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Journal of Algorithms & Computational Technology Vol. 4 No. 1 121

Application of Particle Swarm

Optimisation to Evaluation of

Polymer Cure Kinetics Models

T. Tilford1,*, M. Ferenets2, J. E. Morris3, A. Krumme4,

S. Pavuluri5, P. R. Rajaguru1, M. P. Y. Desmulliez5

and C. Bailey1

1University of Greenwich, Greenwich, London SE10 9LS,

United Kingdom

2Eesti Innovatsiooni Instituut, Sepapaja 6, Tallinn 11415, Estonia

3Portland State University, Portland, OR 97207-0751, USA

4Tallinn University of Technology, Ehitajate tee 5,

Tallinn 19086, Estonia

5Heriot-watt University, Edinburgh, EH14 4AS, United Kingdom

Email: T.Tilford@gre.ac.uk

Received 12/05/2008; Accepted 14/04/2009

ABSTRACT

A particle swarm optimisation approach is used to determine the accuracy

and experimental relevance of six disparate cure kinetics models. The cure

processes of two commercially available thermosetting polymer materials

utilised in microelectronics manufacturing applications have been studied

using a differential scanning calorimetry system. Numerical models have

been fitted to the experimental data using a particle swarm optimisation

algorithm which enables the ultimate accuracy of each of the models to be

determined. The particle swarm optimisation approach to model fitting

proves to be relatively rapid and effective in determining the optimal

coefficient set for the cure kinetics models. Results indicate that the single-

step autocatalytic model is able to represent the curing process more

accurately than more complex model, with ultimate accuracy likely to be

limited by inaccuracies in the processing of the experimental data.

1. INTRODUCTION

The Controlled Collapse Chip Connection package (also referred to as a

‘flip-chip’) is a means of connecting a semiconductor die to a printed circuit

board and, hence, to other components in an electronics assembly. Such

*Corresponding author. Email: T.Tilford@gre.ac.uk

packages are widely used to package high performance semiconductor

devices and enable high density assemblies essential to modern electronics

devices. A flip-chip package consists of a silicon die with a large number of

metallic pads arranged on the upper surface. Small solder balls are deposited

on these pads. The device is then flipped over such that pads and solder balls

face downward. The assembly is then placed onto a circuit board that has

matching pads and bonded onto the board. After bonding, the space between

the circuit board and the silicon die is usually filled with a thermosetting

polymer material known as ‘underfill’ to provide mechanic support to the

silicon die. The whole package is then covered with a second thermosetting

polymer material encapsulant material to protect it from environment factors

such as moisture, humidity and to provide electrical insulation. Fig 1

diagrammatically represents a cross section of a flip-chip package. The

underfill and encapsulant materials are applied in a liquid state and are

heated to induce the cure process. Incomplete cure of the underfill or

encapsulant can be detrimental to the thermomechanical properties and can

lead to failure modes such as de-lamination between material interfaces.

There is therefore a requirement to ensure that the materials are fully cured.

In order to determine the degree of cure of polymer materials, two

experimental approaches are typically used. Differential Scanning Calorimetry

(DSC) equipment can be used to analyse the energy flux from a sample as it is

heated. Alternatively, Fourier Transform InfraRed (FTIR) spectroscopy can be

used to determine the molecular composition of a test sample. However, neither

of these approaches is applicable to in-situ measurement of degree of cure in

practical applications.

122 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Underfill Board pad Circuit board

Encapsulant

Silicon die

Die pad

Solder ball

Figure 1. Cross section of a flip chip package.

2. POLYMER CURE PROCESS

Thermosetting polymers materials are a subset of polymers, in which the

interlinking of the polymer chain is irreversible. Polymer materials consist of a

very large number of basic structural units (known as monomers) which are

interconnected to form a large macromolecule. The fundamental structure

typically consists of a long chain of monomers which become entangled and

interlinked. The manner in which this interlinking occurs at the mesoscopic

scale has a significant influence of the macroscopic material properties. The

cross-linking process is initiated through addition of heat energy which causes

the polymer chain to link into a highly complex three-dimensional structure.

Multiple polymer chains can interlink together, resulting in union of many

polymers molecules. The process of molecular interlinking is referred to as

‘curing’. The ‘degree of cure’ is a measure of the proportion of the interlinking

process that a material has undergone. It is possible to determine the exact

molecular state of a polymer using techniques such as Fourier Transform

InfraRed Spectroscopy (FTIR) and Raman spectroscopy. As the curing of

thermoset materials is generally exothermic, the degree of cure is often linked

to the proportion of energy released by the material. This relation was initially

proposed by Sourour [1]. The proportion of heat released by a polymer material

can be determined through use of Differential Scanning Calorimetry (DSC) and

this technique has been most widely adopted in the analysis of polymer curing.

Alarge body of research focuses on attempting to describe the curing

process, in particular, the rate of cure, in terms of material temperature. Initial

work on thermosetting polymers was based on the work of Svante Arrhenius [2]

who proposed that the rate of reaction (k) in a chemical system could be

described empirically by k =Aexp(E/RT), in which A was a rate constant, E is

an activation energy, R is the gas constant and T is the material temperature.

The relation, which was initially proposed by Jacobus Henricus van ’t Hoff five

years prior to Arrhenius’ interpretation of it, underpinned the early

understanding of chemical reactions. In order to determine the rate of cure of a

polymer material using Arrhenius’ equation, an estimate of the rate constant A

and the activation energy E are required.

3. CURE KINETICS MODELS

Many variations of the basic Arrhenius model have been proposed and assessed.

Areview of cure kinetics methods has been published by Yousefi et al [3], while

an overview of modelling is presented by Morris et al [4]. The most general

formulation of cure kinetics models is the nth order phenomenological model.

Journal of Algorithms & Computational Technology Vol. 4 No. 1 123

The degree of cure can either be determined algebraically for constant heating

rates or through numerical integration. If αis defined as the degree of cure, the

basic assumption of all models is that the reaction rate can be expressed in terms

of the temperature dependent chemical ‘rate constant’, K, which is a function

of temperature, and a function, f(α), of reactant concentration at absolute

temperature, T, such that

(1)

Reaction rate parameters A and activation energy E are assumed to be

characteristic constants of the polymer, and R = 8.31 J/K.mole. Cure models

vary in the assumed form of f(α), with two predominant forms: the nth order

form and the autocatalytic form. The nth order models assume f(α) =(1 – α)n

and, in these cases, one can find αanalytically for constant T, (i.e. isothermal

cure) using equations 2 to 4 for 1st order, 2nd order and nth order models,

respectively.

(2)

(3)

(4)

The autocatalytic models consist of the single-step model given in equation 5,

the double step given in 6 and the modified double step, given in 7. The nth

order models are based on the simple notion that the reaction rate is

proportional to the un-reacted reagent mass available. The single step

autocatalytic model, given in equation 5, is based on the concept that the

reaction proceeds at the boundary of reacted and un-reacted material and is

activated by an exothermic reaction. Note, however, that dα/dt = 0 for α = 0,

which is non-physical and leads to “starting” problems.

The double-step auto-catalytic model (equation 6) is designed to solve this

problem, but the single n-exponent does not suggest two independent reactions.

However, it is the only model with more than a single chemical rate constant, i.e.

all others implicitly assume a single chemical curing reaction, or at least a

δα

δαα

tKnKt

nn

=−

(

)

∴=−+ −

(

)

(

)

−−

1111

11/( )

δα

δαα

tKKt=−

(

)

∴=−+ −

111

1

2[]

δα

δαα

tKe

Kt

=−

(

)

∴=−

−

(

)

11

δα

δαα

tKf f=⋅

(

)

=⋅ ⋅

(

)

Aexp

-E

RT

124 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

single rate controlling reaction across the full temperature range of interest. The

modified double-step, given in equation 7, provides two reaction rates, but with

a single activation energy.

(5)

(6)

(7)

4. TRADITIONAL MODEL-FITTING APPROACHES

Initial work to determine cure kinetics model coefficient sets for thermosetting

polymer materials was carried out by Ozawa et al [5–8] with additional

contribution from other researchers, e.g. [9]. This seminal research focused on

the thermal analysis of materials to determine reaction rates. The Ozawa

method for obtaining cure kinetics parameters, which is well outlined in [10],

assumes that the degree of cure is proportional to the reaction exotherm. DSC

analysis of the material is performed at a minimum of two disparate constant

heating rates. A relation proposed by Doyle et al [11] defined an approximation

for the cure rate for E/RT < 20. Ozawa utilised the Doyle approximation to

form a set of equations for multiple heating rates in which the actual cure

function cancels out. This enables the values of the activation energy and

subsequently the rate constant to be determined. The accuracy of the model is

reliant on the accuracy of the Doyle approximation, but has been utilised

readily since its proposal. The Ozawa approach can be used to determine the

Activation Energy, E, from the relationship given in 8, in which βis the

heating rate, R is the gas constant and Tpis the peak temperature. A popular

alternative to the Ozawa method is the Kissinger equation [12] which

activation energy and rate constant, A, can be determined from relations given

in 9 and 10.

(8)

(9)

ER T

T

p

p

=− ∂

(

)

(

)

∂

(

)

ln /

/

β2

1

ER

Tp

=

(

)

1 502 1

.

ln

/

∆

∆

β

δα

δαα

tKy y y y

mn

=+ − +=()()

12 12

11;

δα

δαα

tKK

mn

=+ −()()

12 1

δα

δαα

tKmn

=−()1

Journal of Algorithms & Computational Technology Vol. 4 No. 1 125

(10)

5. PARTICLE SWARM OPTIMISATION

The cure kinetics models can be applied to a range of thermosetting polymer

materials. The model coefficients are varied in order to fit a model to a

particular material. A number of approaches have been proposed for

determining the optimal coefficient set for polymer materials. However,

traditional approaches rely on a number of assumptions which may not be

applicable to rapid cure processes. To overcome these issues, a particle swarm

optimisation algorithm has been developed to determine optimal model

coefficients in response to sets of experimentally derived cure data. This

approach has been used previously for cure kinetics studies by Pagano et al [13]

and by Ourique et al [14], although not applied to complex multi-component

materials such as those assessed in this study.

Particle Swarm Optimisation (PSO) [15] is a stochastic optimisation

approach based on the concept of ‘swarm intelligence’. The PSO algorithm

defines a large number of particles occupying multi-dimensional space. Each of

the hyperspace dimensions relates to a cure model parameter, so the position of

each particle relates to a set of model parameters. The accuracy of the model at

each coefficient set can be determined by comparing the cure process predicted

by the model with experimentally derived data. Once the accuracy, typically

referred to as ‘fitness’, has been determined, the velocity of the particles is

updated based on the fitness of each particle relative to its own fitness compared

to the optimal value found by any of the particles in the swarm. The

optimisation algorithm then proceeds in an iterative manner, with each particle

considering its own optimal position as part of the velocity update, until it is

decided that a sufficiently converged solution has been obtained.

The steps required in the particle swarm algorithm are:

Initialise particle positions, particle velocities, local fitness and global fitness

Iterative progression while k ≤kmax

xUx x

v

f

f

iii

i

i

global

∈

=

=∞

=∞

(min , max )

0

)

)

AEERT

RT

p

p

=

(

)

βexp /

2

126 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Calculate fitness for each particle

Update particle optimum position if

Update global optimum position if

Update particle velocity

Update particle position

In this description, there are i particles and j hyperspace dimensions, with xi

being the position vector of particle i and vibeing the velocity vector of particle i.

The fitness, fi, of each particle is based on a root mean square error metric of

the difference between the transient cure behaviour of the model, αmand the

experiment data, αe. The global optimum and particle optimum positions are

designated g

and x

irespectively, the particle optimal fitness is ƒ

iwhile the global

optimal fitness is ƒ

global. Algorithm coefficients

ω,

c1and c2represent particle

momentum, particle optimum attraction and global optimum attraction,

respectively, while r1and r2are random values

∈

U(0,1). The algorithm proceeds

iteratively from k =1 to k =kmax.

The values used in the analyses, which are detailed in table I, differ from the

values used for typical particle swarm analyses [16, 17]. This is due to the

poorly posed nature of the problem, with large proportions of the design

hyperspace resulting in no curing of the material and only a small region of the

hyperspace leading to reasonable fitness values. In response, the number of

particles used in this study is somewhat greater than typical, in an attempt to

ensure that a number of the particles lie in the region of the hyperspace resulting

in reasonable fitness values at initialisation. Additionally, the particle optimum

xxv

iii

=+

vvcrxxcrgx

ii ii i

=+ −

(

)

+−

(

)

ω11 2 2

))

ff

iglobal

<)

)

gx

i

=

ff

ii

<)

)

xx

ii

=

ftt

dT

i

me

=∂

∂

−∂

∂

∞

∫αα

22

0

Journal of Algorithms & Computational Technology Vol. 4 No. 1 127

Table I. Particle Swarm Optimisation parameters.

Parameter Symbol Value

Number of particles i 100000

Number of hyperspace dimensions j 6

Number of iterations k 20

Particle optimum attraction c10.02

Global optimum attraction c22.00

Particle momentum ω0.05

attraction and particle momentum have been significantly reduced from typical

values as this was found to stabilise the solution procedure.

The progression of the particles from their initial uniform distribution over the

hyperspace to an optimal point is illustrated in figures 2 to 7 on the following

page. The case is a first order model in which only two parameters (activation

energy E and rate constant A) are considered. The particles rapidly move toward

the optimal values in each of the dimensions and then slowly move toward the

two dimensional optimal point. If the algorithm was allowed to continue, all

particles would eventually occupy a single point in the hyperspace. The

algorithm has been implemented in Fortran 90 and parallelised with OpenMP

[18] directives to run on a 16 core AMD Opteron 8354 system. Runtimes

obviously vary depending on the experimental data, the model analysed and the

runtime setting but analysis of the accuracy of the single step model against five

experimental data sets takes approximately 1027 seconds.

128 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 2. Particle distribution; k = 1.

Figure 3. Particle distribution; k = 2.

Journal of Algorithms & Computational Technology Vol. 4 No. 1 129

Figure 4. Particle distribution; k = 3.

Figure 5. Particle distribution; k = 5.

Figure 6. Particle distribution; k = 18.

130 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 7. Particle distribution; k = 500.

6. DIFFERENTIAL SCANNING CALORIMETRY ANALYSIS OF TEST MATERIALS

In order to determine the actual cure behaviour of the sample materials, a series

of experimental analyses were required. The objective of the experimental

programme was to determine the variation in the degree of cure of the test

materials with time and temperature. Two disparate materials were analysed:

the first material being a commercially available encapsulant material Henkel

Hysol® E01080, the second was an underfill material Henkel FP4511.

Differential scanning calorimetry (DSC) is an easy method for determining

resin cure kinetics. DSC measures the heat flow into or from a sample as it is

heated, cooled and/or held isothermally. For thermosetting resins like epoxies

the technique provides information on glass transition temperature, Tg, onset of

cure, heat of cure, maximum rate of cure, completion of cure and degree of cure.

APerkin Elmer DSC-7 system was used for the analysis of the samples in this

study. Small amounts (typically 3 to 10 mg) of the sample materials were placed

in aluminium crucibles and an empty crucible was used as a reference.

Measurements were made in nitrogen atmosphere with a flow rate of 20 ml/min.

All materials used in these experiments are one component epoxy resins.

This means that epoxy and its hardener are already mixed together, but the

curing is prevented by keeping the mixture at low temperature (at +5 oC for

EO1080 and at –40 oC for FP4511). The cure profiles recommended by the

material manufacturer are presented in Table II.

One of the aims of this study was to assess the applicability of these models

for microwave curing processes. The series of temperature profiles, listed in

tables III and IV, were developed for the tests, with high curing rates intended

to mimic temperature profiles characteristic of microwave curing.

Journal of Algorithms & Computational Technology Vol. 4 No. 1 131

Table II. Curing methods of tested materials suggested by Henkel.

Material Suggested temperature, oCCuring time

Encapsulant material 110 degrees 2 hours

(Henkel EO1080) 140 degrees 1.5 hours

150 degrees 20 min

Underfill material 150 degrees 2 hours

(Henkel FP4511)

Table III. Temperature profiles for encapsulant tests.

Sample number Start Temp Final Temp Ramp rate

°C °C °C/min

120300 200

220300 150

320300 100

40300 50

50300 10

Table IV. Temperature profiles for underfill tests.

Sample number Start Temp Final Temp Ramp rate

°C °C °C/min

10300 20

20300 10

30250 20 (750 s)

0 (180 s)

40250 30

132 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

7. RESULTS

DSC analysis of the sample materials was performed in order to provide energy

flux data. As the material consists of a mix of polymers, fillers and solvents, the

energy flux during cure is a complicated compound result of multiple reactions.

The DSC data therefore required processing to separate the exothermal polymer

cure reaction from the tertiary reactions. The pure DSC data for encapsulant

sample 1 is presented in figure 8 as the blue line, with the exothermal energy flux

as the red line and a user-defined baseline in green. The variation in the energy

flux from the material with time is quite complex, due to both physical and

experimental factors. The rapid variations in energy flux evident at approximately

5 seconds and at 115 seconds are likely to be due to operation of the DSC

equipment while the smaller dip in energy flux between 30 and 40 seconds is

likely to be due to an unidentified physical reaction. The much larger reduction

in energy flux curve between approximately 50 and 115 seconds is assumed to be

the exothermal energy release due to the polymerisation process. The baseline is

defined to be linear between the points at which polymerisation is considered to

commence and terminate. The exothermal energy flux is defined as the difference

between the pure data and the baseline during the polymerisation process. The

degree of cure at any given time can be determined from the cumulative energy

released compared with the total reaction energy. The variation of degree of cure

with time for sample 1 is given in figure 9 while the cure rate is given in figure 10.

The energy fluxes cure degree and cure rate variations have been determined for

four further encapsulant samples and for five underfill samples.

The numerical models have been used to predict transient development of

degree of cure for the time-temperature profiles used during the DSC tests. The

rate of cure has also been determined by the models and compared with

experimental data. Figures 11 to 20 show transient variation in degree of cure

and in cure rate predicted by the cure kinetics models in comparison with the

experimentally derived data. Note the disparity in time scales for the samples

due to disparate heating rates. The procedure has been repeated for the Henkel

FP4511 underfill samples, numerical comparisons for the four samples plotted

in figures 21 to 28. Table V and VI present optimal model coefficients for all

models used for each of the materials. The cumulative error for each of the

Journal of Algorithms & Computational Technology Vol. 4 No. 1 133

50

40

30

20

10

0020

Energy flux (mW)

−10

60

40 60 80 100 120

Ti ( )

140

Figure 8. Energy flux for encapsulant test sample 1.

Figure 9. Variation in degree of cure for encapsulant test sample 1.

0.0

020

Degree of cure

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

40 60 80 100 120 140

134 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 10. Variation in cure rate for encapsulant test sample 1.

0.00

020

Cure rate

0.06

0.05

0.04

0.03

0.02

0.01

40 60 80 100 120 140

Figure 11. Comparison between experimental data and numerical solutions of cure

process for encapsulant test sample 1.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00204060 80100 120 140

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Journal of Algorithms & Computational Technology Vol. 4 No. 1 135

Figure 12. Comparison between experimental data and numerical solutions of cure rate

for encapsulant test sample 1.

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.000204060 80

Time (s)

100 120 140

Cure rate

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 13. Comparison between experimental data and numerical solutions of cure

process for encapsulant test sample 2.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00204060 80100 120 140

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

136 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 14. Comparison between experimental data and numerical solutions of cure rate

for encapsulant test sample 2.

0.06

0.05

0.04

0.03

0.02

0.01

0.000204060 80

Time (s)

100 120 140

Cure rate

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 15. Comparison between experimental data and numerical solutions of cure

process for encapsulant test sample 3.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 020406080100120 140 160 180 200

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Journal of Algorithms & Computational Technology Vol. 4 No. 1 137

Figure 16. Comparison between experimental data and numerical solutions of cure rate

for encapsulant test sample 3.

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0.00 020406080

Time (s)

100 120 140 160 180 200

Cure rate

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 17. Comparison between experimental data and numerical solutions of cure

process for encapsulant test sample 4.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 050100 150 200 250 300 350 400

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

138 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 18. Comparison between experimental data and numerical solutions of cure rate

for encapsulant test sample 4.

0.020

0.018

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000 050100 150 200 250 300 350 400

Time (s)

Cure rate

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 19. Comparison between experimental data and numerical solutions of cure

process for encapsulant test sample 5.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0 200 400 600 800 1000 1200 16001400 20001800

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Journal of Algorithms & Computational Technology Vol. 4 No. 1 139

Figure 20. Comparison between experimental data and numerical solutions of cure rate

for encapsulant test sample 5.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0 200 400 600 800 1000 1200 16001400 20001800

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 21. Comparison between experimental data and numerical solutions of cure

process for underfill test sample 1.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0 100 200 300 400 500 600 800700 1000900

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

140 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 22. Comparison between experimental data and numerical solutions of cure rate

for underfill test sample 1.

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0.000 0 100 200 300 400 500 600 700 800 900 1000

Time (s)

Cure rate

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 23. Comparison between experimental data and numerical solutions of cure

process for underfill test sample 2.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0 200 400 600 800 1000 1200 16001400 20001800

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Journal of Algorithms & Computational Technology Vol. 4 No. 1 141

Figure 24. Comparison between experimental data and numerical solutions of cure rate

for underfill test sample 2.

0.0040

0.0035

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000 0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (s)

Cure rate

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Figure 25. Comparison between experimental data and numerical solutions of cure

process for underfill test sample 3.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0 100 200 300 400 500 600 800700 1000900

Time (s)

Degree of cure

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

142 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 26. Comparison between experimental data and numerical solutions of cure rate

for underfill test sample 3.

0.008

0.007

0.001

0.006

0.005

0.004

0.003

0.002

0.000 0 100 200 300 500 600 700 800 900 1000400

Time (s)

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Cure rate

Figure 27. Comparison between experimental data and numerical solutions of cure

process for underfill test sample 4.

1.0

0.9

0.3

0.8

0.7

0.6

0.5

0.4

0.2

0.1

0.0 050100 150 200 250 350300 450 500400

Time (s)

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Degree of cure

model – material combinations are presented in figures 29 and 30. This is the

sum over each of the samples of the error metric used in the particle swarm

optimisation and is given in equation 11.

(11)

Error tt

dT

me

nsam

=∂

∂

−∂

∂

∞

∫αα

22

0

1

pples

∑

Journal of Algorithms & Computational Technology Vol. 4 No. 1 143

Table V. Model coefficients for encapsulant data.

Model E1 A1 n m E2 A2 y1

1st order 39109.56 1075.40 1 - - - -

2nd order 38947.38 875.53 2 - - - -

3rd order 39071.93 1436.98 3 - - - -

Single step 32493.47 1428.87 2.072 1.075 - - -

Double step 111995.69 322.90 1.761 0.915 33607.75 1295.48 -

Modified double 43519.70 2497.20 0.759 0.683 - - 0.8007

Table VI. Model coefficients for underfill data.

Model E1 A1 n m E2 A2 y1

1st order 44426.07 1658.29 1 - - - -

2nd order 42793.06 874.61 2 - - - -

3rd order 40893.34 544.72 3 - - - -

Single step 36588.74 517.42 1.333 0.827 - - -

Double step 83933.39 542.86 1.208 0.679 39838.32 1000.76 -

Modified double 38237.14 643.49 1.133 1.078 - - 0.0991

Figure 28. Comparison between experimental data and numerical solutions of cure rate

for underfill test sample 4.

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000 050100 150 200 250 350300 450 500400

Time (s)

Experimental data

1st order

2nd order

3rd order

Single step

Double step

Modified double step

Cure rate

144 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models

Figure 30. Error metric for the numerical models fitted to underfill data.

2.0

1.8

1.6

1.4

1.2

1.0

0.8

Error metric

0.6

0.4

0.2

0.0

1st order 2nd order 3rd order Single step Double step Modified double

step

Figure 29. Error metric for the numerical models fitted to encapsulant data.

3.5

3.0

2.5

Model error

2.0

1.5

1.0

0.5

0.0

1st order 2nd order 3rd order Single step Double step Modified double

step

Journal of Algorithms & Computational Technology Vol. 4 No. 1 145

8. CONCLUSIONS

The primary findings of this study are that particle swarm optimisation

techniques can be used to fit cure kinetics models to experimentally derived data

and that the process is rapid and effective. Furthermore, the development of

increasingly complex models does not necessarily lead to improvements in

accuracy. The model fitting approach would appear to underpin the accuracy of

the model, with the single step autocatalytic model providing a more accurate

estimate of cure kinetics process than the more complex double step models.

The primary source of error in this study is considered to be the inability of

the model to consider the complex multi-component nature of the material

under test. Materials such as those tested in this study contain multiple

compounds with undergo a number of disparate reactions during the heating

process. Ideally, each of the discrete reactions would be modelled separately

and then integrated together to form an amalgamated model. The baseline curve

is defined in an attempt to separate a single cure reaction from the remaining

processes. The uncertainties inherent in the definition of the baseline curve are

likely to result in errors exceeding those derived from the inaccuracies of the

cure kinetics models.

Further work is required to more accurately assess the cure process using Fourier

Transform InfraRed spectroscopy. This will enable the proportion of primary,

secondary and tertiary amines, enabling accurate separation of the exothermal

processes from the reactions of the filler and solvents present within the material.

ACKNOWLEDGMENTS

The authors wish to acknowledge funding and support from the European Union

Framework 7 programme (FP7-SME-2007-2), contract number 218350 and

additional support from our partners, Kepar Electronica S.A., Camero di

Commercio Industria, Artigianato e Agricoltura di Milano, Mikrosystemtechnik

Baden-Württemberg e.V., the National Microelectronics Institute, ACI-ecotec

GmbH & Co. KG, Industrial Microwave Systems Ltd. and Ribler GmbH.

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146 Application of Particle Swarm Optimisation to Evaluation of

Polymer Cure Kinetics Models