerolactone is
on the side of the cyclic ester (five- and six-membered rings), the equilibrium of ε-caprolactone
(seven-membered ring) favours the
formation of polyester chains (see Fi-
gure 3.10). O O
n HO R1 OH + n HO C R2 C OH
dicarboxylic acid
3.1.2 Structure of diol
– n H2O
polyesters O O
n HO R1 O C R2 C OH
3.1.2.1 Formation of monoester
linear polyesters
– (n–1) H2O
The formation of polyester resins re-
quires the presence of polyfunctional O O O O
building blocks, i.e. with at least two HO R1 O C R2 C O R1 O C R2 C OH
functional groups. If a diol reacts n–1

polyester
with a dicarboxylic acid, the first en-
tity formed is a single ester, as de Figure 3.12: Formation of polyester molecules con-
scribed above (see Chapter 3.1.1). taining n moles diols and n moles dicarboxylic acids
This single ester still contains both
reactive groups. The OH group of the
single ester can react with the carb-
oxyl group of a dicarboxylic acid or
n HO OH + n HOOC COOH
of another single ester, while the
carboxyl group of the single ester can – n H2O
react with the OH group of a diol or
n HO COOH
of another single ester. This “head-to-
tail” reaction yields chains of oligo- – (n – 1) H2O
mers and finally polymers, i.e. linear
polyesters (see Figure 3.11).
If n moles of diol react with n HO COOH
moles of dicarboxylic acid, theoreti- n–1
cally, linear polyester molecules
with an average number of (n-1) Figure 3.13: Simplified way of depicting the
structural units (monoesters) are formation of linear polyesters
23
formed, all of which are terminated with both types of functional groups (see Figure 3.12).
There is the further possibility that there will be a mixture of molecules bearing two ter-
minal OH groups and molecules bearing two carboxyl groups.
In the past, a simplified way of representing such structures was found and this is
now encountered in the literature as well. Polycarboxylic acids are symbolised by rings
and lines, because most polycarboxylic acids in polyesters intended for coatings are
aromatic compounds. By contrast, the usually aliphatic polyols are represented simply
by lines. The number of line ends represents the functionality of the building blocks. The
simplified representation for the formation of linear polyester molecules is shown in
Figure 3.13.
3.1.2.2 Formation of branched polyesters

Whenever building blocks contain more than two functional groups, branched polyesters
will be formed (see Figure 3.14).
While there is a possibility during the formation of linear polyesters that molecules of
different structure will be formed, the likelihood of this is all the greater during the forma-
tion of branched polyesters. A multitude of structural isomers are formed. Their number
increases even further because some of the polyester molecules formed are terminated
solely with OH groups while others are terminated solely with carboxyl groups. In addition,
the number of structural isomers
increases exponentially as the mole-
cular weight of the polyesters in-
n HO OH + n HOOC COOH
creases. It is theoretically possible
OH
– n H2 O for linear molecules to result from
the preparation of polyesters based
n HO COOH on building blocks bearing more
OH than two functional groups. How
– 6 H2 O ever, the quantity of linear molecu-
les will decline as the polyester mo-
OH OH lecules become larger. The idea that
HO COOH
polyesters based on building blocks
OH OH
bearing more than two functional
OH OH groups of different reactivity will
HO OH primarily yield linear molecules
OH needs to be viewed critically. A clas-
COOH sic example of such an idea is the
preparation of polyesters with the
aid of glycerol as building block.
Figure 3.14: Incipient formation of branched polyesters This postulates that linear polyes-
24
ters are the first to form and that branched polyesters form only at the end of the reaction,
if at all. Of course, the primary OH groups of glycerol will react faster than the secondary
ones and, what is more, there are two of them. However, as soon as some of the primary
OH groups have formed esters, the concentration of secondary OH groups grow and will
be available for reactions that yield branched polyesters.
The use of building blocks of different functionalities (two and more) increases the
potential number of structural isomers of polyester molecules.
3.1.2.3 Ring closure during formation of polyesters?

Publications [21–23] deal in great detail with the theory behind the formation of cyclic struc-
tures during polyester formation. Well known is that five- or six-membered rings are for-
med easily, because of the resulting bonding angles. The suggestion is made that rings will
have stressed bonding angles until the number of atoms exceeds fourteen. The conclusion
is then drawn that the formation of ring structures by polyester molecules may explain
why observed molecular weights and molecular weight distributions deviate from theore-
tical predictions. However, it must be remembered that polyester chains have numerous
potential tertiary structural isomers that would make “head-to-tail” arrangement rather
improbable. Furthermore, esterification requires certain kinetic conditions and successful
collisions between functional groups (this is described by the reaction rate constant k' for
esterification.) Thus, the formation of large rings during the production of polyesters is
improbable.
In this book, it is assumed rings are not formed during the preparation of polyesters.
However, there are building blocks which tend to enter into ring-forming reactions. For
example, oxalic acid and ethylene glycol form dioxolandione in high yield; and 4-hydroxy-
butyric acid react to yield an internal ester, γ-butyrolactone. Such building blocks are not
used in the production of polyesters.
It is reported [24] that GC analysis confirms that the preparation of polyethylene te-
rephthalate by transesterification of dimethyl terephthalate with ethylglycol in an inter-
mediate step gives rise to cyclic molecules consisting of two molecules terephthalic acid
and two molecules of ethylene glycol. The amount of such molecules allegedly worsens
the properties of the otherwise high molecular weight polyester, which serves as a fibre
raw material. Preparing a model of this molecule shows that it is implausible or nearly
impossible to assemble the atoms to form such a structure (due to the planar structure
of the terephthaloyl group and the repulsion effect of π-electrons on aromatic rings).
3.1.2.4 Crosslinking during polyester formation

Under certain stoichiometric conditions, branched polyesters can consist of very large mo-
lecules whose molecular branches bear terminal OH groups and carboxyl groups. If such
molecules react even further, the branched molecules can become crosslinked. Crosslinked
25
molecules are also formed when large branched molecules become joined together by
multiple reactions among small molecules. The crosslinked networks can contain smaller
molecules within them. When crosslinked polyester molecules are forming, the viscosity
of the reaction melt suddenly changes. The melt reverts to a gel of very high viscosity and
a marked yield point. Crosslinked polyesters cannot be melted and are not soluble in any
solvents although they might be swellable. Thus, they cannot be transformed into a usable
condition.
Since work on the development of polyesters focused initially on producing the high
est-possible molecular weights, a great deal of time was spent on elucidating the conditions
for crosslinking (gelation), with numerous experiments aimed at predicting the degree of
condensation which would induce gelling (gel point).
3.2 Determination of and limitations on

the size of polyester molecules
3.2.1 Dependencies regarding molecular weight
In the early days, the first polyesters and especially alkyd resins [25] were formulated according
to empirical rules. Subsequently, in the early 1930s, they attracted the attention of theoretical
chemists. Carothers and Flory played key roles in establishing the theoretical conditions needed
for the preparation of polyesters. Studies back then focused on the development of high mole-
cular weight polymers that would be suitable for plastics and fibres. However, initial attempts to
prepare such high molecular weight polyesters for such applications failed.
It was realised at an early stage that the molecular weight of polyesters based on bifunc-
tional building blocks depends on the molar ratio of the components and on the degree of
condensation. This led to the development of what are now well-established equations for
the formation of A—B linear polymers.
The molecular weight (Mn) of polyesters or the number of structural units (q) exhibits a
dependence on the quotient of the number of functional groups at a given time during the
reaction (νCOOH or νOH) and the number of functional groups at the start of the reaction
(nCOOH or nOH) for a polyester consisting of equal moles of diol (n1) and dicarboxylic acid
(n2). By structural unit here is meant a combination of one molecule of dicarboxylic acid
and one molecule of diol or – by extension – a combination of one mole of polycarboxylic
acid and the attached polyol molecules. For bifunctional building blocks, the number of
structural units exhibits a dependency on the quotient of the initial number of moles of
building blocks (n1 or n2) and the number of functional groups at a given time during the
reaction (νOH or νCOOH), i.e. the degree of polycondensation, as shown in the following
equations:
26
Determination of and limitations on the size of polyester molecules
Equation 3.3
This dependency is presented in Figure 3.15.

The second dependency is given by the use of a molar excess of one building block.
Normally, polyesters contain an excess of polyols. In that case, the number of structural
units in a polyester molecule (q) exhibits a dependence on the quotient of the number of
moles of diol (n1) and the number of moles of dicarboxylic acid (n2), as soon as the number
of residual free carboxylic (νCOOH) groups tends towards zero and the degree of condensa-
tion tends towards 1.
Figure 3.15: Number of structural units as a function of degree of polycondensation in linear

polyesters
27
Equation 3.4
This dependency is presented in Figure 3.16:

A corresponding result is obtained when there is an excess of carboxylic acid or COOH
groups, as shown in the following equation:
Equation 3.5
The attempts to achieve the highest-possible molecular weights led to the study of poly-
esters containing building blocks which have more than two functional groups. The most
commonly observed effect then was gelation or the formation of infinitely large molecules,
Figure 3.16: Number of structural units of linear polyesters as a function of molar excess of diols
28
i.e. molecules which grow until they occupy the full volume of the reaction mixture. Such
molecules are crosslinked. Consequently, many tests were then focused on establishing the
degree of condensation at which gels form, i.e. the gel point.
3.2.2 Derivation of gel-point equations

In 1935, Carothers [26] proposed an equation which describes the degree of polyconden-
sation (p) as being dependent on the difference between the initial number of molecules
of reactants (n1,2) and the number of polyester molecules formed (np) at a given time du-
ring the reaction in proportion to the total number of initial functional groups:
Equation 3.6
Other authors also use this equation [27–31].

In the event that the quotient of the number of building blocks and the number of
polyester molecules formed tends towards infinity, i.e. the numerous molecular building
blocks have formed a few, very large polyester molecules, the degree of polycondensation
(p) becomes critical (pc). The critical degree of polycondensation is defined as the quotient
of twice the number of all building block molecules (n1,2) and the sum of all functional
groups (n1,2 · F1,2), as shown in the following equation:
Equation 3.7
This Carothers equation means that, at the gel point (p=pc=1), the number-average mole-
cular weight of polyesters tends towards infinity.
Equation 3.8
By contrast, Flory [32 and as described in 27, 28, 30, 31, and 36] found that gelation takes place at much
lower degrees of condensation than is assumed in the Carothers equation. He proposed
that, at the gel point, the weight-average molecular weight tends towards infinity. In other
words, only some of the polyester molecules strive to occupy the full reaction volume, with
a great many molecules remaining smaller. He defined a statistical term (α) to describe the
29
probability of two polyester molecules reacting together to form bridges. At the gel point,
this term becomes critical (αc).
The term (αc) is that value at which there is a better-than-even chance of more than one
bridge being built between two chains of polyester molecules, leading to a fractional quan-
tity of infinite network molecules (see next equation):
Equation 3.9
Later authors tried to quantify the value of Flory’s critical degree of condensation (αc) by
introducing further statistical terms and others ran trial series to study the influence exert
ed by building block functionality.
Thus, Stockmayer [34] developed an extension of the Flory equation by adding the
statistical effect of building block functionality, as shown in the following equation:
Equation 3.10
Jonason [33] included the modification of polyesters with monocarboxylic acids (alkyd resins)
and obtained comparable results. His critical degree of polycondensation (pc) varies with the
square root of the quotient of the functional groups (nOH/nCOOH), divided by a term including
the ratio of the number of carboxyl groups of monomer carboxylic acids (n3) to the number
of carboxyl groups of polycarboxylic acids (n2 · F2), as shown in the following equation:
Equation 3.11
Kilb [35] tried to explain the differences between his test results and the predictions of
Flory and Stockmayer in terms of intermolecular reactions that vary with the length and
stiffness of polyester chains and the instantaneous concentration of polymer and functio-
nal groups in a given reaction volume.
Bernardo [27] provided a good overview of the development of all these models and
equations. He compared the theoretical gel point yielded by the various equations with the
results he obtained in a large test programme for producing alkyd resins.
The biggest difficulty with the calculation of gel points of branched polyesters was the fact
that – and this is the basis of Flory’s theory – only a small number of the growing polyester
30
molecules strived to reach an infinite size at the gel point. The reason was the different
molecular weight distributions and the reactivity of the monomers and polyester units in
the reaction.
Flory’s theory, in which he compared the growth of polyester resins (polycondensation)
with the polymerisation reaction undergone by unsaturated monomers, predicted that the
curves for the molecular weight distribution of polyesters would be relatively flat.
However, Kilb [35], Korschak [37] und Bresler [38] found that the polycondensation pro-
ducts, contrary to Flory’s prediction, had relatively steep Gaussian molecular weight distri-
butions. They postulated that the gel points of polyester and alkyd resin formulations were
intermediate between the values postulated by Flory and by Carothers [40]. This view is still
found in current literature [41].
The reason is – as was subsequently discovered – that during the process of polyconden-
sation, an equilibrium exists not only between the acids and alcohols and the esters and water,
but also between the various polyester molecules, which can undergo transesterification.
The bulk of the polyesters synthesised in the various trial programmes exhibited a
tendency towards average molecular size, a tendency which is influenced primarily by the
functionality of the reactants, but also by the reactivity of the functional groups and by
the molecular weight of the polyesters, as would be expected. Surprisingly, the reaction
conditions are of minor importance.
If it is difficult to calculate the molecular weights of branched polyesters, it is even
harder to do the same calculation for alkyds under the additional influence of the mono-
carboxylic acids.
Patton [39] modified Carothers’ gel equation (Equation 3.7) to include the number of
moles of monocarboxylic acids (n3). By performing a great many calculations on the alkyds
produced, he found that the ratio of the number of moles of reactants to the number of
acid equivalents must be 1.000 to 1.005 if an alkyd formulation is not to lead to gelation.
This ratio is called the Patton (or alkyd) constant (KPatton).
Equation 3.12
Patton’s constant remains the basis of several published trial programmes and is still used
by many developers and producers of alkyd resins to formulate alkyds. As this equation is
based on stoichiometric ratios, its scope is not wide enough to cover all alkyd resins, especi-
ally those of relatively low molecular weight and high numbers of residual OH groups Patton
himself recommended some restrictions and extensions to his equation and Sunderland [36]
tried to improve it. Dyck [29] gives an overview of the development of gel-point equations
from Carothers and Flory to Patton to Sunderland.
31
3.3 Methods of calculating average

molecular weights of polyesters
3.3.1 Factors that influence molecular weights
With regard to the question as to which of the aforementioned equations are suitable for
describing polyesters and which of them might lend themselves to systematic trial pro-
grammes, a few critical observations need to be made.
Although the gel point is an important limit in any attempt to prepare different poly-
esters or alkyd resins, it is even more interesting to be able to calculate the molecular
weights of any of the polyesters or alkyd resins being considered.
Naturally, it is important for air-drying alkyds to attain the highest-possible molecular
weight, because the crosslinking reaction with atmospheric oxygen is relatively slow. Con-
sequently, the step from resin polymers to polymers with coating properties necessarily
has to be small. In most cases, the resin chemist must bear two things in mind. First, the
soluble resin which he has to put together must contribute to good application properties
in the coating system. Second, the resin must be able to form sufficiently (ideally: infini-
tely) large molecules in the coating film.
In recent decades especially, much development work has focused on high-solid sys-
tems. One way to obtain a higher solids content in a coating system is to lower the molecu-
lar weight of the vehicle resins. Previously the goal was to maximise the molecular weight
of polyester or alkyd resins (as a way of quickly achieving sufficiently high crosslinking and
high performance) which ran up against the difficult task of avoiding gelation and defining
gel points. The goal for high-solid systems would be to create polyesters and alkyd resins
that have good crosslinking properties but the smallest-possible molecular weight.
3.3.2 I nfluence of molar ratios of polyol and poly

carboxylic acid molecules on molecular weight
Key to the average size of polyester molecules are the molar ratio of the building blocks in
the polyester formulation and the degree of polycondensation (see Figures 3.15 and 3.16).
Figure 3.17 shows that polyesters formulated with diols and dicarboxylic acids have diffe-
rent possible structures than those formulated with triols and dicarboxylic acids. However,
there is no difference in the size of the polyester molecules, i.e. the number of structural
units (q), provided that the same molar ratios of polyol to dicarboxylic acid are used. Figu-
re 3.17 can be extended to polyols of higher functionality, polyol mixtures of different
functionality, and to alkyd resins as well, without change to the definition that the number
of structural units depends on molar ratios.
32
Methods of calculating average molecular weights of polyesters
Such a molar approach is described by Kraft [42], who used it to calculate and formulate
alkyd resins produced with different reactants using equal molar quantities of polycarbo-
xylic acid and polyol but did not derive a calculation method for molecular weights.
Finally, U. and H. Holfort [43] arrived at an equation for calculating the number-average
molecular weight of polyesters from the molar ratio of polyol (n1) to polycarboxylic acid
(n2). Most real polyesters and alkyds are made with a molar excess of polyol. If it is assumed
that polyesters are made only by esterification and transesterification, and no additional
models of polyester molecules

q diols + dicarboxylic acids triols + dicarboxylic acids n1/n2
1 2.00
2 1.50
3 1.33
4 1.25
5 1.20
1.00
Figure 3.17: Number of structural units (q) of polyesters as a function of the molar ratio of
polyols to dicarboxylic acid (n1/n2), regardless of the functionality of the polyols
33
intermolecular reaction occurs during the process, the equation for determining the number
of structural units (q) can be extended as a function of the molar ratio of bifunctional reactants
to polyols of all functionalities (F1 ≥ 2), provided that there is an excess of polyol (n1 ≥ n2).
The molar ratio of polyol to dicarboxylic acid is defined – in contrast to Patton’s cons-
tant – as a constant (kM) for obtaining the number of structural units (q) and ultimately
calculating the molecular weight (MP) of polyesters (see the following equation):
Equation 3.13
On the assumption that the preparation of polyesters involves esterification and transes-
terification only and that the formation of ring molecules is negligible, the number of
Figure 3.18: Number of structural units (q) as a function of the polycondensation constant
(kM) and the number of polyol functional groups (F1)
34
structural units of a polyester (q) can be defined as a function of the molar ratio of polyols
(n1; regardless of their functionality F1 ≥ 2) to – initially – dicarboxylic acids (n2, F2 = 2):
Equation 3.14
Under these conditions, the curve of the function of the number of structural units (q)
is congruent for all polyols, regardless of their functionality, as shown in Figure 3.18.
Where the dicarboxylic acid is in excess (n2 > n1), the number of structural units (q) is
dependent on the value of the polycondensation constant (kM) and the functionality of the
polyols. In the case of diols (F1 = 2), an excess of dicarboxylic acids is just as limiting as an
excess of polyols. The curve of the number of structural units for (kM) values between 0
and 1 is the mirror-image of the curve for (kM) values between 1.00 and 2.00. At
(kM) = 1.00, both sections of the curve tend towards infinity. For polyols of functionality
higher than 2 (F1 > 2), there are different ranges for (kM) below 1.00 where the number of
structural units (q) is not defined, or they tend towards infinity: crosslinked polyester
molecules are present. Only when the dicarboxylic acid is in high excess is the number of
structural units limited. These conditions are described in the following equation.
Equation 3.15
The model of the polyester segment

in Figure 3.19 shows the crosslinked
structure of a polyester comprising
three moles of triol and four moles of
dicarboxylic acid (kM = 0.75). The re-
petition segment which is depicted
has two pendant carboxylic acid
groups and three pendant hydroxy
groups. These allow this structure to
become fully crosslinked which is in- Figure 3.19: Polyester segment of triol and
dicated by the arrows. dicarboxylic acid and (kM) = 0.75
35
The conditions for crosslinked molecules are shown in the following equation.
Equation 3.16
The range of values for constant (kM) over which the number of structural units (q) of
polyesters tends towards infinity lies between 0.50 and 1.00 for triols and dicarboxylic
acids and between 0.33 and 1.00 for tetrols and dicarboxylic acids. Usable polyesters will
result only if all their molecules have (kM) values outside these ranges.
To include the possibility of branching not only by triols, tetrols etc, but also by poly-
carboxylic acids, the definition of constant kM needs to be expanded to include a term to
account for the functionality of po-
lycarboxylic acids (F2 ≥ 2).
Polycarboxylic acids with F2 ≥ 2
will require one more mole of polyol
for every functionality over 2. So,
the additional moles of polyol can be
calculated by multiplying the moles
of carboxlic acids (n2) by the functio-
nality of these carboxylic acids (F2)
Figure 3.20: Molecular models of polyesters of but subtracted by 2 (moles of polyol
dicarboxylic acid and tricarboxylic acid having the already accounted for in linear poly-
same number of structural units (q = 4). merisation). This number is subtract
ed from the total amount of polyols
(n1) when calculating KM (Equation
3.17). In other words, a given num-
ber of moles of polyols (n1) must be
Figure 3.21: A model representing all possible Figure 3.22: Simplified model of all
polyester molecules possible polyester molecules
36
reduced by the additional number of COOH groups (over two) on the polycarboxylic acids
if the latter have more than two such groups per molecule. To illustrate, Figure 3.20 shows
two polyester molecules with equal numbers of structural units (q = 4). The first consists of
four moles of dicarboxylic acid and five moles of diol. The second consists of four moles of
tricarboxylic acid and nine moles of diol, as each of the four moles of tricarboxylic acid re-
quires a mole of diol for limiting the molecular weight and maintain a value of q = 4. If it
did not have these additional polyol molecules, further polymerisation could take place and
q would be greater than 4.
The extension of the definition of the polycondensation constant is presented in the
following equations:
Equation 3.17
For the constant (kM), the numerator is extended by the term {- (F2 - 2)} per mole of poly-
carboxylic acid (n2). For dicarboxylic acids, of course, the term is zero (F2 = 2). In the case
of mixtures of polycarboxylic acids, the average of the functionality can be used for the
calculation, or the term can be disaggregated into factors. The definition of the constant
(kM) can now be used for all polyester building blocks (F1 ≥ 2, F2 ≥ 2).
Where hydroxycarboxylic acids are employed, they are considered to be a combination
of polyol and polycarboxylic acid without ester group. Hydroxycarboxylic acids are coun-
ted both as polyols and polycarboxylic acids, and the equation given above (Equation 3.17)
applies.
If all polyesters consist of molecules without rings or crosslinks, they can be considered
“open branched” and it is possible to draw linearised molecular models. All polyesters
consist of structural units of polycarboxylic acid and attached polyol plus the excess of
attached polyol. Such a model is shown in Figure 3.21.
Figure 3.22 shows the simplified version of the model, without the functional groups.
This general model, which serves for all types of polyesters, makes it very easy to cal-
culate molecular weights. The molecular weight of a polyester (MP) is the molecular weight
of the structural unit (MS) times the number of structural units (q) plus the molecular
weight of one molecule of polyol (M1), so as to block the last carboxylic acid chain end also
by a polyol molecule. (see the following equation):
Equation 3.18
37
Since generating structural units as well as coupling them to a polymer will liberate water
this needs to be corrected for. This is done by defining a corrected mass (M’2) for the
carboxylic acid containing moiety (Equation 3.19). The molecular weight of the structural
unit (MS) is then as the corrected mass of the carboxylic acid containing moiety (M’2) and
of the associated moles of polyol ([F2 - 1] · M1).
Equation 3.19
Substitution of equations 3.17 (q) and 3.19 (MS) in 3.18 yields the following equation for
determining the molecular weight of polyester molecules.
Equation 3.20
If the second term (M1) is extended by the denominator (kM – 1), the equation can be re-
arranged as a fraction:
Equation 3.21
The term (kM + F2 – 2) is, after rearrangement of the equation for the polycondensation
constant (kM; Equation 3.17), equivalent to the molar ratio of polyol to polycarboxylic acid
(n1/n2). The result is an equation (3.22) for calculating the molecular weights of polyesters
based on the number of moles of building blocks (n1, n2), their molecular weights (M1, M'2)
and the polycondensation constant (kM).
Equation 3.22
Rewriting 3.22 yields an equation whose numerator represents the yield of a polyester
composed of a given number of moles of reactants (n1, n2).
Equation 3.23
38
The yield mass (A) of any polyester composed of n1 and n2 moles of building blocks can thus
be used to calculate the molecular weight of the polyester, as shown in the following equation.
Equation 3.24
All terms in the equations are average values. In the case of the usually complex polyester
compositions, however, it makes sense to disaggregate the terms n1 · M1 and n2 · M’2 into
factors, as shown in the following equations:
Equation 3.25
The calculation of the molecular weights of polyesters in the manner just presented returns
average molecular weights, in this case the number-average molecular weight (Ṁn).
3.3.3 alculating the influence of the degree

C
of condensation on the molecular weight
All definitions and equations relating to the calculation of the molecular weights of poly-
esters prepared with an excess of polyols presuppose that the degree of condensation has
a value of one, meaning that the number of instantaneous residual carboxyl groups (νCOOH)
is approaching zero. In reality, poly-
esters and alkyd resins rarely have
exhausted every possibility for for-
ming ester groups, with the result
HO OH
that usually some carboxyl groups
remain. Each residual carboxyl
q=6
group forms a chain-end in the poly-
OH
ester molecule. For all polyesters,
there are two limiting effects on HO COOH
chain length: the excess of polyol
and the residual carboxyl groups. q=6
Figure 3.23 compares two polyes- OH
ters having the same number of
structural units (q = 6), the first of Figure 3.23: Excess of polyol or residual carboxyl
which was prepared by reaction of groups – an alternative limitation on molecular size
39
6 moles of dicarboxylic acid with 7 moles of polyol, while the second was prepared from
6 moles of dicarboxylic acid and 6 moles of polyol, in this case resulting in one residual
carboxyl group, which forms an alternative chain-end.
If carboxyl groups in polyesters act to form a chain-end in a similar manner as an excess
of polyol, the equation of the constant (kM) can be extended by the number of residual
carboxyl groups to arrive at the constant (k’M), as shown in these equations:
Equation 3.26
As residual carboxyl groups (limited degree of condensation) can act as an alternative to

an excess of polyol, this equation also has validity for polyesters that do not contain an
excess of polyol (n1 ≤ n2).
Equation 3.27
Figure 3.24: General model of linearised polyesters
Figure 3.25: Simplified form of a general linearised polyester
40
This equation is therefore valid for all polyesters containing n1 moles of polyols and n2
moles of polycarboxylic acids and any number of residual carboxyl groups νCOOH ≥ 0.
The general linearised polyester model is shown in Figure 3.24.
Or, in a more simplified form shown in Figure 3.25.
The molecular weight calculation is similar to that calculation which ignores the degree
of polycondensation (described earlier). There are only minor differences, concerning the
residual carboxyl groups, as shown in the following equations:
Equation 3.28
The term f2 is the quantity of residual carboxyl groups (νCOOH), per mole of polycarboxylic
acid.
Equation 3.29
The term k'M + F2 - f2 - 2 in the numerator is replaced by the quotient of the number of
moles of polyol and the number of moles of polycarboxylic acids (n1/n2) resulting from the
rearrangement of the equation for k'M (using Equations 3.26 and 3.29).
Equation 3.30
The residual carboxyl groups (νCOOH) can be calculated with the aid of the acid value (AV)
for the polyester, as shown in the following equation. The acid value is expressed in terms
of the current yield mass of polyester (AAV).
Equation 3.31
The resulting equation for the molecular weight of the polyester (by combining Equation
3.28 and 3.30) applies to all polyesters consisting of n1 moles of polyol of molecular weight
M1 and n2 moles of polycarboxylic acid of molecular weight M’2.
41
Equation 3.32
This equation also applies when there is no excess of polyol (n1 ≤ n2), i.e. the limitation on
the molecular weight is due solely to the degree of condensation.
The yield mass for a given acid value (AAV) is higher than the theoretical yield mass
(A0) of the condensation reaction that went to completion by an amount of water corre-
sponding to the number of carboxyl groups that are still free:
Equation 3.33
Replacing the number of residual carboxyl groups (νCOOH) by Equation 3.31, gives us:
Equation 3.34
Rewriting Equation 3.34 and solving for AAV results in:
Equation 3.35
Substituting Equation 3.35 in 3.32 gives us the following expression for the molecular
weight:
Equation 3.36
By introducing the known constant terms this leads to:
Equation 3.37
42
Table 3.1: Calculation of a model polyester with a limited degree of condensation

n•F n building element M [g/mole] n • M [g] wt.%
–1.40 0.70 isophthalic acid 166 116.2 49.72
–0.60 0.30 adipic acid 146 43.8 18.74
+1.00 0.50 neopentyl glycol 104 52.0 22.25
+0.60 0.30 propylene glycol 76 22.8 9.76
+0.75 0.25 trimethylol propane 134 33.5 14.33
sum 268.3 114.81
–0.08 νCOOH
+0.43 νOH
1.92 water 18 34.6 14.81
yield 233.7 100.00
What can easily be seen is that if the acid values are quite low, the second bracketed term
in the denominator can be reduced to “1”. This provides enough precision.
3.3.4 Sample calculations of molecular weights

and related characteristics
The following examples show how the calculation methods are used.
a) A polyester is to be prepared by making 0.70 moles of isophthalic acid, 0.30 moles of
adipic acid, 0.50 moles of neopentyl glycol, 0.30 moles of propylene glycol and 0.25
moles of trimethylol propane react until 96 % of the carboxyl groups have been con-
verted into ester groups.
The number of functional groups (n · F) for carboxyl groups are assigned negative values
while the number of OH groups are assigned positive values. This is only done to indicate
that these numbers need to be subtracted due to reactions taking place. The acid value and
the OH value are computed as follows:
Equation 3.38
43
Table 3.2: Calculation of a model polyester of given acid value

n•F n building elements M [g/mol] n • M [g] wt.%
–1.50 0.75 isophthalic acid 166 124.5 46.10
–0.75 0.25 trimellitic anhydride 192 48.0 17.97
+1.30 0.65 neopentyl glycol 104 67.6 25.31
+0.70 0.35 hexane diol-1,6 118 41.3 15.64
+0.70 035 ethylene glycol 62 21.7 8.12
sum 303.8 113.48
+0.45 νOH
2.00 water 18 36.0 13.48
yield A0 267.1 100.00
yield AAV 268.4
The average molecular weight is:
Equation 3.38

a) A polyester consisting of 0.75 moles of isophthalic acid, 0.25 moles of trimellitic an-
hydride, 0.65 moles of neopentyl glycol, 0.35 moles of 1,6-hexanediol and 0.35 moles
of ethylene glycol is condensed until the acid value is 15 mg KOH/g.
The yield mass is then:
Equation 3.40
The quantity of residual free carboxyl groups is:
Equation 3.41
The number of residual OH groups increases accordingly:
Equation 3.42
44
Table 3.3: Systematic trial plan for studying the variation in molecular weights and degree
of branching in polyesters
trials a b c d e
building elements [mole]
isophthalic acid 0.65 0.65 0.65 0.65 0.65
adipic acid 0.35 0.35 0.35 0.35 0.35
neopentyl glycol 0.80 0.85 0.90 0.90 0.80
trimethylol propane 0.30 0.30 0.30 0.20 0.40
KM (AV = 0) 1.10 1.15 1.20 1.10 1.20
acid value [mg KOH/g] 10 10 10 10 10
molecular weight [g/mol] AV = 10 1716 1295 1049 1701 1058
The resulting OH value is:
Equation 3.43
The polycondensation constant (k’M) is:
Equation 3.44
The value of 0.25 in the numerator is the molar quantity of trimellitic anhydride and is
obtained by factorising the term n2 · (F2 - 2).
The (number) average molecular weight then computes to:
Equation 3.45
Or, based on the yield mass for an acid value of 0 mg KOH/g, the average molecular weight
computes to:
Equation 3.46
45
Table 3.4: Calculation of the characteristics of a polyester based on analytical data

wt.% building elements M [g/mol] m/M = n n•F
45.3 phthalic anhydride 148 0.3061 –0.6122
11.2 adipic acid 146 0.0767 –0.1534
n2 0.3828 –0.7656
10.6 glycerol 92 0.1152 +0.3456
9.0 hexane diol-1,6 118 0.0763 +0.1526
23.9 neopentyl glycol 104 0.2298 +0.4596
n1 0.4213 +0.9578
100.0 sum +0.1912
8.3 water 18 0.4595
91.7 yield (AV = 0)
Equation 3.47
If the third term in the denominator is ignored, the molecular weight is 1571 g/mol, indi-
cating as mentioned previously that for small acid values, this term can be set to “1”.
The number of moles of polycarboxylic acid was deliberately taken to be 1.00 in both
calculations. This simplifies the calculations and makes them easier to illustrate. It makes
sense to adopt the same approach in trial plans, as shown in the following example:
a) Trial plan to study the variation in molecular weight and degree of branching of polyesters
prepared from isophthalic acid, adipic acid, neopentyl glycol and trimethylol propane:
In practice, trials a to c will show the influence of differences in molecular weight on the
properties of the polyesters (viscosity, application characteristics). Comparisons of trial a
with trial d, and trial c with trial e will show the influence exerted by the degree of branch
ing on these properties.
The calculations presented above can also be used for polyesters described by formu-
lations based on percentage mass, e.g. analytical data. The calculation process is shown in
the following example.
a) An analysis of a polyester yields 45.3 wt.% phthalic anhydride, 11.2 wt.% adipic acid,
10.6 wt.% glycerol, 9.0 wt.% 1,6-hexanediol, and 23.9 wt.% neopentyl glycol. The total
is 100 wt.% which is the initial weight – not the yield mass of the polyester. The acid
46
number of the polyester is found to be 18 mg KOH/g. The calculation of the molar

composition and quantities of functional groups is presented in Table 3.4.
0.3061 moles of phthalic anhydride and 0.0767 moles of adipic acid can cleave at most
0.45596 moles of water by condensation, equivalent to 8.3 parts by weight. For an acid
value of 18 mg KOH/g, the residual quantity of carboxyl groups (νCOOH) computes to:
Equation 3.48
The polycondensation constant (k’M) is then calculated as follows:
Equation 3.49
The yield mass at the given acid value is 0.53 parts by weight (0.0296 · 18) larger than at
the acid value of 0 mg KOH/g; i.e. the mass is 92.23 parts by weight. The average molecu-
lar weight of this polyester is then calculated as follows:
Equation 3.50
And the OH value is given by:
Equation 3.51
Both the calculated average molecular weight and the calculated OH value can be correlat
ed with corresponding analytical results (GPC, OH value determination). Naturally, it must
be borne in mind that the analytical data may contain systematic errors and deviations
arising from the analytical process. The results have been presented here with greater
precision than is necessary – for illustration purposes.
47
3.4 Molecular weight distribution

of polyesters
3.4.1 Definitions of average molecular weights
The definitions and calculations described above (k'M, q', MP,AV) deal with average values.
However, polyesters produced industrially with the usual process always consist of mix
tures of molecules of different size and are hence polydisperse.
Now, it might be thought that, at the start of polyester preparation, the building blocks
are consumed via the formation of individual esters in a reaction that subsides over time.
However, smaller polyester molecules (M1) are built up before the individual esters are
formed. These smaller molecules are consumed over time through the formation of larger
polyester molecules (M2, M3). Very large molecules (M4) are formed slowly, but continu-
ously. The tendency is for polyester molecules of average size (M2) to form. All studies
show that molecules of average size are formed preferentially due to transesterification
reactions in which smaller molecules react with larger molecules to form average-size
molecules.
The overall reaction involves a plethora of dynamic equilibria between polyester mo-
lecules of different sizes. At the planned degree of condensation (acid value), these equilibria
are frozen, yielding a mixture of polyester molecules of different sizes. The reaction se-
quence is shown in Figure 3.26.
Figure 3.26: Model of the formation of polyester molecules of different molecular weights over time
48
Molecular weight distribution of polyesters
The mixture of polyester molecules of different size – the molecular weight distribution
– can serve as the basis for deriving average molecular weights. There are different defini-
tions of average molecular weight.
The number-average molecular weight (Ṁn) is the quotient of the sum of the molecular
weights of the individual number fractions of molecules (ni · Mi) and the total number of
polyester molecules (ni).
Equation 3.52
The weight-average molecular weight (Ṁw) is the quotient of the sum of the molecular
weights of the individual weight fractions of molecules (mi · Mi) and the total mass of the
polyester (mi). And since the weight of the polyester molecules (mi) is the product of the
number of molecules and their molecular weights (ni · Mi), this leads to the following
equation:
Equation 3.53
Since higher molecular weight fractions contribute more to the weight-average molecular
weight (MW), the MW is always larger than the number-average molecular weight (MN).
The larger the difference between the two averages, the broader is the molecular weight
distribution. The quotient of both values for the average molecular weights is defined as
the dispersity (DM) of the molecules. The larger the value for dispersity, the broader is the
molecular weight distribution.
Equation 3.54
A hypothetical example serves to illustrate the various terms.

Consider a model polyester which consists of structural units prepared from 0.75
moles of isophthalic acid, 0.25 moles of adipic acid, 0.75 moles of neopentyl glycol, and
0.25 moles of trimethylol propane and which has a molecular weight of 236.5 g/mol. The
excess of polyol in the mixture has a molecular weight of 111.5 g/mol. The limitation on
49
Table 3.5: Molecular weight distribution of a polyester model

q ni Mi ni • Mi = mi ni • Mi²
1 0 347 0 0
2 1 585 585 342,225
3 6 821 4,926 4,044,246
4 10 1,058 10,580 11,193,640
5 12 1,294 15,528 20,093,232
6 14 1,531 21,434 32,815,454
7 15 1,767 26,505 46,834,335
8 15 2,004 30,060 60,240,240
9 14 2,240 31,360 70,246,400
10 13 2,477 32,201 79,761,877
11 12 2,713 32,556 88,324,428
12 11 2,950 32,450 95,727,500
13 10 3,186 31,800 101,505,960
14 9 3,423 30,807 105,452,361
15 8 3,659 29,272 107,106,248
16 7 3,896 27,272 106,251,712
17 6 4,132 24,792 102,440,544
18 5 4,369 21,845 95,440,805
19 4 4,605 18,420 84,824,100
20 3 4,842 14,526 70,334,892
21 3 5,079 15,237 77,388,723
22 2 5,315 10,630 56,498,450
23 2 5,552 11,104 61,649,408
24 1 5,788 5,788 33,500,944
25 1 6,025 6,025 36,300,625
sum 184 485,763 1,548,318,349
number-average molecular weight = 485763/184 = 2640 g/mol, weight average molecular weight = 1548318349/485763 = 3187 g/mol,
dispersity = 3187/2640 = 1.21
the molecular weight arising from the degree of condensation is neglected in this exercise
(AV = 0). Table 3.5 shows the number of structural elements (column 1), the number of
molecules (arbitrary, column 2), their molecular weights (column 3), the mass fraction
50
(column 4) and the mass fraction times the molecular weight (column 5). The quotient of
the sum of column 4 and the sum of column 2 yields a number-average molecular weight
(Ṁn) of 2640 g/mol. The quotient of the sum of column 5 and the sum of column 4 yields
the weight-average molecular weight (Ṁw) of 3187 g/mol. The dispersity is therefore 1.21.
The measured dispersity of polyesters produced according to this recipe is actually higher,
at around 3. The reason is that the formation of polyesters by conventional preparation
methods leads to some molecules which have molecular weights above 10,000 g/mol.
The values in Table 3.5 are plotted in Figure 3.27.
3.4.2 GPC analysis

Currently, gel permeation chromatography (GPC) is the most widely employed method for
analysing the molecular weight distributions of polymers, including polyesters and alkyds.
To assist with interpreting the results of GPC analysis of polyesters and alkyd resins, the
method is briefly described below. The results quoted were obtained on an “Agilent 1000”
[formerly Hewlett Packard] and software from PSS (Polymer Standards Service).
GPC analysis works on the principle that polyester molecules of different molecular weight
in solutions of very low concentration have different hydrodynamic volumes. The stationary
Figure 3.27: Plot of molecular weight distribution of a model polyester
51
phase is a gel of a crosslinked copolymer of styrene and some divinyl benzene that contain
pores with different sizes. The polyester molecular coils have different retention times that
vary with their hydrodynamic volume. Smaller molecules enter the pores more easily and
therefore remain longer on the column than larger molecules.
The solvent (mobile phase) is usually tetrahydrofuran, but other solvents like dimethyl-
formamide may sometimes be used in the case of sparingly soluble polymers. The polymer
test solution has a concentration of about 5 g/l and is metered automatically from a loop
onto the columns. A flow rate of typically 1 ml/min is maintained and the temperature is
kept strictly constant, e.g. at 35 °C. After passage through the columns, the concentration
of polymer in solution is determined continuously, commonly with a refractometer, which
determines the refractive index (RI) as a function of concentration. However, UV detectors
may also be used.
The refractometer quantifies the changes in refractive index at short time intervals and
interprets the changes as mass concentrations. The detected mass concentrations are then
expressed in terms of the total mass. Each molecular weight has a different elution time. The
relationship between elution time and molecular weight is established by calibration against
standards, usually polystyrene standards are used. These standards contain polystyrene
Figure 3.28: Calibration curve showing the dependence of the molecular weight (mass fraction)
of polystyrene standards on elution time
52
Table 3.6: Composition of the model polyesters, theoretical molecular weights and analytical
results
polyester
building elements /1 /2 /3 /4 /5
phthalic anhydride 0.500 0.500 0.500 0.500 0.500
adipic acid 0.500 0.500 0.500 0.500 0.500
neopentyl glycol 0.750 0.650 0.600 0.550 0.525
hexane diol-1,6 0.750 0.650 0.600 0.550 0.525
characteristic values:
kM (AV = 0) 1.500 1.300 1.200 1.100 1.050
acid value (measured) 4.6 5.0 6.2 4.5 4.9
OH value (calculated) 200.2 132.2 94.7 50.8 28.6
average molecular weights:
molecular weight, calculated 548 818 1113 2030 3353
molecular weight, osmometric 680 984 1150 1903 2570
molecular weight, GPC 744 944 1274 1488 2089
samples of well-defined molecular weight and a particularly narrow molecular weight dis-
tribution. A typical calibration curve is shown in the Figure 3.28. The equation that is ob-
tained by fitting a curve through the datapoints is:
Equation 3.55
Thus, the dependence of the decimal logarithm of the molecular weight (mass fraction) on
the elution time follows a polynomial of degree three. Further standards are available for
other polymers (e.g. polymethyl methacrylates).
Comparisons of analytical results with theoretical values need to make allowance for
the analytical conditions used which may cause variations between the two.
For instance, the elution times may have been calibrated with polystyrene standards.
However, the coils formed by polyesters in tetrahydrofuran will very likely differ from
those formed by polystyrene and therefore have different hydrodynamic volumes. Hence
the elution times may therefore differ for the same molecular weights. This applies espe-
cially to branched polyesters since branched molecules will have different coiling beha
viour than linear molecules. For this reason, it is the polystyrene equivalent of the molecu-
lar weight which is measured for polyester molecules, not the actual molecular weight.
53
Another deviation may be software-related. The algorithm has to determine the median
concentration at every time interval, not at the edge of the interval. Otherwise, the calcu-
lations of the high molecular weight fractions could lead to lower values for these fractions
which in turn would lead to lower average molecular weights. To minimise this effect is
important to maintain a high sampling frequency.
The use history of the actual GPC columns also plays a role, as they are often used for
all kinds of analyses. This has the effect of lowering the reproducibility of GPC determin-
ations. It therefore makes sense to perform comparative measurements in series.
To illustrate the scope for deviation, model polyesters of different molecular weight
were prepared, and the theoretical values were compared with the analytical results. Be
sides GPC analysis, osmotic determinations were carried out. The composition of the mo-
del polyesters, the theoretical molecular weights and the analytical results are presented
in Table 3.6 and illustrated in Figure 3.29.
The figures in Table 3.6 and the curves in Figure 3.29 show very good correlation
between theory and analytical results at low molecular weights. At higher theoretical mo-
lecular weights, the data diverge, increasingly so at higher molecular weights. The GPC
Figure 3.29: Comparison of theoretical molecular weights with analytical results
54
curve deviates much more substantially than the osmometric curve. From this deviation it
cannot be concluded that the calculated values are wrong, but rather the conditions of the
analytical methods have to be examined.
3.4.3 Influences on the molecular weight distribution

Ever since Flory [32] postulated that, at the gel point, only a fraction of polyester molecules
strive to become infinitely large and Stockmayer [34] tried to quantify this, there have been
numerous trials aimed at predicting the molecular weight distribution or the weight-
average molecular weight of polyesters. Korshak [37] and Bresler [38] found that the mole-
cular weight distribution of polyesters is always narrower than statistically calculated by
Flory and Stockmayer.
Figure 3.30: Integral distribution curve of linear polyesters of different number-average

molecular weights
55
Again, there were many attempts to extend the above-mentioned definitions and the equa-
tions that were subsequently formulated or to at least calculate the weight-average mole-
cular weight. The first step, of course, was to analyse the factors that influence the mole-
cular weight distribution. This was done by examining the results of numerous polyester
syntheses.
The key influencing factors were found to be the:
– targeted number-average molecular weight
– degree of branching in the polyester
– reactivity of the functional groups of the building blocks
Other factors, chief among them the reaction conditions, play a very minor role. Thus, the
reaction temperature has no particular influence and the polycondensation process can be
interrupted, without changing the molecular weight or the molecular weight distribution
at a given degree of condensation. There are only a few exceptions to this.
Influence of the targeted number-average molecular weight

If the targeted number-average molecular weight increases, the dispersity of the polyester
molecules also increases, and the molecular weight distribution becomes broader. The
Figure 3.31: Comparative scope for esterification and transesterification of growing polyester
molecules as a function of degree of branching.
56
reason is that growth of polyester molecules takes place not in distinct steps, but continu-
ously, as shown in Figure 3.26. The larger molecules, too, grow continuously, but transes-
terification has the effect of regulating for average molecular weights.
Figure 3.30 shows the integral distribution curves for linear model polyesters of different
number-average molecular weight which were prepared as described in Table 3.6 before. The
diagram clearly indicates that the higher the average molecular weight, the broader is the
molecular weight distribution. While the polyester of polycondensation constant (kM) 1.50
contains molecules having molecular weights of up to 5,000 g/mol, the polyester with a (kM)
of 1.30 has molecular weights of up to 7,000 g/mol, with a (kM) of 1.20 extending to 9,000
g/mol, a (kM) of 1.10 reaching 10,000 g/mol and a (kM) 1.05 attaining 20,000 g/mol.
Influence exerted by the degree of branching

The degree of branching in polyesters has a pronounced effect on the molecular weight
distribution. The more extensive the branching, the broader is the molecular weight dis-
tribution. The reason is that the regulatory effect of transesterification declines as poly-
ester molecules become more branched. The scope for molecular growth outweighs the
probability of transesterification. This is illustrated in Figure 3.31.
The degree of branching, which is so crucial to the molecular weight distribution, is
calculated as follows:
Equation 3.56
Figure 3.32: Model polyesters of different degrees of branching
57
Table 3.7: Polyesters of equal number-average molecular weights but different degrees of

branching
polyester
building elements (mol) /6 /7 /8 /9
phthalic anhydride 0.700 0.700 0.700 0.700
adipic acid 0.300 0.300 0.300 0.300
MPPD-1,3 1.150 1.000 0.850 0.700
trimethylol propane 0.000 0.150 0.300 0.450
characteristic values
kM (AV = 0) 1.150 1.150 1.150 1.150
acid value (mg KOH/g, measured) 13.6 13.8 15.2 13.0
OH value (mg KOH/g, calculated) 74.4 104.8 136.4 164.5
degree of branching (mol/kg) 0.00 0.54 1.08 1.62
average molecular weight (g/mol)
number-average, calculated 1276 1271 1233 1296
number-average, GPC 1152 1200 1251 1275
weight average, GPC 2245 2856 3529 3938
dispersity 1.95 2.38 2.82 3.09
viscosity (ICI-P+C-visc., 23 °C)
60 % in MPA (mPa·s) 180 235 345 370
65 % in MPA (mPa·s) 375 520 710 850
This equation factors in all the branching permutations for the building blocks, i.e. the sum
of all building blocks having a functionality greater than 2. Like other characteristic values,
the degree of branching (v) is expressed in terms of unit mass and indicates the number
of potential branching points in 1000 g of polyester.
This is illustrated in Figure 3.32 for two polyester models. If the first consists of 5 moles of
phthalic anhydride, 5 moles of neopentyl glycol, and 1 mole of trimethylol propane, and if the
acid value approaches 0, the number-average molecular weight is 1304 g/mol and the potential
degree of branching is 0.77 mol/kg. In contrast, the second polyester, consisting of 5 moles of
phthalic anhydride, 4 moles of neopentyl glycol, and 2 moles of trimethylol propane, has a
number-average molecular weight of 1334 g/mol, and a degree of branching of 1.50 mol/kg.
To demonstrate the influence of the degree of branching (v) on the molecular weight
distribution, polyesters of different branching but the same number-average molecular
weight were prepared and analysed. The polyesters consisted of phthalic anhydride, adipic
acid, and different quantities of methylpropyl 1,3-propanediol (1,3-MPPD) and trimethylol
58
propane (TMP), but with an equal excess of polyols. 1,3-MPPD was chosen because its
molecular weight of 132 g/mol is close to the molecular weight of TMP at 134 g/mol. Thus,
the different structural units have nearly the same weight. The formulations for the four
polyesters (series 1) are presented in Table 3.7.
GPC analysis shows that the number-average molecular weights match the theoretical
values with sufficient precision. The molecular weight distribution curves for the polyes-
ters are shown in Figure 3.33.
From Table 3.7 it can be seen that a higher degree of branching leads to a higher weight
average molecular weight. In Figure 3.33 this effect is visualised by broader molecular
weight distributions, increased fractions of higher molecular weight polyester and lower
curve maxima. The use of polyols of very similar molecular weights yields structural units
of very similar molecular weights. Thus, it is possible to match the individual peaks for
polyester molecules of lower molecular weight in the GPC diagram to the theoretical values.
Figure 3.33: Molecular weight distribution of polyesters of equal number-average molecular

weights but different degrees of branching
59
Table 3.8: Polyesters of the same average molecular weight but different degrees of branching
polyester /10 /11 /12 /13
building element
phthalic anhydride 1.000 1.000 1.000 1.000
adipic acid 0.540 0.480 0.430 0.380
MPPD-1,3 0.540 0.480 0.430 0.380
trimethylol propane 0.120 0.240 0.340 0.440
characteristic values
kM (AV = 0) 1.200 1.200 1.200 1.200
acid value (mg KOH/g, measured) 11.8 11.2 10.3 11.0
OH value (mg KOH/g, calculated) 121.1 144.4 163.0 182.8
degree of branching (mol/kg) 0.45 0.89 1.25 1.60
average molecular weight (g/mol)
number-average, calculated 1042 1062 1088 1081
number-average, GPC 940 982 998 1037
weight average, GPC 1629 1847 1991 2228
dispersity 1.733 1.881 1.995 2.149
viscosity (60 % in MPA) (mPa·s, 23 °C) 160 205 285 315
The logarithmic value for the first peak is 2.58, corresponding to 380 g/mol. This correlates
well with the theoretical value of 390 g/mol for the molecular weight of one mole of a
mixture of polycarboxylic acid and two moles of polyol – the smallest possible molecule
in the distribution.
When polyesters are being prepared in practice, it is important to monitor the growth in
size of the polyester molecules. The degree of condensation is usually determined by
measuring the acid values. It would be very time consuming to follow the molecular weight
development by GPC. Instead, a faster indication can be obtained by measuring the visco-
sity of a specific test solution or of the reaction melt. Both allow the weight-average mole-
cular weight to be quantified.
The viscosity of polyester solutions correlates with the weight-average molecular
weight. The theoretical background for this was laid down in the equations of Einstein and
Staudinger [45] these state that the viscosity of solutions of very low concentration is de-
pendent on the coil volume of diluted polymer. And the coil volume is dependent on the
weight-average molecular weight of the polymer.
60
To eliminate the interactions of the molecular coils amongst themselves, the viscosity is
taken to be the intrinsic viscosity ([η], Staudinger index). The intrinsic viscosity is the extra-
polation of the specific viscosity as a function of polymer concentration for a concentration
of 0.
The specific viscosity is the difference between the solution viscosity and the solvent
viscosity divided by the solvent viscosity. The intrinsic viscosity is then the weight-average
molecular weight with exponent (α) multiplied by a specific viscosity constant k[η] (see
Mark-Houwink equation [46]):
Equation 3.57
η = viscosity of solution
η0 = viscosity of solvent
ηsp = specific viscosity
c = solution concentration
[η] = intrinsic viscosity
k[η] = viscosity constant
Ṁw = weight-average molecular weight
α = exponent of intrinsic viscosity
The exponent α as well as the viscosity constant k[η] are dependent on the interaction
between polymer and solvent. The values are therefore substance- and temperature-spe-
cific. For polyester solutions in typical analytical solvents (e.g. tetrahydrofuran), the vis
cosity constant (k[η]) assumes values between 0.01 and 0.04. The exponent (α) has values
between 0.70 and 0.90 (at 20 °C). It is not possible to derive the two terms theoretically;
they must be determined experimentally for various polymers and solvents.
However, for practical polyester preparation, it is sufficient to monitor the viscosity of
a test solution and to set a value for the end of the polycondensation process. This value
is then used for continuous, reproducible production of that polyester.
A second series of model polyesters illustrates the relationship between molecular
weight distribution, degree of branching and viscosity. To this end, polyesters were syn-
thesised from phthalic anhydride, neopentyl glycol, 1,6‑hexanediol, and trimethylol propa-
ne, with the same excess of polyol and the same degree of condensation, and therefore
with approximately the same molecular weight (number-average molecular weight) but
different degrees of branching (see Table 3.8).
61
The table shows that the number-average molecular weights obtained by GPC analysis are
somewhat lower than the theoretical values. Obviously, the result of GPC is influenced by
the building blocks that compose the polyesters.
Figure 3.34 shows how the dispersity and viscosity vary with the degree of branching
in both series of polyesters.
A nearly linear relationship exists between dispersity and degree of branching for both
series and it increases substantially with increase in the degree of branching. A higher
number-average molecular weight clearly leads to higher dispersity (series 6–9: num-
ber-average molecular weight about 1280 g/mol; series 10–13: number-average molecular
weight about 1070 g/mol).
The viscosity of the polyester solutions increases exponentially as a function of the
degree of branching. The difference in the viscosity curves of both series is explained by
the different molecular weights of both polyesters.
A plot of viscosity – representing the weight-average molecular weight – against the
polycondensation constant (k'M ≥ 1.00) and the degree of polyester branching – shown
Figure 3.34: Dispersity and viscosity of polyesters as a function of degree of branching
62
here as the average functionality of the polyols – reveals a region where the viscosity tends
towards infinity. The polyesters there have gelled. The higher the functionality of the
polyols and thus the higher the degree of branching, the larger is this region. Although the
number-average molecular weight is still of finite dimension (k'M ≥ 1.00), the polyester has
gelled. This substantiates Flory’s postulate that gelation takes place if the weight-average
molecular weight, but not the number-average molecular weight, tends towards infinity.
This universal statement is illustrated in Figure 3.35.
This condition is also illustrated in Figure 3.18. The curves there for the polyconden-
sation constants (kM) represent average values. If the curve is extended to include polyes-
ters made from triol and dicarboxylic acid, the molecular weight distribution of such poly-
esters has the effect of more or less broadening the region for the constant. If, as a result
of the broad molecular weight distribution, these regions extend into the region where the
polycondensation constant is below 1.00 (kM < 1.00), the outcome is some molecules
Figure 3.35: Viscosity of polyester solutions as a function of polycondensation constant and

polyol functionality
63
which are tending towards infinite molecular weight, i.e. the polyester has gelled (in accor-
dance with Flory's postulate). Polyesters can naturally then be obtained only if the values
of the polycondensation constant for all molecules exceed 1.00 (kM > 1.00). In other words,
it is necessary to increase the planned average number of structural units and thus also
the expected number-average molecular weight, so that all molecules fulfil the condition
kM > 1.00. That is the reason why all practical highly branched polyesters (if they are pre-
pared by the conventional polycondensation process) have lower number-average molecu-
lar weights, as shown in Figure 3.36.
Figure 3.36: Molecular weight distributions as regions of the polycondensation constant;

preconditions for the feasibility of branched polyesters
64
Table 3.9: Composition, characteristic values and solution viscosities of model polyesters

of different molecular weights and different degrees of branching
polyester
building
element
[mol] /6 /7 /8 /9 /14 /15 /16 /17 /18 /19
phthalic 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
anhydride
adipic acid 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
MPPD-1,3 1.15 1.00 0.85 0.70 0.72 0.53 0.28 0.20 – 0.11
trimethylol – 0.15 0.30 0.45 0.30 0.52 0.82 0.95 1.20 1.02
propane
characteristic values:
K'M 1.22 1.22 1.23 1.21 1.09 1.12 1.17 1.22 1.27 1.20
q' 4.6 4.6 4.4 4.7 11.3 8.1 6.0 4.5 3.7 5.1
acid value 13.6 13.8 15.2 13.0 14.8 15.6 13.5 14.6 13.7 13.5
[mg KOH/g]
OH value 74 105 136 165 88 147 224 266 328 273
[mg KOH/g]
v [mol/kg] 0.00 0.54 1.08 1.62 1.15 1.96 3.01 3.40 4.19 3.69
molecular weights:
(g/mol)
Mn 1276 1271 1233 1296 2936 2143 1644 1253 1060 1407
(calculated)
Mn (GPC) 1152 1200 1251 1275 2257 1883 1693 1121 1408 1619
Mw (GPC) 2245 2856 3529 3938 30829 14154 10527 2927 5317 8746
dispersity 1.95 2.38 2.82 3.09 13.70 7.52 6.22 2.61 3.78 5.40
viscosity 180 235 345 370 2100 1600 1800 920 1600 1900
[mPa·s]
60 % in MPA
To quantify the general statement of Figure 3.35 showing the dependence of the molecu-
lar weight distribution of different polyesters on the polycondensation constant , a fourth
series of model polyesters complementing the polyesters of the second series was prepa-
red. This series is presented in Table 3.9.
The viscosities of solutions of these model polyesters are illustrated in Figure 3.37, in which
the number-average molecular weight, as expressed by the polycondensation constant (kM),
65
and the average number of structural units (q) are plotted against the degree of branching
(v). Any attempt to draw lines of equal viscosity, based on the viscosity values of the poly-
esters, reveals a skewed exponential dependence of the viscosity on the two values. As a re-
sult, it is possible to estimate the feasibility of polyesters due to viscosity limitations.
Influence of the reactivity of the polyester building blocks

Next to the targeted number-average molecular weight and the degree of branching, the
reactivity of the functional groups of the building blocks is the third important influence
on the molecular weight distribution of polyesters.
For example, a comparison of the molecular weight distributions of polyesters based on
either adipic acid or phthalic anhydride but of otherwise identical composition shows that the
adipic polyesters always have a broader molecular weight distribution than their phthalic an-
hydride counterparts. This difference is not necessarily discernible from viscosity values due
to the huge difference in solubility of the products. Adipic acid polyesters have much more
mobile aliphatic polyester chains than phthalic anhydride polyesters, with their relatively stiff,
Figure 3.37: Viscosities of model polyesters as a function of polycondensation constant and

degree of branching
66
aromatics containing chains. However, GPC does reveal the difference. The reason for the
narrower molecular weight distribution of phthalic anhydride polyesters is the differential
reactivity of the two potential carboxyl groups of phthalic anhydride. The anhydride addition
takes place very rapidly, even at low temperatures, to yield individual esters. The second
carboxyl group, which is the outcome of the anhydride addition, reacts slowly due to steric
hindrance of the ortho-structure. In addition, the vicinal ester groups readily support alcoho-
lysis, which forms the basis for a transesterification reaction, which is the reason that average
molecular weights are formed. By contrast, adipic acid has two carboxyl groups of similarly
high reactivity. The esters formed are not so readily amenable to transesterification.
As another example, consider polyesters composed of trimellitic anhydride and of tri-
mesic acid but of otherwise comparable composition. The trimellitic anhydride polyesters
always have a narrower molecular weight distribution than their trimesic acid counter-
parts. Also, the viscosity of trimellitic anhydride polyester solutions is much lower than
that of trimesic acid polyester solutions of otherwise identical composition and the same
number-average molecular weight. This also applies if the two compounds constitute only
a fraction of the polycarboxylic acids in a polyester formulation.
These differences also become noticeable when polyesters based on isophthalic acid
are compared with those based on phthalic anhydride. As isophthalic acids have melting
temperatures greater than 300 °C, during polyester production at conventional tempera-
tures of 180 to 240 °C, they react only at the particle interfaces. Once isophthalic acid
molecules available at the interface go into solution, both carboxyl groups may react at the
same time by esterification. This boosts the growth of polyester molecules. By contrast,
phthalic anhydride reacts in two steps, as described above, to yield polyester molecules of
narrower molecular weight distribution.
A further example is the production of polyesters from terephthalic acid compared
with that of polyesters from dimethyl terephthalate by transesterification. Dimethyl te-
rephthalate polyesters initially have a much narrower molecular weight distribution than
their free terephthalic acid counterparts, because terephthalic acid is less reactive and has
a higher melting point than isophthalic acid, as described above. As dimethyl terephthala-
te is readily available in the molten phase at common reaction temperatures, the two ester
groups can react at different times. Again, as both polyesters consist of the same building
blocks, the differences in molecular weight distributions can be recorded by measuring the
viscosity of solutions. However, it must be remembered that the degree of transesterifica-
tion of dimethyl terephthalate is difficult to establish because acid values cannot be mea-
sured. It may be assumed that, after sufficient reaction time, the molecular weight distri-
butions of both polyesters approach one another due to the transesterification equilibrium.
Given that the width of the molecular weight distributions is dependent on the specific
reactivity of the building blocks, there is no way, based on the information described ab-
ove, to incorporate the parameters governing the molecular distribution into a universal
67
equation. The limitation of the area of polyesters (see Figures 3.35 and 3.37) which are
not feasible since they would gel, even though their number-average molecular weight is
finite, is variable for different polyester formulations.
Therefore, what remains is only the definition of the number-average molecular
weight, calculated from the excess of polyol and the degree of condensation and the esti-
mate of the molecular distribution curves which are influenced by the value of the expec-
ted number-average molecular weight, the degree of branching, and the reactivity of func-
tional groups of building blocks.
Clearly, the reaction conditions for preparing polyesters play only a minor role as far
as the molecular weight distribution is concerned. The reason is that the gradients of the
reaction rate for esterification and transesterification are nearly the same under the com-
mon reaction conditions (180 to 240 °C) of polyester production. If a polyester is prepared
one time at 200 °C and another at 240 °C under the same degree of condensation, the
outcome will be a solution of the same viscosity and the same molecular weight distribu-
tion. However, it takes much longer to achieve the projected degree of condensation at 200
°C than at 240 °C. It seems that only at very high temperatures does the relative transes-
terification reaction rate surpass the relative esterification reaction rate. However, that
favours the formation of average polyester molecules. These facts explain why polyester
preparation can be interrupted in the laboratory without impairing the reproducibility of
molecular weight and molecular weight distribution, followed by viscosity at the same
degree of condensation.
To emphasise the importance of the transesterification reaction in the control over the
molecular weight distribution – which has been neglected by previous authors – consider
the following examples.
Example a)
Variations on the preparation of branched polyesters of terephthalic acid
Since the 1950s, relatively low-molecular weight, branched polyesters prepared from di-
methyl terephthalate, glycerol and ethylene glycol have been used for heat-stable electrical
insulation coatings (wire enamels). Usually the polyesters are prepared by transesterifica-
tion, starting with the aforementioned raw materials and adding transesterification cata-
lysts (e.g. tetrabutyl titanate) at temperatures of about 240 °C.
It is possible to synthesise the same polyesters from polyethylene terephthalate (PET).
Polyethylene terephthalate is an important raw material for fibres and film and has a mo-
lecular weight of more than 20,000 g/mol. It consists of esters of terephthalic acid and
ethylene glycol in the precise ratio of 1 : 1 (i.e. kM −> 1.00). If the polyethylene terephtha-
late is made to react with an appropriate excess of polyol, the same polyester as described
earlier can be produced. Here, too, it is necessary to use transesterification catalysts and
to carry out the reaction at temperatures of about 240 °C for a long time. The two poly-
68
Table 3.10: Comparison of the preparation of polyester from raw materials and from
polyethylene terephthalate
polyester 1 polyester 2
building
element M n n•M m-% n n•M m-%
dimethyl 194
1.00 194.0 85.9 – – –
terephthalate
PET (192)x – – – 1.00 192.0 85.0
ethylene 62
1.10 68.2 30.2 0.10 6.2 2.8
glycol
glycerol 92 0.30 27.6 12.2 0.30 27.6 12.2
sum 289.8 128.3 225.8 100.0
methanol 32 2.00 64.0 28.3 – – –
yield 225.8 100.0 225.8 100.0
esters have nearly the same viscosity, a fact which points to the same molecular weight
and molecular weight distribution.
This is evidence that polyesters can be prepared not only from monomeric building blocks,
but also from high molecular weight polyesters. It is also a clear indication of the effect
that transesterification has on regulating average molecular weights. At one time, it seemed
a good idea to use polyester film scrap in the preparation of polyesters (several patents
describing this method exist). However, the process failed to find acceptance. The reason
was that the bulky scrap was difficult to handle, and it was not readily absorbed into the
relatively small quantity of polyol.
The two formulations are shown in Table 3.10 for comparison.
Example b) “Repairing” of polyesters undergoing gelation

Gelation of polyesters may happen in the laboratory if the polyol is not present in sufficient
excess or if the degree of branching is too high. More problematic is gelation that takes
place in production due to metering errors. In such cases, the viscosity rises rapidly and a
gel forms. According to Flory’s theory, the gel consists of only a few molecules striving to
become infinitely large. The bulk consists of small molecules. If the gel is sufficiently mo-
bile (due to adequate agitation energy), it is possible to add polyol and so to destroy the
gel state. This works because the added quantities of polyols can react by transesterifica-
tion and can so reduce the size of the very large molecules. To achieve this, the tempera-
ture needs to be raised. In production, it takes courage to heat the reaction mixture of such
gelling polyesters to raise the temperature once excess quantities of polyol have already
69
been added. However, this is the only way to “repair” the polyester. If the added polyol is
already contained in the polyester, this approach will often “save” the batch of polyester
being prepared and produce a suitable batch that meets the specifications. At the very
least, this will prevent the cold, solidified gel from having to be “dug out”. Here, too, the
transesterification reaction is an important part of the regulating effect.
3.5 Formation and structure

of alkyd resins
3.5.1 Special aspects of the preparation of alkyd resins
As mentioned earlier, alkyd resins are prepared by reaction of polycarboxylic acids, polyols
and monocarboxylic acids. Common alkyd resins contain phthalic anhydride as the poly-
carboxylic component, polyols bearing more than two (average value in the case of mixtu-
res) hydroxyl groups (both primary and secondary), and monocarboxylic acids containing
primary and, rarely, secondary carboxyl groups.
There is a particular reason for this selection. When alkyd resins are produced from a re-
action mixture of these three ingredients, the phthalic anhydride reacts first with OH
groups to form an ester and a vicinal carboxyl group. This carboxyl group reacts relatively
slowly with further OH groups due to steric hindrance of the vicinal ester group and the
π-electron system of the aromatic ring. Therefore, the remaining OH groups react much
faster with the carboxyl groups of the monocarboxylic acids (especially if they are primary
types). Only then does the second carboxyl group of phthalic anhydride react with OH
groups, yielding larger molecules, the polyester backbone of alkyd resins. The process is
shown in Figure 3.38.
Naturally, the preparation of alkyds is not a genuine step-wise process. The steps
shown are merely intended to indicate the different reaction velocities of the building
blocks. The reactions take place in parallel but with markedly different velocity gradients.
Of particular importance is the fact that the polyols form esters with the monocarboxylic
acid faster than they participate in polyester chain growth of the alkyd resins.
If, at the end of alkyd resin production, the measured acid values are relatively low, the
remaining carboxyl groups stem from the partial phthalic esters and not the monocarbo-
xylic acids. This has been confirmed by GC analysis, which found only partial esters and
free phthalic anhydride after preparation of alkyd resins.
This fact has consequences for the molecular weight distribution of alkyd resins. If they
are prepared from dicarboxylic acids other than phthalic anhydride, different molar ap-
proaches are required. For example, the polycondensation constant needs to be increased
if isophthalic acid is used or if adipic acid is employed in addition to phthalic anhydride.
70
Formation and structure of alkyd resins
The reason is that both carboxyl groups of these dicarboxylic acids have nearly the same
reaction velocity, in contrast to the two potential functional groups on phthalic anhydride.
The use of tertiary monocarboxylic acids, too, requires specific measures. The reason is
that the intermediate reduction in OH group functionality through the formation of mo-
nocarboxylic esters is not as effective as when phthalic anhydride is used. Such differences
become noticeable when primary carboxyl groups are compared with secondary carboxyl
groups, e.g. caprylic acid with 2-ethylhexanoic acid.
3.5.2 Calculation of molecular weights of alkyd resins

Particularly in American literature [44], monocarboxylic acids acting as building blocks for
alkyd resins are known as chain stoppers. This term is mainly used in connection with ben-
zoic acid. However, if the term is understood as implying a limitation on molecular size, it
is not really appropriate. Alkyd resins are always formulated in such a way that the number
of hydroxyl groups (nOH) from the polyols (n1) is larger than the number of carboxyl groups
Figure 3.38: Step-wise formation of alkyd resins
71
(nCOOH) from the polycarboxylic acids (n2) and the monocarboxylic acids (n3), as shown in
the following equation.
Equation 3.58
Alkyd resins always contain an excess of hydroxyl groups (νOH), one of the reasons being
to avoid residual free monocarboxylic acids, which would impair the resin properties. If
alkyd resin molecules contain not only free hydroxyl groups (νOH) but also free carboxyl
groups (νCOOH), they can keep growing. This will occur totally independently of the quan-
tity of monocarboxylic acid. Therefore, the same rules, definitions and calculations apply
to alkyd resins as to polyesters. Thus, the same equations for calculating the polyconden-
sation constant (kM) and the number of structural units of polyester can also be used for
alkyd resins, as shown in the following equation:
Equation 3.59
However, the molecular weight of the structural units (M'S) of alkyd resins may differ
considerably from that of the structural units of unmodified polyesters, because allowance
needs to be made for the quantity of monocarboxylic acid (n3). For example, the structural
unit of an alkyd resin consisting of the residue of phthalic acid, pentaerythritol and 1.6
moles of a monocarboxylic acid (molecular weight 280 g/mol) has a molecular weight of
685 g/mol. It is thus three times as large as a structural element of a polyester prepared
from isophthalic acid and neopentyl glycol (234 g/mol). For the molecular weight of a
structural element of alkyd molecules, the following applies.
Equation 3.60
The number-average molecular weight of alkyd resins is calculated as follows:
Equation 3.61
72
Table 3.11: Formulation of sample alkyd resin and calculated characteristic values

M n•M
n•F n building elements [g/mol] [g] wt.%
–2.00 1.00 phthalic anhydride 148 148.0 41.02
+3.09 1.03 trimethylol propane 134 138.0 38.26
–0.65 0.65 isononanoic acid 158 102.7 28.47
sum 388.7 107.75
+0.44 νOH
1.65 water 18 29.7 7.75
yield A0 359.0 100.00
yield AAV 360.7
which can be re-written as:
Equation 3.62
The numerator of the quotient then represents the yield mass of an alkyd resin consisting of
n1 moles of polyol, n2 moles of polycarboxylic acid and n3 moles of monocarboxylic acid. The
equation also applies if there is no excess of polyol (n1 ≤ n2). In such cases, limiting of the
molecular weight is accomplished by restricting the degree of condensation. However, in such
cases, the number of all hydroxyl groups then needs to be higher than the number of carboxyl
groups (nOH > nCOOH). From the yield mass and with allowance for the actual acid value (AAV),
the number-average molecular weight and the related data are calculated as follows.
Equation 3.63
For alkyd resins, too, the calculation methods will now be illustrated with an example.
An alkyd resin is to consist of 1.00 mole of phthalic anhydride, 1.03 moles of trimet-
hylol propane, and 0.65 moles of isononanoic acid. The condensation reaction is carried
out until the acid value is 15 mg KOH/g. The formulation and the calculated characteristic
values are presented in Table 3.11.
73
The yield mass at a given acid value is calculated as follows:
Equation 3.64
The number of unreacted carboxyl groups is given by the following equation
Equation 3.65
That value increases the quantity of residual OH groups as follows:
Equation 3.66
The OH value is then given by:
Equation 3.67
The polycondensation constant (k'M) is:
Equation 3.68
The number-average molecular weight then computes to:
Equation 3.69
3.5.3 Molecular weight distribution of alkyd resins

Theoretically, the same factors which influence the molecular weight distribution of poly-
esters apply to alkyd resins. These are mainly the targeted number-average molecular
weight and the reactivity of the different functional groups of the building blocks. However,
74
unlike in the case for unmodified polyesters, the quantity of monocarboxylic acid plays an
important role in regulating the molecular weight distribution. As described in Chapter
3.6.1, the monocarboxylic acids act in situ, as it were, to reduce the functionality of polyols.
The extent of this effect, i.e. the quantity of the monocarboxylic acid, has a significant
bearing on the molecular weight distribution. When a greater excess of polyol OH groups
(νOH) in the alkyd resin molecule has reacted with monocarboxylic acid, the possibility of
disproportionate alkyd molecule growth has been reduced. The degree of branching of
alkyd resins may be derived from the degree of branching for the polyester backbone by
subtracting the quantity of monocarboxylic acid in the numerator of the equation:
Equation 3.70
The degree to which an excess of polyol has reacted with monocarboxylic acid is equivalent
to the degree of branching in unmodified polyesters. Thus, the degree of branching (v) in
polyesters is replaced by the degree to which an excess of polyol has reacted with mono-
carboxylic acid (b), as defined in the following equation:
Equation 3.71
(b) is the fraction of monocarboxylic acid (n3) on the excess residual OH groups (νOH). And
the excess residual OH groups are the balance of all the functional groups of polyols and
the polycarboxylic acids in the alkyd backbone.
Equation 3.72
If the degree of condensation is incorporated into the definition (b'), the number of excess
OH groups (νOH) increases by the number of residual carboxyl groups (νCOOH), giving rise
to the following equation for the degree to which an excess of polyol has reacted with
monocarboxylic acid:
Equation 3.73
75
The influence of b' on the molecular weight distribution can be illustrated with viscosity
curves for alkyd resins. Figure 3.39 shows how the viscosity of triol alkyd resins varies with
the polycondensation constant (k'M; representing the number-average molecular weight)
and the number of moles of monocarboxylic acid per mole of triol (n3/n1; representing b').
These curves are equivalent to the curves showing the dependency of the viscosity of
unmodified polyesters on the polycondensation constant and the average functionality of
polyols (representing the degree of branching), as shown in Figure 3.35.
The higher the degree of reaction (b`), the narrower is the molecular weight distribu-
tion and the lower is the solution viscosity. For alkyd resins, too, there is a region where
the alkyds still have finite number-average molecular weights but are gelled, because some
molecules tend towards infinity.
In the description of the preparation of alkyd resins (see Chapter 3.6.1), it was stressed
that the difference in reactivity of the two potential carboxyl groups of phthalic anhydride
ensures that the in situ reduction in functionality of polyols through occupation with mo-
Figure 3.39: Viscosities of triol alkyd resins as a function of polycondensation constant and

degree of reaction of triols with monocarboxylic acid
76
nocarboxylic acids is very efficient. This influence exerted by the building blocks is very
important.
If alkyd resins are formulated not with the usual phthalic anhydride but rather with isopht-
halic acid, it is necessary to lower the expected number-average molecular weight, i.e. to
increase the polyol excess (k'M) or to lower the degree of condensation by raising the acid
value. Otherwise the feasibility limit might be reached, i.e. the alkyd resin would gel at
comparable degree of condensation.
This is all the more true for alkyds prepared from adipic acid instead of phthalic anhydri-
de. Thus, it is practically impossible to prepare an alkyd resin from adipic acid, pentaerythritol
and monocarboxylic acid at low values for the polycondensation constant (k'M) – not even at
the maximum-possible degree of reaction of the monocarboxylic acid with the excess of poly-
ol. The competition between the two reactive carboxyl groups on the adipic acid and the
carboxyl groups on the monocarboxylic acid gives rise to gelled – partially crosslinked –
polyesters of pentaerythritol and adipic acid, along with residual free monocarboxylic acid. If
the monocarboxylic acid is esterified first with pentaerythritol and the adipic acid is added in
a second stage, the changes of obtaining usable alkyd resins is increased.
Also, the reactivity of the monocarboxylic acids is important for the molecular weight
distribution of alkyd resins. For example, comparison of an alkyd containing phthalic an-
hydride, polyol and isononanoic acid (3,5,5-trimethyl hexanoic acid, with primary carboxyl
group), with another containing 2-ethylhexanoic acid (secondary carboxyl group) and with
a third containing neodecanoic acid (mainly 2,2,3,5-tetramethylhexanoic acid, tertiary
carboxyl group) shows that the alkyd made with 2-ethylhexanoic acid has a broader mole-
cular weight distribution than its isononanoic counterpart. Whether or not an alkyd resin
containing neodecanoic acid is feasible at all depends on the size of the polycondensation
constant (k'M). The reason is that the in situ decrease in polyol functionality varies with the
differential reactivity of the carboxyl groups of monocarboxylic acids. Naturally, the reac-
tion of excess functional groups of the polyols with monocarboxylic acids does not neces-
sarily lead to linear alkyd molecules. Reaction of monocarboxylic acids may give rise to
side chains as well as end groups. The
possible structures which can occur in
alkyd resin molecules are presented in
Figure 3.40.
Therefore, if conventional methods
are used in their preparation, alkyd resins
with a high degree of reaction with excess
polyol are the only way to achieve polyes-
ter molecules which are highly branched
and have relatively high molecular weights. Figure 3.40: Possible structures in alkyd resin
molecules
77
Figure 3.41 shows a typical molecular distribution curve for an alkyd resin consisting of
phthalic anhydride, pentaerythritol and a fatty acid with a high degree of reaction with excess
polyol (b').
At high average molecular weight, the distribution curve is relatively broad. The mass
fraction of very large molecules falls away, because the functional groups on the large
molecules have lower reactivity for kinetic reasons.
3.6 Functionality of polyesters

and alkyd resins
The most important application of polyesters and alkyd resins lies in coating systems which
form films by crosslinking. Crosslinking takes place by reaction of the functional groups of
polyesters and alkyd resins with themselves or with crosslinking partners. The crosslinking
reactions mainly occur via the OH groups, although some occur via the carboxyl groups.
Figure 3.41: Typical molecular distribution curve for an alkyd resin of phthalic anhydride,
pentaerythritol and fatty acid, with high degree of reaction with excess polyol
78
Functionality of polyesters and alkyd resins
The content of functional groups in polyesters and alkyds is found via the acid and OH
values or the OH-% content. The corresponding calculation equations are expressed in
terms of unit mass of polyesters and alkyd resins.
Equation 3.74
Equation 3.75
Equation 3.76
When it comes to the estimation of crosslinking efficiency, it is not the mass concentration
of functional groups (e.g OH groups and the OHV) which is decisive, but rather the func-
Figure 3.42: Correlation between OH value, degree of branching and the number-average mo-
lecular weight and functionality of polyester or alkyd resin molecules
79
tionalities (F) of the reacting partners: crosslinking molecules as well as the resins used.
For example, a linear polyester with two terminal OH groups may lead to totally different
molecular networks than a branched polyester containing four OH groups per molecule.
Nonetheless, the two can have the same OH value. It is therefore instructive to review the
relationships between OH value or acid value and the functionality of polyesters (FP,OH and
FP,COOH), the degree of branching and the number-average molecular weight. Calculation
equation is provided below:
Equation 3.77
Figure 3.42 describes the relationship between OH value (OHV), the degree of branching
(ν) and the number-average molecular weight and functionality of the polyester or alkyd
resin molecules.
Accordingly, a polyester with an OH value of 112 mg KOH/g and a number-average
molecular weight of 1000 g/mol contains two OH groups per mole and is linear, and its
degree of branching is 0 mol/kg. If the molecular weight were 2000 g/mol and the OH value
Figure 3.43: Dependence of functionality of polyester molecules on the OH value and the
number-average molecular weight
80
Exceptions and their influence on the molecular weight distribution
remained unchanged, the molecule would contain four OH groups and the degree of branch
ing would be 1.00 mol/kg. For a number-average molecular weight of 3000 g/mol, the
polyester contains six OH groups and the degree of branching is 1.50 mol/kg.
This relationship can be plotted as the dependence of the functionality of polyester mole-
cules on the OH value and the number-average molecular weight (see Figure 3.43).
3.7 xceptions and their influence

E
on the molecular weight distribution
All the definitions and calculations presented so far apply to polyesters and alkyd resins
prepared by esterification and transesterification at elevated temperatures and by distil-
lation of a low-molecular reaction product. It was pointed out that the number-average
molecular weight and the degree of branching or reaction of excess OH groups deter
mine the molecular weight distribution, in addition to the influence exerted by specific
building blocks. Accordingly, it is impossible for highly branched polyesters or alkyd
resins with a low degree of reaction of excess polyol with monocarboxylic acid to achie-
ve high number-average molecular weights, because the molecular distribution func
tions are so broad that some of the molecules will strive to become infinitely large, i.e.
become crosslinked.
However, there are special processes which give rise to polyesters by means of
step-wise addition. The goal here is to achieve polyesters of high functionality but
narrow molecular weight distribution. These processes are time consuming and there
fore very expensive.
Several accounts describe the preparation of dendrimer polyesters [47].
For example, one reports the esterification of polyfunctional compounds by biosynthe-
sis [48] at low temperatures. The resulting polyesters are claimed to have narrower molecu-
lar weight distributions than polyesters of the same composition that have been prepared
in the conventional way.
Another describes the preparation of polyesters from polyols or polyol derivatives and
dimethylol propanoic acid in a step-wise process to form spherical molecules of high func-
tionality and narrow molecular weight distribution [49]. That is impossible by conventional
esterification. For one thing, the tertiary carboxyl group is very difficult to esterify. Fur
thermore, polyesters cannot be formed in separate steps. Dimethylol propanoic acid can
also form esters with itself and there is also the possibility of transesterification. The result
will be a polyester with a random distribution of different molecules.
Such polyesters can be synthesised by a series of addition reactions at low temperatu-
res. One example is an anhydride addition followed by addition of epoxy compounds [50].
81
Finally, there is the possibility of a step-by-step reaction. This starts by masking the OH
groups of dimethylol propanoic acid with benzaldehyde. The resulting aldol adduct is then
made to react with pentaerythritol in the presence of a special catalyst to yield a tetra-ester.
The tetra-ester is unmasked, yielding the first generation of dendrimer polyesters. The
three reaction steps are repeated to produce further generations of dendrimer polyesters,
with larger molecules of higher functionalities [51]. These were initially prepared with
high-solid products in mind. However, such polyesters are very expensive to prepare, and
the future will tell whether the preparation of high-solid coatings in this way is worth the
effort.
3.8 xplanation of symbols

E
in definitions and equations
Symbols
n number of moles of building blocks or of functional groups
(type indicated by an index)
ν number of current or residual functional groups (type indicated by an index)
F functionality of a building block molecule (average value in the case of mixtures
M molecular weight of building blocks or polyesters or alkyd resins
(type indicated by an index)
A yield mass of a polyester or an alkyd resin
q number of structural units in a polyester or an alkyd resin, expressed in terms of
the number of moles polycarboxylic acid and the associated polyols and, where
indicated, of monocarboxylic acid
p degree of polycondensation (see Equation 3.4)
K calculation constants (KM polycondensation constant for molar ratios)
KPatton Patton or alkyd constant)
v degree of branching – number of potential branches in a polyester or an alkyd resin,
expressed in terms of 1000 g
b occupation of the excess OH groups on the polyester backbone of an alkyd resin
by monocarboxylic acidsw
D dispersity – measure of the molecular distribution
AV acid value
OHV OH value
OH-% OH-% content
Indices
1 expressed in terms of polyols and their derivatives
82
Index of equations
2 expressed in terms of polycarboxylic acids and their derivatives

3 expressed in terms of monocarboxylic acids and their derivatives
OH expressed in terms of OH groups
COOH expressed in terms of carboxyl groups
n calculated on the number of moles
m calculated on the mass
M calculated on the molecular weight
AV calculated on the acid value
s expressed in terms of the structural unit of polyesters and alkyd resins
c critical value of a term
Apostrophe as exponent:
K'M KM with allowance for the degree of condensation
M'2 M2 without condensation product (residue of polycarboxylic acid)
M'3 M3 without condensation product (residue of monocarboxylic acid)
M'P MP, with allowance for the degree of condensation
b' b, with allowance for the degree of condensation
3.9 Index of equations

Equation 3.78 page 27
Dependence of number of structural units of linear polyesters on the degree of condensation
Dependence of number of structural units of linear polyester on the molar ratio of polyol
and polycarboxylic acid
Equation 3.80 page page 28
Dependence of number of structural units of linear polyester on the degree of condensation
83
Equation 3.81: Degree of polycondensation (Carothers)
Critical degree of polycondensation (Carothers)
Critical degree of branching (Flory)
Critical degree of polycondensation (Stockmayer)
Critical degree of polycondensation (Jonason)
Polycondensation constant (Patton)
Molar ratios as polycondensation constant
84
Index of equations
Dependence of number of structural units on the molar ratio of polyol and polycarboxylic
acid, irrespective of polyol functionality
Dependence of number of structural units on the molar ratio of poly carboxylic acids and
polyol, irrespective of polycarboxylic acid functionality
Polycondensation constant for all polyols and polycarboxylic acids
Molecular weights for all polyesters
Calculation of molecular weights of polyesters of n1 and n2 building blocks
Conversion of calculation of molecular weights of polyesters of n1 and n2 building blocks
Conversion of calculation of molecular weights of polyesters of n1 and n2 building blocks
85
Calculation of molecular weight of polyesters based on the mass yield
Development of mean values for moles of polyols
Polycondensation constant after allowance for degree of condensation
Polycondensation constant after allowance for degree of condensationdegree
Calculation of molecular weight of polyesters by given acid value
Yield mass as function of acid value
Yield mass as function of acid value
86
Index of equations
Calculation of molecular weight of polyesters based on yield mass
Number-average molecular weight
Weight-average molecular weight
Value of molecular inconsistency (dispersity)
Degree of branching of polyesters
Preconditions for the composition of alkyd resins
Molecular weight of the structural units of alkyd resins
87
Calculation of molecular weights of alkyd resins
Calculation of molecular weights of alkyd resins
Calculation of molecular weights of alkyd resins based on the yield mass
Degree of branching of alkyd resins
Occupation of excess hydroxyl groups on alkyd backbone
Occupation after allowance for the acid value
88
Index of equations
Calculation of acid values
Calculation of OH values
Calculation of OH-% content
OH functionality of polyester molecules
89

3 Formation and Structure of Polyesters and Alkyd Resins: 3.1 Reactions That Produce Polyesters

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3 Formation and Structure of Polyesters and Alkyd Resins: 3.1 Reactions That Produce Polyesters

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Reactions that produce polyesters

3 Formation and structure of

3.1 Reactions that produce polyesters

Ulrich Poth: Polyester and Alkyd Resins

Figure 3.2: Overall equilibrium equation for esterification and saponification

equilibrium needs to be shifted to

role in the preparation of alkyd re-

3.1.1.3 Reaction catalysis

Figure 3.5: Acid catalysis at the start of esterification

3.1.1.5 Epoxy addition O

electrophilic carbonium ions. The

condary OH groups (see Figure 3.8). O

Under the influence of strong

Several other reactions are known in + epoxide

The high enthalpy of formation of

Figure 3.10: Ring-opening reaction of ε-caprolactone

Figure 3.11: Initiation and formation of linear polyester molecules

functional groups. If a diol reacts n–1

3.1.2.2 Formation of branched polyesters

3.1.2.3 Ring closure during formation of polyesters?

3.1.2.4 Crosslinking during polyester formation

3.2 Determination of and limitations on

This dependency is presented in Figure 3.15.

Figure 3.15: Number of structural units as a function of degree of polycondensation in linear

This dependency is presented in Figure 3.16:

3.2.2 Derivation of gel-point equations

Other authors also use this equation [27–31].

3.3 Methods of calculating average

3.3.2 I nfluence of molar ratios of polyol and poly­

models of polyester molecules

The model of the polyester segment

3.3.3 alculating the influence of the degree

As residual carboxyl groups (limited degree of condensation) can act as an alternative to

Figure 3.24: General model of linearised polyesters

Figure 3.25: Simplified form of a general linearised polyester

Rewriting Equation 3.34 and solving for AAV results in:

By introducing the known constant terms this leads to:

Table 3.1: Calculation of a model polyester with a limited degree of condensation

3.3.4 Sample calculations of molecular weights

Table 3.2: Calculation of a model polyester of given acid value

The average molecular weight is:

The quantity of residual free carboxyl groups is:

The number of residual OH groups increases accordingly:

The resulting OH value is:

The polycondensation constant (k’M) is:

Table 3.4: Calculation of the characteristics of a polyester based on analytical data

number of the polyester is found to be 18 mg KOH/g. The calculation of the molar

The polycondensation constant (k’M) is then calculated as follows:

And the OH value is given by:

3.4 Molecular weight distribution

A hypothetical example serves to illustrate the various terms.

Table 3.5: Molecular weight distribution of a polyester model

3.4.2 GPC analysis

Figure 3.27: Plot of molecular weight distribution of a model polyester

Figure 3.29: Comparison of theoretical molecular weights with analytical results

3.4.3 Influences on the molecular weight distribution

Figure 3.30: Integral distribution curve of linear polyesters of different number-average

Influence of the targeted number-average molecular weight

Influence exerted by the degree of branching

3.1.2.2 Formation of branched polyesters

3.3.2 I nfluence of molar ratios of polyol and poly

Equation 3.79 page 28

Equation 3.80 page page 28

Equation 3.81 page 29

Equation 3.82 page 29

Equation 3.83 page 30

Equation 3.84 page 30

Equation 3.85 page 30

Equation 3.86 page 31

Equation 3.87 page 34

Equation 3.88 page 35

Equation 3.89 page 35

Equation 3.90 page 37

Equation 3.91 page 37

Equation 3.92 page 38

Equation 3.93 page 38

Equation 3.94 page 38

Equation 3.95 page 39

Equation 3.96 page 39

Equation 3.97 page 40

Equation 3.98 page 41

Equation 3.99 page 42

Equation 3.100 page 42