AB L
x L
y
2L
z
2sin L
k
: (6.15)
The transitionbetween both structures occurs, if F =0,resultingin
L
z
L
k
=
4sin
: (6.16)
ThecrossoverregimebetweenthetwostructuresintheL
z {L
k
diagramisbetweenthetwolines
with slope1=4 (for !0) and slope=8(for !=2). Bothlines are marked ing. 6.12 for
comparison, showing agreement with the simulations.
This result is also conrmed by experiments of an AC demixing process in thin lms on
an ABC{triblock copolymer brush, cf. the experiments of Fukunaga et al. [Fuk00] described
in sec. 1.4. The dierent geometry in the experiment yields only a dierent constant in the
righthandsideofeq.(6.16). Moreover,intheexperimentthesubstrateitselfconsistsofpolymer
block of type A and type B, therefore eectively only one interfacial tension
AB
has to be
considered and the ratio L
z
=L
k
can be determined directly.
4 6 10 8 20 30 40 50
6 8 10 20 30 40 50
L z
L ||
partial structure
full structure
Figure 6.12: Regions inthe L
z {L
k
plane, where the domainpattern inthe slab is fully structured
() and only partially structured (), in double logarithmic representation. The dividing lines are
given by the two limiting cases of eq. (6.16), L
z
= 1=4L
k
(straight line) and L
z
= =8L
k
(dashed
line).
6.4.2 Pattern Directed Spinodal Decomposition
In thissection weconsider thekinetics ofasystem similartothe onedescribed inthe previous
section, but for xed fraction f
A
=0:5. The initialconditions are asin sec. 6.3.3.
The intermediate scattering function
I
y (k
y
;z;t)= Z
1
0 dk
x I
k (k
k
;z;t); (6.17)
with I
k (k
k
;z;t) given in eq. (6.10) characterizes the domain formation with respect to the
pattern{direction.
t = 1000 MCS (a)
0 0.1 0.2 k y 0.3 0.4 0.5 0.6 0
1 2 3 4 5 6
z
1.0 10 -3 2.0 10 -3 3.0 10 -3
I y (k y ,z)
t = 10000 MCS (b)
0 0.1 0.2 k y 0.3 0.4 0.5 0.6 0
1 2 3 4 5 6
z
1.0 10 -3 2.0 10 -3 3.0 10 -3 4.0 10 -3
I y (k y ,z)
Figure 6.13: Intermediate scattering function I
y (k
y
;z;t) for a system of L
k
= 25 and L
z
= 6 at
times (a) t= 1000 MCS and (b) t =10000 MCS. The patterned structure is stable throughout the
whole lm.
The situation corresponding to a full structure in equilibrium is given in g. 6.13, where
I
y (k
y
;z;t) is shown for a thin slab of thickness L
z
= 6 and pattern periodicity L
k
= 25,
at two times. At early times (t = 1000 MCS, see g. 6.13a), a dominant structure arises
near the patterned substrate on the length scale k 1
y
= L
k
=2. This structure propagates in
z{direction until it reaches the opposite wall (t =10000 MCS, see g. 6.13b). At later times,
this structure is stable, as one would expect from g. 6.12. We determined no further change
inI
y (k
y
;z;t) untilt =30000 MCS.
Figure 6.14 shows I
y (k
y
;z;t) for a lm of thickness L
z
= 25 and pattern periodicity
L
k
= 12:5. This case corresponds to a partial structure in equilibrium. The pattern{induced
demixing is much faster than the lateral spinodaldecomposition. The pattern{induced
struc-t = 1000 MCS (a)
0 0.2 0.4 k y 0.6 0.8 1 1.2 0
5 10 15 20 25
z
2.0 10 -5 4.0 10 -5 6.0 10 -5 8.0 10 -5
I y (k y ,z)
t = 30000 MCS (b)
0 0.2 0.4 k y 0.6 0.8 1 1.2 0
5 10 15 20 25
z
2.0 10 -5 4.0 10 -5 6.0 10 -5
I y (k y ,z)
Figure 6.14: Intermediate scattering function I
y (k
y
;z;t) fora system ofL
k
=12:5 and L
z
=25 at
times (a) t= 1000 MCS and (b) t =30000 MCS. Lateral domaincoarsening occurs away from the
vicinityof thepatterned wall. Near thewallthestructure changes onlyweakly.
(a)
L || = 6
0 0.3 0.6 k y 0.9 1.2 1.5 0
2.5 5 7.5 10 12.5
z
1.0 10 -5 2.0 10 -5 3.0 10 -5 4.0 10 -5 5.0 10 -5
I y (k y ,z,t)
(b)
L || = 12.5
0 0.2
0.4 0.6
k y 0.8 0
2.5 5 7.5 10 12.5
z
1.0 10 -4 2.0 10 -4 3.0 10 -4 4.0 10 -4
I y (k y ,z,t)
(c)
L || = 25
0 0.2
0.4 0.6
k y 0.8 0
2.5 5 7.5 10 12.5
z
1.0 10 -4 2.0 10 -4 3.0 10 -4 4.0 10 -4 5.0 10 -4
I y (k y ,z,t)
Figure 6.15: Intermediate scattering function I
y (k
y
;z;t) for early demixing times t = 1000 MCS
and lmthicknessL
z
=12:5. The patternperiodicityis(a) L
k
=6,(b)L
k
=12:5 and (c)L
k
=25.
0 0.5 1 1.5
0 5 10 15 20 25
I y (2 π /L || , z, t) [10 -4 ]
z
50000 MCS 30000 MCS 10000 MCS 3000 MCS 1000 MCS
Figure6.16: IntermediatescatteringfunctionI
y (2=L
k
;z;t)asafunctionoftime. Thelmthickness
isL
z
=25and thepattern periodicityL
k
=50.
ture for small values of z leads to a much larger corresponding peak in I
y (k
y
;z;t), g. 6.14a,
than the disordered lateral domainpattern emerging for larger z. The ordered periodic
struc-ture propagates intothe lmover a nitedistance only(z 6,see g. 6.14b). For latertimes
(t=30000 MCS), the amplitude of the lateral decomposition waves is comparableto the
am-plitudeofthepattern{inducedwaves. However, the peakinI
y (k
y
;z;t)occursatsmallervalues
of k
y
. Thus, the periodicdomainstructure is stablenear the patterned wall, but further away
the domain patterncoarsens.
Such a behavior, i.e. a rapid formation of a patterned equilibrium structure in thin lms,
but alateral domaincoarsening afterwards inthick lms, has alsobeen observed innumerical
treatmentsoftheCahn{Hilliardequationwithappropriateboundaryconditions[Kar98,Kie99]
and in experiments [Kar98]. Moreover the equilibrium structures that have been found in
ref. [Roc99], i.e. large scale phase separation far away from a nanoscopically patterned wall,
but a remainingstructure near the wall, cf. sec. 1.4, agree with the present results.
In both cases considered so far the periodicity L
k
was comparable to or larger than the
bulk demixing length
m
. The question arises of what happens for L
k
m
. In this case,
we expect that the patternedsurface induces structures,which are toosmalltodrive spinodal
decomposition because the interfacial energy of such structures is higher than the free energy
gained by decomposition.
To investigatethe interplay between the twolength scales L
k
and
m
we consider aslab of
thickness L
z
= 12:5 and vary L
k
. We note that strictly speaking,
k;m
is the relevant length
scale for lateral decomposition in a slab. However, for L
z
=12:5 the demixing length
k;m is
close tothe bulkvalue
m
,cf. sec. 6.3.3.
Figure 6.15shows corresponding lateralstructure factors I
y (k
y
;z;t)for t =1000 MCS and
forL
k
=6;12:5;and25. Inthe caseofL
k
=6
m
,g.6.15a,the peakscorrespondingtothe
surface induced structure and the lateral modes are comparable. The demixing process is not
dominated by the surface{induced pattern. Also, the induced structure develops much more
slowly compared to the case of L
k
= 12:5 '
m
, shown in g. 6.15b. In this case, the peak
corresponding tothe ordered structureis afactor ofabout 10higher,and the lateraldemixing
modes arethusalmostinvisible. ForL
k
=25
m
adoublepeakstructureemerges. This can
be interpreted in terms of a pattern directed spinodal decomposition wave, i.e. if one looked
at the concentration uctuations Æc
A
(y) = % 0
A
(y) c
A
in the yz{plane and labeled positive
and negativeregionsofÆc
A
(y) blackand whiterespectively, onewould see acheckerboard{like
pattern [Kie99].
Tosee moredirectlyhowinthelastcasesuchapatternpropagatesintothelm, weplotted
in g. 6.16 the intermediate scattering function I
y (2=L
k
;z;t) as a function of z for various
timestforasystemwithL
z
=25andL
k
=50. Thispicturedescribeshowthe surface{induced
periodicdomain patterns grow in perpendicular direction. Since our lmthickness is limited,
we did not tryto quantify the propagationof this kind of kinetics further. Forvery large lm
thicknesses wewould expectthe usualLifshitz{Slyozovdomaingrowthtooccuratlargetimes.
In this work we described long{time kinetic properties of polymeric systems. Particularly
we considered the case of a binary polymer blend conned between two walls that could be
homogeneousor heterogeneous.
Since the long{time kinetics can hardly be reached within models that resolve the chain
structure of the polymers, coarse{grained models have to be used. We employed two kindsof
treatments: First,weusedadescriptionbasedontimedependentGinzburg{Landautheory,i.e.
onthe Cahn{Hilliardequation(CHE)thatdescribesthe kineticsofthe systemsviaanonlinear
partial dierential equation for the order parameter eld. Secondly, we treated the systems
in the framework of soft particle models. In this case one polymer is mapped onto one soft
particlewithinternaldegrees offreedom. Thechainstructure isnot resolved, butthe polymer
is modeled by a monomer density for a given state { or shape { of the chain. The kinetics of
the system is suppliedby adiscrete time MonteCarlo algorithm.
AgoodphenomenologicaldescriptionofphaseseparationprocessesisgivenbytheCHEthat
inthe case ofconnements issuppliedby appropriateboundary conditions. From the analytic
solution of the linearized problemthat is valid for short times, a criterion for the suppression
ofsurface directed spinodaldecompositioninthin lmscouldbe derived. Accordingtothis no
SDWoccurs if thelmthickness issmallerthan halfthe criticalwavelength, L<L
crit
==k
c .
A numerical solution of the CHE is complicated by the fact, that the corresponding line
system is sti. This strongly restrictsthe size of the possible time{steps for explicit methods
in order to avoid numerical instabilities. This problem was solved by applying an implicit
time{steppingscheme that allows toexplore the whole kinetics ofphase separation until
equi-librium. The eÆciency of the method is particularly of importanceconsidering the new time
scales arisinginconned systems. Especiallythe perioddoublingprocess occurringin thelate
stages of quasi one dimensional phase separation could be explored. However, the implicit
method involves the inversion of huge matrices inthe solution of coupled nonlinear equations
by Newton's method and therefore isat the moment practicablyintwo dimensions only.
Therefore soft particle models have been developed, based on ideas by Murat and
Kre-mer[Mur98 ]. Herebythefreeenergyfunctionalofthesystemisdecomposedintoan
intramolec-ular and an intermolecular part, the rst being determined by the probability for the internal
states of the particle, the second by the intermolecular interaction energy of the monomer
densities. However, instead of using self{avoiding chains as an input, as in ref. [Mur98], in
the present work Gaussian chains providethe underlyingchain model. The self{interactionof
the chains, i.e.interactionof dierent monomersof one polymer,is treatedconsistentlyonthe
same footingas the interaction of monomers belongingtodierent polymers.
The main advantage of this approachis that the model nowfullls the basic scalingrelations
of polymer systems without the necessity toadjust the interaction parameter as a function of
chain length. Particularly, no simulations of explicit chains are therefore needed in the setup
of the model. Further on, for Gaussian chains the monomer densities can be approximately
described by simplefunctions,allowingforaneÆcientanalyticalcalculationofthe overlapand
making the model highlyportable.
Whiletheinputquantitiesaregivenforsomepossiblesoftparticlemodels,i.e.fora
descrip-tion as spheres, ellipsoids and dispheres, the simulationswere done for the Gaussian ellipsoid
model(GEM). Within the GEM ithas been shown that not only many features characteristic
for polymeric systems can be found, for example, the correlation hole in a homogeneous melt
or the orientation and deformation of the particles' conformations at phase boundaries, but
moreover the model is very eÆcient, such that the late stages of phase separation processes
can bereached.
Finally, conned systems were studied with the GEM. For a binary blend conned
be-tween two homogeneous neutralwallsan increase of the lateral demixing length
k;m
and the
corresponding demixing times with decreasing slab thickness was found for L
z
<
m
. This
supports the results based onthe CHE, but now in a more realistic model, and suggests that
experimentalstudies of this eect shouldbe promising.
The case, where one of the two walls is structured, corresponds to experiments that try
to transfer the pattern of the substrate on the domain pattern. Two dierent equilibrium
patterns are found. Either the structure propagates through the whole lm, leading to a
striped demixingpattern(\fullstructure"), orthesurface induced structureoccurs onlyinthe
vicinity of the wall(\partialstructure"). Which patternemerges, depends on the ratio L
z
=L
k
only. This is explainedby a considerationof the involved surface energies.
Dierent equilibrium patterns lead to dierent kinetic pathways for spinodal
decomposi-tion insuch a lm. In the case of anequilibrium patternwith full structure, pattern directed
spinodal decomposition is the dominant process. The periodic ordered pattern freezes after
reaching the opposite wall. On the other hand, the penetration depth of the pattern directed
spinodal wave remains nite and lateral domain coarsening takes place in the late stages of
decomposition. In the case of small pattern periodicity (L
k
<
m
), pattern directed spinodal
decompositionis less pronounced, asthepatterninducesunfavorable structureswith toolarge
interfaces.
Insummary, weshowed that thekindof softparticlemodels developedinthis work supply
a useful and very eÆcient tool, to study polymer systems on large time and length scales.
While a direct connection between soft particles and Gaussian chains exists, no simulations
of systems of chain models are needed in the implementation. The models are especiallywell
suited, tostudy phase separation processes.
Therearemany applicationsthatcan bethoughtof. Firstofall,itshouldbeinterestingto
study diblock copolymers with the disphere model. In such systems it is diÆcult to simulate
even the phase diagram with chain models, because a possible incommensurability between
the periodicity of the microphase separated structure and the system size can lead to strong
nite size eects [Bin94, Bin99]. Therefore large systems are needed in the simulations that
can be provided by soft particle models. The inuence of the morphology on the long{time
still poorly understood froma theoretical point of view. In this respect simulations based on
the disphere modelmay help toidentify the underlying processes.
From amore conceptualpointof view, itisimportanttoexamine thehierarchy ofpossible
multisphere models that havebeen proposed in sec. 3.6. This would not only leadto adeeper
understanding of the second coarsening step, but provide a more direct connection to chain
models and make a remapping, i.e. a substitution of soft particles by chain molecules, easier.
Moreover, it could be thought of basing a thermalizationprocedure for explicit chain models
on this hierarchy. Large scale structures could hereby be thermalized on a very coarse scale,
while smallscale structures could be treatedon aner scale.
Here wegive the discretization of the Cahn{Hilliard equation (2.23) and the boundary
condi-tions eqs. (2.26a,b) and their static version eqs. (2.28a,b). Since the CHE(2.23a) is validalso
atthe boundary, eqs. (2.26a,b) are rewritten by
@
for all times t. In this section we suppress the time argument wherever it is convenient.
Denotingthenumberofgrid pointsbyN
x
1)in slab geometry. The space grid is
dened by
the set of interiorgrid pointsis dened by
and the set of boundary grid points isgiven by
Æ
Forthe grid functionswe use the notation
i;j
(t):=(ix;jy;t);
i;j
(t):=(ix;jy;t): (A.5)
To deduce an appropriate semi{discrete model for the underlying continuous problem, the
spatial derivatives of somefunction f =f(x;y)are replaced by standard symmetricdierence
formulae of order 2. In terms of the discrete f
i;j
Here we dened := x=y. The discrete versions of the continuous eqs. (2.23) yield a
N
x N
y
dimensional system of coupledordinary dierentialequations
@
). In the determination of @
t
, virtual variables occur, corresponding to grid points that are not in
x;y
. These
virtual variableshave tobereplaced by the boundaryconditions. In the bulkcase the virtual
variablesin eq.(A.8a) can bereplaced by the discrete periodicboundaryconditions
In the caseof aslab eq. (A.10)is replacedby thediscretizationof the no{ux condition(2.25)
Note, that the splittingof the originalCHEinto twoequations withan explicitrepresentation
of leads to a convenient use of the no{ux condition to eliminatethe virtual variables. To
determine the virtual variablesin eq. (A.8b), we discretize the second boundary condition. In
the bulkcase this leads to
In the case of a slab, we have to discretize eqs. (A.1a,b). We start with the lower boundary
(y =0)and obtain
Solving this equation for
i; 1
and using eq. (A.11) yields
Note, that by periodicity
1;0
. Inserting eq. (A.15) in eq. (A.8b)
for j =0 yields
i;0
with the matrix
A I+
where I denotes the identity matrix. The inhomogeneity is given by
An analogous calculation forthe upper boundary (y =N
y
1)yields
A
SinceAispositivedenite,eqs.(A.17,A.20)haveauniquesolutionforevery ;xand
s
>0.
Theinverse matrix A 1
can bedeterminedbeforetime integration by Choleski{decomposition
LL
In the case of quasi{staticboundary conditions (eqs. (2.28a,b) corresponding to
s
!1), no
system of equations has to be solved (A = I). Thus
j;0
vanish ineqs. (A.19, A.21).
To summarize, the problem to be solved numerically consists of the ordinary dierential
equation (A.8a), together with the replacements (A.9, A.10) for the chemical potential in
the bulk case and (A.9, A.11) in the slab case. The variables for the chemical potential
i;j
themselvesare given ineq.(A.8b) withthe replacements(A.12, A.13) inthebulkcase andthe
special representations of
i;0
and
i;N
y 1
given by eqs.(A.17, A.20) inthe slabsituation. The
system of dierentialequations iscompleted by the initialvalue condition
i;j
(0) =(ix;jy;0): (A.23)
B.1 Averages of Gaussian Chains
HerewepresentmomentsoftheeigenvaluesSasobtainedbyMonteCarlosimulation. TableB.1
displays results for chains of length N = 1000 averaged over 10 7
dierent realizations. The
data for the radius of gyration R
G
are given for the sake of comparison. The deviations from
the exact values for N !1 are due tostatisticalerrors and nite chain lengths.
Table B.1: Moments ofthe eigenvaluesS asdetermined byMonteCarlo simulation.
hR 2
G
i=N 0.16685 hR 4
i=N 0.12758 hS
2
i=N 0.028708 hS 2
i=N 0.010568 hS 2
0.0038714 hS 3
0.00032239 hS 3
2 i=N
3
4:620610 5
0.0013717 hS 3
3 i=N
3
1:933810 6
B.2 Parameters of the Distribution Functions
The determination of the parameters of P
R
momentsoftheheuristicfunctionsarecalculableanalytically. Thereforewedenetheconstants
a
i are exact
in the limit N ! 1. With this denition and the well known results hR 2
=540, cf. [Yam71],we are led tothe two equations
a
At this point we also give a more quantitative test of P
R (R
2
G
) by comparison of the higher
momentsofR 2
G
. The asymptoticexact resultforthethirdmomentishR 6
G i=N
3
=631=68040
0:009274,whilewendhR 6
G i=N
3
=0:009280. ForthefourthmomenttheexactresulthR 8
G i=N
4
=
1219=4082400:002986 has tobecompared with hR 8
G i=N
4
=0:003000.
For P
(S
)the same procedure leads to
hS
SincenoexactanalyticalresultforthemomentshS m
iexists toour knowledge,thesequantities
aredeterminedbyMonteCarlosimulationsofGaussianchainsoflengthN =1000. Theresults
are given in tab. B.1.
B.3 Improved Formula for the Monomer Density
For the sake of completeness, we present animproved heuristic formula for the density of the
x
A least squares tof the constants yieldsu
2
=0:674 and 0
2
=0:748.
B.4 Improved Calculation of P
C
Here, we present an improved approximation for P
C
). On this behalf, we make
the ansatz
P
whichisinspiredbythecumulantexpansion. ThecoeÆcients 2
aretobedetermined
by the conditional momentsdened ineq. 4.26.
It is possible, to determine approximatively higher moments than the second one from
eq. (4.19),
In the determinationof the fourth and sixth momentsome integralsare neglected, that could
not be solved exactly. We checked the accuracy of the expressions (B.5,B.6) by Monte Carlo
simulations and found a satisfying agreement, the relativeerror being smaller than about 0:1
in the regions of high probabilityP(R 2
A
;R 2
B
),justifying the result a posteriori.
ThenthecoeÆcients 2
aregiven intermsofthe conditionalmoments(4.27,B.5,
B.6). Astobeexpectedforthiskindoftreatment(acutocumulantexpansion),thepolynomial
term is negative for large u
AB
and thus also P
C
. This failure can be compensated
by increasing the degree of the polynomial by one and requiring, that this polynomial has a
minimum,wherethe originalpolynomialgoestozero,thusdeterminingthe coeÆcienta
3 . The
B.4 Improved Calculation of P
C (r
AB
;R
A
;R
B )
spurious error for large u
AB
can be minimizedby dividing the polynomial by a suitable term,
leadingto
P
C (r
2
AB
;R 2
A
;R 2
B )=C
r
r 2
AB
2
3
2
0
3
2
exp
3
2 u
AB
1+a
1 u
AB +a
2 u
2
AB +a
3 u
3
AB
1+a
3 u
3
AB
: (B.7)
AswearemainlyinterestedindeterminingF
intra
,thedeterminationofthenormalizationfactor
C is of minor importance, because it leads to an additive constant only. (If necessary, it can
be determined eithernumerically orin terms of amodied hypergeometricfunction.)
In principle, a systematic polynomialexpansion of this kind is possible, the coeÆcients a
i
being determined by higher conditional moments.
C.1 Ellipsoid{Ellipsoid Interaction
Inthissectionwederiveanexplicitexpressionfortheinteractionbetweentwoellipsoids,dened
ineq. (3.5) in the case that the monomer densities are given by eq. (3.17).
When inserting eqs. (3.17, 3.16) into eqs. (3.13, 3.5) the calculation of F
inter
reduces to a
sum of nine terms of the form
Z
where the functions (j)
were dened ineq.(3.17) andspecied intab.3.2. Moreover, w (j)
dependingonwhichof thethree termsineq. (3.17a)contributesto (j)
(y ). With
the denitions
expression (C.1) reads
1
it follows that
The quadratic form(C.6) can berewritten as
T(y)=
which yields
Z
C.2 Ellipsoid{Wall Interaction
The integrals in eq. (6.2) with a potential of the form eq. (6.3) and monomer densities, that
are essentially sums and products of Gaussian functions, see eq. (3.17), can be cast into the
more general form already treated in the previous section C.1. The six resulting integrals of
eq. (6.2) are of the form