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This is to certify that the

thesis entitled

Configurational Bias Monte Carlo Simulations of Phase
Segregation in Networked Block Copolymers

presented by

Kent Ivan Palmer

has been accepted towards fulfillment
of the requirements for

M.S. degreein Chemical Engineering

 

 

. Mafl professor
5/2/00

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
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DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11/00 W359.“

Configurational Bias Monte Carlo Simulations of Phase Segregation in
Networked Block Copolymers

BY

Kent Ivan Palmer

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

2000

ABSTRACT

Configurational Bias Monte Carlo Simulations of Phase Segregation in
Networked Block Copolymers

By

Kent Ivan Palmer

Crosslinked block copolymers are used as adhesives in fiber-reinforced composite
materials manufactured for automotive applications. Good adhesion between the
polymer matrix and fibers in the interphase region is required for the structural integrity
of these materials. Experimental evidence indicates that superior adhesion is obtained
when phase segregation occurs between the two matrix phase block copolymers. It is
desirable to predict the conditions under which phase segregation is expected to occur.
Configurational-bias Monte Carlo simulations of networked, two-component,
trifunctional block copolymers were carried out to examine phase segregation in these
materials. The effects of three principal parameters on phase segregation were examined:
the weight fractions of the two components, the crosslink length, and the ratio of the
square-well interactions. The molecular simulation results confirmed trends observed in

laboratory measurements.

To my wife, Tiffany

And my parents, Robert and Judy Palmer.

ACKNOWLEDGMENTS

I would like to thank my advisor Dr. Christian M. Lastoskie. His guidance, patience and

support are deeply appreciated.

Thanks to Dr. Lawrence T. Drzal and Dr. Carl T. Lira for their insight and support as

committee members.

Thanks also to Dr. Juan J. DePablo and Dr. Fernando Escobedo for the use of their

Fortran Code. Their studies provided a helpful tool for this work.

I am also grateful to Robert Dombrowski for providing a second view, thought provoking

discussions and making many hours in the lab bearable.

I thank the Chemical Engineering Department at Michigan State University for their

support, as well as the opportunity to perform this work and advance my education.

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LIST OF TABLES VII
LIST OF FIGURES VIII
LIST OF FIGURES VIII
LIST OF ABBREVIATIONS XII
LIST OF ABBREVIATIONS X11
1. INTRODUCTION 1
2. GENERAL BACKGROUND 3
2.1 PREVIOUS EXPERIMENTAL WORK .............................................................................. 3
2.2 PREVIOUS NETWORK SIMULATION ............................................................................. 4

3. THEORTICAL CONSIDERATIONS 7
3.1 FLORY-HUGGINS THEORY .......................................................................................... 7
3.2 MOLECULAR SIMULATIONS ...................................................................................... 10

4. METHODS 13
4.1 EXPERIMENTAL SETUP ............................................................................................. 13
4.2 RADIAL DISTRIBUTION FUNCTIONS ........................................................................... 18
4.2. 1 DETERMINATION OF PHASE SEGREGA TION- -- 20
4.2.2 CALCULATED RDF CURVES 20

5. RESULTS & DISCUSSION 22
5.1 GENERAL OBSERVATIONS ........................................................................................ 22
5.1.1 LENGTH OF RUNS 22
5.1.2 RDF BEHA VIOR DUE TO SQUARE- WELL MODEL 24

5.2 EFFECT OF HARD SEGMENT CONTENT ..................................................................... 26
5.2.1 TONE 301, 50% HARDSEGMENT CONTENT.....- 26
5.2.2 TONE 301, 65% HARD SEGMENT CONTENT -- - -28
5.2.3 TONE 301 CONCLUSION. - ..... 29
5.2.5 TONE 305, 50% HARD SEGMENT CONTENT- - 32
5.2.6 TONE 305, 63% HARD SEGMENT CONTENT 34

5.2. 7 TONE 305 CONCLUSION- 35
5.2.8 TONE 310, 13% HARD SEGMENT CONTENT 36
5.2.9 TONE 310, 37% HARD SEGMENT CONTENT--- 38

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.2.10 TONE 310, 50% HARD SEGMENT CONTENT 40
5.2.11 TONE 310 CONCLUSION - -- 41

5.3 EFFECT OF TONE SIZE .............................................................................................. 42
5.3.1 Low HARD FRACTION - 42
5.3.2 50% HARD FRACTION 42
5.3.3 HIGH HARD FRACTION-_ _- 42

5.4 EFFECT OF CROSS-LINKING .................................... ‘ .................................................. 4 3
5.4.1 TONE 305, 50% HARD FRACTION, INCOMPLETE CROSS-LINKING ................... 43
5.4.2 TONE 310, 38% HARD FRACTION, INCOMPLETE CROSS-LINKING ................... 45
5.4.3 TONE 310, 50% HARD FRACTION, INCOMPLETE CROSS-LINKING ................... 47

5.5 EFFECT OF e-RATIOS ................................................................................................ 49
5.5.1 e-RATIO 0F 1 49
5.5.2 E-RATIO 0F 10 - ....... 51
5.5.3 NODE-NODE DISTRIBUTION FOR VAR YING E-RA TIOS- - 52

5.7 COMMENT ON QUALITATIVE ANALYSIS ................................................................... 53
5.8 SUMMARY OF RESULTS ............................................................................................ 55

6. CONCLUSIONS 60
7. RECOMMENDATIONS FOR FUTURE WORK 61
APPENDIX A 63
APPENDIX B 76
APPENDIX C 78
APPENDIX D 79
APPENDIX E 82
BIBLIOGRAPHY - 89

 

vi

LIST OF TABLES

TABLE 3-1 THEORETICAL AND EXPERIMENTAL PHASE SEGREGATION

 

 

 

 

 

RESULTS 9
TABLE 4-1 FULLY CROSS-LINKED SYSTEM SPECIFICATIONS 16
TABLE 4-2 PARTIALLY CROSS-LINKED SYSTEM PARAMETERS .................... 17
TABLE 5-1 SUMMARY OF LENGTH OF RUNS 23
TABLE 5-2 SUMMARY OF PHASE SEGREGATION OBSERVED 55
TABLE D-l SUMMERY OF AVERAGE ENERGY PER SITE 79
TABLE D-2 SUMMERY OF SYSTEM PACKING FRACTIONS 80

 

TABLE D-3 CROSS-LINK LENGTH AND PACKING FRACTION
CORRELATION 81

 

vii

LIST OF FIGURES

FIGURE 4-1. REPRESENTATION OF VARYING TONE SIZES IN MODEL ..... 13
FIGURE 4-2. REPRESENTATION OF VARYIN G CROSS-LINK LENGTH ..... l4

 

 

 

 

FIGURE 4-3. FULLY EXTENDED FULLY CROSS-LINKED SYSTEM. ............... 15
FIGURE 5-1. GENERAL RDF, TONE 305, 17% HARD CONTENT, e-RATIO =

0.1 24
FIGURE 5-2. RDF, TONE 301, 50% HARD CONTENT, e-RATIO = 0.1 ................... 26
FIGURE 5-3. RDF, TONE 301, 65% HARD CONTENT, e—RATIO = 0.1 ................. 28
FIGURE 54. RDF, TONE 305, 17% HARD CONTENT, e-RATIO = 0.1 ................. 30
FIGURE 5-5. RDF, TONE 305, 50% HARD CONTENT, e-RATIO = 01 32
FIGURE 5-6. RDF, TONE 305, 63% HARD CONTENT, e-RATIO = 0.1.. ................ 34
FIGURE 5-7. RDF, TONE 310, 13% HARD CONTENT, e—RATIO = 0.1 .................. 36
FIGURE 5-8. RDF, TONE 310, 37% HARD CONTENT, e—RATIO = 0.1.. ................ 38
FIGURE 5-9. RDF, TONE 310, 50% HARD CONTENT, e—RATIO = 0.1 ................ 40
FIGURE 5-10. RDF, TONE 305, 50% HARD CONTENT, e—RATIO = 0.1,
INCOMPLETE CROSS-LINKING 43
FIGURE 5-11. RDF, TONE 310, 38% HARD CONTENT, e—RATIO = 0.1,
INCOMPLETE CROSS-LINKING 45
FIGURE 5-12. RDF, TONE 310, 50% HARD CONTENT, e-RATIO = 0.1,
INCOMPLETE CROSS-LIN KING 47
FIGURE 5-13. RDF, TONE 310, 50% HARD CONTENT, e-RATIO = 1 .................. 49
FIGURE 5-14. RDF, TONE 310, 50% HARD CONTENT, e—RATIO = 10 ........ 51

FIGURE 5-15. NODE-NODE RDF, TONE 310, 50% HARD SEGMENT WITH
VARYING e-RATIOS 52

 

FIGURE 5-16. SNAPSHOT: TONE 301, 50% HARD SEGMENT CONTENT ....... S3

viii

FIGURE 5-17. SNAPSHOT: TONE 301, 65% HARD SEGMENT CONTENT ....... 54
FIGURE 5-18. EFFECT OF CONSTANT TONE SIZE FOR TONE 310 ............. .. 56
FIGURE 5-19. EFFECT OF CONSTANT TONE SIZE FOR TONE 305 ................... 57
FIGURE 5-20. EFFECT OF CONSTANT HARD SEGMENT CONTENT — 50% 58

FIGURE 5-21. CORRELATION OF CROSS-LINK LENGTH WITH PACKING
FRACTION 59

 

FIGURE A-l. RDF, TONE 301, 50% HARD CONTENT, e-RATIO = 0.1,
DIFFERENT CHAINS 63

 

FIGURE A-2. RDF, TONE 301, 50% HARD CONTENT, e-RATIO = 0.1, SAME
CHAINS 63

 

FIGURE A-3. RDF, TONE 301, 65% HARD CONTENT, e-RATIO = 0.1,
DIFFERENT CHAINS 64

 

FIGURE A-4. RDF, TONE 301, 65% HARD CONTENT, e-RATIO = 0.1, SAME
CHAINS 64

 

FIGURE A-S. RDF, TONE 305, 17% HARD CONTENT, e-RATIO = 0.1,
DIFFERENT CHAINS 65

 

FIGURE A-6. RDF, TONE 305, 17% HARD CONTENT, e-RATIO = 0.1, SAME
CHAINS ' 65

 

FIGURE A-7. RDF, TONE 305, 50% HARD CONTENT, e-RATIO = 0.1,
DIFFERENT CHAINS 66

 

FIGURE A-8. RDF, TONE 305, 50% HARD CONTENT, e-RATIO = 0.1, SAME
CHAINS 66

 

FIGURE A-9. RDF, TONE 305, 63% HARD CONTENT, e-RATIO = 0.1,
DIFFERENT CHAINS 67

 

FIGURE A-10. RDF, TONE 305, 63% HARD CONTENT, e-RATIO = 0.1, SAME
CHAINS _ 67

 

FIGURE A-ll. RDF, TONE 310, 13% HARD CONTENT, e-RATIO = 0.1,
DIFFERENT CHAINS 68

 

FIGURE A-12. RDF, TONE 310, 13% HARD CONTENT, e-RATIO = 0.1, SAME
CHAINS 68

 

ix

FIGURE A-l3. RDF, TONE 310, 37% HARD CONTENT, e-RATIO =

DIFFERENT CHAINS

0.1,
69

 

FIGURE A-l4. RDF, TONE 310, 37% HARD CONTENT, e-RATIO =

CHAINS

0.1, SAME
69

 

FIGURE A-15. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

DIFFERENT CHAINS

0.1,
70

 

FIGURE A-l6. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

CHAINS

0.1, SAME
70

 

FIGURE A-l7. RDF, TONE 305, 50% HARD CONTENT, e-RATIO =

INCOMPLETE CROSS-LINK, DIFFERENT CHAINS

0.1,
71

 

FIGURE A-18. RDF, TONE 305, 50% HARD CONTENT, e-RATIO =

0.1,
71

 

INCOMPLETE CROSS-LINK, SAME CHAINS

FIGURE A-l9. RDF, TONE 310, 38% HARD CONTENT, e-RATIO =

INCOMPLETE CROSS-LINK, DIFFERENT CHAINS

0.1,
72

 

FIGURE A-20. RDF, TONE 310, 38% HARD CONTENT, e-RATIO =

INCOMPLETE CROSS-LINK, SAME CHAINS

0.1,
72

 

FIGURE A-Zl. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

INCOMPLETE CROSS-LINK, DIFFERENT CHAINS

0.1,
73

 

FIGURE A-22. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

INCOMPLETE CROSS-LINK, SAME CHAINS

0.1,
73

 

FIGURE A-23. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

DIFFERENT CHAINS

1.
74

 

FIGURE A-24. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

CHAINS

l, SAME
74

 

FIGURE A-25. RDF, TONE 310, 50% HARD CONTENT, e-RATIO = 10,

DIFFERENT CHAINS

75

 

FIGURE A-26. RDF, TONE 310, 50% HARD CONTENT, e-RATIO =

CHAINS

10, SAME
75

 

FIGURE E-l. SNAPSHOT: TONE 305, 17% HARD SEGMENT CONTENT ......... 82

FIGURE E-2. SNAPSHOT: TONE 305, 50% HARD SEGMENT CONTENT ... ...... 83

FIGURE E-3. SNAPSHOT: TONE 305, 63% HARD SEGMENT CONTENT ......... 84

FIGURE E-4. SNAPSHOT: TONE 310, 13% HARD SEGMENT CONTENT ....... 85
FIGURE E-5. SNAPSHOT: TONE 310, 37% HARD SEGMENT CONTENT ......... 86
FIGURE E-6. SNAPSHOT: TONE 310, 50% HARD SEGMENT CONTENT ......... 87

xi

Nrdf

Rin

Rout

AU
IJ(r)

VBox

LIST OF ABBREVIATIONS

Definition

radial distribution function

pair number at radius r

total number of pairs

Boltzmann constant

Monte Carlo

number of sites in system

number Of times radial distribution function called
acceptance probability

pressure

radius

inner radius Of Shell

out radius of Shell

temperature

change in internal energy
interaction energy Of a given radius

simulation box volume

GREEK

inverse product of kg and T

square-well potential depth

xii

maximum limit of energy well range

site diameter

xiii

1. INTRODUCTION

Cross-linked, or networked, polymers are found in a larger variety of materials, including
therrnoplastics, therrnosetting resins and elastomers. They find application in traditional
fields like the textile and automotive industries, as well as newer areas such as
biomedical applications. The automotive industry is the largest US. consumer of
elastomeric polyurethanes (Kroschwitz, 1997 ), which belong to the family of cross-linked
block co-polymers. Within this industry cross-linked polyurethanes are used in such
applications as adhesives in fiber reinforced composite materials (e.g., bumper beams)
and sealing materials (e. g., gasket applications). Studies (Dawson & Shortall, 1982; Kau,
Baer, & Huber, 1989; Schwarz, Critchfield, Tackett, & Tarin, 1979; Yang & Lee, 1987)
have Shown that good adhesion between the polymer matrix and fibers in the interphase
region is required for structural integrity of these composite and sealing materials.
Agrawal and Drzal (1995a, 1995b, 1996) investigated the adhesion’between polyurethane
and glass using both experimental methods and adapted Flory-based thermodynamic
calculations. Their study sought to identify correlations between the polyurethane
structure and adhesion to glass surfaces. They were able to identify phase segregation as
a major controlling factor Of adhesion. The polyurethanes in their study consisted of
varying diisocyanate content and three triols, the latter differentiated by their molecular
weight. Both increased diisocyanate content and trio] molecular weight where shown to
increase the degree of phase segregation, and this in turn favored adhesion.

Since phase segregation affects adhesion, it is of interest to investigate the phase

segregation behavior in the matrix phase. The aim of this work is to investigate, using

molecular modeling, the effect of three principle parameters on phase segregation in the
polymer matrix:

1. Hard Fraction Content

2. Cross-Link Length (or Tone molecular weight)

3. Ratio of interaction potentials
These parameters will be investigated by using isobaric-isothermal (NPT) Monte Carlo
(MC) simulations to equilibrate fully cross-linked polymer networks. A further feature of
this work is molecular simulations of dense, highly cross-linked systems. These results
can then be compared with experimental data. Previous Flory-base calculations have had
to resort to phase diagrams to investigate block copolymers. The work Of this will
investigate if molecular modeling provides an efficient method to predict phase

segregation behavior.

2. GENERAL BACKGROUND

2.1 PREVIOUS EXPERIMENTAL WORK

As part of their work on polyurethane-glass adhesion Agrawal and Drzal’s work (1995a,
1995b, 1996) prepared polymers using polycaprolactone-based triol, toluene
diisocyanate (TDI), and l-4-butane diol (BDO) as chain extender. The triols varied in
molecular weight and were identified by their trade names, in order of decreasing
molecular weight, as Tone 0310, Tone 0305 and Tone 0301. The triol name refers to the
Union Carbide trade name Tone®.

Agrawal and Drzal adopted Macosko's (1989) terminology of referring to TDI and BDO
as hard segment and the trio] component as soft segment. These descriptions refer to the
high and low glass transition temperatures of the hard and soft segments respectively.
Hard segment related transition temperatures obtained by Agrawal and Drzal (1995a)
through differential scanning calorimetry experiments revealed that higher hard
copolymer content increased the degree of phase segregation at a greater rate in the case
of higher molecular weight soft segments than lower molecular weight segments. Tensile
and iOSipescu shear experiments also showed that higher tone molecular weight increased
the degree of phase segregation. Near-infrared spectroscopy and Fourier transform
infrared spectroscopy indicated that hard segment content also increased phase
segregation development.

Additional experiments by Agrawal and Drzal (1995b), conducted with angular
dependent X-ray photoelectron spectroscopy (ADXPS), detected an inter-phase region

between the glass substrate and polyurethane matrix. ADXPS data Showed that both the

composition and phase segregation of the matrix influenced the composition of the inter-
phase region. To investigate the inter-phase region in terms of its surface free energy they
conducted calculations using a method described by Eberhardt (1966) combined with an
additive function proposed discussed by Van Krevelen (1990). These were compared
with calculations based on contact angle measurements.

Agrawal and Drzal compared these calculations with data obtained from block-shear
measurements and found a linear relationship between the polar surface free energy
component and adhesion values. This higher polar surface free energy component was
observed in phase-separated polyurethanes in which phase segregation had lead to
butanediol/butandediol-derived moeities preferentially segregating to the polyurethane
surface. This observation lead Agrawal and Drzal to conclude that "the mechanism Of
adhesion between the polyurethanes and the glass surface could be through the formation
of an interphase region in which hydrogen bonding between the butanediOl-rich
interphase region and the hydroxylated glass surface places a key role." (1996)
Experiments with pretreated glass furthermore lead them to assume that the effect of

ionic or covalent bonding at the polyurethane/glass interphase was negligible.

2.2 PREVIOUS NETWORK SIMULATION

Binder’s (1995) review of work in molecular polymer simulations describes three
methods of modeling cross-linked polymers.

The first system is a randomly linked (Duering, Kremer, & Grest, 1991; Grest & Kremer,
1990a, 1990b; Lay, Sommer, & Blumen, 1999; Plischke & Barsky, 1998; Schulz &

Sommer, 1992; Sommer, 1994; Sommer, Schulz, & Trautenberg, 1993) network. This

structure is created by randomly cross-linking an equilibrated melt and most closely
resembles radiation cross-linked or vulcanized polymers. A quantitative description in
this case is difficult though due to variations in parameters such as cross-link length,
chain length, and dangling chain ends.

The second type is an end-linked network. In this case an equilibrated monodisperse
melt is kinetically cross-linked at the ends, either by defining a certain percentage of
chain ends as multi-functional sites which can bond with more than two sites (Duering,

Kremer, & Grest, 1993, 1994; Grest, Kremer, & Duering, 1992, 1993; Kenkare, Hall, &

Khan, 1999; Kenkare, Smith, Hall, & Khan, 1998) or using cross linkers (H6121,

Trautenberg, & GOritz, 1997; Trautenberg, Sommer, & GOritz; 1995). The advantage of

this system is that all cross-links have the same length.

While the two types described consist of partially connected networks, the third type
represents the most idealized system, the fully connected network. This network type
provides the highest degree parameter control. Fully connected networks have both been
used to study simple networks (Escobedo & de Pablo, 1996, 1997a), as well as gel
swelling behavior (Escobedo & de Pablo, 1997a, 1997b) by including solvent molecules
in the latter case.

Lay et a1. (1999) and Escobedo and de Pablo (1996, 1997a, 1997b) conducted computer
simulations specifically involving fully connected Structures. Lay et a1. conducted
lattice-based bond-fluctuation MC moves on Stochastically cross-linked diblock
copolymers to investigate structural behavior based on A-B repulsion parameters. These
results were compared with those of a cross-linked homOpOlymer and networks with a

diamond topology, both with comparable cross-link density. This was accomplished by

randomly cutting cross-linking strands in the diamond topology case. They Observed that
the fully connected lattice required a higher number of dangling ends than the randomly
linked structure to Show similar results.

Escobedo and de Pablo conducted atherrnal and square-well NPT MC moves on a defect-
free continuum diamond-like structure. They investigated P-V behavior and attractive
interaction effects on system packing fraction. As part of their work they developed and
examined the capability of extended continuum configurational-bias (ECCB) (Escobedo
& de Pablo, 1995b), "cluster" (Escobedo & de Pablo, 1997b), "hole" (Escobedo & de
Pablo, 1995b) and ‘slab’ (Escobedo & de Pablo, 1995a) moves to handle cross-linked
networks with trifunctional and tetrafunctional nodes. The importance of their work lies
among other things in the methods developed to simulate highly cross-linked polymer
networks.

In general, research of fully connected networks has been limited to homopolymers,

whereas randomly linked systems have been used in copolymer research.

3. THEORTICAL CONSIDERATIONS

3.1 FLORY-HUGGINS THEORY

Agrawal and Drzal performed Flory-Huggins theory (Flory, 1953) based calculations and
compared these results with experimental Observations. This theory is the most
commonly used theoretical approach to polymer blends (Nath, McCoy, Curro, &
Saunders, 1995) and uses a dimensionless quantity, the Flory interaction parameter x, to
describe the interaction energy between the polymers.

_ VR
R*T

 

Z (5H ‘63) (1)

Eq. (1) shows that the Flory interaction parameter x is a function of a reference volume
VR, temperature T, the universal gas constant R and solubility parameters 85 and 5... for
soft and hard segments respectively. The solubility difference in polymers is the
important driving force for phase segregation (Macosko, 1989). This solubility
difference can be estimated using x (Van Krevelen, 1990).

Agrawal and Drzal Obtained x-values using both a group contribution method (Van
Krevelen, 1990) and listed values. Phase segregation occurs when x exceeds the critical

interaction parameter are. The critical interaction parameter is described by Eq. (2):

l l l
=— +
ZC 2[ NA r—NB]

N A and N3 represent the number Of polymer repeat units. Equations (1) and (2) predict

 

(2)

that phase segregation will occur as the solubility difference between solubility

parameters increases and as chain lengthy increases. The Flory-Huggins method of using
x and xc to predict phase segregation applies to polymer blends.

Polyurethanes being block copolymers necessitated Agrawal and Drzal to resort to phase
diagrams developed by Benoit and Hadziioannou (1988) for multiblock copolymers to
obtain XC- These calculations allowed Agrawal and Drzal to predict phase segregation

OCCUI'I'CIICC.

TABLE 3-1 THEORETICAL AND EXPERIMENTAL PHASE SEGREGATION RESULTS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I 30:

 

 

 

Wmer j Hard Segment Phase Segregation
(Wt%) ' FISry'TiIeBr; ' ' ' " Ex-peI-imeII-t- '
310 22 no no
310 37 yes no
310 47 yes maybe
310 51 yes yes
310 67 yes yes
305 36 no no
305 47 yes no
305 60 yes yes
305 68 yes yes
301 47 nO no
301 61 no no
68 no no

 

 

* Note. Flory Theory results are taken from Agrawal & Drzal, 1995b. Experimental

results are taken from Agrawal & Drzal 1996.

Table 3—1 shows the results Obtained by Agrawal & Drzal. They Observed good
qualitative agreement between Flory-Huggins calculations and experimental results,
though differences regarding the onset of phase segregation were observed.
Theoretical work using modified Flory-Huggins theories have been conducted by de
Gennes (1979) and Vargas and Barbosa (1998). Their work predicts that partial phase
segregation occurs at a transition temperature, total phase segregation being prevented

due to the cross-linking.

3.2 MOLECULAR SIMULATIONS

The development of rudimentary computers during Word War II first made molecular
simulations possible. Molecular modeling manipulates models of comparatively small
number of molecules to create different configurations, termed ensembles. The averages
of these ensembles are comparable to observed experimental properties, which
themselves are an average of behavior at the molecular level behavior.

These averages can be calculated either as time averages, as in the Molecular Dynamic
(MD) approach, or as ensemble averages, as in the Monte Carlo (MC) approach. Both
MD and MC represent the two main areas of molecular simulation. MD solves the
classical equations of motion of the system particles. The resulting molecular mechanical
properties are time averaged. MC simulations create a large number of system
configurations. The weighted contribution of each configuration to the ensemble average
is determined by its occurrence probability. MC allows more efficient equilibration than

MD, whereas MD allows the collection of kinetic data.

10

MC simulations ensembles are generated by moving an arbitrarily chosen particle to a
randomly determined location. The trial move is then accepted or rejected according to an
importance sampling scheme proposed by Metropolis, Rosenbluth, Rosenbluth, Teller
and Teller (1953). In this scheme Eq. (3) produces an acceptance probability p.

p = eXIX-MU) . (3)
In Eq. (3) AU is the change in energy between the two ensembles, B the inverse of the I
product of the Boltzmann constanth and absolute temperature T (i.e. l/[kBT]). A
generated random number is then compared with p. If the random number is smaller than
p the move is accepted. This method is sufficient for small molecules. The probability of
moving a whole molecule successfully using the method by Metropolis et al. (1953)
decreases as the molecule size increases. This lead to 'work by Rosenbluth and
Rosenbluth (1955) with polymer configurations, which provided the basis for the
configurational-bias (CB) MC method developed by Siepman and Frenkel (1992).
CBMC allows the efficient modeling of polymer chains on a lattice.
Further work by de Pablo, Laso and Suter (1992) was able to expand this concept to the
continuum environment. They termed their method the continuum configuration-bias
(CCB) MC. Both CBMC and CCBMC involve cutting of a randomly chosen polymer
chain end and regrowing the end site by site. In dense systems only relatively short
molecule segments can be successfully regrown using CBMC and CCBMC, thus making
simulation of chain middle segments difficult and unlikely. Both CB methods can not be
applied to ring polymers.
The Crank-shaft move allows the movement of a Single middle chain site and has been

used successfully by various authors (Kumar, Vacatello, & Yoon, 1988; Li & Chiew,

11

1994). Escobedo and de Pablo (1995b) devised a method by which an arbitrary section
of a chain molecule, including intra—chain and trifunctional segments, can be cut and
regrown with chain closure assured. They termed this method extended continuum
configurational bias (ECCB) MC. This method is used in this work to equilibrate the
system thermally. The ‘slab’ method developed by Escobedo and de Pablo (1995a) is
used to equilibrate the system mechanically. In this case the volume move is performed
by arbitrarily selecting a Slab of the Simulation box, changing its width and repositioning

the sites within the slab to satisfy connectivity restraints.

12

4. METHODS

4.1 EXPERIMENTAL SETUP

Simulations were conducted on 2 types of systems. Both systems consisted of strings of
sphere, each sphere referred to as a “site” and representing a 90 g/mol molecular weight
unit. These sites were created using the “united atom” model, according to which intra-
molecular structures and interactions are combined into an overall structure represented
by a sphere of a given diameter. The experimental basis for this research was the work
conducted by Agrawal & Drzal (1995a, 1995b, 1996), hence the model is based on the
polyurethane copolymers used in their work. For this work butane diol [HO-CH2-CH2—
CHz-CHz-OH] was represented by one sphere and toluene diisocyanate

[CH3(C5H3)(NCO)2] was represented by two spheres.

 

 

Tone 301 Tone 305 Tone 310

 

 

 

Figure 4-1. Representation of varying Tone sizes in Model

13

The polycaprolactone-based tn'Ols Of varying molecular weight were modeled according
to Figure 4-1. The sites with three neighbors are termed nodes. For the purpose of this
work the Tone 0301 , Tone 0305 and Tone 0310 polymers are referred to as Tone 301,
Tone 305 and Tone 310 respectively and “Tone” collectively. The Strings of soft sites
(hashed spheres in Figure 4-1) ‘going out’ from a node are termed node arms.

Of the two types of systems modeled, the first was fully connected (i.e. no dangling or
lose ends). This system type was built by placing nodes on a diamond lattice and cross-
linking the nodes with freely jointed hard-sphere sites using ECCB type moves. The
lattice structure was modified so that each node was trifunctional, as opposed to the usual
diamond structure with four neighbors. The Sites had a fixed bond length, but no
restriction on bond angels. All cross-links were the same length and contained the same
fraction of hard and soft sites. The cross-link length depended both on Tone molecular
weight and hard segment content. Figure 4-2 shows how cross-link length varies with

varying hard segment content.

 

 

 

 

 

 

 

 

Low HS% High 118%

 

 

 

 

 

Figure 4-2. Representation of varying cross-link length.

14

In Figure 4-2 the nodes are represented by white spheres, hard sites by black spheres and
soft node arm sites by hashed spheres. As the hard segment content changes, the number
of hashed spheres remains constant and the count of black spheres increases. This
corresponds to the experimental situation of constant tone size with varying hard segment
content. Varying the Tone size on the other hand in the model is equivalent to keeping
the number of black spheres constant and varying the number of shaded spheres in each
node arm. The building process insured that both this type of system was fully connected
(i.e. not dangling ends) and that all nodes were trifunctional. Figure 4-3 shows a View of

a fully extended fully cross-linked system from one side of the simulation box.

 

(‘1‘ (if. “(f “IO.

 

 

 

 

Figure 4-3. Fully extended fully cross-linked system.

The clear spheres correspond to soft sites, the black spheres to hard sites in the cross-

links. The diamond lattice structure is apparent in Figure 4-3. Once the Structure was

15

built, all sites (including nodes) were free to move, as long as bonding constraints (i.e.
constant bond length) were respected. The system was not restricted by a fixed lattice
structure.

TABLE 4-1 FULLY CROSS-LINKED SYSTEM SPECIFICATIONS
Tone 301 Tone 305 Tone 310

 

 

17 % Hard Segment Cont. 13 % Hard Segment Cont.

1836 Sites in System 2484 Sites in System

 

37 % Hard Segment Cont.

1024 Sites in System

 

50 % Hard Segment Cont. 50 % Hard Segment Cont. 50 % Hard Segment Cont.

I 1836 Sites in System 928 Sites in System 1312 Sites in System

 

65 % Hard Segment Cont. 63% Hard Segment Cont.

2484 Sites in System 1216 Sites in System

 

 

 

 

Table 4—1 shows the details of the fully cross—linked systems modeled in this work.
The system parameters where chosen to allow comparison with samples prepared in the
work by Agrawal & Drzal (1995a, 1995b, 1996). In all cases in Table 4-1 the ratio of
soft to hard energy well depth (ass/Shh) was 0.1. The choice of this system also allowed
the effect of increasing hard segment content and tone Size (cross-link length) to be
investigated. Further system statistics can be found in Appendix B.

Special cases of fully cross-linked systems were used to investigate the effect of
varying the energy-well depth ratio. The Tone 310, 50% hard segment content system
was modified, SO that the esJehh-ratio was 1 and 10. These two systems were then

compared with the original case shown in Table 4-1.

16

 

The second type of system modeled was partially connected. This system type was
created by placing nodes randomly in a Simulation box, placing soft sites to create node
arms and randomly linking the arms with hard sites. The building process insured that at
least one hard site was contained in each cross-link. This type of system contained
dangling ends and a distribution of cross-linked and dangling end lengths.

TABLE 4-2 PARTIALLY CROSS-LINKED SYSTEM PARAMETERS
_‘ I

 

 

 

 

 

 

 

 

   

; Tone Type Hard Segment Average Average Total number
Content (%) number of sites number of sites sites in system
in cross-links in lose ends
0305 50 5.5 7.2 989
0310 38 7.5 6.6 1066
0310 50 7.4 9.8 1071

 

 

 

 

 

Table 4-2 shows statistics for the partially connected systems, also termed modified
systems. The system parameters where chosen to allow a comparison with similar cases
among the fully cross-linked systems (see Table 4-1). Further statistics on the modified
systems can be found in Appendix B.

In all systems Isothermal-isobaric (NPT) Monte Carlo (MC) simulations were conducted
using periodic boundary conditions to Simulate bulk behavior. As afore mentioned,
‘Slab’ and ECCB moves developed by Escobedo and de Pablo (1995a, 1995b) were used
to equilibrate the system mechanically and thermally.

The NPT simulations were run at 373 K and 1 atm, corresponding to polyurethane curing
conditions in experiments conducted by Agrawal & Drzal (1995a). Detailed calculations
Of reduced parameters are found in Appendix C.

Modeling was conducted with the square-well energy function as defined in Eq. (4):

17

    

=4»: OSISAO (4)

In Eq. 5(4), r is the distance between site centers, 0' the site diameter, 8 the energy well
depth and (A-1) the width of the well. The hard-hard interaction em. = 1.0 corresponds to
the 8 calculated in Appendix C. The soft-soft interaction 8,, has a value 1/10th of that of
8”,, Le. 8,, = 0.1. The hard-soft interaction eh, is defined by the Lorentz-Berthehold rule,
which states that ch: = ehhess.

All systems were initially equilibrated with a ratio of ECCB to volume moves of
1:1. AS packing fraction and system energy curves Showed a significant decrease in
slope the ratio of ECCB moves to volume moves was switched to 2,500:1 or 10,000:1 for

more efficient sampling.

4.2 RADIAL DISTRIBUTION FUNCTIONS

Phase segregation of hard and soft segments was investigated with radial distribution
functions (RDF). The RDF is a subset of pair correlation functions in which molecular
interactions are described by spherical potentials. The RDF is obtained by Eq. (5),

gm = 45’wa (5)

HSUM 375(16):: — R; )Nrdf

 

where H(r) is the number of pairs in a shell with a central radius r, VBox the volume of
the Simulation box, HS'UM the total number of pairs, and Nrdf the number of times the RDF

is measured during the run. The Shell for which H(r) is counted is described by R0... and

R,“ , the outer and inner radius respectively. These values are defined as r +/- 0.056. In

18

the case of polymers the RDF does not include nearest bonded neighbors in its
calculations. These neared bonded neighbors are always at the same distance from a
given site, and thus their inclusion in the RDF would both not provide any new
information and possibly obstruct noteworthy results.

The RDF is a measure of the correlation between molecule pairs and the resulting
structure of the fluid. In a physical sense, the RDF represents the likelihood relative to
the bulk number density of finding a second molecule at a given distance r from the
center of a molecule. Values of the radial distribution function g(r)=l indicate no
correlation between the molecules. This indicates that the bulk density and local number
density are identical and hence the presence of one molecule does not affect the position
of another molecule. If g(r)¢1, then the first molecule affects the likelihood of finding
the second molecule in the area. Specifically if g(r)>l, then the probability is enhanced
with regards to the average density that a site lies at that distance. Conversely if g(r)<l

there is a decreased probability.

19

4.2.1 DETERMINATION OF PHASE SEGREGATION

Thermodynamics teaches that the mixing process is a function of both energy and
entropy. These factors are considered in Monte Carlo simulations where the number of
possible ensembles affects the configurations sampled and the energetic interactions
contribute to the acceptance criteria. From an energetic viewpoint phase segregation
occurs when interactions between like-copolymers become larger then between unlike
copolymers and an ‘un-mixing’ or phase segregation occurs. The RDF graph reflects this
phenomenon of phase segregation when both the soft-soft and hard—hard curves have

greater magnitude than the hard-soft curve.

4.2.2 CALCULATED RDF CURVES

Systems are evaluated using three types of radial distribution functions: soft-soft, hard-
hard, and hard-soft. In the soft-soft case the variable H(r) of Eq. (5) only counts pairs of
soft sites. The other two RDFS only count pairs consisting of the site types indicated by

the curve name.

Four additional RDF types, listed in Appendix A, are used in certain cases to clarify the
relative contribution of sites on the same node or segment vs. sites on different nodes or
segments. In this case the variable H(r) of Eq. (5) only counts pairs of:

1) soft sites on the same node (i.e. nodal Site and node arms)

2) hard sites on the same segment

3) soft sites on different nodes
4) hard sites on different segments

20

The HSUM of points 1 and 3 is the total number of all soft pairs, NOT the number of soft
pairs on different nodes or on the same node. The same applies to the hard site

measurements.

21

5. RESULTS & DISCUSSION

5.1 GENERAL OBSERVATIONS

5.1.1 LENGTH OF RUNS

Table 5-1 on the following page shows a summary of the systems simulated and
corresponding number of ECCB and volume moves. In all cases the partially linked
systems require a smaller number of volume moves compared to the fully cross-linked
systems to calculate an equilibrated system. In two cases this also applies to the ECCB
moves. The exception is the Tone 305 - 50% hard content system. In this case more
ECCB moves were necessary. One must consider though that volume moves are
computationally more expensive. It follows that the partially linked system presents a

model that allows a more efficient calculation of the equilibrated structure.

22

TABLE 5-1 SUMMARY OF LENGTH OF RUNS

Number of
Volume Moves

Hard Segment Number of Number of

Content (%)

ECCB Moves Sites in
System

 

 

 

 

 

Fully Cross—
Linked Systems

 

301

3.08E+05

l .76E+08

1836

 

301

2.03E+05

4.52E+07

2484

 

305

2.63E-I-05

l .74E+07

1836

 

305

6.01E+05

7.30E+07

928

 

305

3.36E+05

1.11E-I-08

1216

 

310

1 .89E+05

1.53E+08

2848

 

310

5.07E+05

2.26E+08

1024

 

310

 

 

4.18E+05

 

2.92E+08

 

1312

 

Partially Cross-
Linked Systems

 

305

3.13E+05

1 .65E+08

 

310

3.19E+05

9.53E+07

 

310

 

 

2.61E+05

23

 

1.83E+08

 

 

5.1.2 RDF BEHAVIOR DUE To SQUARE-WELL MODEL

 

 

 

 

 

 

 

 

 

 

 

 

 

RDF, Tone 301, 65% hard content, em“, = 0.1
3 I
2.5 : M
2 f I
1 I - __ _
0.5 f
O : r I l 1 I 1 I U I I 1
0 2 4 6 8
do

 

 

 

Figure 5-1. General RDF, Tone 305, 17% hard content, e-ratio = 0.1

Figure 5-4 Shows a general RDF of a given system (e. g. Tone 305, 17% hard content) and
serves as illustration for the following discussion. The general curve correlates all sites,
without respect for hard or soft sites. All RDF curves Show a g(r)=0 for O-values below
1.0 0'. Most of the RDF curves Show a peak at 1.0 0' and a local maximum followed by a
local minimum at 1.5 O’. This behavior is a result of the energy model chosen for the
simulations: the square-well potential model. At values smaller than 1.0 o the hard
spheres cannot overlap and have an infinite repulsion. The energy well with a depth of 8
between 1.0 o and 1.5 0 leads to energetically favorable interactions of site pairs within

this radius and normally larger g(r) values. Sites within this well form a cluster around

24

the site, creating what is commonly referred to as a first coordination shell. This cluster
Of Sites physically prevent other Sites from occupying space near the coordination shell,
leading to the local minimum around 1.5 O'. The curve falls off after the 1.0 0' peak, but
peaks around 1.5 0, right before falling to the local minimum. This is explained by the
Sites in the region preferentially falling’ into the energy well if possible. In the case of
fluids these local structures, and associated short-range correlations, determine the
uniqueness of the thermodynamic properties.

Whereas the RDF behavior to the left of 1.5 o is explained by the potential model, the
behavior to the right of 1.5 O is a result of the composition and structure of each
individual run. The discussion of each of these runs in detail follows. In the following
discussion soft sites are identified as being associated with nodes. Hard sites are
identified as being associated with segments, the segment referring to the central section
Of the cross-link that they occupy.

In the following discussion, the term “short-range” will refer to distances less than 1.5 O',

and “long-range” to distances larger than 1.5 0’.

25

5.2 Effect of Hard Segment Content

5.2.1 TONE 301 , 50% HARD SEGMENT cONTENT

 

RDF, Tone 301, 50% hard content, 2...... = 0.1

 

 

 

 

 

 

 

 

5
I — soft-soil
4 ‘ -— hard-soft
3 —hard-hard
3 i

 

 

 

 

 

 

 

 

 

Figure 5-2. RDF, Tone 301, 50% hard content, e-ratio = 0.1

In this system the soft Sites on a given node are no further then 2 o from each other,
hence the Sharp drop after 2 O' signifies the transition from intra-nodal to inter-nodal
correlation. The weak correlation at long range corresponds to the distance of soft sites
on different node arms. This follows from the hard segment length of three sites. The
weak short-range behavior stands in contrast to Figure A-2, which indicates that the

majority of sites on a given node arm lie at short-range distance. This is explained by the

26

fact that for Tone 301 the nodes lie at most 2 o from each other. The correlation in
Figure A-2 is a result of the bonded nature Of the system, not energetic interactions.

The greater correlation of the hard-hard curve, compared to the soft-soft curve, is due to
the stronger hard-hard interaction. The absence Of a sudden drop after 2 O' in the hard-
hard curve represents correlation beyond the 2 0’ range between different hard segments,
since hard sites on the same segment are at most 2 O' apart. Figures A-1 and

A-2 indicate that a substantial amount Of interaction in the square-well range is due to
sites from different segments. As the hard segments align with each other they position
the nodes on average 3 0’ to 4 O' from each other. This fact is indicated by the soft-soft
curve behavior within this range.

The hard-soft curve in Figure 5-2 does not indicate phase segregation. Though the hard
sites appear to interact strongly with each other, the system as a whole does not phase

segregate.

27

5.2.2 TONE 301, 65% HARD SEGMENT CONTENT

 

RDF, Tone 301, 65% hard content, 3...... = 0.1

 

 

 

 

5 1
i —soft-soft
4 i -— hard-soft
‘ —-hard-hard

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5-3. RDF, Tone 301, 65% hard content, e—ratio = 0.1

The soft-soft peak at 1 o is Shown in Figure A-3 and A4 to result mostly from node arms
on the same node moving within this range to each other. The peak of the same curve at
2 O’ is the result Of. sites on the same node being no further than 2 o from each other. The
3 o to 5 0' range curve behavior Shows the physical limitation imposed on nodal
movement by a hard segment of 5 Sites.

Figures A-3 and A-4 Show the hard-hard short-range behavior to be more strongly
influenced by hard sites on the same hard segment when compared with the Tone 301,

50% hard segment case. This is due to the greater number of hard Sites, three vs. five, in

28

each segment. A major contribution to the curve at this short range still comes from hard
sites on different segments, so that here too the hard segments align with each other.
Figure A-4 also shows that there is no hard site correlation between sites on the same
segment beyond 3 O'. The hard segment appears to have contracted substantially. The
longer-range correlation in this case is again due to different segments interacting with
each other.

The relation of the curves to each other here also shows no phase segregation.

5.2.3 TONE 301 CONCLUSION

In both cases the stronger interaction between the hard segments leads to the hard
segments folding on themselves and their alignment with each other. This behavior
determines the placement of the nodes in relationship to each other. As the cross-link
increases in number of sites, the node arms on the same node show more freedom to

move together. In both cases no phase segregation was observed.

29

5.2.4 TONE 305, 17 % HARD SEGMENT CONTENT

 

RDF, Tone 305, 17% hard content, em... = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

5 -I
—Soft-soit
4 I -—hard-soft
j —hard-hard
32
‘6 5
2 4
0 1 2 3 4 5 6 7 8
r/o

 

 

 

Figure 5-4. RDF, Tone 305, 17% hard content, e—ratio = 0.1

Figure A-6 shows that soft sites on both the same node and different nodes contribute
equally to the 1.5 O' soft-soft peak in Figure 5—4. The 2 0 peak though is dominated by
contributions from sites on different nodes. This shows that the node arms have not
folded on themselves to an appreciable extent. Had the arms folded on themselves the
soft curve in Figure A-6 would fall to 0 sooner than at the 30 observed.

The hard segment in this case consists of one hard site, so that the 1.5 6 peak is due

exclusively to hard Sites from different cross-links. This shows that where hard sites

30

came within range of the energy-well during the simulation, the interaction energy is
sufficiently strong to keep the hard sites at this range. This interaction is sufficiently
strong to overcome the large number of soft sites that could prevent interaction through
stearic hindrance.

This case is not phase segregated.

31

5.2.5 TONE 305, 50% HARD SEGMENT CONTENT

 

RDF, Tone 305, 50% hard content, em, = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5:
4* —soft-soft
I -o—hard-soft
3 V ——hard-hard
‘6 :
24 “
1- ...
oi.— ....... 5.55.......-.LL....L..-.,. .....
0 1 2 3 4 5 6 7 8

 

 

 

Figure 5-5. RDF, Tone 305, 50% hard content, e—ratio = 0.1

Figure 5-5, in correlation with Figures A-7 and A-8, shows that the only appreciable soft-
soft correlation is between soft sites on different nodes.

The hard segment in this case consists of five sites. The absence of correlation around 4
O' and strong correlation at shorter ranges indicate that the hard segments have folded on
themselves. Figures A-7 and A-8 confirm this. The appendix figures do show that the
hard sites from different segments contribute the most to the overall hard RDF curve. The

g(r) < 1 values around 5 o are due solely to a lack of interactions between sites from

32

different segments. This is a result of hard segments folding on themselves and aligning
themselves, thus interacting overall at short range.

Phase segregation is shown to have occurred.

33

5.2.6 TONE 305, 63% HARD SEGMENT CONTENT

 

RDF, Tone 305.63% hard content, 8...... = 0.1

 

 

 

 

 

 

 

 

 

 

 

5-
j —soft-sofl
4. -—hard-soft
I —hard-hard
3
S
m '1
2 A
i
1-
0“ T T IIMTI—ITIIITIfTTfi‘TTTIIIIIIIII]

 

  

 

0 1 2 3 4 5 6 7 8

[/6

 

 

 

Figure 5-6. RDF, Tone 305, 63% hard content, e—ratio = 0.1

The soft-soft short-range behavior is shown by Figures A-9 and A-10 to be contributed to
in approximately equal proportions by sites on the same and different nodes. The sites on
different nodes contribute more strongly to the soft-soft correlation in this case than in the
Tone 305, 50% hard segment content case. This indicates that the nodes have a greater
degree of freedom in this case.

The hard-hard curve only showing correlation up to a range of 4 0 indicates a segment

that has folded on itself. This follows from a hard segment length of 8 sites. This is

34

substantiated by figure A-10. Figure A-9 shows that the segments have aligned with each
other. The below-1.0 values at longer range are due to the hard segments aligning and
interacting at Short range.

Phase segregation is indicated in this case.

5.2.7 TONE 305 CONCLUSION

The low hard segment content case shows the strength of the hard site interaction
compared to the soft site interaction. In cases of larger hard segments the stronger
interaction between the hard segments leads to contraction of the hard segments and their
alignment with each other. As the cross-link increases the Tone 305 node arms on the
same node also show, compared to the Tone 301 node arms, a greater degree of freedom.
In the low hard segment content case the fraction of hard sites is so low as to not allow
phase segregation. The other two Tone 305 cases Show increased phase segregation as

hard segment content increases.

35

5.2.8 TONE 310, 13% HARD SEGMENT CONTENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RDF, Tone 310, 13% hard content, 2...... = 0.1
5 -I
i —soft-sofl
4 . -—hard-soft
; —hard-hard
3 I
S 3
2 .,
1
1 : ..—
O f fl I I I 1 fr 1 M I I r f T r T n
o 1 2 3 4 5 6 7
rlo

 

 

 

Figure 5-7. RDF, Tone 310, 13% hard content, e—ratio = 0.1

The soft-soft curve behavior is comparable to the Tone 305, 17% case in Figure 5-3. The
soft curves in Figures A-6 and A-12 are also comparable. As mentioned with the Tone
305, 17% hard segment case the node arms here also do not appear to fold on themselves
to an appreciable extent.

This system is another case of the cross-link containing one hard site out of a total of
seven. Here, as in the Tone 305 -— l7 % hard segment case, hard sites that manage to

come within the range of the square-well potential during the simulation are retained at

36

this range through the strong hard-hard interaction potential. The interaction energy is
sufficient to overcome the relatively number of soft-soft interactions.

The hard-soft curve indicates that not phase segregation has occurred. The hard-soft and
soft-soft curves are identical after 1.5 0 due to the fact that any interaction beyond the 1.5
0' range is soft-soft and hence this contribution dominates the hard-soft curve. The
Figure 5-4 hard-soft curve shows similar behavior. The effect is more pronounced in the
Tone 310 case due to higher ratio of soft sites in the cross-link. The Tone 305 case has 4

soft Sites out of a total of 5, the Tone 310 case has 6 soft sites out of a total of 7.

37

5.2.9 TONE 310, 37% HARD SEGMENT CONTENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RDF, Tone 310, 37% hard content, em“, = 0.1
5
j — soft-soft
4 ‘ + hard-soft
\ — hard-hard
,
3 d
a E
2 ..
1 i
O ‘ j j T fl I D T I
0 2 4 6 8
rlo
Figure 5-8. RDF, Tone 310, 37% hard content, e—ratio = 0.1

The soft sites do not interact to an appreciable extent. The small peak at 2 0' appears to be

a result of the soft sites that neighbor the node being within this range due to connectivity
constraints.

The hard-hard curve in Figure A-14 shows that most segments have folded on

themselves. This follows from the hard segment in this system consisting of 4 sites.

Some segments still have sites interacting at ranges larger then 2 0 compared with the

38

 

largest range possible of 3 o at full extension. Comparing hard curves in Figures A-13
and A-14 with each other shows that hard sites on different segments mostly contribute to
the peaks in Figure 5-7. This shows that in addition to folding on themselves the hard
segment of different cross-links align. The majority of the hard sites interact at short
range leading to an absence of correlation at longer range. The curve drop below 1.0 at
larger o-values supports this observation.

The hard-soft curve indicates that phase segregation may have occurred in some regions

and not in other areas.

39

5.2.10 TONE 310, 50% HARD SEGMENT CONTENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RDF, Tone 310, 50% hard content, em“, = 0.1
5
—-soft-soft

4 —--hard-soft

1 \ —hard-hard
3 V

a 'l

2 * /\
1 i
0 L- S 5 . . . ......... . fl .

o 1 2 3 4 5 6 7 8

[/6

 

 

 

Figure 5-9. RDF, Tone 310, 50% hard content, e-ratio = 0.1

The soft-soft behavior in Figure 5-9 is comparable to the soft curve results in the two
previously discussed Tone 310 systems. Here also the soft sites do not interact or
correlate to a noticeable extent.

Figure A-16 shows that the fraction of hard-hard interactions contributed to by sites on
the same segments are larger than in the Tone 310, 37% hard segment case. This
indicates that the segments have folded on themselves to a greater extent than in lower

hard segment systems. Figure A-15 shows that the hard-hard interactions of hard sites on

40

different segments still dominate the overall correlation, thus indicating that segments of
different cross-links align. The drop of the Figure 5-8 hard curve below 1.0 at longer
range is once again a result of the majority of hard sites on different segments interacting
at short range. This is substantiated by the hard curve behavior in Figure A—lS. .

The hard-soft curve shows that phase segregation has occurred.

5.2.11 TONE 310 CONCLUSION

Here also the low hard segment content case shows the capability of the strong hard site
interaction potential to overcome the steric hindrance of a larger number of soft sites. In
spite of this strong interaction the system’s hard fraction is to small to lead to phase
segregation.

In the case of higher hard segment content hard segments fold on themselves and align
with other hard segments, thus creating regions with relatively high concentrations of
hard sites. In all three Tone 310 cases the soft curve RDFs alone did not indicate that
increasing cross-link length increased the node’s degree of freedom. This is in contrast to
Tone 301 and Tone 305, where an increase in the degree of freedom was observed.

Tone 310 showed a marked dependence of phase segregation on hard segment content.

The degree of phase segregation increased as hard site fraction increased.

41

5.3 EFFECT OF TONE SIZE

5.3.1 Low HARO FRACTION

The Tone 305 — 17 % hard segment content and Tone 310 — 13% hard segment content
systems show very similar behavior. Phase segregation did not occur in either case. The
only noticeable effect Of the tone size is the extent tO which the soft-soft sites dominate
the overall soft-hard correlation. At larger tone molecular weight the soft and hard-soft

curve overlap sooner (see Figure 5-4 and Figure 5-7).

5.3.2 50% HARD FRACTION

In all 50% hard segment content cases the soft-soft behavior remains similar. The hard-
hard correlation increases as Tone size increases. The degree Of phase segregation also
increases as Tone size increases. This indicates that the hard-hard interaction plays an
important roleiin bringing about phase segregation. The added degree Of freedom Of

longer cross-links also appears to be a contributing factor.

5.3.3 HIGH HARD FRACTION

At high hard fraction increasing Tone molecular weight contributes to a greater degree Of

phase segregation.

42

5.4 EFFECT OF CROSS-LINKING

5.4.1 TONE 305, 50% HARD FRACTION, INCOMPLETE CROSS-LINKING

 

RDF, Tone 305, 50% hard content, em“, = 0.1,
incomplete cross-link

 

 

 

 

 

 

 

 

5 . —soft-soft
-— hard-soft
4 q — hard-hard

 

 

 

 

 

 

 

 

 

 

 

Figure 5-10. RDF, Tone 305, 50% hard content, e—ratio = 0.1,
incomplete cross-linking

The two Tone 305 - 50% hard fraction systems, fully cross-linked and partially cross-
linked, are comparable. The fully linked system contains 928 sites, including 448 soft
sites and 480 hard sites. The partially linked system has 469 soft sites and 520 hard sites,

totaling 989 sites.

43

The soft curves in both cases are similar, though the partially linked system shows
stronger correlation at 2 0'. Comparisons Of the soft-soft curves indicate a greater degree

Of freedom in the partially linked system.

The hard-soft curve in Figure 5-10 indicates that phase segregation occurs tO a greater
extent in the partially cross-linked system than in the fully cross-linked system (see
Figure 5-5).

Figures A-7, A-8, A-17 and A-18 show comparable soft site behavior. The hard sites
On the same chain show weaker interaction, those on different chains stronger interaction
in the partially cross-linked system than in the fully linked system. The chains in the
fully linked system appear tO fold upon themselves to a greater degree compared with the
partially linked system. On the other hand the partially linked system shows a greater
degree Of alignment between different chains than does the fully connected system.
These Observations indicate a greater range Of possible configurations in the partially
cross-linked system. Two factors contribute to this situation. First, the smaller degree Of
cross-linking presents fewer constraints due tO bonding for the remaining sites.

Secondly, the dangling ends have a greater range Of space to be placed in during a move.

Overall this leads to a more efficient relaxation Of the system (see Table 5—1).

5.4.2 TONE 310, 38% HARD FRACTION, INCOMPLETE CROSS-LINKING

 

RDF, Tone 310, 38% hard content, smug = 0.1
incomplete cross-link

 

 

 

 

 

 

8 .
7 3 -—sofl-soft ,
- -—hard-soft
5 —hard-hard “'—

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5-11. RDF, Tone 310, 38% hard content, e—ratio = 0.1,
incomplete cross-linking

The fully cross-linked Tone 310 - 37% hard content and partially linked Tone 310 - 38%
hard content system also show similar system size and hard site content. The fully linked
system has 640 soft sites and 384 hard sites, totaling 1024 sites. The partially linked, or
modified, system contains 1066 sites, with 660 soft and 406 hard sites.

The soft-soft and hard-hard curve in Figure 5-11 show greater correlation than the soft-
soft and hard-hard curve in Figure 5-8. The modified system’s hard curve also shows

correlation over a greater range than does the unmodified system.

45

The hard-soft curve’s position relative to the other curves in the fully linked system
does not show a conclusive proof for phase segregation. In the partially linked system
' the hard-soft curve definitely indicates phase segregation.

Figures A-l3, A—14, A-19 and A-20 show comparable soft site behavior. The hard site
behavior is comparable to the Tone 305 system discussed above. The partially linked
system in this case also shows a lesser degree of hard segments folding on themselves
and a greater degree of alignment between different chains.

Here also the partially cross-linked system has a greater sampling space. This allows this
system to accomplish equilibrium more efficiently (see Table 5-1). In all three systems
another factor may also have some effect. In the fully cross-linked system the nearest
bonded neighbors, which are not counted in the radial distribution function, prevent other
sites from coming near a given site. In the partially linked system the dangling ends
allow sites from other chains to move into interactions at locations that are occupied by
neighbors in the fully linked system. The lack of this form of steric hindrance may

contribute to the greater hard curve values observed.

46

5.4.3 TONE 310, 50% HARO FRACTION, INCOMPLETE CROSS-LINKING

 

RDF, Tone 310, 50% hard content, em“, = 0.1
Incomplete cross-link

 

 

 

 

 

 

 

5 4
i —soft-soft
4 —-— hard-soft
} — hard-ha rd
3 i

 

 

 

 

 

 

 

 

 

Figure 5-12. RDF, Tone 310, 50% hard content, E—ratio = 0.1,
incomplete cross-linking

The partially lined Tone 310 - 50% hard content system consists of a total of 1071 site,
including 510 soft sites and 561 hard sites. The fully linked system with the same hard
site content has 640 soft site and 672 hard sites, totaling 1312 sites. The size difference
between fully cross-linked and modified system is the greatest in this case.

The soft curves in Figures 5-12 and Figure 5-9 show similar soft site behavior. The
hard curve in Figure 5-12 on the other hand indicates a greater degree of correlation over

a greater range than in Figure 5- 9.

47

As in the previous two case descriptions the hard-soft curve indicates a greater extent of
phase segregation in the partially linked system than in the fully linked system.
Figure A-15 and A-21 also indicate that the chains of the partially linked system fold on
themselves to a lesser degree than in the fully linked system. The modified system does
show a greater degree of alignment between the different Chains.

As discussed before the partially cross-linked system exhibits a larger space in which to
place displaced sites. This is also indicated by a more efficient calculation of the

equilibrated system (see Table 5-1).

48

5.5 EFFECT OF e-RAnos

5.5.1 E-RATIO OF 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RDF, Tone 310, 50% hard content, em“, = 1.0
5
-—soft-solt
4 ‘ -—hard-soft _
I — hard-hard
3 .
a I
.
2 -
1
O “ fir I Tfi r fiT T T I T T T—T'T fl T
o 1 2 3 4 5 6 7 a
[lo

 

 

 

Figure 5-13. RDF, Tone 310, 50% hard content, e—ratio = l

The RDF in Figure 5-13 shows a system in which the energy well depth of all sites is
equal. In essence this system represents a homOpOlymer. The hard-hard and soft-soft
sites are identified in name only and the location of the sites in the cross-link: the soft
sites in the node and node arms, the hard sites in the connecting segments between the
nodes. No phase segregation is apparent. The hard segments appear to have a slightly

higher structural correlation. This can be ascribed to a slightly greater degree of freedom

49

of the cross-link center segments than the nodes. The segments only have two points of
attachment, the nodes three points. This somewhat greater degree of freedom is
expressed in the higher correlation. This system indicated that the phenomenon of phase
segregation is largely due to the difference in energetic interactions between hard and soft

sites. Structural effects are negligible.

50

5.5.2 e-RATIO OF 10

 

RDF, Tone 310, 50% hard content, em“, = 10

 

 

 

 

 

 

 

5 .I
3 —soft-soft
4 ‘ -- hard-soft ,
j — hard-hard
3 I

 

 

 

 

 

 

 

 

 

 

 

Figure 5-14. RDF, Tone 310, 50% hard content, e—ratio = 10
In this system the node sites have an energy well-depth ten times that of the segment

sites. The soft sites have the higher attractive energy and this is reflected in the larger
correlation in the radial distribution function. The segment sites show a lower degree of
correlation reflecting the lower energy potential.

Figure 5-14 clearly indicates that phase segregation has occurred. This is partly expected
due to the difference in energetic interaction between the different sites and partly due to

the cross-link length.

51

5.5.3 NODE-NODE DISTRIBUTION FOR VARYING 8-RATIOS

Figures 5-9, 5-13 and 5-14 compare a system with the same cross-link length, but
differing energetic interactions. Figure 5-9 and 5-14 appear to be similar since in both
cases phase segregation occurs, though the curve magnitudes differ. Figure 5-15 shows

an important structural difference.

 

 

 

Node-Node RDF, Tone 310,
50% hard segment, em: + 0.1
— 1
5 _ 10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5-15. Node-Node RDF, Tone 310, 50% hard segment with varying E-ratios

Figure 5-15 shows the radial distribution function for the nodes in each of the three Tone
310, 50% hard segment systems. The curves show that as the e-ratio becomes greater the
average distance between the nodes decreases. An increase in the e-ratio means that the

soft-soft attraction increases in comparison to the hard—hard interaction. Figure 5-15

52

indicates that in Figure 5-9 the nodes are spread apart, whereas in the case of Figure 5-14

the nodes are clustered.

5.7 Comment on Qualitative Analysis

Another possible method to determine phase segregation is the evaluation of snapshots
taken of the simulations box. Figures 5—16 and 5-17 were taken using visualization

software (IRIS Explorer 4.0) on a 801 workstation.

 

 

 

 

 

Figure 5-16. Snapshot: Tone 301, 50% hard segment content

53

 

 

 

 

 

Figure 5-17. Snapshot: Tone 301, 65% hard segment content

In these snapshots the light spheres represent nodes and node arms, the dark sphere
represent the hard segments of the cross-links.

Comparing Figures 5-16 and 5-17 shows that determination of phase segregation based
on snapshots is mostly a subjective exercise. Figure 5-17 for example may appear phase
segregated, when in fact the system is not phase segregated. Hence an objective method,
such as radial distribution functions, is needed and applied in this work. Further snap

shots can be found in Appendix E.

54

5.8 Summary of Results

TABLE 5-2 SUMMARY OF PHASE SEGREGATION OBSERVED
Experimental Modeling Experiment Flory
Hard Fract. Hard Fract.

38 37 NO Yes
48
52 50 , Yes

17

67
33
44
57

65
43
57
64

 

Table 5-1 compares the Observations Of this work with experimental results and
Flory Theory predictions taken from the work of Agrwal and Drzal (1995a, 1995b, 1996).
In general the results agree and indicate that configurational-bias Monte Carlo
simulations can successfully predict physical behavior of networked block—copolymers.
Thus this method shows a number of possible applications, such as the validation of

obtained results, or the investigation of contributing factors to various physical behaviors.

55

 

 

 

 

 

 

 

 

Constant Tone Size - 310
x
_ \x c
-0.50 \ .2
‘6
§ — 0.20 a A
2 “as
I.l.l -1.50 — .E
x
0
(U
D.
-2.501T111eriavv 0.00
0% 30% 60%
Hard Segment Content
— — - average energy —— 11

 

 

 

 

Figure 5-18. Effect of Constant Tone Size for Tone 310

Figure 5-18 shows a the effect of constant tone size on packing fraction and average
energy per site (denoted ‘energy’ in Figure 5-18) in the case of the Tone 310 system. The
average energy per site is obtained by dividing the system’s total energy by the number
of sites. The figure shows that as hard segment content increases the average energy
increases in magnitude. This is a result of the higher number of hard sites, which have
higher energetic interaction.

The packing fraction is obtained by summing the volume of all sites and
dividing the total by the simulation box volume. The packing fraction in the Tone 310

case also increases as hard segment fraction increases. This observation correlates with

the higher average energy. As the average attractive energy increases between sites, the

chains are pulled closer together, thus increasing packing fraction.

Figure 5-19 shows the same trends for the case where the tone size is kept

constant with the Tone 305 system.

 

Constant Tone Size 305

 

 

 

 

0.00 ~ —r 0.30
-1.00 3
>.
E’
o r- 0.15
:
l.I.I
-2.00 —
-3.00 , . r . r . . fl . 0.00
0% 45% 90%

Hard Segment Content

 

- — - average energy "—71

 

 

 

Packing Fraction (Tl)

 

 

Figure 5-19. Effect of Constant Tone Size for Tone 305

57

 

Constant Hard Segment Content - 50%

 

 

 

0.00 0.25
-0.50 - / —_ 0.20 E
C
.2
5 -1.00 ~ -~ 0.15 §
3 It:
... -150- - —~— 0.10 2
\~~“ .%
-2.004 ‘T~~~-_: -~ 0.05 E
-2.50 — i 0.00

Tone 301 Tone 305 Tone 310

 

 

 

 

[— - -average energy '1 J

 

Figure 5-20. Effect of Constant Hard Segment Content - 50%

For the case that hard segment content is held constant and tone size varied Figure 5-20
shows that the average energy increases in magnitude as the Tone size increases. This is
explained by the higher number Of hard sites per cross-link that can interact with eaCh

- other. A part of the increase is also due to the greater number of hard-soft interactions.

The packing fraction is also shown to increases with larger tone size.

58

 

 

 

 

 

 

Cross-Link Length vs. Packing Fraction
0.3
i
E ..
35
E 0.2 _ i
*6
2
IL ..
g’ -
5 0.1 -
0
{B
o.
0 1 l *1
0 5 10 15
Cross-Link Length (No. of Sites)

 

 

 

Figure 5-21. Correlation of Cross-Link Length with Packing Fraction

Figure 5-21 shows comparable packing fractions for given cross-link lengths. The values
are taken from all systems investigated in this study. The exception to this general trend
is a Tone 301 case. The Tone 301 - 50% hard segment content and Tone 305 - 50% hard
segment content systems both have a cross-link length of 5 sites. The Tone 301 case
though shows a packing fraction nearly double that of the Tone 305 case. This is
explained by the higher number Of hard sites per cross-link in the Tone 301 case. This
higher number leads to a stronger interaction between sites and hard segments, thus
increasing the packing fraction. In the Tone 305 case the cross-link only has one hard

site. Tabulated values for Figures 5-18 through 5-21 can be found in Appendix D.

59

6. CONCLUSIONS

Block copolymer phase segregation is favored by three main factors: (1) longer cross-link
length (i.e. larger Tone size), (2) higher hard segment content and (3) greater ratio of
interaction energies. The modeling of this work shows agreement with trends observed
in previous theoretical (Flory Theory) and experimental work by Agrawal and Drzal.
Phase segregation was also shown to be largely controlled by energetic interaction,

though some negligible structural effect is observed.

Partially cross-linked systems show similar behavior to the fully cross-linked
counterparts. Greater mobility and larger degrees of phase segregation are observed in the
modified systems. Here also larger tone sizes and greater hard segment content increases
the degree of phase segregation. The partially cross-linked system’s equilibrated
structure was calculated in a more efficient manner than the fully cross-linked system,

while obtaining similar results.
This work Obtained results comparable to those of experimental work, thus indicating that

modeling provides a method of predicting macroscopic behavior of block copolymers

while investigating contributing factors at the molecular level.

60

7. RECOMMENDATIONS FOR FUTURE WORK

Previous work by Agrawal and Drzal showed an interphase region between the matrix
and glass surface. This work investigated the bulk phase of the polyurethane matrix. A
future line of work may be the investigation of the interphase region. This would entail
the simulation of systems studied in this work next to a hard continuos surface. The
nature of the future work would be of a heterogeneous nature, compared with the

homogeneous nature of this study.

The equilibrated structure of the partially cross-linked systems was shown to be more
efficiently calculated than the fully cross-linked system, while displaying similar results
to the fully cross-linked system. The partially linked system also had a greater
resemblance to the physical cases investigated in previous works. Future work may

include investigating a larger number of partially cross-linked systems.

61

APPENDICES

62

 

APPENDIX A

 

RDF, Tone 301 , 50% hard content, g-ratlo = 0.1

— soft-soft, different nodes
—-— hard-hard, different segments

 

rlo

 

 

Figure A-l. RDF, Tone 301, 50% hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 301 , 50% hard content, g-ratlo = 0.1

 

 

 

 

 

 

 

1.2
1 — soft-soft, same node
—-— hard-hard, same segment
0.8
g

 

l\
0.2 r \

 

 

 

 

 

Figure A-2. RDF, Tone 301, 50% hard content, e-ratio = 0.1, same chains

63

 

RDF, Tone 301, 65% hard segment, g-ratlo = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

4.5
4 -— soft-soft, different nodes
3 5 I + hard-hard, different segments
3 is
g 2.5

 

 

 

 

 

 

 

 

0.5

 

 

 

 

flo-

 

 

Figure A-3. RDF, Tone 301, 65% hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 301 , 65% hard segment, g-ratlo = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2
1 l
k — soft-soft, same node
0.8 -— hard-hard, same segment
0.4
1A
0 K— - - f
0 1 2 3 4 5 6 7

 

 

Figure A-4. RDF, Tone 301, 65% hard content, e-ratio = 0.1, same chains

 

 

 

 

 

RDF, Tone 305, 17% hard segment, g-ratlo = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.5
4
3.5 — soft-soft, different nodes
3 —-— hard-hard, different segment
g 2.5
2
1.5
. ,2} M...
0.5
0 « T . . r .
o 4 s 10

rlo-

 

 

Figure A-S. RDF, Tone 305, 17 % hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 305, 17% hard segment, e-ratlo = 0.1

 

 

 

 

 

 

 

 

1.2
— soft-soft, same node
1 —-— hard-hard, same segment “—
0.8
g 0.6

 

0.4 L
0.2

 

 

 

 

 

Figure A-6. RDF, Tone 305, 17% hard content, e-ratio = 0.1, same chains

65

 

RDF, Tone 305, 50% hard segment, g-ratlo = 0.1

 

 

4.5

 

 

 

 

 

 

4 — soft-soft, different nodes __

3.5 -- hard-hard, different segments .——

 

 

2.5

gr)

 

 

1.5

 

 

 

 

 

 

0.5 r

 

 

rlo

 

 

 

Figure A-7. RDF, Tone 305, 50% hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 305, 50% hard segment, g-ratio = 0.1

 

0.7

 

 

0.5 — soft-soft, same nodes
—-— hard-hard, same segment

 

 

 

 

0.5

 

0.4 —

g(r)

 

0.3

 

 

 

0.2

 

0.1

 

 

 

rlo-

 

 

 

Figure A-8. RDF, Tone 305, 50% hard content, e-ratio = 0.1, same chains

66

 

 

RDF, Tone 305, 63% hard segment, g-ratlo = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5
4.5
4 -—soft-soft, different nodes ' ~—
3-5 —-— hard-hard, different segments *—
3
g 2.5 '5‘
2
1.5
1 law—
0.5
O 1
0 1 2 3 4 5 5 7

[10'

 

 

Figure A-9. RDF, Tone 305, 63% hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 305, 63% hard segment, e-ratio = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2
1
_ — soft-soft, same node
0-8 —-— hard-hard, same segment
A
'3 0.6 k
0.4
0.2
O M , _——1
o 1 2 3 4 5 6

 

r/o'

 

 

Figure A-10. RDF, Tone 305, 63% hard content, s-ratio = 0.1, same chains

67

 

RDF, Tone 310, 13% hard segment, g-ratio = 0.1

 

 

4.5

 

 

3.5 — soft-soft, different nodes .—

 

 

 

3 —-— hard-hard, different segments ——

 

 

 

2.5

Jr)

 

 

1.5

 

 

 

 

0.5

 

 

 

 

rlo

 

 

 

Figure A-ll. RDF, Tone 310, 13% hard content, E-ratio = 0.1,
different chains

 

RDF, Tone 31 0, 13% hard segment, g-ratio = 0.1

 

1.2

 

 

 

— soft-soft, same node

 

—-— hard-hard, same segment

 

 

0.8
0.6 r ,1 1

0.4

0.2
04—4——

 

 

 

90)

 

 

 

 

 

 

 

Figure A-12. RDF, Tone 310, 13% hard content, e-ratio = 0.1, same chains

68

 

 

RDF, Tone 310, 37% hard segment, g-ratio = 0.1

 

 

4.5

 

 

4 — soft-soft, different nodes ‘—

 

 

 

3.5 ' —-— hard-hard, different segments ——

 

 

 

2.5

9(7)

 

 

 

1.5

 

 

 

 

0.5

 

 

 

0 1 2 3 4 5 6 7
rlo

 

 

Figure A-l3. RDF, Tone 310, 37% hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 310, 37% hard segment, g-ratio = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2
1
— soft-soft, same node
03 -- hard-hard, same segment
A
i 0.6
0.4
0.2
o -H% ,
0 1 2 3 4 5 5 7

rlo

 

 

 

Figure A-l4. RDF, Tone 310, 37 % hard content, e-ratio = 0.1, same chains

69

 

RDF, Tone 31 0, 50% hard segment, g-ratlo = 0.1

 

 

4.5

 

 

 

 

 

 

4 — soft-soft, different nodes 15——
3.5 —-— hard-hard, different segments ——

 

2.5

d?)

 

 

 

 

1.5 l

 

 

 

 

 

 

0.5 I -

O 'l r T l T T T
O 1 2 3 4 5 6 7
rlo

 

 

 

Figure A-lS. RDF, Tone 310, 50% hard content, e-ratio = 0.1,
different chains

 

RDF, Tone 31 0, 50% hard segment, g-ratio = 0.1

 

1.2

 

 

‘ -— soft-soft, same node

0.8 K
0 6 “

 

-— hard—hard, same segment

 

 

 

 

90')

.. “$1

 

 

0.2

 

 

 

 

0 1 2 3 4 5 6 7
[10'

 

 

 

Figure A-16. RDF, Tone 310, 50% hard content, e-ratio = 0.1, same chains

70

 

RDF, Tone 305, 50% hard segment, g-ratlo = 0.1 ,
incomplete crossllnk

 

 

 

 

 

 

 

 

 

 

 

4.5
4 — soft-soft, different nodes '—
3.5 l —-— hard-hard, different segments ——

3

g

 

 

 

 

 

 

 

 

 

 

0.5 f
o " f T T

l‘lo'

 

.l

 

 

 

Figure A-l‘7. RDF, Tone 305, 50% hard content, e-ratio = 0.1, incomplete cross-
link, different chains

 

RDF, Tone 305, 50% hard segment, g-ratio = 0.1 ,
incomplete crossllnk

 

 

 

 

 

 

 

 

 

1.2
1
— soft-soft, same node
0.8
—-— hard-hard, same segment
g 0.5

{\x—q

 

 

 

 

0.4 \
0.2 I i l
O 44 b
O 1 2 3 4 5 6 7
fig

 

 

 

Figure A-18. RDF, Tone 305, 50% hard content, e-ratio = 0.1, incomplete cross-
link, same chains

71

 

RDF, Tone 310, 38% hard segment, g-ratio = 0.1 ,
incomplete crossllnk

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 ‘l
5 — soft-soft, different nodes e
—-— hard-hard, different segments
A 4
h
‘61
3
2
1
0 L . . 4 . . . 4
0 1 2 3 4 5 s 7 a
r/o'

 

 

Figure A-l9. RDF, Tone 310, 38% hard content, e-ratio = 0.1, incomplete cross-
link, different chains

 

RDF, Tone 31 O, 38% hard segment, g-ratio = 0.1 ,
incomplete crossllnk

 

1.2

 

 

— soft-soft, same node
-— hard-hard, same segment

 

0.8
0.6 ,
k

 

 

 

 

g(r)

 

 

 

 

Figure A-20. RDF, Tone 310, 38% hard content, e-ratio = 0.1, incomplete cross-
link, same chains

72

 

RDF, Tone 310, 50% hard segment, g-ratlo = 0.1 ,
incomplete crossllnk

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5
4.5
4 ‘ — soft-soft, different nodes 15——
3.5 l + hard-hard, different segments ~—
3
g
2
1.5
1 f
0 5 f
O f 1
0 1 2 3 4 5 6 7

flo-

 

 

 

Figure A-Zl. RDF, Tone 310, 50% hard content, e-ratio = 0.1, incomplete cross-
link, different chains

 

RDF, Tone 310, 50% hard segment, g-ratlo = 0.1 ,
incomplete crossllnk

 

 

 

 

 

 

 

 

 

1.2
1
— soft-soft, same node
0.3
+ hard-hard, same segment
g 0.5

 

0.4 K
0.2

 

 

 

 

rlo-

 

 

 

Figure A-22. RDF, Tone 310, 50% hard content, e-ratio = 0.1, incomplete cross-
link, same chains

73

 

RDF, Tone 310, 50% hard segment, g-ratlo = 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5
4.5
4 — soft-soft, different nodes
3-5 —-—hard-hard, different segments L—
3
g 2.5
2 .
1.5 1
1
0.5
o 1 2 3 4 5 5 7
rlo

 

 

Figure A-23. RDF, Tone 310, 50% hard content, e-ratio = 1,
different chains .

 

RDF, Tone 310, 50% hard segment, g-ratlo = 1

 

 

 

 

 

1.2
1 -— soft-soft, same node ‘—
0 a _ -— hard-hard, same segment __

 

 

 

g(r)

0.6 A
0 4

 

... \

 

 

 

 

0 1 2 3 4 5 6 7
rlo-

 

 

 

Figure A-24. RDF, Tone 310, 50% hard content, e-ratio = 1, same chains

74

 

RDF, Tone 310, 50% hard segment, g-ratlo = 10

 

 

4.5

 

 

4 I — soft-soft, different nodes
3.5 —-— hard-hard, different segments

 

 

 

 

 

 

2.5

1.5 v
. LL

0.5

d?)

 

 

 

 

 

 

 

 

 

flo-

 

 

 

Figure A-25. RDF, Tone 310, 50% hard content, e-ratio = 10,
different chains

 

RDF, Tone 301 , 50% hard segment, g-ratlo = 10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.4
1.2 I
N — soft-soft, same node
1 -— hard-hard, same segment
0.8
C
‘5
0.6
0.4
0.2
o g b
0 1 2 3 4 5 6 7 8
fig

 

 

 

Figure A-26. RDF, Tone 310, 50% hard content, e-ratio = 10, same chains

75

APPENDIX 8
Listing of Additional System Parameters
Number of sites in each Cross-Link of fully linked systems:

Tone 301, 50% hard segment 2 soft sites, 3 hard sites, 5 sites total

Tone 301, 65% hard segment 2 soft sites, 5 hard sites, 7 sites total
Tone 305, 17% hard segment 4 soft sites, 1 hard site, 5 sites total
Tone 305, 50% hard segment 4 soft sites, 5 hard sites, 9 sites total
Tone 305, 63% hard segment 4 soft sites, 8 hard sites, 12 sites total
Tone 310, 13% hard segment 6 soft sites, 1 hard site, 7 sites total
Tone 310, 37% hard segment 6 soft sites, 4 hard sites, 10 sites total
Tone 310, 50% hard segment 6 soft sites, 7 hard sites, 13 sites total

Distribution of cross-link and dangling end lengths in partially linked systems:

Tone 305, 50% hard segment Longest Cross-Link: 10 sites
Shortest Cross-Link: 5 sites
Average Cross-Link: 5.5 sites
Longest Dangling End: 11 sites
Shortest Dangling End: 3 sites

Average Dangling End: 7.2 sites

76

Tone 310, 38% hard segment

Tone 310, 50% hard segment

Longest Cross-Link: 11 sites
Shortest Cross-Link: 7 sites
Average Cross-Link: 7.5 sites
Longest Dangling End: 11 sites
Shortest Dangling End: 3 sites

Average Dangling End: 6.6 sites

Longest Cross-Link: 10 sites
Shortest Cross-Link: 7 sites
Average Cross—Link: 7.4 sites
Longest Dangling End: 16 sites
Shortest Dangling End: 6 sites

Average Dangling End: 9.8 sites

77

APPENDIX C

Calculation of reduced Temperature and Pressure
k, = 1.3806 .10-.. . 315(5- Boltzmann Constant

P = 1 atm chose atmospheric pressure, since Agrawal & Drzal
conducted experiments at this Pressure

T = 363 K curing occurred at 90°C

Choice of site size: 0' = 3.8 Angstrom (A) methylene beads
0' = 4.812 A n-C4Hlo [square well]
0 = 5.339 A n-C4Hm [Lennard -—Jones]
0' = 5 .27 A Benzene [Lennard — Jones]
0' = 5.22 A Benzene [Sutherland]
O’ = 4.938 A Benzene [Kihara]

O chosen: 0 = 5 A

Choice of energy well depth:

81kg = 387 K n-C4H10 [Square well]

Elks = 309.74 K n-C4Hm [Lennard —Jones]
81kg = 440 K Benzene [Lennard — Jones]
Elks = 1070 K Benzene [Sutherland]

Elks = 975.37 K Benzene [Kihara]

An E-value of e = 400 K*k is Chosen as a mean between the values for n-C4Hlo and
Benzene. Butane and benzene represent two significant backbones of the polyurethanes
investigated by Agrawal & Drzal (19953). The chosen E. corresponds to the hard-hard

interaction energy (3”,) used in this work.

 

3
reduced Pressure: PR = fl— = 2.294 010’3
8
reduced Temperature: TR = k” . T = 0.908
8

78

APPENDIX D

TABLE D-l SUMMERY or AVERAGE ENERGY PER SITE

Hard Segment
Content (%)

Energy per Site

Standard
Deviation

 

50

-l .572

T 1.54E-02

 

65

-2.274

T 2.96E-02

 

17

-0.228

T 2.42E-02

 

50

-l.914

T 1.64E-02

 

63

-2.772

T 2.87E-02

 

13

-0.201

T 1.70E-02

 

37

-1.232

T 1.67E-02

 

50

 

 

79

-2.094

 

T 1.18E-02

 

TABLE D-2 SUMMERY 0F SYSTEM PACKING FRACTIONS

Hard Segment
Fraction (%)

Packing Fraction

Standard
Deviation

 

50

0.186

T 1.22E-03

 

65

0.138

T 9.83E-04

 

17

0.103

T 2.56E-04

 

50

0.206

T 4.9lE-03

 

63

0.278

T 6.65E-03

 

13

.113

T 8.56E-04

 

37

.216

T 5.27E—03

 

50

 

 

80

.231

 

T 2.20E-03

 

Hard
Segment
Content (%)

No. of Soft
sites per
Cross-Link

TABLE D-3 CROSS-LINK LENGTH AND PACKING FRACTION CORRELATION

Total
Cross-Link

No. of Hard
sites per
CroLss-Link

Packing
Fraction

 

50

Length
5

2

3

 

65

7

2

5

 

17

5

l

 

50

 

63

 

13

 

37

 

 

50

 

 

81

 

 

 

APPENDIX E

 

  

    

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Co‘s?“ . 031'
®§$‘{%m¥3
,. 7‘ ‘OI‘SB'O’C. 3'
>4 r:- ‘555' 3
' ..l. .-.x

        

 

 

 

Figure E-l. Snapshot: Tone 305, 17% hard segment content

82

 

 

 

 

 

Figure E-2. Snapshot: Tone 305, 50% hard segment content

83

 

 

 

 

 

Figure E-3. Snapshot: Tone 305, 63% hard segment content

84

 

0 y . .s
4 » - ”£53m...

182361;"?

 

 

 

 

Figure E-4. Snapshot: Tone 310, 13% hard segment content

85

 

 

 

 

 

Figure E-S. Snapshot: Tone 310, 37% hard segment content

86

 

 

. .M‘.
or xi... a...

.\ ,
.\

any.)

 

 

 

Figure E-6. Snapshot: Tone 310, 50% hard segment content

87

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88

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92

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