Kinetics of Polymeric Systems
Vergroberte Modelle zur Beschreibung
der Kinetik von Polymersystemen
Dissertation
zur Erlangung des akademischen Grades
des Doktors der Naturwissenschaften (Dr. rer. nat.)
an der Universitat Konstanz
Fachbereich Physik
vorgelegt von
Frank Eurich
Tag der mundlichen Prufung: 18. Juli 2002
Referent: Prof. Dr. Philipp Maa
Referent: Prof. Dr. Peter Nielaba
Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über <http://dnb.ddb.de> abrufbar.
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Diese Arbeit befasst sich mit der Beschreibung des kinetischen Verhaltens von Polymersys-
temen auf halbmakroskopischen Zeit{ und Langenskalen. Grundlage der Beschreibung sind
hierbeiaus mikroskopischen Konzepten abgeleitete, vergroberte Modelle. Ein besonderes Au-
genmerkliegtaufderEntmischungskinetikvonbinarenPolymermischungenineingeschrankten
Geometrien. Dabeiwerden sowohlhomogene, alsauchheterogeneSubstrate untersucht. Letz-
tere sind von besonderem technologischen Interesse, da man mit ihrer Hilfe versucht, gezielt
Polymerstrukturen im m und sub{m Bereich herzustellen. Solche Strukturen konnten zum
Beispieldazudienen,mikroelektronische Schaltkreise aus organischen Halbleiternaufzubauen.
Verwendung ndenkonnten sieweiterhinalspolymereMaskeninderLithographiebeiderHer-
stellung von Prozessoren oder als Materialien mit besonderen optischen Eigenschaften. Eine
bereits realisierteAnwendung ist eine verbesserte Antireexbeschichtung.
DieDiusionvonlangkettigenPolymerenwirddadurchverlangsamt,dassineinemdichten
System die Kettenmolekule Verschlaufungen bilden. Die charakteristische Zeitskala, auf der
sich diese Verschlaufungen auosen, skaliert mit der dritten Potenz der Kettenlange. Daher
kann dasLangzeitverhalten vonPolymersystemen kauminSimulationenvonModellenerreicht
werden, welche die Kettenstruktur der Molekule berucksichtigen. Wir verwenden daher eine
grobere Beschreibung der Systeme.
Zunachst betrachten wir Systeme im Rahmen der zeitabhangigen Ginzburg{Landau The-
orie. Fur den Fall einer binaren Mischung entspricht diesem Zugang die Beschreibung des
Systems durch die Cahn{Hilliard Gleichung, welche eine nichtlineare, partielle Dierential-
gleichung furdas Ordnungsparameterfeld darstellt. Der Ordnungsparameter istin diesemFall
dieDierenzderentsprechendenKonzentrationenbeiderKomponenten. DasKurzzeitverhalten
des Systems kann im Rahmen der Theorie durch eine Stabilitatsanalyse beschrieben werden,
diebezuglichdes Anfangszustandes des linearisierten Problemsdurchgefuhrt wird. Aus dieser
Analyse leitenwir einKriteriumfurdieUnterdruckung vonoberacheninduzierten spinodalen
Wellenin dunnenFilmen ab.
In einer fruheren Stabilitatsanalyse der Cahn{Hilliard Gleichung [Fis98b] wird das Auf-
treten von neuen charakteristischen Langen{ und Zeitskalen bei Entmischungsprozessen in
eingeschrankten Geometrien vorhergesagt. In sehr dunnen Filmen kann dabei die spinodale
Entmischung unterdruckt werden. Um diese Vorhersagen zu testen, wird die Cahn{Hilliard
Gleichung in zwei Dimensionen numerisch gelost. Dass das entsprechende Liniensystem der
Gleichung \steif" ist, fuhrt zu einer Beschrankung der moglichen Zeitschritte bei der Anwen-
dung von expliziten Verfahren zur numerischen Integration der Gleichung in der Zeit. Durch
die Anwendung eines impliziten Verfahrens konnen diese Beschrankungen umgangen und die
gesamteKinetikdesSystemvondenAnfangenderPhasenseparationbishinzumGleichgewicht
verfolgt werden. Dies ist besonders fur den uns interessierenden Fall einer binaren Mischung
zwischen zwei Wandenwichtig. Zum einen wachst diecharakteristische Zeitskalabeider Ent-
mischungmiteinerVerringerungderFilmdickean,zumanderen ndetzu spatenZeiteneinex-
trem langsamer, quasi{eindimensionalerPeriodenverdopplungsprozess statt. Die Vorhersagen
der Stabilitatsanalysewerden durchdienumerischen Studienbestatigt. Eine
Ubertragung des
Verfahrens auf dreidimensionale Systeme ist allerdings aufgrund der momentan verfugbaren
Computerkapazitatennichtpraktikabel.
Eine realistischere Beschreibung der Systeme geben Modelle, indenen ein Polymer als ein
weiches TeilchenmitinnerenFreiheitsgraden beschrieben wird. Eine Moglichkeit,Polymersys-
teme auf diese Weise zu beschreiben, wurde von Murat und Kremer vorgeschlagen [Mur98].
Im Prinzip werden bei dieser Art von Modell die moglichen Zustande einer Polymerkette
durch einen oder mehrere Parameter charakterisiert, zum Beispiel durch die Eigenwerte des
Tragheitstensors. AllePolymeremitdergleichenCharakteristik,d.h.dengleichenParametern,
werden als aquivalente weiche Teilchen betrachtet. Die Kette selbst wird durch eine mittlere
Monomerdichte bei einer gegebenen Charakteristik beschrieben. Fur ein System aus solchen
weichen Teilchen wird das Funktional der Freien Energie in einen intramolekularen und einen
intermolekularenAnteilsepariert. BeidiesemAnsatzwirdderintramolekulareAnteildurchdie
WahrscheinlichkeitfurdasAuftretenbestimmterTeilchenformenfestgelegt,derintermolekulare
Anteil durch das
Uberlappintegral der Monomerdichten. Da Polymere oft durch vergroberte
Kettenmodellebeschriebenwerden,beidenenmehrereMonomereaufeineektivesKettenglied
abgebildetwerden,kannmandieseArtderBeschreibungalseinenzweitenVergroberungsschritt
auassen. DieKinetikdesSystemsweicherTeilchenwirddurcheinenMonteCarloAlgorithmus
deniert.
Als zu Grunde liegendes Kettenmodel fur die Beschreibung von Polymeren durch weiche
Teilchen wahlen wir die Gau'sche Kette. Fur diese werden in einer detaillierten Studie die
fur den Aufbau der Modelle notwendigen Eingangsgroen bestimmt. Dies sind zum einen
die Wahrscheinlichkeitsverteilungen fur das Auftreten einer bestimmten Form, zum anderen
die entsprechenden bedingten Monomerdichten fur eine gegebene Form. Es zeigt sich, dass
durch geeignete Skalen{ und Separationsansatze die Komplexitat dieser Funktionen deutlich
reduziert werden kann. Inwieweit diese Ansatze eine Naherung darstellen, wird ausfuhrlich
diskutiert. Schlielich werden geschlossene Ausdrucke fur die Eingangsgroen angegeben, die
beiihrerImplementierungdasModellweicherTeilchensehreÆzientmachen. DieSimulationen
des Systems weicher Teilchen werden furdas Ellipsoid{Modelldurchgefuhrt.
InderArbeitvonMuratundKremerstelltedasBead{SpringModelldaszuGrundeliegende
Kettenmodelldar [Mur98]. Ein wesentlicher Aspekt dieser Arbeit ist dieAnbindung des Mo-
dells weicher Teilchen an Kettenmodelle. Es wurde gezeigt, dass ein entsprechendes dichtes
System weicher Teilchen wesentliche Eigenschaften von Polymersystemen aufweist. Das kor-
rekte Skalenverhalten desGyrationsradiusindichten Systemenkonnteallerdingsnurdurchdie
Einfuhrung eines von der Kettenlange abhangigen Wechselwirkungsparameters erreicht wer-
den. Dieser wurde durch Anpassung der Teilchengroe an Ergebnisse von Simulationen des
Bead{Spring Modells bestimmt.
Die Wahl der Gau'schen Kette hat den Vorteil, dass das Model unter Berucksichtigung
eines Selbstwechselwirkungsterms die wesentlichen Skalenrelationen der Polymerphysik in na-
turlicher Weise, d.h. ohne dieAnpassung eines Wechselwirkungsparameters, erfullt. Insbeson-
dere sind auch keine Simulationen von Kettensystemen fur eine vollstandige Denition des
Modells erforderlich. Das Skalenverhalten des Gyrationsradius wird in Abhangigkeit von der
Kettenlange und der Teilchendichte richtig wiedergegeben, allerdings mit Florys Resultat fur
denExponenten. DasModellistnichtnureÆzient,sodassdasLangzeitverhaltenderPhasen-
separation in dreidimensionalen Systemen beschrieben werden kann, sondern weist auch viele
charakteristische Eigenschaften von Polymersystemen auf, so zum Beispiel die Existenz eines
Korrelationslochs,sowie Umorientierungenund Deformationender EllipsoideanPhasengrenz-
achen.
SchlielichwerdenPolymersystemeineingeschranktenGeometrienimRahmendiesesEllip-
soid{Modells beschrieben. Die Ergebnisse fureine binare Mischung zwischen zwei homogenen
undneutralenWandenbestatigtqualitativdieVorhersagenderCahn{HilliardTheorie,namlich
einWachstumderlateralenEntmischunglangeundderentsprechendenZeitenmitabnehmender
Filmdicke. Eine Unterdruckung der spinodalen Entmischung wie von der Cahn{Hilliard The-
orie vorhergesagt wird allerdings aufgrund der in diesem System vorhandenen thermischen
Fluktuationennicht gefunden.
ZurUntersuchungderEntmischungskinetikaufheterogenen SubstratenwirdderFallbetra-
chtet, dasseine derbeidenWandemiteinemperiodischen Streifenmusterstrukturiertist. Ver-
schiedeneBereichedesSubstratsbevorzugenunterschiedlicheKomponenten. Wirndenhierbei
zweiunterschiedliche Gleichgewichtsstrukturen. EntwederwirddasSubstratmustervollstandig
aufdieDomanenstrukturubertragen,oderdasvonderOberacheinduzierteMusterdringtnur
einStuckweitinden Filmein. Der
Ubergangzwischenden beidenStrukturenndetbeieinem
bestimmten Verhaltnis von Periodizitat und Filmdicke statt, was durch eine Betrachtung der
Grenzachenenergienerklart werden kann.
Betrachtet man den Verlauf des Entmischungsprozesses, so werden die unterschiedlichen
Gleichgewichtsstrukturen aufverschiedene ArtundWeiseangenommen. Furden Fall,dassdie
Oberachenstruktur vollstandig abgebildet wird, ist die Ausbildung einer musterinduzierten
spinodalen Welle{ in Analogie zur oberacheninduzierten spinodalen Welle { der zu Grunde
liegende Prozess. Die periodische Ordnung des Domanenmusters friert ein, sobald die gegen-
uberliegende Oberache erreicht ist. Ist die Eindringtiefe der Struktur dagegen endlich, so
ndet zusatzlich eine laterale Vergroberung der Domanenstruktur statt. Fur den Fall, dass
diePeriodizitatdes Musters kleiner istalsdiecharakteristische Entmischungslange des \unge-
storten"Systems,wird derEinudesSubstratmustersaufdieDomanenstrukturreduziert,da
demSystemdurchdieOberacheenergetischungunstigeStrukturenmitzuvielenGrenzachen
aufgepragt werden.
List of Figures XI
List of Tables XIII
Notation XV
1 Introduction 1
1.1 Polymer Systems . . . 1
1.1.1 IdealChains . . . 2
1.1.2 Excluded Volume Eects . . . 5
1.1.3 DenseSystems of Polymer Chains . . . 7
1.1.4 Dynamicsof Polymer Chains . . . 7
1.2 Phase Separation . . . 10
1.2.1 Flory{Huggins Theory . . . 10
1.2.2 Ginzburg{Landau Theory . . . 13
1.2.3 Self{Consistent{Field Theory . . . 14
1.3 DemixingProcesses . . . 15
1.3.1 SpinodalDecomposition inBulk Systems . . . 16
1.3.2 DemixingProcesses in Conned Geometries . . . 18
1.4 Experiments with Polymer Blends . . . 19
1.5 TheoreticalApproaches . . . 24
1.6 Goals of this Work . . . 27
2 Cahn{Hilliard Theory 29 2.1 Linearized Problem . . . 29
2.1.1 Particular Solutionfor the Linearized Problem . . . 30
2.1.2 The Case sl !1 and g s =0 . . . 32
2.2 NumericalSolution . . . 34
2.2.1 Bulk Solutionsin Two Dimensions . . . 35
2.2.2 Solutionsfor aTwo DimensionalSlab . . . 40
2.3 Limitationsof the Cahn{Hilliard Theory . . . 43
3 Soft Particle Models 45 3.1 Second Coarse{Graining Step . . . 45
3.2 Choiceof the Chain Model . . . 48
3.3 SoftSphere Model . . . 49
3.4 Soft EllipsoidModel . . . 50
3.5 Disphere Model . . . 52
3.6 Multisphere Models . . . 54
4 Geometry of the Gaussian Chain 57 4.1 Probability Distributions . . . 57
4.1.1 Radius of Gyration . . . 58
4.1.2 Eigenvalues of the Gyration Tensor . . . 60
4.1.3 Center of Mass Distance and Radiiof Gyration . . . 63
4.2 Conditional MonomerDensities . . . 66
4.2.1 Spherical Density . . . 66
4.2.2 EllipsoidalDensity . . . 68
4.2.3 CylindricalDensities . . . 74
4.3 Origin of the BimodalForm . . . 75
4.4 Summary . . . 77
5 GEM: Bulk Systems 79 5.1 Homogeneous Melt . . . 79
5.1.1 Static Properties . . . 79
5.1.2 Kinetic Properties . . . 84
5.2 Binary Mixture . . . 86
5.2.1 Coexistence Curve . . . 86
5.2.2 Chain Dimensionsin a Blend . . . 87
5.2.3 SpinodalDecomposition . . . 88
6 GEM: Slab Geometry 91 6.1 WallInteractions . . . 91
6.2 Eects of Hard Walls onHomogeneous Systems . . . 92
6.2.1 Concentration of Monomersand of Centers of Mass . . . 93
6.2.2 Orientation . . . 95
6.2.3 Deformation . . . 95
6.2.4 Discussion . . . 95
6.3 Binary Blends in Thin Films . . . 99
6.3.1 Dependence of the Phase Diagramon the Film Thickness . . . 99
6.3.2 Orientationat Phase Boundaries . . . 100
6.3.3 SpinodalDecompositionbetween Neutral Walls . . . 102
6.3.4 Surface Directed SpinodalDecomposition. . . 104
6.4 Structured Surfaces . . . 106
6.4.1 Structural Phase Diagram . . . 107
6.4.2 PatternDirected SpinodalDecomposition . . . 110
7 Summary and Outlook 115
A Line System for the CHE i
B Properties of Gaussian Chains v
B.1 Averages of Gaussian Chains. . . v
B.2 Parameters of the Distribution Functions . . . v
B.3 Improved Formulaforthe MonomerDensity . . . vi
B.4 Improved Calculationof P C (r 2 AB ;R 2 A ;R 2 B ) . . . vi
C Calculation of Integrals ix C.1 Ellipsoid{EllipsoidInteraction . . . ix
C.2 Ellipsoid{WallInteraction . . . x
C.3 FourierTransformof the MonomerDensity . . . xi
C.4 LateralFourierTransform of the Density . . . xii
D GEM: Algorithm xv D.1 DataStructure . . . xv
D.2 Algorithm . . . xvi
D.3 Standard InitialConditions . . . xvi
Bibliography xvii
1.1 Coarse{grainingof a polymer chain . . . 3
1.2 Mean square displacement of a bead in apolymer melt . . . 10
1.3 Phase diagramand free energy of a binary mixture . . . 11
1.4 Illustrationof the lattice polymer modelused in Flory{Huggins theory . . . 12
1.5 Surface directed spinodaldecomposition,experimentalresults of [Bru92] . . . . 20
1.6 Interference of surface directed spinodal decomposition waves, experimentalre- sults of [Kra93] . . . 22
1.7 Transferringsubstrate structure todomain structure, after[Wal00]. . . 23
1.8 Hierarchy of coarse{graining . . . 26
2.1 Snapshots of domainpatterns, bulk-system . . . 36
2.2 Structure functionI(k;t) characterizing domainpatterns . . . 37
2.3 Correlationfunction g(r;t) . . . 38
2.4 Timedependence of characteristicdomain sizes . . . 39
2.5 Snapshots of domainpatterns, slab geometry . . . 41
2.6 Lateral intermediate scattering function I k (k x ;y;t), analytical and numerical results . . . 42
2.7 Quasione dimensionaldomain coarsening: period{doublingprocess . . . 43
2.8 Timedependence of characteristiclateral domain sizes, slab system . . . 44
3.1 Second coarse{graining step for the soft ellipsoid model . . . 46
3.2 Schematicillustrationof adiblock copolymer . . . 53
4.1 Scaledprobability function P R (R 2 G ) . . . 59
4.2 Scaledprobability functions P (S ) . . . 61
4.3 Comparisonof P (S )P (S )with P(S ;S ) . . . 62
4.4 Conditionalprobability P(r 2 AB ;R 2 A ) and jointprobability P(r 2 AB ;R 2 A ) . . . 65
4.5 Spherical monomer density %^ r (u;R G ) . . . 68
4.6 Conditionalmonomer densities %^ (x =R ;S ) . . . 70
4.7 Eective scalingfunctions %~ (x =R ) . . . 71
4.8 Comparisonof %~ (x =R )~% (x =R ) with %~ (x =R ;x =R ) . . . 73
4.9 Conditionalmonomer density % A (x ;r AB ) . . . 75
5.1 R 2 G (N) for a dilutesystem of ellipsoids . . . 80
5.2 R 2 G (N) for a densesystem of ellipsoids . . . 81
5.3 Test of de Gennes' scaling relation. . . 82
5.4 Scaling behaviorof P
R (R
2
G
) ina densesystem . . . 82
5.5 Scaling of the correlation hole . . . 83
5.6 Mean square displacementin a densesystem . . . 84
5.7 Autocorrelation function C R (t)of R 2 G . . . 85
5.8 Scaling of the phase diagram. . . 86
5.9 Shrinkingof polymers inminority . . . 87
5.10 Intermediatescattering functionI(k;t) for abinary blend . . . 88
5.11 First moment of I(k;t) . . . 89
6.1 Monomer density c(z) and center of mass density c cm (z) in aslab . . . 93
6.2 Monomer density c(z) for various L z . . . 94
6.3 Orientationof ellipsoidsat the wall . . . 96
6.4 Deformation of ellipsoidsatthe wall . . . 97
6.5 Dependence of the phase diagramon L z . . . 99
6.6 Behavior of ellipsoidsat aphase interface . . . 101
6.7 Lateral intermediate scattering function I k (k k ;z;t) for a binary blend . . . 103
6.8 First moment of I k (k k ;z;t) . . . 104
6.9 Surface directed spinodaldecomposition, monomer concentration c B (z) . . . 105
6.10 Possible structures on a patternedsubstrate . . . 106
6.11 Normalizedcenter of mass density c cm to distinguish dierent patterns . . . 108
6.12 Structural phase diagram. . . 109
6.13 Pattern directed spinodaldecomposition . . . 110
6.14 Pattern directed spinodaldecomposition and lateral domaincoarsening . . . 111
6.15 Interplay of lateral demixinglength and pattern periodicity . . . 112
6.16 Pattern directed spinodaldecomposition wave . . . 113
3.1 Parameter dening P
R (R
2
G
) and P
(S
) . . . 50
3.2 Constants dening %~ (u ) . . . 51
4.1 CorrelationcoeÆcients to test ansatz eq.(4.13) . . . 63
4.2 CorrelationcoeÆcients to test ansatz eq.(3.16) . . . 74
6.1 System sizes for homogeneouslms . . . 93
6.2 System sizes for lms of binary mixtures . . . 98
6.3 System sizes for lms of binary mixtures onstructured surfaces . . . 107
B.1 Moments of the eigenvalues S . . . v
Throughoutthis workwedenote spatialindicesby lowercaseGreek lettersand particleindices
by lowercase Latin letters. Indices referring to a grid are also denoted by lowercase Latin
letters. Quantitiesreferringtocomponentsofamixtureorcopolymeraremarked byrespective
uppercase Latin indicesA;B;C or X.
If the components are not explicitely denoted by respective indices, matrices are usually
denoted by calligraphic letters and vectors are typed in bold. If a vector symbol is not typed
bold, then usually the normof the respective vector is meant.
The coordinates in the laboratory system are denoted by y (y
1
;y
2
;y
3
) (x;y;z), the
coordinates in the principalaxis system of a chain oran ellipsoidby x(x
1
;x
2
;x
3 ).
Averages of a quantity Q are denoted either by hQi or by
Q .
List of Abbreviations
AFM atomicforcemicroscopy
BDF backward dierentiationformulae
CDS celldynamicalsystem
CHE Cahn{Hilliardequation
CPU centralprocessing unit
DSCFT dynamicself{consistent{eldtheory
EPD externalpotentialdynamics
GEM Gaussianellipsoidmodel
MCS MonteCarlo step
ODT octadecylthiol
OM opticalmicroscopy
PB polybutadiene
PEP poly(ethylene{propylene)
PMMA polymethylmethacrylate
PS polystyrene
PVP polyvinylpyridine
RIS rotationalisomericstate
SAM selfassembled monolayer
SCFT self{consistent{eld theory
SDW surface directed spinodalwave
TOF{ERD time{of{ightelastic{recoil{detection
TOF{FRES time{of{ightforward{recoil{spectrometry
List of Symbols
hAi average asphericity
A;B rst and second component of amixture or block copolymer
A
n
;B
n
;A
n;r
;B
n;r
Fourier coeÆcients
B
2
(T) second virialcoeÆcient
C
R (t);C
O
(t) autocorrelationfunctions of shapeand orientation
D diusion constant
D() order parameter distribution
E
1
typical energy of acovalent bond
E
2
typical contact energy between twomonomers
F free energy, free energy functional
F
b [];F
s
[] bulkand surface part of the free energy functional
F
inter
;F
intra
;F
wall
intermolecular,intramolecular,and wall interaction part
of the free energy functional
G(y;t) correlation function of the order parameter
G
k (y
k
;z;t) lateral correlationfunction
G
0 (y;y
0
) pair correlation function
G
cm
(r) correlation function of centers of mass
H(:) step function
I(k;t) intermediate scattering function
I
k (k
k
;z;t) lateral intermediate scattering function
K() characteristic function of P
R (R
2
G )
K
n
(:) modied Besselfunction of n'th order
L system size
L
k
periodicity of substrate pattern
L
crit
criticalthickness for SDW formation
L
p
contour length ofa polymer chain
L
t
contour length ofthe primitivepath of the tube
L
D
average domainsize
M;M
A
;M
B
number of polymers or ellipsoids
N+1 number of beads of a Gaussianpolymer chain
N
A
;N
B
number of beads of components A and B
N
A
;N
B
number of beads of a block
N
e
entanglementlength
N
p
+1 number of monomers ina polymer
P(:) probability distributionof a quantity
P(:;:) joint probability distribution
P(:;:) conditional probabilitydistribution
~
P(:) scaling functionfor a probability distribution
Q
A
;Q
B
partitionfunction of a singlechain of type A or B
R end{to{end vector of a polymer chain
R rotationmatrix
R (t) inverse of k
n (t)
R
0
(t) rst zero of g(r;t)
R
0;k
(t) rst zero of G
k (y
k
;0;t)
R
e
rootmean square end{to{end distance
R
G
radius of gyration
R
G
average radius of gyration
R 2
G
random change of R 2
G
R
A
;R
B
radius of gyration of a block
R
p
S
extensionof ellipsoid with respect toprincipalaxis
R
Gk
;R
G?
lateraland perpendicularradius of gyration with respect to awall
S(q) structure function
S
; SS
radiusof gyration tensor and itseigenvalues
S
randomchange of S
T; T
c
temperature, critical temperature
U(u ) eectivebond potentialof a Gaussianchain
V; @V volume, surface
V
w
(z) wallpotential
W
A
;W
B
auxiliaryelds inself{consistent eld theory
W
A (
A );W
B (
B
) quantities needed in the calculation of P
AB (r
AB
;R 2
A
;R 2
B )
Z partitionsum
a(T); b parametersin Ginzburg{Landau freeenergy density
b rootmean square bond lengthof a Gaussianchain
c monomer concentrationin apolymericsystem
c
cm
concentration of centers ofmass
c
?
crossovermonomer concentration from the dilute tothe semi{dilute
regime
d
t
tube diameter
f freeenergy, freeenergy density
f(u) scalingfunction
f
b
; f
s
bulkand surface free energy density
f(fy
i g); f
G (fy
i
g) partitionfunction of asingle polymer, a Gaussiancoil
f
A
;f
B
fractionof chains of one component
f
i
forceonbead i
g numberof beads ina blob
g parameter inf
s
describing modication of interactions
g
s
scaledsurface parameter
g(r;t) circularaveraged correlationfunction G(y;t)
h parameter inf
s
describing external surface eld
h
l
scaledsurface parameterin linearized CHE
j (y;t) interdiusive current
k
c
criticalwave vector
k
n
(t) nthmomentof I(k;t)
k
n
discretewave vectors
k
1;k
(t) lmaveragedrst momentof I
k (k
k
;z;t)
k
B
Boltzmannconstant, k
B
=1;38066210 23
J/K
l Kuhn'ssegment length
~
l persistence length
l
w
wall interaction range
m mass of a bead
p(u) bond probability distribution for Gaussianchains
p
if
transitionprobability
r center of mass
r randomdisplacementof the center of mass
r
AB
vector between the centers of mass of two blocks
s distance between two monomers along the contour of a polymer
s contour parameter of apolymer coil
t time
t time discretizationlength
u scaling variable
u
i
bond vector between two beads i and i 1
v
i
bond vector between two monomers i and i 1
v
0
excluded volume of one monomer or bead
v
e
(r) eective potentialbetween monomers
w
A
;w
B
external eld inthe saddle{pointapproximation of the SCFT
x;y spatial discretizationlengths
y
i
position of the i'th bead of a polymer coil
z lattice coordination number
z
i
position of the i'th monomer ofa polymer
(:) gammafunction
b
bulkkinetic coeÆcient
s
kinetic coeÆcient atthe surface
sl
scaled kinetic coeÆcient
s
in linearized CHE
temperature, dened by B
2
()=0
inertia moment
polymer concentration
?
crossover polymer concentration fromdilute to semi{diluteregime
A
;
B
auxiliaryelds in self{consistent eld theory
ratio of x and y
correlation coeÆcients
Æ Dirac'sÆ{functionor Kronecker's Æ
Æ; Æ
w
AB{interaction, wall{interaction mismatch parameter
^=k
B
T interaction between two monomers
^
w
interaction between wall and monomer
?
crossover interaction fromGauss to Flory typebehavior
friction coeÆcient
viscosity
(y;t) noise eld
contact angle
ij
angle between two bond vectors i and j
m
bulkdemixing length
(y;t) chemical potential
scalingexponent
;
b
bulkcorrelation length
crossoverlength from meaneld type toIsing typebehavior
m
blob size, screening length insemi{dilute solution
%(y ;X) conditionalmonomer density
% 0
(y) totalmonomer density
% 0
i
(y) actualmonomer density of particlei in alaboratory xed
coordinatesystem
^
% (u;X) test function forscaling of the monomer density
~
% (u) eectivescaling function of the monomer density
% (x) average monomer density
(k ) Fouriertransform of % 0
(y)
k (k
k
;z) lateralFourier transform of % 0
(y)
b
;
s
measure for the interfacial tension inthe bulk and atthe surface
sl
scaled
s
inlinearized CHE
XY
surface tensionof XY{interface
characteristic timescale of the CHE
1
Rouse time
d
disentanglementtime
e
entanglement time
0
elementaryMonte Carlo time
S
;
O
characteristic timesfor shape and orientation changes inthe GEM
(y;t) orderparameter eld of a binary mixture
average orderparameter value
Æ(y;t) perturbationof the order parametereld
p
(z;t);
hom (y
k
;z;t) particular and homogeneoussolution of linearized CHE
c
criticalvalue of the orderparameter
1
;
2
orderparameter at coexistence
s1
;
s2
orderparameter at the spinodal
A
;
B
monomer concentrationsof components
~
A
;
~
B
microscopicmonomer density of components
Flory{Hugginsparameter
!
b
(k) bulkdispersion relation
Materials based on polymers exhibit a vast variety of dierent properties. These range from
simple rubber that can be found on every desk, to DNA in which the genetic information
of all creatures is encoded. Due to this fact, it is clear that the physics of polymers is a
broadresearcheld,which,despiteftyorsixty yearsofintensivestudiesandmanyimportant
progresses,entailsmanyproblems thatuptonoware onlypoorlyunderstoodorhavenot been
investigated in detail. The large variety of the properties of polymer materialsis achance for
materialscientiststotailorsubstanceswith certaindesiredproperties. Anoptimizationofsuch
propertiesrequires an understanding of the physical processesinvolved.
In thiswork wewillfocus onthemodelingof homopolymermelts andpolymer blendsfrom
aphenomenologicalpointofview, butwewillkeep alinktomicroscopicmodels. Withthehelp
of these coarse{grained models we will describe demixing processes of polymer blends in slab
geometry with patterned substrates. These processes are of importance in experiments that
trytotranslateasubstrate patternintoapolymerpattern[Bol98,Kar98,Roc99,Fuk00]. Such
polymer patterns might, for example, be used to fabricate microelectronic circuits consisting
of organicsemiconductors or polymer resists for lithographic semiconductor processing.
In the following sections we shortly present the necessary background in polymer physics
and phase separation processes. Thereafter we show some experimental results that, apart
fromother aspects, motivatedthis work. Finally weaddress the issue of practicalproblems in
the theoretical and computationaldescription of such systems.
1.1 Polymer Systems
Polymers are molecules, which consist of many monomer units that are chemically linked
together to form a chain. The most simple polymers are homopolymers ( A )
N
A
, where one
chemical monomer unit A is repeated N
A
1 times. Polymers, that are build of dierent
monomers, are called copolymers. The dierent monomers can have an arbitrary order along
the chain. For example, DNA consists of four types of monomers and their order along the
chain encodes the genetic information. A polymer, that consists only of two homopolymers
( A )
N
A
and( B )
N
B
linkedtogetheratone endofeachother,iscalledadiblockcopolymer
( A )
N
A
( B )
N
B
. Triblock copolymers are build of three homopolymer blocks and so on.
At this point we only note, that polymers may not only dier in chemical composition, but
alsoin topology. There are ring polymers, where the two chain ends are bound together, star
polymers, where the ends of some polymer chains are attached to one point, polymer combs,
brushes and whole polymer networks, where many polymers are cross linked randomly, the
most prominentexample of which isrubber. However, these more complicated topologieswill
not beconsidered in this work.
Homopolymers exhibita varietyof macroscopicstates. Perhaps the mostsimple isadilute
solution of homopolymers in asolvent, i.e. the distances between the polymers insolution are
so large, that they do not interact with each other. This state can be considered as an ideal
\gas"state. Onincreasingtheconcentration,therewillbeacertainmonomerconcentrationc
? ,
wheretheinteractionbetween thepolymersbecomesimportant,but thevolumefractionof the
polymers is stillmuch less than the volume fraction of the solvent. Those solutions are called
semi{dilute. The solution is called concentrated, when the volume fractions of both polymer
and solvent are comparable. Concentrated solutions or pure polymer substances can exist in
crystalline, glassy, high elastic or viscoelastic states. While the crystalline state is in some {
but not all { properties similar to the crystalline state of low{molecular{weight substances,
the glassy, highelastic,and viscoelasticphase statescorrespond tothe \liquid"state ofsimple
uids. At low temperatures many polymer liquids have such a high viscosity, that their ow
is invisible during reasonable observation times. They form a glass and many plastics are
actually glassy polymers. Above the glass transition temperature the mobility of individual
links increases strongly. If the relative motion of whole chains is constricted due to chemical
cross{linksorremainingglassyregionsthatformeectivecross{links,thepolymerisinastate
characterized by a high elasticity. Otherwise, if the relative motion of the macromolecules is
possible,eitherbecausetherearenochemicalcross{linksorbecausethetemperatureissohigh
thatnoglassydomainsexist,thepolymerbeginstoowonlargetimescales,butretainselastic
behavior onshort time scales. This is the viscoelastic state, alsocalled a polymer melt.
Despite the obvious complexity of such molecules, let alone the complexity of systems
composed of such molecules, there has been great success in describing the various states
of polymers theoretically by adequate models. The reason for this is, that it is possible to
identifytwoessential\large"parametersforsuchmodels. Therstisthechain lengthL
p . The
second oneis the ratiobetween theenergies of covalent bonds E
1
5eV between neighboring
monomers and the contact energies between non{neighboring monomers or monomers and
solvent molecules E
2
0.1 eV. Since E
1
=k
B
T 1 at room temperature and E
1
=E
2 1,
neither thermal uctuations nor direct interactions at contact can break a covalent bond.
Therefore the monomers are xed along the chain and the chain length L
p
is constant. All
interactions,that are not associated withcovalentbonds are called volume interactions.
The following parts of this section summarize important results of the theory of polymer
physics. Thereader,whoisinterestedinmoredetails,isreferredtorefs.[Gen79,Doi88,Gro94].
In arst stepwepresent theresults forideal chains. Then weturn tovolumeinteractions and
show, thatsingle chainsare notideal. Wethen discuss,whypolymersindensesystemsbehave
like ideal chains. We conclude this section with a short summary of the dynamic behavior of
polymer chains indense systems.
1.1.1 Ideal Chains
We consider a polymer consisting of N
p
+1 identical monomers at positions z
k
, where k =
0;:::;N
p
. All models of ideal chains have in common that volume interactions are neglected,
but they usually dierinthe probability distributionof the bond vectors v
k z
k z
k 1 , with
k = 1;:::;N
p
. This probability distribution depends essentially on eective bond potentials
between the monomers, that can be derived by quantum mechanical calculations, and the
v R
v
v u v
u N N
1 p
N −1 p i
1
y
y
y k 0
1
y N
y N−1
v 2
u k
Figure 1.1: The gure illustratesthe mapping of a polymer to a Gaussian chain. The polymer is
describedbyN
p
bondvectorsv
i
and theend{to{endvector R. The Gaussianchainisrepresentedby
N+1beadsat positionsy
k
thatare connectedbythermalspringsalong thevectors u
k .
temperature.
In random walk theory it can be shown, that under quite general conditions correlations
along the chain decay exponentially [Gro94]. The orientational correlation function hv
i v
j i
is given essentially by the average cosine of the angle
ij
between the two links i and j and
depends for a homopolymer only on the distance s between the two links along the chain. If
the length ofthe bond vectors is(almost)constant, whichis avery good approximation, then
the orientation correlationfunction reads
hv
i v
j i'v
2
hcos
ij
(s)i=v 2
exp
s .
~
l
: (1.1)
Thisequation denes thesocalledpersistencelength
~
l. It isameasure ofthemaximum length
overwhich apolymer remainsstraight.
A quantity characterizing the size of a polymer coil is the root mean square end{to{end
distance R
e
p
hR 2
i, where the end{to{end vector of a polymer chain isgiven by
R= Np
X
k=1 v
k
: (1.2)
Byusingeq.(1.1)one ndshR 2
i(L
p
=
~
l)
~
l 2
L
p
~
l =N
p v
~
lforN
p
1,whereL
p
isthe contour
length of the polymer chain. Thus for ideal chains R
e
p
N
p
holds. Because the persistent
length
~
l can not bedetermined experimentally,usually Kuhn'ssegment length l dened by
hR 2
i=L
p
l; (1.3)
is used to characterize the exibility of macromolecules and the decay of correlations. The
Kuhnlengthlisofthesameorderofmagnitudeasthepersistencelength
~
l. Ageneralapproach
to determine hR 2
i approximately in terms of microscopic model parameters and thus Kuhn's
length l is the rotational isomeric state theory (RIS)[Flo74].
Because the correlations between the bonds decay exponentially and the second moment hv 2
i
exists, it follows from the central limit theorem of probability theory, that if L
p
l the
probability distribution of the end{to{end vector is Gaussian
P
N
p (R)=
3
2hR 2
i
3=2
exp
3R 2
2hR 2
i
: (1.4)
The second moment hR 2
i is given by eq. (1.3), therefore l is the only microscopic parameter,
that enters this distribution. The same argument,that applies to the distribution of the end{
to{end vector, applies to all vectors between two points z
i
and z
j
along the chain, as long as
their distance s along the chain is large in comparison to
~
l. If we are only interested in the
largescale properties ofpolymers,itisthereforealwayspossibletodoacoarse{grainingonthe
scale of l,i.e.to describean idealpolymer chainconsisting of N
p
monomers by N <N
p beads
at positions y
k
with an equal distance s l along the chain. This kind of coarse{graining
is illustrated in g. 1.1. The beads are connected by bonds u
k
= y
k y
k 1
, that satisfy a
Gaussian distribution
p(u)=
3
2b 2
3=2
exp
3u 2
2b 2
: (1.5)
We denoted the root mean square bond length p
hu 2
i between beads in the chain by b. This
model iscalled the standard Gaussianmodel of the polymer chain and we emphasize, that all
large scale properties of all ideal chain models can be described in terms of this model. The
beads can beinterpreted to represent a certainpart of the macromolecule, while the bonds u
provide eective connections between the beads. Given the number of beads N, the eective
bond lengthb for aspecial chain modelcan be determined by the condition
hR 2
i=Nb 2
=L
p
l; (1.6)
cf. eq. (1.3). In order to have an \optimal" description of a polymer in terms of a Gaussian
chain, N has tobe maximal,while not violatingthe condition sl. Forexample itwould be
possible toset the eective bond length equalto Kuhn'ssegment length,i.e. b =l. Therefore
one would have to determine either l (numerically or by RIS theory) or give a more direct
criterium, at which distance s the distributionfunction of vectors z
i z
j
is Gaussian.
The distribution functionof a Gaussianchain with N beads isgiven by
f
G (fy
i g)=
3
2b 2
3N =2
exp 3
2b 2
N
X
i=1 (y
i y
i 1 )
2
!
; (1.7)
becauseits bonds are uncorrelated. Introducing the potential
U(u)=k
B T
3
2b 2
u 2
; (1.8)
we nd, that eq. (1.7) can be interpreted as the equilibrium Boltzmann distribution of ideal
particlesinterconnectedbythermicsprings. Itisconvenienttorewritethedistributionfunction
by taking the continuum limit. Parameterizingthe coilwith the variable s, with (0 s1),
substitutingy
i+1 y
i
by @y=@s and replacing the sum by anintegral,we can rewriteeq. (1.7)
by the Wiener measure
f
G
[y(s)] /exp
3
2hR 2
i Z
1
0 ds
@y
@s
2
!
: (1.9)
Finally, we note that another important quantity characterizing the overall size of a polymer
coilis given by the radius of gyration
R 2
G
= 1
N+1 N
X
k=0 (y
k r)
2
; (1.10)
dened by the average squared distances of the beads fromthe center of mass
r= 1
N+1 N
X
i=0 y
i
: (1.11)
Forlarge N 1 the average radius of gyrationis for Gaussianchains given by
hR 2
G i
1
6 hR
2
i= 1
6 Nb
2
: (1.12)
1.1.2 Excluded Volume Eects
While ideal chains are a good starting point for the understanding of polymer physics, they
neglectthe interactionsbetweenthe monomers,thatdonot arisefromthechemicalbonds,but
the \contacts" of non{neighboring monomers. These volume interactions involve strongshort
range repulsions of the monomers and long range attractions. Excluded volume eects are
typically of universal nature and therefore independent on the specic chain structure. Thus
itispossibletochoose a mathematicallyconvenient modelfortheir description. Thestandard
Gaussianmodelis oftenused, where spherically symmetricbeads are connected by links with
Gaussian correlations and interact via aneective potential v
e
(r) that mightfor example be
of the Lennard{Jones type. If one neglects the attractive part of the interaction, which is
possible athigh temperatures, thenthe short range repulsion of v
e
(r) mighthavethe formof
a Æ{function. In this eective potential interactions with a solvent may be incorporated, too.
Thesimplestrealizationtostudy excludedvolumeeects experimentallyisthe caseof adilute
polymer solution.
Macroscopicallythevolumeinteractionsaredescribedbythethermodynamiccharacteristics
of a system of disconnected beads. The partition function (of a single chain) splits into two
parts, one accounting for volume interactions, the other for the eective Gaussianbonds
f(fy
i
g)=exp X
i6=j v
e jy
i y
j j
!
f
G (fy
i
g); (1.13)
with f
G (fy
i
g) given by eq. (1.7). As long as the monomer concentration in the system re-
mainsmoderate,the volumeinteractions are reduced torelativerarelinkcollisionsand can be
described by a virialexpansion. The second virialcoeÆcientis given by
B
2 (T)=
1
2 Z
d 3
r
1 exp v
e (r)
: (1.14)
Bysplittingv
e
(r)intoanattractiveand arepulsivepart[Doi88],wend thatforhighenough
temperatures
B
2
(T)=v
0
T
T
; (1.15)
where v
0
is the excluded volume of one particle (bead) and a certain temperature, where
the second virial coeÆcient vanishes. For exible chains v
0
is essentially determined by the
eectivelink lengthv
0 b
3
. It canbeshown, that inadilute polymersystem binary collisions
are dominant, i.e.they occur N times more frequently, than triple collisions. Therefore triple
(and higher) collisions do not aect the conformation of the chain and in the special case
B
2
(T) 0 (or T ) volume interactions can be neglected and the chains are ideal. For
B
2
(T)>0the chains swell andconversely shrink in caseB
2
(T)<0. Thusthe {temperature
separates two regimes of good and bad solvent. The properties of a polymer coil in a good
solvent are universal, in the sense, that they only depend on the contour length L
p
, Kuhn's
segment length l and the second virial coeÆcient B
2
. More precisely, they depend on the
dimensionless parameter B
2
=b 3
. In a poor solvent (but not too far from the {temperature)
the coilproperties remain mostly universal, but the third virialcoeÆcient becomes important
also. In the following we will no longer consider poor solvents and attractive forces, but
restrict ourselves to the case of athermal chains, i.e. consider only repulsive interactions. For
our purposes this isnot a majorrestriction.
Themoststrikingeect ofexcludedvolumeisthatthetypicalsizeofapolymerchainscales
no longerwith p
N, but with a higherpower
hR 2
iN 2
: (1.16)
While the exact value of is unknown, renormalization group calculations and Monte Carlo
simulations yield 0:588 in d = 3 dimensions. However, there is a simple argument by
Flory, that gives an excellent estimate of . According to this argument, two eects have
to balance each other: The excluded volume tends to swell the coil, while the elastic forces
shrink it. The elastic part of the free energy is of entropic nature and given by F
el (R ) =
k
B T lnP
N
(R )k
B T 3R
2
=2Nb 2
, cf. eqs.(1.4, 1.6). Neglectingprefactors and estimating the
energy of the excluded volume by U
V
(R ) k
B T v
0 c 2
R 3
, where c N=R 3
is the average bead
concentration within the area of the polymer,the freeenergy is given by
F(R )=k
B T
3R 2
2Nb 2
+v
0 N
2
R 3
: (1.17)
Minimizing eq. (1.17) with respect to R yields R
e
N
3=5
. However, this kind of calculation
benetsfromthecancellationoftwoerrors: First,intheansatzfortheexcludedvolumeenergy
allcorrelationsbetween theparticlesare neglected, thereforethe interactionenergy isstrongly
overestimated. Secondly, the elastic energy is also overestimated, because it is a function of
R 2
=hR 2
i, rather thanof R 2
=Nb 2
. Furtheron, thereare quantities that are simply inconsistent
with this simple theory, for example the end{to{end distance probability distribution of a
polymerwithexcludedvolumeeects. Becauseasinglepolymercoilisaverylooseandstrongly
uctuating system, itis not surprising, that this kindof mean{eld argument provides not in
all aspects a gooddescription.
1.1.3 Dense Systems of Polymer Chains
Until now we shortly summarized the properties of single polymer chains or dilute polymer
systems. Now we regard the opposite case, a concentrated polymer solutionor polymer melt.
Againwewanttoestimatethesizeofapolymerchainby amean{eldargument. Therefore we
consider one single chain C
s
in a dense system surrounded by other chains. As argued in the
previous section, the interaction of the chain C
s
with its own monomer density eld willlead
toaswelling ofthechain. Onthe otherhandinadensepolymersystem theuctuationsofthe
totalmonomerconcentrationsareveryweak. Thereforethemonomerdensityofallotherchains
isreduced inthe region of C
s
and thusleads toa compression ofthe coil. Therefore both, the
expansionforcesduetoself{interactionofC
s
andthecompressionforcesduetotheinteractions
withsurroundingpolymers, balanceeachotherandthe chain remainsideal [Gen79]. Thiskind
of self{consistent{eld argument is valid, because in this case uctuations are negligible and
therefore mean{eldtheory applies.
Fromamoremicroscopicpointofviewtheidealbehaviorofchainsinameltmeansthatthe
excludevolumeeect between thebeadsofC
s
isscreened. Tounderstandthis werstconsider
themoregeneralsituationofarbitrarymonomerconcentrationc. Startingfromthedilutelimit
the chains will begin to overlap at c
?
N=R
3
e
N
1 3
. In the semi{dilute region (c > c
? )
the system consists of an interpenetrating network of chains that has some average mesh size
m
. For
m
the following two statements are certainly correct: First, for c>c
?
the mesh size
depends only on the monomer concentration c but not on the chain length N. Secondly, at
the crossover concentration the mesh size is of the order of
m R
e
N
. Therefore we have
m bN
(c=c
? )
v=(1 3)
in the semi{dilute region.
The polymers are virtually undisturbed on scales smaller than
m
. Such an undisturbed
partof achain iscalledablob and thenumberofbeads ofablob g isgiven by
m
=bg
. From
this denition and the scaling relation above it follows that g c 3
m
. Therefore the semi{
dilute solution is essentially a closely packed { dense { system of blobs. If now we consider
the polymer as a chain of N=g blobs, i.e. we look atthe polymer on a scale muchlarger than
m
,the polymer willshow ideal behavior. The volume interactions are screened on the length
scale
m
,uctuations onlargerscalesare suppressed. Theaverage end{to{enddistance can be
estimated by R 2
e
2
m
N=g and nallyyields
R 2
e
(c)Nb 2
c 1=4
c
?
c: (1.18)
Ofcoursethenotionofblobsbecomesmeaninglessassoonasthemeshsizebecomescomparable
to the link size
m
b. In this case the volume interactions are totallyscreened, uctuations
suppressed anda meaneld descriptioncorrect. Thus asecondcrossover concentrationexists,
that can be estimated from the breakdown of meaneld theory and amounts toc
??
v
0
=b 6
.
1.1.4 Dynamics of Polymer Chains
So far we have considered only time independent properties of polymer coils. Now we take a
shortlookatthecharacteristicpropertiesofthedynamicsofthosemacromolecules. Werestrict
the discussion to polymer melts, where hydrodynamic interactions are screened.
The simplest and most common modelconcerning polymer dynamics is the Rouse model.
IntheframeworkofthismodeloneconsidersanidealGaussianchain,neglectingalltopological
constraintsandhydrodynamicinteractions. TheequationofmotionforthebeadsofaGaussian
chain on a time{scale much larger than typical atomistic relaxation times is then given by a
Langevin equation
m d
2
y
i
dt 2
=f (ch )
i +f
(fr)
i +f
(st)
i
: (1.19)
The termsinthis equationhavethefollowingmeaning: Thechainforceisgiven by the respec-
tiveterms of the potential U in eq. (1.8)
f (ch )
i
=k
B T
3
b 2
(y
i+1 2y
i +y
i 1
): (1.20)
The friction due to the solvent and other beads is assumed to be proportional to the bead
velocity
f (fr)
i
=
dy
i
dt
; (1.21)
where we dened the friction coeÆcient . The stochastic force f (st )
accounts for thermal
excitationsof the beads orthe solvent. Its averageis zero, otherwise the forcewould lead toa
drift. Theprobabilitydistribution off (st )
isassumed to beGaussianand delta{correlatedwith
respect to time, spatialcomponents and dierent beads, thus
D
f (st)
i E
=0 and D
f (st )
i (t)f
(st )
j (t
0
) E
=2Æ
ij Æ
Æ(t t 0
): (1.22)
This kind of stochastic force is consistent with the uctuation{dissipation theorem. As the
friction acting on the beads is much stronger than the inertia forces, the term / d 2
y =dt 2
is
neglected. Therefore the set of dierentialequations tobe solved is
dy
i
dt
=k
B T
3
b 2
(y
i+1 2y
i +y
i 1 )+f
(st)
i
; (1.23)
with obvious modicationsat the chain ends for i=0 and i=N. This linear set of equations
can be cast into a diusion type equation in the continuum limit and solved by Fourier tech-
niques, i.e.anexpansionof the solutionintoRouse modes. The solutionyieldsacharacteristic
set of relaxationtimes [Gro94]
1
= N
2
b 2
3 2
k
B T
;
p
=
1
=p 2
; p=1;2;:::;N; (1.24)
where
1
is the largest relaxation time and grows quadratically with N. The mean square
displacementof the beads is proportional tothe square rootof time for times t
1 ,
[y
i
(t) y
i (0)]
2
'
12k
B T b
2
t
1=2
(1.25)
For times t
1
the behavioris governed by the diusion of the center of mass r. The mean
square displacementof r isfor alltimes t given by
[r(t) r(0)]
2
=6Dt; with D= k
B T
N
: (1.26)
This is a normal diusion process with a diusion constant D / N 1
. Although the Rouse
model is very simple and is not valid for the description of the real dynamics of polymers in
solution,theshort timebehaviorofpolymerchains indense systems,canbeanalyzedinterms
of the Rouse model.
A major shortcoming of the Rouse model concerning the dynamics in a polymer melt is
the neglect of topologicalconstraints, that arise from the interactions between the chains and
restrict the motion of the beads. Contacts between two polymers that hinder their motion
signicantly,are called entanglements, but most contacts are non{entangled. These entangle-
ments lead to the formation of an eective tube around the contour of the macromolecule,
with atubediameterd
t
that signicantlyexceeds the size ofamonomer. Dening the average
number of links between two entanglements by the phenomenological parameter N
e
, the tube
diameteris determined by d
t
p
N
e
b. The contour of the axis of the tube is called its prim-
itive path. As the tube follows the contour of the polymer but only on a coarser scale d
t , the
primitivepath is coiled in a Gaussian state, too. The contour length L
t
of the primitivepath
is given by L
t
Nd
t
=N
e
Nb=
p
N
e .
In the reptation model one considers a polymer motion, that resembles a diusive creep
along the tube. The friction
t
for such amotion is N times larger than the friction acting on
a single bead
t
N. According to the Nernst{Einstein relation the diusion coeÆcient is
given by D
t
=k
B T=
t
. Therefore we can estimate the time
d
needed for a polymer to leave
the original tube to which itwas conned at t=0by
d
L 2
t
D
t
b 2
N 3
k
B TN
e
: (1.27)
Thistime
d
iscalleddisentanglement timeordisengagementtime,becauseonlargertimescales
entanglementsdonolongerrestrictthedynamicbehavior. Theself{diusioncoeÆcientforthe
reptation model is given by D
self R
2
e
=
d k
b TN
e
=N 2
yielding astronger N{dependence as
inthe Rouse model. Thusdue toentanglementeects the diusion of longpolymersis slowed
down dramatically.
Thereisadiscrepancybetweenthepredictionofthereptationmodelandexperimentaldata
concerningthe N{dependence of
d
. Whilethe reptationmodelpredicts
d
N
3
,experiments
yield
d
N
3:4
. This discrepancy might be due to the fact, that in experiments a broad
crossoverregimeisobservedthatshouldonpracticallyunobservabletimesnallyyield
d
N
3
.
Neweranalytical resultsincludingreptation and contour lengthuctuations of the tubeshow,
that such acrossover actually can occur in the reptation model, too [Min98].
The overall behavior of the mean square displacement of a bead, as a function of time is
depicted in g. 1.2. For very short times the motion of the monomers is not restricted by
the entanglements. Therefore the Rouse model applies. From the mean square displacement
according to the Rouse model eq. (1.26), and the tube diameter d
t
p
N
e
b, it is possible to
estimate the entanglement time
e d
4
t
=k
B Tb
2
. This is the rst time when entanglements
begin to restrict the dynamics. Afterwards the polymer relaxes by one dimensional Rouse
dynamics along the tube, until the time exceeds
1
. In this time regime a result analogous
to eq. (1.25) in one dimension holds for the mean square displacements along the contour,
yy ii (t) ( 〉 〈 − )² (0) ln
d t 2
N b d 1/ 2 t N b 2
τ τ
e 1 d
ln t
1/4 1/2
1/2
1
τ
Figure 1.2: Schematicbehaviorofthemean squaredisplacementasafunctionof timeina polymer
melt, after [Doi88]. Three characteristic times,theentanglement time
e
,theRouse time
1
and the
disentanglement time
d
are marked in, as well as the respective mean square displacements. The
numbers state the slope of h(y (t) y(0)) 2
i in doublelogarithmic representation, corresponding to a
powerlaw.
h(s
i (t) s
i (0))
2
it 1=2
. BecausethetubeitselfformsaGaussiancoilonascalelargecompared
to d
t
, it follows that h(y
i
(t) y
i (0))
2
i d
t hjs
i
(t) s
i
(0)ji d
t t
1=4
. Thereafter the behavior
is governed by the creeping diusion along the tube. Finally, on time{scales larger than the
disentanglement time
d
the entanglements restrict no longer the diusion. Throughout the
rest of this work we are onlyinterested inprocesses, that occur ontimes t
d .
1.2 Phase Separation
Considering a simple system of two components A and B there are two macroscopic states
in which the system can be found, a homogeneous { mixed { state and an inhomogeneous {
demixed { state. The phase diagram describes which state is found with respect to external
macroscopicvariables. Thelineseparatingthetwostatesinthephasediagramiscalledbinodal
or coexistence curve. Parts of the phase diagram, in which the system is thermodynamically
unstable, are subdivided by the spinodal line, see g. 1.3for aschematic illustration.
1.2.1 Flory{Huggins Theory
A simple description of the phase separation of binary polymer blends in terms of a lattice
model was given by Flory and Huggins, see e.g. [Bin94]. This model covers many essential
aspects and isoften used asa reference, therefore we describe itbriey.
In this model all lattice sides are occupied either by an A{monomer or a B{monomer,
cf. g. 1.4. Only nearest neighbor monomers interact. The interaction energies are given by
T c
T f
binodal
spinodal
φ T
φ 1 φ s1 φ c φ s2 φ 2 φ
f b ( φ , T f )
unstable meta-
stable
meta- stable
Figure 1.3: Phasediagram and freeenergy ofa binarymixture.
^
AA
;^
BB
and ^
AB
according to the kindof neighbors. The connectivity of the chains leads to
a restriction with respect to the possibilities of depositing monomers on the lattice and thus
toamodicationof the entropy of the system with respect to simple\atomic"mixtures. This
kindof modelleads tothe following meaneld expression for the free energy of the system
f
FH
k
B T
=
A
N
A ln
A +
B
N
B ln
B +
A
B
: (1.28)
Here
A
=M
A N
A
=(M
A N
A +M
B N
B
)and
B
=M
B N
B
=(M
A N
A +M
B N
B
)denotethemonomer
concentration of the respective components. The number of A and B polymers isdenoted by
M
A
andM
B
,therespectivechainlengthsaregivenbyN
A
andN
B
. Themixtureisincompress-
ible(
A +
B
=1). The rst two parts of expression (1.28) denotethe entropy of mixing. For
large chain lengths N
A
;N
B
1 this entropy is small, thereforebinary blends of long polymer
chains demix easily. The lastterm in eq.(1.28) accounts forthe energydue tothe interaction
ε AB ε BB
ε AA
Figure 1.4: Illustrationof thelattice polymermodelusedinFlory{Hugginstheory.
of the dierent components. The dimensionless Flory{Huggins parameter is
(T)= z
k
B T
^
AB
^
AA +^
BB
2
; (1.29)
where z denotes the coordination number of the lattice. This is the original form of for
the lattice model, but more generally can be treated as a phenomenological parameter.
Experiments yieldtypically
(T)=C
1 +
C
2
T
; (1.30)
and values in the range 10 4
< < 10 1
. In the following we restrict our discussion to the
case N
A
=N
B
=N+1and N 1, suchthat N+1'N.
The binodal can be calculated from eq. (1.28) by Maxwells construction leading to the
followingimplicit equation
N(T) =2arctanh() (1.31)
where for convenience we introduced the order parameter =
A
B
. This equation has
one single solution corresponding to a stable minimum in the case (T) <
c
= 2=N at the
critical concentration
c
= 0. Otherwise ( >
c
) three solutions exist, where the solution
= 0 yields a maximum, the other two solutions
1
;
2
have to be determined numerically
and yield stable minima. The stable solutions determine the binodal in dependence on or
T. The system decomposes into two homogeneous parts with concentrations
1
and
2 . The
size of the two phases is given by the lever rule. The coexistence curve depends only on the
parameterN(T)ascanbeseen directlyfromeq.(1.31). Particularly,the criticaltemperature
T
c
corresponding to (T
c
)
c
= 2=N is proportional to the chain length, T
c
N. The
spinodalcan be determined explicitelyby the condition @ 2
f
FH
()=@
2
=0 yielding
sp
c
= 1
1
2 or
T
sp
T
c
=1
2
; (1.32)
where inthe last step we used (T)/1=T as ineq. (1.29).
An alternative and generic formof a freeenergy is given by the Landau expression
f
GL ()
k
B T
= 1
2 a(T)
2
+ 1
4 b
4
: (1.33)
It is possible to identify a(T) and b in terms of and N, inregions of the phase diagramnot
too faraway from the critical point. Expanding eq. (1.28)to fourth order in yields
a(T)= 1
2
[(T)
c
] ; b = 1
3N
: (1.34)
1.2.2 Ginzburg{Landau Theory
Intheprevioussectionwedescribedequilibriumpropertiesofabinarypolymerblend. However,
when we thinkabout ademixing process, where the system starts froma homogeneous phase
andseparatesintotwophases,inhomogeneousstateswilloccur. Therefore theorderparameter
willbealocalquantity (y),where(y) denotes aconcentrationaveragedover asmallpart
of space. It is possible to dene a constrained partition sum Z[(y)] where all microscopic
states consistent with the order parameter eld (y) are integrated out. From this denition
a functional of the free energy F[(y)] = k
B
T lnZ[(y)] can in principle be derived. Since
a direct calculation of F[(y)] is not possible, phenomenological expressions are used, as for
example,
F
GL [(y)]
k
B T
= Z
V d
d
y
f
b ()+
1
2
b (r)
2
: (1.35)
The squared gradient term accounts for contributions of interfacial tensionto the free energy,
provided that the spatialvariationsof are small. For the bulk free energy density f
b
() one
can, under the assumption of local equilibrium, use the Landau expression (1.33), but now
with a spatialvarying order parameter.
This kind of description is very general and valid for a much broader class of materials.
The chain length N entering via the eective parameters a(T) and b, eq. (1.34), is the only
parameter specic for polymers. Therefore one cannotexpect, that the results can yield more
than a qualitative description of polymer systems. To improve this kind of description the
Flory{Huggins expression (1.28) isused for the bulk freeenergy. Further on, a dependence of
the prefactorof the squaredgradient term
B
onthe actual density istaken intoaccount, in
suchaway thatinthelimitof longwavelengths andweakuctuationsÆthe inversestructure
