Prediction of the Formulation Dependence of the Glass 

Transition Temperature for Amine-Epoxy Copolymers 

Using a Quantitative Structure-Property Relationship Based 

on the AM1 Method 

by Jason A. Morrill, Robert E. Jensen, Phillip H. Madison, and 

Cary F. Chabalowski 

ARL-TR-3137 February 2004 

Approved for public release; distribution is unlimited.
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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3137 February 2004 

Prediction of the Formulation Dependence of the Glass 

Transition Temperature for Amine-Epoxy Copolymers 

Using a Quantitative Structure-Property Relationship Based 

on the AM1 Method 

Jason A. Morrill 

National Research Council Postdoctoral Fellow 

Robert E. Jensen and Cary F. Chabalowski 

Weapons and Materials Research Directorate, ARL 

Phillip H. Madison

Oak Ridge Institute for Science and Education

Approved for public release; distribution is unlimited. 
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Prediction of the Formulation Dependence of the Glass Transition Temperature for 

Amine-Epoxy Copolymers Using a Quantitative Structure-Property Relationship 

Based on the AM1 Method 

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6. AUTHOR(S) 

Jason A. Morrill,* Robert E. Jensen, Phillip H. Madison,† and Cary F. Chabalowski 

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*National Research Council Postdoctoral Fellow 

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14. ABSTRACT 

A designer Quantitative Structure-Property Relationship (QSPR), based upon molecular properties calculated using the AM1 

semi-empirical quantum mechanical method, was developed to predict the glass transition temperature (Tg) of amine-cured 

epoxy resins based on the diglycidyl ether of bisphenol A. The QSPR (R2

 = 0.9977) was generated using the regression 

analysis program, Comprehensive Descriptors for Structural and Statistical Analysis (CODESSA). By applying an ad hoc 

treatment based on elementary probability theory to the quantitative structure-property relationship analysis, a method was 

developed for computing bulk polymer Tg for stoichiometric and non-stoichiometric monomeric formulations. A model 

polymer was synthesized and found to validate our model predictions. 

15. SUBJECT TERMS

AM1, semi-empirical, QSPR, amine, epoxy, polymer, glass transition 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

Robert E. Jensen 

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iii

Contents 

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. Procedure 3

2.1 General Computational Procedure ..................................................................................3 

2.2 Construction of Polymer Models ....................................................................................3 

2.3 Molecular Modeling and Derivation of QSPRs ..............................................................7 

2.4 Prediction of Tg for Non-Stoichiometric Monomeric Ratios ..........................................8 

2.5 Validation of Tg Predictions for Non-Stoichiometric Monomeric Ratios.......................9 

3. Results and Discussion 10

3.1 QSPR Derivation...........................................................................................................10 

3.2 External Validation of QSPR ........................................................................................15 

4. Conclusions 16

5. References 17

Distribution List 21 
iv

List of Figures 

Figure 1. Repeat unit (b) resulting from the events used by Miller and Macosko (16) to 

derive a set of recursion relations for solving the probability that a randomly chosen 

monomeric subunit leads to a finite chain. ................................................................................5 

Figure 2. Plot of the probabilities that a randomly chosen monomer will have m out of f of its 

branches leading to the infinite polymer network as determined from equation 9. The 

values plotted are for m = 4 (), m = 3 (c), m = 2 (y), and m = 1 (¡). ....................................7 

Figure 3. Branching patterns used to predict the stoichiometric dependence of Tg for PACM 

with predicted contributions to the polymer Tg. The top row represents the epoxy (B) 

excess structures and the middle row represents the amine (A) excess structures. ...................9 

Figure 4. Predicted (c) and experimental ()Tg values for PACM-cured DGEBA with 

varied stoichiometric ratios of monomers................................................................................15 

List of Tables 

Table 1. Data for the training set of amine-cured epoxy polymers from the best fourparameter correlation equation (R2

 = 0.9977)..........................................................................11 

Table 2. Correlation equation parameters and associated significance values appearing in the 

best four-parameter correlation (R2 = 0.9977).........................................................................11 

Table 3. Equations having the highest coefficients of determination and F values following a 

heuristic analysis......................................................................................................................14 
v

Acknowledgments 

Jason Morrill would like to thank the National Research Council for the postdoctoral fellowship 

that supported this work. Phillip Madison was supported in part by an appointment to the 

Research Participation Program at the U.S. Army Research Laboratory (ARL) administered by 

the Oak Ridge Institute for Science and Education through an interagency agreement between 

the U.S. Department of Energy and ARL. Appreciation is also extended to the ARL Major 

Shared Resource Center for providing outstanding computational facilities. 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

In describing the tendency of a polymer to absorb energy, the glass transition temperature (Tg) is 

a property of fundamental importance. It is the temperature above which thermal motions of 

atoms about their equilibrium positions in a polymer increase the free volume enough to allow 

long-range coupled motions of larger groups of atoms along the backbone of the polymer (1). At 

the Tg, the increase in the mobility of the polymer backbone gives rise to a change from a glassy 

state to a more fluid rubber-like state. The change in the backbone rigidity of a polymer 

consequently leads to a change in a host of the physical properties of the polymer. Among these 

properties are density, mechanical moduli, dielectric coefficients, viscosity, and gaseous or liquid 

absorption (2). Toward designing new polymeric materials and predicting their overall physical 

performance, it is useful to have a means by which the Tg of a polymer can be reliably predicted. 

Numerous researchers have found that Tg values for linear polymers can be estimated on the 

basis of quantitative structure-property relationships (QSPRs) developed from the structure of 

the repeat units making up the polymer. Many such approaches have relied on the additive 

nature of the repeat units in contributing to the Tg value of the polymer and are termed group 

additivity methods. One of the best-known examples of the group additivity approach is that of 

van Krevelen (3). The van Krevelen approach is purely empirical consisting of a weighted sum 

of scalar quantities associated with functional groups commonly occurring in polymers. This 

method provides a rapid and computationally inexpensive approach to the estimation of Tg

values, but has been criticized due to its limited extensibility to systems composed only of 

functional groups that have been previously investigated (4). 

Bicerano (5) extended the group additivity concept to a more generally applicable model 

including solubility parameters and topological considerations that are independent of specific 

functional groups. These quantities were used as descriptors of the cohesive forces and chain 

stiffness of polymers as these properties relate to the Tg. Using quantities of this type, a 

relationship with a coefficient of determination of 0.9749 was obtained for a dataset consisting of 

320 polymers, thus accounting for 95% of the variance in the experimental Tg values. The model 

overestimated the Tg values for a series of polymers consisting of phthalimide groups. The lack 

of performance of the model for a specific class of polymers is likely due to it being based on a 

large and widely varied training set of polymers. This is a well-known phenomenon experienced 

in QSPR studies in which superior correlations are obtained from congeneric series of molecular 

species and has led to the development of so-called “designer” correlation equations (6–9). 

Koehler and Hopfinger extended the group additivity approach to include terms describing 

intermolecular interactions, such as hydrogen bonding, in order to relate the contribution of 

interchain interactions to polymer Tg values (10). This approach made use of a probe to describe 

non-bonded electrostatic and steric repulsions, via a molecular mechanics treatment. A linear 
2

combination of backbone and side chain flexibility parameters from Boltzmann distributions of 

conformational energy states was also used to predict Tg values. Several correlations were 

obtained, all with coefficients of determination in excess of 0.9. 

A designer correlation equation was developed by Camelio et al. (9, 11) to investigate the role of 

bulky substituents in determining the Tg of substituted methacrylates and acrylates. The Camelio 

QSPR model, referred to as the energy, volume, and mass (EVM) model, was developed to 

describe the effects of bulky substituents in terms of energetic barriers to rotation along the 

polymer backbone as affected by the proximity of the center of mass and volume of substituents 

to the polymer backbone. In order to give a more accurate picture of the volume of the “free” 

space around the polymers, Camelio randomized the conformations of the polymer models by a 

molecular dynamics/molecular mechanics simulation. A standard deviation of Cartesian 

coordinates of atoms along the polymer backbone was then used to define the volume of an 

elliptical cylinder around the polymer backbone. The other terms used in the correlation 

analyses included intramolecular energies, intermolecular van der Waals energies, and repeat 

unit molecular weights. Correlations obtained using these descriptors gave coefficients of 

determination >0.90 for several groups of polymers (8, 10). 

In contrast to previous models that are based on specifically chosen molecular descriptors, 

Katritzky and coworkers created a generally applicable QSPR development program, called 

Comprehensive Descriptors for Structural and Statistical Analysis (CODESSA) (12). 

CODESSA is typically used in conjunction with a molecular modeling package to analyze 

hundreds of computed molecular descriptors and then sort them into multivariate linear equations 

based on the statistical correlation of the descriptors to a property of interest. This approach has 

been shown in numerous studies to result in superior correlations for a wide variety of molecules 

and several physical and biological properties. Katritzky used CODESSA to predict the Tg for 

88 linear homopolymers using a five-parameter equation that gave a coefficient of determination 

of 0.946 (4). 

All of the discussed approaches to predicting Tg values were developed on the basis of linear 

homopolymers despite the importance of crosslinked copolymers in commercial applications. 

The focus on linear homopolymers is likely due to the complexity associated with constructing 

an atomistic model of the branching that exists in a networked copolymer. This complexity, 

when coupled with the added computational cost of modeling a polymer structure large enough 

to incorporate various degrees of crosslinking, has previously made simulations of the effect of 

crosslinking on polymer properties such as Tg extremely computationally expensive, even at 

modest levels of theory. Despite this difficulty, attempts have been made by researchers to 

model the effect of the extent of crosslinking on Tg. Galy used a group additivity model to 

predict Tg values for non-stoichiometric amine-cured epoxy networks (13). Bicerano employed 

a relation that involved a correction to maximal experimental Tg values as a function of the 

number of rotatable bonds and average number and molecular weight of repeat units between 

crosslinks (14). And, Beloshenko related the mean statistical molecular mass between crosslinks 
3

and compression moduli (both obtained from experimental data) to the Tg for an 

amine/anhydride cured epoxy crosslinked copolymer (15). These approaches all require the 

existence of experimental data or previously determined group additivity parameters for a 

polymer of interest in order to make a prediction of Tg for that same polymer. 

Our interest is in the rational design of amine-cured bisphenol A epoxy copolymers with 

improved impact and shock resistance. Empirically, it is known that substitution of aromatic 

backbones of polymers of this type with cycloaliphatic species yields increased localized 

molecular mobility in the glassy state of the polymer. These short-range relaxation mechanisms 

yield increased high strain rate impact and shock resistance. However, this approach often leads 

to inferior performance in other critical polymer properties, such as the coefficient of thermal 

expansion. For successful application a true balance of properties is necessary, and using 

traditional empirical methods to discover an optimal copolymer formulation may never be 

achieved due to the elevated number of formulations that must be probed. Herein, we describe 

our results for the prediction of the behavior of the Tg with respect to the stoichiometric ratio of 

monomers in a crosslinked amine-cured model epoxy polymer. 

2. Procedure 

2.1 General Computational Procedure 

In general, our computational procedure involves the development of a designer QSPR for 

prediction of the glass transition temperatures of amine-cured polymers based on diglycidyl ether 

of bisphenol A (DGEBA). In a second step, the QSPR is used to predict the contribution of 

repeat units, each built to mimic crosslink points having varied degrees of branching in a 

networked copolymer. The contributions of each of the crosslink points are then multiplied by a 

weighting factor obtained from a previously developed method (16) to describe the probability of 

finding a finite chain in a polymer network. Finally, the individual contributions are summed to 

arrive at a composite predicted Tg for any given stoichiometric ratio. This approach retains the 

group additivity concept proposed by van Krevelen (3), but is fundamentally different. The 

group contributions used here are not tabulated values; rather, they are predicted solely on the 

basis of computable molecular descriptors, found to be predictive of Tg, and the appropriate 

choice of a polymer repeat group. 

2.2 Construction of Polymer Models 

Toward simplifying the polymer structures, representative polymer models were constructed for 

a set of 13 amine-cured epoxy copolymers for which Tg data exists in the literature. The 

monomers in the dataset consisted of DGEBA cured with one amine curing agent. The curing 

agents consisted of 11 diamines and 2 monoamines and were formulated in a 1:1 stoichiometry 

of functional groups, which is defined as one amine N-H bond per DGEBA oxirane ring. Since 
4

the literature data corresponded to stoichiometric formulations it was desirable to build models 

having relative numbers of monomeric subunits that are equivalent to their respective literature 

stoichiometries. 

It is generally observed for systems of this type that a maximally crosslinked polymer results 

from equivalent stoichiometries of functional groups (17). Further, maximal crosslinking results 

in an infinitely large and hyperbranched macromolecule (16, 18). Choosing a repeat unit for 

such a complex polymer is not as straightforward as it is for linear homopolymers or semiregularly repeating linear copolymers. Fortunately, the nature of branching in polymers of the 

type with which we are concerned has been studied previously and this work led to our choice of 

a suitable repeat unit to use for the basis of our QSPR model. 

To predict the concentration of elastic chains and crosslink densities for calculating molecular 

weight averages of hyperbranched polymers the work of Flory (18) was simplified by Miller and 

Macosko (16). Their approach retained Flory’s ideal network assumptions: 

• all functional groups of the same type are equally reactive; 

• all functional groups react independently of one another; and 

• no intramolecular reactions occur in finite species. 

Their derivation was used to describe random stepwise copolymerization of a four-functional 

monomer with a two-functional monomer and is analogous to the copolymerization of a primary 

diamine (four-functional) and a diglycidyl ether (two-functional). The derivation begins with 

the reaction of the four-functional monomer (A4) with a two-functional monomer (B2) (figure 1). 

The probability of A4 reacting with B2 such that a finite chain results ( ) out P FA is given by the law 

of total probability: 

( )( ) ( )

 P(F A does not react)( ) P A does not react

P F P F A reacts P A reacts

4 4

out

A

4 4

out

A

out

A

+

=

,

 (1) 

where the first term is the conditional probability that A4 leads to a finite chain given that A4 has 

reacted and the second term is the conditional probability that A4 leads to a finite chain given 

that A4 has not reacted. For the copolymer model in this study, equation 1 takes the form 

( ) ( ) ( ) A A

in

B

out P FA = P F p + 1 1- p , (2) 
5

Figure 1. Repeat unit (b) resulting from the events used by 

Miller and Macosko (16) to derive a set of 

recursion relations for solving the probability that a 

randomly chosen monomeric subunit leads to a 

finite chain. 

where the probability that returning from B2 is a finite chain, ( ) in P FB , is equivalent to the 

probability that A4 leads to a finite chain given that A4 has reacted, and pA is the fraction of A4

monomers having reacted, or the probability that A4 reacts. The second term in equation 2 

simply states that the probability that A4 leads to a finite chain given that A4 has not reacted is 

unity and that the probability A4 does not react is the fraction of A4 monomers that do not react 

(1 – pA). Because B2 is a two-functional monomer the probability of returning from a finite 

chain at B2, ( ) in P FB , must be equivalent to probability of leading out to a finite chain at B2, 

( ) out P FB , thus giving rise to equation 3: 

( ) ( ) out

B

in P FB = P F . (3) 

The probability that leading out from B2 results in a finite chain is analogous to leading out from 

A4 and finding a finite chain and results in equation 4: 

( ) ( ) ( ) B B

in

A

out P FB = P F p + 1 1- p . (4) 

For another A4 monomer adding to the growing chain, the probability that leading in from that 

monomer is a finite chain is equivalent to the probability that leading out from that monomer 

results in a finite chain. Leading out from that monomer depends on its functionality, and if all 

functional groups are assumed to be equally reactive, then the probability of leading out to a 

finite chain will be the product of individual probabilities of leading to finite chains at each 

functional group, thus 

( ) ( )

3 out

A

in P FA = P F . (5) 

 (a) (b) 
6

If assumptions are made on the particular values that pA and pB are allowed to take, then 

equations 2–5 become a system of four equations in four unknowns. 

The necessary reaction steps for setting up these equations have led to a repeat unit for a fully 

crosslinked amine-cured epoxy copolymer that bears a connectivity like that shown in figure 1. 

To solve the system of four equations, Miller and Macosko (16) assumed two cases: one for 

amine excess (in which the epoxide is fully reacted) and one for epoxide excess (in which the 

amine is fully reacted). The stoichiometric ratio, r, of monomeric functional groups is defined as 

A

B

p

p

r = . (6) 

For the case of amine excess, all of the epoxide groups are assumed to have reacted, thus pB = 1

and pA = 

r

1 . Substituting these values and solving the system of equations yields 

( ) 2

1

4

3 P F r

2

1

out

A ⎟ − ⎠

⎞ ⎜

⎝

⎛ = − . (7) 

For the case of epoxide excess, all of the amine N-H bonds are assumed to have reacted thus, 

pA = 1 and pB = r. Repeating the substitution and again solving the system of equations gives 

( ) 2

1

4

3

r

1 P F

2

1

out

A ⎟ − ⎠

⎞ ⎜

⎝

⎛ = − . (8) 

These probabilities were then substituted into the binomial distribution function 

out m

A

out f -m

m, f A P(F ) [1 P(F )]

m

f P(X ) − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = (9) 

to arrive at the probability that an f-functional monomer (diamine) chosen at random in the 

infinite polymer network would have m out of f of its functional groups leading to the infinite 

network. For the case of amine excess (r < 1), equation 7 was substituted into equation 9 and for 

the epoxide excess case (r > 1), equation 8 was substituted into equation 9. For each case, the 

resulting values obtained from the binomial distribution function were plotted in figure 2. 

The values are plotted for f = 4 (diamine) and m having values from 4 to 1, representing the 

fractional portions of structure types ranging from the fully crosslinked repeat units (m = 4) to 

those repeat units having one branch (m = 1) leading to the infinite polymer network. As shown 

in figure 2, as the formulation deviates from being stoichiometric the portion of the polymer 

having four branches leading to the infinite polymer network decreases with subsequent increase 

in the other branched species. 
7

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

 Functional Group Stoichiometric Ratio (r = Epoxide/Amine)

Weight of Branching Species

Figure 2. Plot of the probabilities that a randomly chosen monomer will have m out of f of 

its branches leading to the infinite polymer network as determined from equation 

9. The values plotted are for m = 4 (), m = 3 (c), m = 2 (y), and m = 1 (¡).

In deriving a relationship for the crosslink density, Miller and Macosko (16) used the 

probabilities obtained from equation 9 to obtain the relative weights of “effective crosslinks” of 

degree m. In their work, the distinction is made as to what constitutes a crosslink or branching 

point in the infinite network. For example, a four-functional monomer is considered an 

“effective crosslink” if at least three out of four of its functional groups lead to the infinite 

polymer network. If either one or two of the functional groups lead to the infinite network, then 

the monomer is considered to be either a point on a dangling end of the polymer or a point 

between effective crosslinks. We are concerned with the relative weights of all repeat unit types 

contributing to the Tg of the networked polymer, thus we do not differentiate between effective 

crosslinks and other components. 

2.3 Molecular Modeling and Derivation of QSPRs 

In the CODESSA framework, several descriptors depend explicitly upon the three-dimensional 

conformation of the molecule for which they are being computed. Examples of such descriptors 

are the principal moments of inertia and the shadow indices, both of which are computed on the 

basis of the principal axes of inertia. In order to avoid the introduction of a bias in terms of such 

descriptors, conformational analysis of the polymer models was performed according to a 

randomization procedure using the program Cerius2

 (19). The randomization procedure was 

conducted by altering the rotatable bonds (single bonds that are not part of a ring or a terminal 

hydrogen bond) to a randomly chosen value within a user-defined window of ±65°. The models 

were then subjected to a maximum of 2500 steps of a universal force field (UFF) (20) energy 

minimization. This random altering and minimization process was applied 100× for each 

polymer model, without an energy cut-off criterion applied to high-energy models. The resulting 
8

lowest energy polymer model for each amine was then optimized using a parallel version of 

Gaussian 98 (21) implementing the AM1 (22) semi-empirical quantum mechanical method 

with the correct AM1 parameters being applied to sulfur atoms (23). Due to the large size of the 

polymer models, characterization of the AM1 optimized structures by means of frequency 

analyses could not be conducted on all models. For those models that were subjected to 

frequency analyses, six near-zero eigenvalues were obtained. For all models, the default 

convergence criterion were used within Gaussian 98. In order to compute the quantum 

mechanical descriptors for statistical analysis using CODESSA (24), the program AMPAC 

with graphical user interface (GUI) (24) was used to carry out single point calculations on all 

preoptimized geometries. The QSPR equation was developed upon these models using the “best 

multilinear regression analysis” method within CODESSA following a similar procedure to that 

described by Katritzky et al. (4). The best correlation equation resulting from this analysis 

was then used to predict the dependence of Tg on the formulation of comonomers of 

bis(p-aminocyclohexyl)methane (PACM) and DGEBA. 

2.4 Prediction of Tg for Non-Stoichiometric Monomeric Ratios 

In order to predict the Tg for non-stoichiometric copolymer formulations, polymer models were 

built to represent various degrees of branching of repeat units in the polymer network. Each of 

the models for prediction was built on the basis of the equations derived by Miller and Macosko 

(16). For all of the diamines in the training set, structures were built with a diamine surrounded 

by four monomers of DGEBA and another amine built onto one of the distal functional groups of 

the epoxide monomers. Such a connectivity not only bears resemblance to the structural repeat 

unit resulting from Miller and Macosko’s recursive technique, but it also retains a 1:1 

stoichiometric ratio of functional groups of comonomers. For predicting the off-stoichiometric 

dependence of the Tg, structural types representing different degrees of branching were built by 

removing, in succession, one of the three epoxides not already having an amine on each end. 

This resulted in four structure types shown in figure 3. 

These models were then subjected to the same randomization and optimization routine described 

for the training set of structures. Following the randomization and optimization, descriptors 

were calculated to predict the contribution of each of the structure types to the Tg value. These 

Tg contributions were then multiplied by the corresponding fractional contributions (from 

equation 9) and the resulting products summed to arrive at a composite predicted Tg for the given 

stoichiometric ratios. For the epoxide excess structures, representation of various degrees of 

branching by the successive removal of one of the epoxide subunits from the repeat unit was 

intuitive. It is well known that for polymers of this type branching is decreased for 

stoichiometric ratios of <1. Representing various degrees of branching for amine excess repeat 

units involved a less intuitive process of trial and error. After several attempts at employing 

various representations of amine excess structures, the suggestion of Miller and Macosko (16) 

that an entanglement of two polymer chains can behave like a crosslink was used (figure 3). The 

entanglements were built by starting with two linear trimers each consisting of an epoxide 
9

Figure 3. Branching patterns used to predict the stoichiometric dependence of Tg 

for PACM with predicted contributions to the polymer Tg. The top row 

represents the epoxy (B) excess structures and the middle row represents 

the amine (A) excess structures. 

monomer with an amine monomer on each of its two ends. The two oligomers were then placed 

along perpendicular lines at a distance such that the Connolly surfaces of the center of the 

epoxide subunits were in approximate contact. The Connolly surfaces were calculated using a 

zero probe radius and plotted using the Cerius2 GUI. The distance between the center carbon 

atoms of the epoxides was then constrained and the system treated with the same randomization 

and molecular mechanics optimization routine just described. The system was then optimized 

without constraints using the AM1 method within Gaussian 98 and descriptors calculated using 

AM1 within AMPAC and CODESSA as previously described. 

2.5 Validation of Tg Predictions for Non-Stoichiometric Monomeric Ratios 

As with the polymers of the training set, the epoxy resin was based upon DGEBA (Epon* 828, 

epoxy equivalent weight [EEW] = 190 g/equivalent). The DGEBA epoxy resin was cured with 

bis(p-aminocyclohexyl)methane (PACM) (EEW = 52.5 g/equivalent), which was acquired from 

Air Products and Chemicals, Inc. The stoichiometry of the DGEBA-PACM polymerization 

reaction was varied to yield epoxy/amine ratios ranging from ~2:1 through 0.6:1 in regular 

intervals, including the stoichiometric ratio of 1:1. The DGEBA-PACM mixtures were then 

oven cured at 160 °C for 2 hr. The Tg values of the reacted samples were measured using a TA 

Instruments 2980 differential scanning calorimeter (DSC). The DSC was ramped to 225 °C at a 

 *

Epon is a trademark of Shell Chemical Co. 
10

constant heating rate of 15 °C/min through two cycles to ensure complete cure and matching 

thermal histories of all of the samples studied. The Tg values used to validate the predictive 

model were then taken from the second heat. 

3. Results and Discussion 

3.1 QSPR Derivation 

The literature and predicted Tg values, and their differences, resulting from the correlation 

equation used to predict the formulation dependence of Tg of an amine-cured DGEBA epoxy 

resin copolymer are listed in table 1. Also listed in table 1 are the abbreviations used for the 

names of the polymers in the training set. The coefficient of determination corresponding to 

this data is R2 = 0.9977 and the leave-one-out cross-validated coefficient of determination is 

R2

cv = 0.9951. These values indicate that there exists a high degree of correlation between the 

predicted and experimental values for the training set of structures and that the training set of 

structures is insensitive to removal of any one particular structure. Given the small size of the 

data set the adjusted coefficient of determination, R2

adj, 

N M

N 1

(y y)

(y y )

R 1 - N

i 1

2

oi

N

i 1

2

oi pi

2

adj −

−

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

−

−

=

∑

∑

=

= (10) 

was calculated. In equation 10, N is both the number of experimental (yo) Tg values from the 

literature and the number of predicted (yp) values, M is the number of descriptors, and y is the 

mean of experimental values from the literature. The value of the adjusted coefficient of 

determination is R2

adj = 0.9970 indicating a high degree of correlation for the training set of 

polymers even after applying a penalty for the relative number of descriptors and structures used 

to derive the correlation equation. The correlation is also characterized by a Fisher value of 

F = 864.0, this value exceeds the 99.9% confidence level*

 (25) indicating a correlation that 

accounts well for the variance in the experimental data. 

The parameters appearing in the correlation equation, including the coefficients to the descriptors 

and the intercept, and coefficients of determination associated with the individual descriptors are 

listed in table 2. The values of the coefficients of determination for each descriptor indicate that 

each descriptor is moderately to highly predictive of Tg for the study set of monomers. The 

values of the coefficients of determination for each descriptor correlated with the set of 

remaining descriptors in the equation indicate a moderate to low degree of collinearity. 

 *

The 99.9% confidence level for 4 degrees of freedom (DOF) in the numerator and 8 DOF in the denominator is 14.39 

according to the CRC Handbook of Chemistry and Physics (25). 
11

Table 1. Data for the training set of amine-cured epoxy polymers from the best four-parameter correlation 

equation (R2

 = 0.9977). 

Amine Curing Agent 

Literature 

Tg 

(°C) 

Predicted 

Tg 

(°C) 

Difference 

(°C) 

Reference 

3,5,5-trimethylcyclohexylamine (TMCA) 84 83.6 −0.4 (26)

Aniline 85 86.2 1.2 (27)

1,3-bisaminomethylcyclohexane (13BAC) 150 149.4 −0.6 (28)

Isophorone diamine (IPD) 165 161.4 −3.6 (29)

4,4'-methylenebis(2,6-diethylaniline) (MDEA) 165 163.4 −1.6 (30)

2,2'-bis(4-(4-aminophenoxyl)phenyl)propane (BAPP) 169 170.5 1.5 (31)

meta-phenylene diamine (MPDA) 169.2 170.9 1.7 (32)

4,4'-methylenebis(3-chloro-2,6-diethylaniline) (MCDEA) 172 171.0 −1.0 (33)

Trimethylene glycol di-p-aminobenzoate (TMAB) 180 183.6 3.6 (34)

4,4'-diaminodiphenyl methane (DDM) 183 183.0 0.0 (35)

Bis(4-(4-aminophenoxyl)phenyl) sulfone (BAPS) 183 181.5 −1.5 (36)

3,3'-diaminodiphenyl sulfone (33DDS) 184 185.3 1.3 (28)

4,4'-diaminodiphenyl sulfone (44DDS) 221 220.3 −0.7 (28)

Table 2. Correlation equation parameters and associated significance values appearing in the best four-parameter 

correlation (R2

 = 0.9977). 

Descriptor/Intercept Coefficient t test R2 a R2 b 

Average nucleophilic reactivity index for a C atom ( NC ) −1.4314 × 105 −39.11 0.80 0.62 

Average bond order of an O atom (OBO) −2.1660 × 103 −23.25 0.24 0.24 

Weighted negative charged partial surface area (WNSA2) 1.0567 × 10−3 13.08 0.44 0.71 

Maximum electron-nuclear attraction for a C-N bond (CN'en) 5.0311 5.26 0.43 0.55 

Intercept 7.8202 × 102 2.25 — — a

Single descriptor correlation coefficient. 

b

Correlation coefficient for correlation with all other descriptors. 

The most significant descriptor appearing in the correlation equation, according to the t test, was 

the average Fukui nucleophilic reactivity index for a carbon atom ( NC ) (37). This descriptor is 

computed according to the following equation: 

( ) ∑ ∑= ∈

=

n

i C HOMO

2

ij

k

j 1

C 1 - E

c

k

1 N . (11) 

In equation 11, cij is the i

th coefficient to an atomic orbital centered on the jth carbon atom, which 

contributes to the highest occupied molecular orbital, where there are n such atomic orbitals and 

k such carbon atoms. Also in equation 11, EHOMO is the energy of the highest occupied 

molecular orbital. The original intent of the Fukui nucleophilic reactivity indices was to describe 

the tendency of a molecule to react as a nucleophile. This descriptor depends upon the 

magnitude of the energy of the highest occupied molecular orbital and the extent to which that 

orbital is localized on the carbon atoms of the polymer model. Analysis of EHOMO with respect to 

Tg resulted in a coefficient of determination of R2

 = 0.2272. This suggests that most of the 
12

correlation of this descriptor is contained within the atomic orbital coefficients. Multiplication of 

NC by the quantity, 1-EHOMO, appearing in the denominator of equation 11 and analyzing the 

product with respect to Tg results in a coefficient of determination of R2 = 0.7914. The 

difference between these two correlations corroborates the idea that the correlation lies in the 

atomic orbital coefficients. Further, the coefficients appearing in the equations resulting from 

the regressions of NC and the described product with Tg both bear a negative sign. This 

indicates that the HOMO is delocalized from the carbon atoms to the heteroatoms with 

increasing Tg values. The shift in contribution to the HOMO from the carbon atomic orbitals to 

the heteroatoms suggests that the rotable bonds involving these atoms are made more rigid by 

delocalizing the HOMO among them, resulting in stabilization of the polymer backbone and 

increased Tg values. Thus, we propose that NC is a measure of the rigidity of the bonds along 

the backbone of the polymers of the training set. 

By the t test the second most significant descriptor appearing in table 2 is the average bond order 

of an oxygen atom (OBO). The AM1 bond orders were calculated within AMPAC from a 

Coulson population analysis of the density matrix elements. For this descriptor, the matrix 

elements correspond to the oxygen atoms in the polymer model. According to table 2, the 

coefficient of determination for this descriptor in a single parameter equation is R2

 = 0.24. While 

all polymer models contain oxygen atoms, analysis of table 1 shows that the three largest Tg

values correspond to amines that contain sulfone functional groups that have two doubly bonded 

oxygen atoms, thus raising the average bond order for these models. We believe that the high 

significance indicated by the t test coupled with the low predictive quality indicated by the single 

parameter coefficient of determination is a consequence of the small size and composition of the 

training set of polymers. This descriptor suggests a donor-acceptor property for the models 

containing the sulfone groups as well as for TMAB, which contains a 4-aminobenzoate moiety. 

The donor-acceptor effect for these polymers would arise from the sulfone and carbonyl 

functionalities drawing electron density from the amino group by induction. The transfer in 

electron density is similar in effect to the delocalization previously described. Here again, the 

increase in electron density around the rotable bonds neighboring the aromatic rings would 

strengthen these bonds, consequently raising the amount of energy required for rotation about the 

bond, thus increasing the Tg for these polymers. Further support for this interpretation can be 

found in the difference between Tg values for 33DDS and 44DDS listed in table 1. Although the 

described inductive effects could contribute to a higher value for the Tg of 33DDS, through some 

degree of p-π delocalization, the amino groups for this diamine are in a meta position with 

respect to the sulfone and this delocalization is probably not nearly as strong as that in 44DDS. 

The para orientation for the functional groups exhibited in 44DDS could lead to further 

stabilization through more delocalization of the nitrogen lone pair into the aromatic ring, 

resulting in an even higher Tg value for this copolymer. 
13

The next most significant descriptor appearing in table 2 is the weighted negative charged partial 

surface area (WNSA2) (26). This descriptor is calculated according to the following formula: 

( )

1000

PNSA2 TMSA WNSA2 = , (12) 

where TMSA is the total molecular surface area and PNSA2 is the negative charged partial 

surface area of the polymer model and is calculated according to the following equation: 

∑ ∑ − ⎟

⎠

⎞ ⎜

⎝

⎛ = i

i

i

- PNSA2 SAi Q , (13) 

where - SAi is the non-occluded surface area of the i

th atom bearing a negative charge from a 

Coulson population analysis and − Qi is the value of the negative charge centered on the atom. 

The surface area of this descriptor is computed as the non-occluded solvent accessible surface 

area after increasing the van der Waal’s radii of the atoms in the molecule by 1.5 Å. This 

descriptor belongs to a class of charged partial surface area (CPSA) descriptors that were created 

by Stanton and Jurs (26) to describe the polar interactions between molecules that give rise to 

their physical and biological properties. Within the context of Tg, WNSA2 is indicative of 

associative effects between polymer chains. The Pearson correlation coefficient for this 

descriptor is r = −0.6644 demonstrating a moderately strong negative correlation with Tg. The 

strong negative correlation indicates an inversely proportional relationship suggesting that 

localization of negative charge onto a smaller surface area of the polymers results in more polar 

interactions between the polymer chains for the higher Tg polymers. The presence of a bond 

order descriptor in the derived QSAR equation also suggests that this descriptor is a measure of 

increased charge localization resulting in shorter stronger bonds within the polymers exhibiting 

higher Tg values. Evidence supporting this reasoning can be found in table 3, in which are listed 

five equations obtained by a heuristic (12) analysis of all descriptors. These equations are 

characterized as being the five most correlated to Tg (within the heuristic method) based first on 

the F statistic and then on the coefficient of determination. The minimum bond order for a 

nitrogen atom appears in three out of five of the equations as do other related descriptors. 

The last descriptor appearing in the correlation equation is the maximum electron-nuclear 

attraction for a carbon-nitrogen bond. This descriptor is computed within the constructs of the 

NDDO approximation of semiempirical molecular orbital theory according to the following 

equation: 

ν

i,N

N

μ,ν C

en μ,ν μ φ R

Z CN' ∑P φ ∈

= , (14) 
14

Table 3. Equations having the highest coefficients of determination and F values 

following a heuristic analysis.

Descriptor/Intercept t test R2 a R2 b 

R2 = 0.9942, F = 344.9 

Number of N atoms 28.85 0.82 0.13 

Minimum bond order (>0.1) of an N atom −10.10 0.35 0.60 

Average valency of an O atom −5.54 0.26 0.40 

Relative number of O atoms −4.51 0.28 0.62 

Intercept 12.75 — — 

R2 = 0.9925, F = 264.3

Maximum total interaction for an N-H bond 9.66 0.80 0.69 

Maximum e-e repulsion for an N atom 13.47 0.73 0.41 

Minimum e-n attraction for a C-C bond 5.16 0.11 0.32 

Maximum 1-electron reactivity index for a C atom −4.33 0.38 0.58 

Intercept −6.51 — — 

R2 = 0.9910, F = 220.0

Maximum total interaction for an N-H bond 8.29 0.80 0.71 

Maximum -e repulsion for an N atom 12.05 0.73 0.50 

Minimum nucleophilic reactivity index for a C atom −3.78 0.22 0.50 

Minimum e-n attraction for a C-C bond 3.67 0.11 0.26 

Intercept −5.00 — — 

R2 = 0.9908, F = 216.0

Number of N atoms 20.83 0.82 0.24 

Minimum bond order (>0.1) of an N atom −7.71 0.35 0.37 

Average valency of an O atom −4.44 0.26 0.43 

Total dipole of the molecule −3.13 0.27 0.42 

Intercept 11.90 — — 

R2 = 0.9905, F = 207.5

Number of N atoms 20.83 0.82 0.23 

Minimum bond order (>0.1) of an N atom −10.24 0.35 0.20 

Maximum electrophilic reactivity index for a C atom −3.11 0.25 0.18 

Minimum e-n attraction for a C-C bond 3.08 0.11 0.22 

Intercept −2.79 — — a

 Single descriptor correlation coefficient. 

b

 Correlation coefficient for correlation with all other descriptors. 

where Pμ,ν represents the density matrix elements corresponding to the carbon atom and 

ν

i,N

N

μ φ R

Z

φ is the electron-nuclear attraction integral for the i

th valence electron centered on 

the carbon atom and interacting with the nitrogen atom, which bears a charge of ZN. This 

descriptor is related to the strength of the carbon-nitrogen bonds in the backbone of the polymer 

models. The lack of significance of this descriptor as evidenced by the t test is likely due to the 

location of the bonds corresponding to this descriptor. Visualization of this descriptor within 

CODESSA by plotting it upon the corresponding bonds of the polymer models indicates that 

these bonds are at the terminal end of the terminal diamine (figure 3). Although the descriptor is 

indicative of bond strength, it is difficult to draw a further conclusion since the presence of the 

terminal amine is a consequence of the chosen connectivity of the polymer models in the training 

set. 
15

3.2 External Validation of QSPR 

To test the extensibility of the derived QSPR, as well as our model for predicting the formulation 

dependence of Tg, we carried out predictions of Tg and compared these values with those from 

our DSC determinations for the PACM/DGEBA formulations previously described. As 

described, the predictions involved building polymer models exhibiting various degrees of 

crosslinking and then using the derived QSPR to predict their individual contribution to the bulk 

Tg for the polymer (figure 3). These predicted values were then multiplied by fractional weights 

obtained by the method of Miller and Macosko (16). The resulting products were then summed 

to arrive at the composite predicted Tg values for each stoichiometric ratio. The results are 

plotted in figure 4. 

0

20

40

60

80

100

120

140

160

180

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Stoichiometric Ratio (r = Epoxide/Amine)

Tg (oC)

Figure 4. Predicted (c) and experimental ()Tg values for PACM-cured DGEBA with 

varied stoichiometric ratios of monomers. 

As shown in figure 4, the trend in Tg vs. stoichiometric ratio is accurately reproduced with a 

maximum corresponding to a 1:1 stoichiometry of functional groups. These results compare 

well with those reported by Williams (33) whose work involved predicting Tg values for nonstoichiometric epoxy-amine polymer formulations. In attempting to describe the influence of 

stoichiometry on Tg, Williams used the Nielsen (36) equation 

T T 3.9 x 10 ν4

g = g0 + , (15) 

where ν is the concentration of elastic chains and Tg0 is the glass transition temperature of the 

uncrosslinked polymer. To arrive at the concentration of elastic chains Williams used the 

method of Miller and Macosko (16). The value for Tg0 was obtained by using a value for ν in the 

Miller and Macosko analysis that corresponds to a 1:1 stoichiometric ratio and a corresponding 

experimental value for Tg. The difference between the work of Williams and that of our own is 
16

that Williams used a Tg value for a given polymer to predict the formulation dependence of Tg

for that same polymer. In contrast, our approach uses Tg values from similar polymers to arrive 

at an equation that can be used to predict formulation dependent Tg values for any similar 

polymer. While our results exhibit a greater degree of error than those of Williams, they are 

qualitatively identical. The greater error in our predictions with respect to experimental results is 

likely due to the fact that they do not directly involve corrections to experimental values. Rather, 

our model involves only computed quantities and as such should be expected to exhibit greater 

error. 

4. Conclusions 

We have derived a designer QSPR capable of predicting the Tg of amine-cured resins of 

bisphenol A diglycidyl ether. The descriptors appearing in the correlation equation lend 

themselves to intuitive interpretation based on the molecular level events that occur when a 

polymer exhibits a change from a glassy state to a rubbery state. For the study set of polymers, 

the Tg values appear to be reflective of the strengths of bonds along the backbone of the polymer. 

The strongly predictive nature of the quantum mechanically derived descriptors presented here is 

a testimony to the role that semiempirical methods play in studying such large molecular 

systems. 

Using an ad hoc probabilistic treatment based on the work of Miller and Macosko (16) we have 

developed a model that accurately predicts the dependence of Tg on the stoichiometric ratio of 

comonomers in crosslinked copolymers. This is the first such model to do so without requiring 

an experimental value for the polymer of interest, and although our model does not exhibit high 

quantitative accuracy, our model qualitatively predicts the correct trend. The generally 

applicable methodology employed here suggests that models for the prediction of other 

properties should be readily accessible. We anticipate that by employing this methodology to 

other design problems researchers will be able to design new materials without the costly need 

for synthesis of novel monomers. 
17

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