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Abstract: operations necessary to impose the condition of self consistency. ... verifying the conditions of self consistency and incompressibility of Equations 2.4 and 2.7, (d) updating the potential ﬁeld aiming for these conditions to satisfy and ﬁnally, once these ...

Chapter 1
Introduction
1.1 Overview
A block copolymer molecule is a polymer molecule wherein more than one homopolymers are
joined together by a covalent chemical bond. A block is an extended contour in the polymer
chain that contains the same kind of monomers. Common examples of the linear block copolymer
molecules include diblocks AB, and triblocks such as ABA and ABC. Here A, B, and C refer to
blocks formed of different kinds of monomers.
Materials formed by self assembly of the block copolymer molecules possess a richness in
their structural behavior, and thereby, in their macroscopic properties. For this reason, block
copolymers have appealed to the interests of both the theorists and experimentalists. The chemical
nature of these molecules can be tuned over a wide range by engineering (a) the relative chemi
cal dissimilarity between the monomers of the individual blocks, (b) the relative placement of the
various blocks in the molecule, and (c) the relative sizes of the various blocks. Since these as
pects eventually control the microstructure of their self assembled phases, synthetic expertise can
greatly facilitate designing block copolymer molecules that can ﬁnd use in various technological
applications. Block copolymers have been successfully used to realize materials with novel and
improved mechanical behavior [14, 25]. They have also been used for applications such as ther
moplastic elastomers [73], surfactants [7], solid state batteries [5], meso and nanofabrications
[1, 12, 30, 35, 58, 59, 78, 79, 80], photonic band gap materials [75, 76], and controlled drug deliv
1
Figure 1.1: The microphase segregation in diblock melts. On the left the high temperature dis
ordered phase is shown. Arrows indicate a decrease in temperature or an increase in segregation
strength characterized by the χparameter.
ery [23, 53, 71]. Structural richness of block copolymers has been observed in both undiluted ﬂuid
state, also referred to as melts, and in solutions. A signiﬁcant effort has gone into devising theoret
ical and computational tools to understand the major thermodynamic forces that dictate their self
assembly, and predicting the microstructure in the self assembled block copolymers, which has
also proved very successful. In this thesis, the formulation and implementation of some theoretical
tools relevant to diblock melts will be presented.
A unique aspect of block copolymer physics is their ability to undergo microphase segrega
tion. Microphase segregation in block copolymers is schematically depicted in Figure 1.1. Block
copolymer melts as well as homopolymer mixtures form a homogeneous phase at sufﬁciently high
temperatures. At some lower transition temperature, the molecular species in a homopolymer mix
ture undergo macroscopic phase separation (also called macrphase separation). However, in block
copolymer melts, macrophaseseparation of two dissimilar blocks is prevented by the permanent
covalent bond that links them together. This leads to segregation of the monomers into different
2
microscopic domains each rich in local density of different types of monomers. The junctions be
tween the blocks chieﬂy reside on the interface of these microdomains. The interfaces between
the microdomains are not necessarily ﬂat. The interfacial energy depends on the monomeric re
pulsion, and a curved interface would result from the tendency to minimize the interfacial energy.
However, a curvature in the interface crowds up the chains on one side, and thereby results into an
increased stretching energy of one of the blocks due to reduced entropy. Therefore, an equilibrium
morphology results from a balance between these two opposing effects.
1.2 Self consistent ﬁeld theory (SCFT)
SCFT is a mean ﬁeld treatment of a system of interacting polymers wherein one assumes that each
chain in such a system effectively behaves like an isolated chain in a mean chemical potential ﬁeld.
This assumption makes the problem more tractable as it reduces it to a study of behavior of a
single chain. In regards to the mathematical structure and the assumptions involved, this theory is
analogous to the Hartree theory of the electronic structure in solids.
In SCFT formalism, every segment of the polymer chains is assumed to live in a ﬁctitious
chemical potential ﬁeld that mimics its interactions with the rest of surrounding monomers. In
general, this chemical potential ﬁeld is subject to the thermal ﬂuctuations resulting from the cor
responding ﬂuctuations in the monomer densities. The mean ﬁeld picture ignores the thermal
ﬂuctuations and assumes that the monomers live in a mean chemical potential ﬁeld. It is also be
lieved to become exact for a dense melt in the limit of inﬁnite chain lengths. The assumed mean
ﬁeld can be spatially homogeneous or inhomogeneous. The FloryHuggins theory that describes
the homogeneous disordered phase corresponds to a spatially homogeneous ﬁeld. In case of the
microsegregated structures in diblocks, the chemical potential mean ﬁeld is inhomogeneous.
In SCFT, it is assumed that the total free energy of an interacting chains system is a functional
of the average monomer density ﬁelds ρ α (r), and can be expressed as the sum of two parts:
F [ρ] = Fid [ρ] + Fint [ρ] (1.1)
3
The ﬁrst term Fid is the free energy of the noninteracting chains in the assumed mean ﬁeld ω,
and the second term Fint is the overall interaction energy which arises due to the local monomer
monomer repulsion. Also, we know that the potential ﬁeld ω couples with the density ﬁeld, such
that
δFid
ωα = − (1.2)
δρα
If Z[ω] denotes the partition function of an isolated chain in the presence of a chemical potential
ﬁeld ω, then Fid , the ideal gas part of the free energy, is obtained by taking its Legendre transform
as follows:
Fid [ρ] = − log Z[ω] − drρα (r)ωα (r) (1.3)
Next, we extremize the free energy functional of Equation 2.1 by setting its functional derivative
with respect to the density ﬁelds equal to zero. This yields the following relation between the
interaction energy and the effective external ﬁeld experienced by the ideal chains:
δFint
ωα = (1.4)
δρα
The Equation 2.4 is the self consistency condition. We deﬁne the interaction energy density f int
which when integrated over the volume, gives F int . The interaction energy density arising due to
local monomer repulsion and the bulk compressibility is taken to have the following form (in units
of kB T ) for a diblock melt:
1 1
fint = χρA (r)ρB (r) + B (ρA (r) + ρB (r) − ρ0 )2 (1.5)
ρ0 2
where ρ0 is the average uncompressed number density of the monomers, χ is a parameter that
measures the monomermonomer repulsion, and B is the bulk modulus of compressibility. The
incompressibility is incorporated by taking a limit of B → ∞ such that the net density at any point
remains equal to ρ0 . The constraint of incompressibility thus leads to a spatially varying pressure
ﬁeld ξ(r) which is independent of monomers, and is deﬁned in the following equation:
ξ(r) = lim B(ρA + ρB − ρ0 ) (1.6)
B→∞
4
In this limit, the self consistency equation of 2.4 assumes the following form:
χρB,A (r)
ωA,B (r) = + ξ(r) (1.7)
ρ0
The equation for the incompressibility condition is
ρA (r) + ρB (r) = ρ0 (1.8)
In terms of the volume fractions φA = ρA
ρ0 and φB = ρ0 ,
ρB
the above condition of incompressibility
becomes φA (r) + φB (r) = 1.
There have been several approaches to obtain the mean ﬁeld solutions for the given set of
parameters. One way to achieve a self consistent solution is by iteratively updating the ﬁelds
numerically. This approach is referred to as the numerical SCFT. However, there are two limiting
regimes which are amenable to analytical treatments. These limiting regimes are known as the
strong segregation regime and the weak segregation regime.
The strong segregation analysis applies to the cases with very high χN values where the chains
on the either side of the interface are strongly stretched resulting into very narrow interface [42,
45, 50, 52, 67]. In particular, the interface width is very small compared with the domain size. On
the other hand, the weak segregation anaysis applies to those cases where χN is small, and as a
result, the interfacial width is comparable to the domain size [33].
The strong segregation theory is not considered in this Thesis at any point, and therefore the
same will not be discussed at all. However, the weak egregation theory and the numerical SCFT
methods will be described in detail since these methods have been used in the Thesis to address
some of the questions in diblock copolymer melts. In the weak segregation limit the density varia
tion is small and as a result free energy can be Taylor expanded in terms of the ﬁeld that represents
the density variation with respect to the homogeneous phase. This is essentially a Landau expan
sion with the density variation chosen as the order parameter. The coefﬁcients in the expansion
are the functional derivatives of the expression in Equation 2.1 with respect to the density ﬁelds.
5
Figure 1.2: The diblock phase diagram of the diblock melt obtained by Leibler using weak segrega
tion theory [33]. The horizontal axis corresponds to the volume fraction of either of the monomers.
The phase diagram is symmetric about the symmetric volume fraction of 0.5.
The formalism of the weak segregation theory is described in detail in Chapter 3. The calculation
of the expansion coefﬁcients for the diblocks which supplements the weak segregation analysis, is
described in Appendix A.
The weak segregation theory was ﬁrst applied to block copolymers by Leibler [33]. Leibler
showed that the phase behavior in diblocks is found to be governed by two independent parame
ters, χN and φ, where χ is a parameter that charatcerizes the repulsion between two chemically
dissimilar monomers, N is the degree of polymerization of the diblock copolymer molecule, and
φ is the volume fraction of one of the monomers. The weak segregation phase diagram obtained
by Leibler is shown in the Figure 1.2. The equilibrium morphologies considered by Leibler in the
diblock melts, also referred to as classical phases, are those of lamellae with onedimensional peri
odicity, cylinders arranged parallely on a twodimensional hexagonal lattice, and spheres arranged
on a threedimensional BCC lattice.
For intermediate segregations, the numerical implementation of the SCFT have been frequently
6
Figure 1.3: The phase diagram of the diblock melts calculated using the numerical SCFT. L, C, S,
and G denote respectively the lamellar, hexagonal cylinders, BCC spheres, and the gyroid struc
tures. The symmetry of the BCC phase belongs to the space group Im3m and the gyroid has the
space group of Ia3d.
employed for melts, blends, and solutions [9, 17, 20, 39, 40, 43, 44, 46, 47, 48, 49, 51, 68]. For
a numerical implementation of the SCFT, one uses the modiﬁed diffusion equation to numerically
obtain the density ﬁelds and the free energy. The modiﬁed diffusion equation and the methodology
of the numerical SCFT is presented in detail in Chapter 4 abnd Appendix B and C.1. In addition to
the classical phases considered by Leibler, various nonclassical phases such as bicontinuous gy
roid network, hexagonally perforated lamellae, and the orthorhombic F ddd networks, which have
been observed experimentally, are also considered as the competing structures in the numerical
SCFT calculations. A numerical SCFT phase diagram for the diblock and the morphologies of the
various phases therein are shown in Figure 1.2.
In the following three sections the three topics that constitute this Thesis are introduced.
1.3 F ddd phase in diblock melts
The ﬁrst topic of this Thesis concerns with the F ddd structure in the block copolymer melts.
’F ddd’ speciﬁes the space group of the periodic orthorhombic structure that has been recently
7
Figure 1.4: Top part shows the phase map of the triblock ISO obtained experimentally [8, 18]. The
lower part is the phase diagram of the same triblock system calculated using the numerical SCFT
[74]. The topology of the various calculated phase regions is seen to agree with the experimentally
obtained phase map.
8
Figure 1.5: The numetrical SCFT phase diagram of the diblock melt calculated by allowing F ddd
to be one of the candidate phases [74]. The F ddd appears stable in the near the critical point of
χN = 10.5 and the volume fraction of φ = 0.5.
found to be one of the equilibrium structures in the triblock systems [8, 18]. The F ddd structure is
also referred to as O 70 because it has an orthorhombic unit cell and the space group F ddd that it
belongs to is the 70th entry in the International Tables for Crystallography. This ﬁnding is ususual
because other known structures in the block copolymer melts such as lamellae, cylinders, spheres,
and gyroid, have also been found in other soft matter systems such as colloids and liquid crystals.
However, this orthorhombic network structure has been reported for the ﬁrst time in any soft matter
system. Top part of Figure 1.4 shows the phase mapping in the triblock system of isopreneb
stryrenebethylene oxide (ISO). A signiﬁcant region near the AC isopleth in the composition
phase triangle is seen to be occupied by the O 70 /F ddd phase [18]. A numerical SCFT calculation
for the ISO triblocks by Morse and Tyler [74] also predicts an equilibrium F ddd structure in the
ISO triblocks as shown in the lower part Figure 1.4. The topology of the phas map obtained by the
numerical SCFT is in agreement with the experiments. Furthermore, Tyler and Morse have also
9
predicted an equilibrium F ddd phase in diblocks melts. The numerical SCFT phase diagram for
diblocks with F ddd considered as one of the candidate structures, is shown in the Figure 1.5. The
equilibrium F ddd structure has a unique unit cell as suggested by the experiments and predicted
by the calculations. The details of this special unit cell will be discussed in detail in Chapter 3.
We note in Figure 1.5 that the range of parameters over which the F ddd structure appears stable
in the diblocks is very close to the critical point. By virtue of being so close to the critical point,
the predicted equilibrium F ddd structure is expected to be very weakly segregated. Therefore it is
possible to analyze this structure in the framework of the weak segragation theory. In Chapter 3,
we ﬁrst present the formalism of the weak segregation theory, and then show that, in the mean ﬁeld
description, the F ddd structure with a special unit cell is indeed a stable structure in any weakly
crystallized material system. In particular for diblock melts, the results from weak segregation
analysis is shown to agree well with the numerical SCFT results.
1.4 Linear response and stability of diblock phases
The second topic of this thesis, presented in Chapter 4, deals with the linear response behavior and
the stability of various diblock copolymeric equilibrium ordered phases, with respect to small plane
wave perturbations. Particularly, we seek to understand how the density ﬁeld of an equilibrium
phase responds to small perturbations in chemical potential ﬁelds. To this end, an approximation
called Random Phase Approximation (RPA) has been applied to estimate the response function
of the ordered phases in the block copolymers. The RPA, originally formulated for studying the
dielectric response functions in solids [56, 57, 60, 61], essentially calculates the self consistent
linear reponse of an equilibrated system. The technique was ﬁrst employed by de Gennes in the
polymer physics [13]. The ﬁrst assumption of this approximation is that the perturbation ﬁeld is
small so that the response remains linear in perturbation. Next, it is assumed that, under an inﬂu
ence of an external ﬁeld, an internal ﬁeld is generated that screens the applied external ﬁeld. As
a result the monomers experience a ﬁeld which is different from the applied ﬁeld. Finally, it is
10
assumed that the reponse function that relates the effective ﬁeld to the local density response is
essentially the one associated with the system of non interacting chains. Thus, calculation of the
response function of a noninteracting chains systems (also referred to as ideal gas response func
tion) becomes a crucial step in the effective response function calculation. The response function
of the noninteracting chains system is calculated by using the perturbation theory. Obtaining the
RPA response function from the noninteracting chains response function involves a set of matrix
operations necessary to impose the condition of self consistency.
The RPA methodology was ﬁrst applied by by Shi et al. to diblock copolymer melts, and these
authors successfully addressed the stability of various classical phases of lamellar, hexagonal, and
BCC. However, their prediction regarding the stability of the gyroid has been controversial. Their
method predicted an unstable gyroid for cases where it is known to be stable from experiments.
This is believed to be a result of an insufﬁciency in their numerical method used for calculating
the noninteracting chains response function. Their method is severely limited by the degrees of
freedom that can be allowed in the system. This makes their method inapplicable to cases of high
segregation strengths and complex structures. We have developed a new method to overcome their
difﬁculty and our results predict a stable gyroid phase where it is expected to be stable.
1.5 Effect of polydispersity
The third topic that is brieﬂy touched upon in this Thesis is the effect of chain polydispersity on
the phase behavior of the diblock melts. The phase diagrams presented earlier in this chapter
correspond to the monodisperse melts. Recent experiments have shown that, on introducing poly
dispersity, diblock copolymer equilibrium phases exhibit some interesting alterations [36]. In par
ticular, the phase diagram for the monodisperse systems, which is symmetric about the symmetric
monomer composition, becomes strongly asymmetric as one of the blocks become polydisperse.
Experiments suggest that if the polydisperse block corresponds to the monomers in minority, then
a phase with a higher interfacial curvature is favored at the same composition. Also, the critical
11
point shifts towards the higher volume fractions of the polydisperse blocks. This behavior has been
observed both in AB diblocks and the ABA triblock systems, where the block B represents the
polydisperse block. Also, adding polydispersity causes an increase in the domain size [36].
Several theoretical and numerical studies have been performed to study the the effect of poly
dispersity on the block copolymer phase behavior [6, 16, 19, 28, 54]. Most recently, Cook and Shi
[11] treated the effect of the polydispersity analytically using a perturbation theory and succeeded
in showing the trends that appear in the phase diagram as a result of polydispersity. In this Thesis,
we perform a numerical calculation on the diblock melts where one of the blocks, A, is assumed
monodisperse and the other block, B, is polydisperse. For the purpose of simplicity in numerical
implementation, we consider such a polydisperse systems as a mixture of many chain species with
varying block lengths of B and volume fractions. We consider a most probable distribution of the
blocklengths of B. The results, presented in Chapter 5, are consistent with the experiments.
12
Chapter 2
Self consistent ﬁeld theory (SCFT)
2.1 The physical premises of SCFT
The self assembly of various chain molecules in a dense polymer melt is assumed to be effected
by an inhomogeneous chemical potential ﬁeld ω α (r). This chemical potential ﬁeld is essentially
a mathematical construct that models the effective local environment of various monomers. It
depends on the local monomer density ﬁelds, which in turn, is dictated by the local χinteraction
between monomers and the condition of incompressibility. Moreover this potential ﬁeld couples
with the monomer density ρα and therefore affects the monomer density ﬁelds as well. Since
both the density and potential ﬁelds are coupled, the desired solution is obtained by imposing
the condition that these two ﬁelds be self consistent. Such a condition is found to arise from the
condition that the free energy be minimum.
SCFT is a mean ﬁeld treatment of the interacting molecular species wherein one assumes
that an interacting multichain system in a ﬂuctuating chemical potential ﬁeld effectively behaves
like a number of noninteracting chains in a non ﬂuctuating mean chemical potential ﬁeld. This
assumption makes the problem more tractable as it reduces it to a study of behavior of a single
chain. It also becomes exact for a dense melt in the limit of inﬁnite chain lengths. The potential
ﬁelds and the density ﬁelds adjust so as to minimize the free energy functional which depends on
these ﬁelds. This minimization leads to a condition on the effective chemical potential ﬁeld that
the noninteracting chains perceive. This condition is called the condition of selfconsistency.
13
In order to mathematically formulate the ideas expressed above, we start with the assumption
that the net free energy of the interacting chains system, which is a functional of the density and
the potential ﬁelds, can be expressed as the sum of two parts:
F [ρ, ω] = Fid [ω] + Fint [ρ] (2.1)
The ﬁrst term Fid is the free energy of the noninteracting chains in the assumed mean ﬁeld ω,
and the second term Fint is the overall interaction energy which arises due to the local monomer
monomer repulsion. Also, we know that the potnetial ﬁeld ω couples with the density ﬁeld, such
that
δFid
ρα = (2.2)
δωα
Therefore, we can express the free energy entirely as a functional of the density ﬁelds only, by
taking a Legendre transform of the Equation 2.1:
F [ρ] = Fid [ρ, ω] + Fint [ρ] − drρα (r)ωα (r) (2.3)
Next, we extremize this functional by taking a functional derivative with respect to the density
ﬁelds, and setting it equal to zero. This yields the following relation between the interaction energy
and the effective external ﬁeld experienced by the ideal chains:
δFint
ωα = (2.4)
δρα
The Equation 2.4 is the self consistency condition. We deﬁne the interaction energy density f int
which when integrated over the volume, gives F int . The interaction energy density arising due to
local monomer interactions and the bulk compressibility is taken to have the following form (in
units of kB T ) for a diblock melt:
1 1
fint = χρA (r)ρB (r) + (ρA (r) + ρB (r) − ρ0 )2 (2.5)
ρ0 2
where ρ0 is the average uncompressed number density of the monomers and B is the bulk modulus
of compressibility. The incompressibility is incorporated by taking a limit of B → ∞ such that
14
the net density at any point remains equal to ρ 0 . The constraint of incompressibility thus leads to
a spatially varying pressure ﬁeld ξ(r) which is independent of monomers. In this limit, the self
consistency equation of 2.4 assumes the following form:
χρB,A (r)
ωA,B (r) = + ξ(r) (2.6)
ρ0
The equation for the incompressibility condition is
ρA (r) + ρB (r) = ρ0 (2.7)
In terms of the volume fractions φA = ρA
ρ0 and φB = ρ0 ,
ρB
the above condition of incompressibility
becomed φA (r) + φB (r) = 1 Having established the criteria of self consistency, one needs next a
method to evaluate the density ﬁeld for any given omega ﬁeld. One uses the statistical mechanics
for this purpose. It is known that average value of any physical quantity for a thermodynamical
system can be obtained from a knowledge of the partition funtion. To this end we ﬁrst deﬁne a
singly constrained partition function q(r, s) which is essentially the statistical weight assigned to
the occurence of such a conﬁguration where the s th monomer of the chain is at the position r. It
can be shown that two relevant quantities, free energy F id and density of the sth monomer, can be
calculated using the singly constrained partition function.
Before we describe the way to obtain the free energy and density of the monomers from the
partition function, we present one of the celebrated equations in polymer physics, known as modi
ﬁed diffusion equation [? ]
∂
( + H)q(r, s) = 0 (2.8)
∂s
which is satisﬁed by the q(r, s) with an initial condition of q(r, s = 0) = δ(r). The Hamiltonian
operator H is given by
b2
H=− α 2
+ ω(r) (2.9)
6
where bα is the statistical segment length of the monomer α that corresponds to the s th monomer.
15
The Fid of a single chain is given by the following equation:
Fid = −kB T ln Q (2.10)
where the quantity Q is the spatially averaged unconstrained partition function deﬁned by the
following equation:
1
Q= drq(r, L) (2.11)
V
Here V is the volume and L is the contourlength of the polymer chain or the degree of polymer
ization. The quantity q(r, s) also satisﬁes the following factorization property:
1
Q= drq(r, s)q † (r, s) (2.12)
V
This is essentially a result arising from the random walk nature of the polymer chain. The quantity
q † (r, s) is the same as q(r, L − s) and carries the meaning of the singly constrained partition
function with contour length measured from the other end of the chain. We solve for both q(r, s)
and q † (r, s) usnig the MDE.
The density of the monomer α is given by the following equation:
1
ρα (r) = dsq(r, s)q † (r, s) (2.13)
QV
where the integration is performed over those s values of the chain which correspond to the
monomer α.
Thus, the overall SCF calculation consists of (a) solving Equation 2.8 for q(r, s) and q † (r, s)
for a given potential ﬁeld ω(r), (b) calculating monomer densities from the Equation 2.13, (c)
verifying the conditions of self consistency and incompressibility of Equations 2.4 and 2.7, (d)
updating the potential ﬁeld aiming for these conditions to satisfy and ﬁnally, once these conditions
are met, (e) evaluating the free energy from the Equations 2.10 and 2.5. The most important and
computationally intensive part of these calcualtions is solving the MDE. There have been two
methods in pratice for solving the MDE: The spectral method, discussed in detail in the Appendix
B, and the pseudospectral method, discussed in Appendix C.1.
16
2.2 Implementation of the numerical SCFT
We now brieﬂy discuss the implementation of the numerical SCFT which is now a well established
technique for predicting the equilibrium morphologies in polymeric systems [Refs]. As mentioned
in the previous section, the three main steps in the numerical SCFT are solving the modiﬁed diffu
sion equation for the given potential ﬁeld, calculating the density, and then updating the potential
ﬁeld with a goal to achieve the self consistency. Solving the initial value problem stated by the
MDE essentially requires efﬁcient techniques to propagate a spatial function along the time like
quantity s, which is the contour length of the chain. This has been realized by two ways in the
block copolymer calculations. The ﬁrst method that we describe in Appendix B is due to Matsen
and Schick [51]. It is called the spectral method which takes advantage of the spatial symmetry of
the concerned strcuture.
The spectral method described in the Appendix B starts with a presumed structure of the mor
phology. The space group of the presumed structure is used to construct the basis functions in real
space that satisfy the symmetry constraints. These basis functions are also known as symmetry
adapted basis functions. The concerned structure is expressed as a chemical potential ﬁeld ex
panded in terms of the symmetry adapted basis functions. After constructing the symmetry adapted
basis functions, the MDE is solved as described in the next subsection, and gives the amplitudes of
the function q(r, s) in the symmetry adapted basis function representation.
The other method of propagating q(r, s) along the chain involves a clever splitting of the
Hamiltonian operator which is sometimes called pseudospectral method [65]. The pseudospectral
propagation scheme is described in the Appendix C.1. This method performs part of the calculation
of q(r, s) on a real space grid and part of it in the reciprocal space. This method exploits efﬁcient
Fast Fourier Transform (FFT) operations.
17
2.3 SCFT in the weak segregation limit
The SCFT can be analytically formulated in the weak segregation limit. In this limit, the density
variations in the inhomogeneous system are approximately sinusoidal sinusoidal. As a result the
Fourier components of the density variations have only one wavenumber. In the weak wegregation
limit, the free energy can be Taylor expanded in the density ﬁelds and the coefﬁcients of expansion
are essentially the functional derivative of the expression in Equation 2.1 with respect to the density
ﬁeld. We discuss the weak segregation formulation in detail in the next chapter.
18
Chapter 3
Landau theory of the orthorhombic
F ddd phase
3.1 Introduction
The strength of the Landau approach to phase transitions lies in its ability to make qualitative
predictions about entire classes of materials, independent of differences in their chemical details [?
]. A Landau theory of crystallization was ﬁrst proposed by Kirzhnits and Nepomnyashchii [? ] and
later by Alexander and McTague (AM)[? ]. The mathematical structure of their theory suggested
some purely geometrical rules for selecting plausible candidate crystal structures for systems that
undergo very weakly ﬁrstorder crystallization transitions. In particular, the theory predicted that
the ﬁrst phase to form upon cooling a liquid of any system with a sufﬁciently weak ﬁrst order
crystallization transition should be a bodycentered cubic (BCC) crystal. More recently, the weak
crystallization theory has been used to predict the equilibrium phases in triblocks [? ] and and
block copolymer thin ﬁlms [? ].
One particularly fruitful application of Landau theory to a crystallization phenomena was the
study by Leibler of the phase diagram of weaklysegregated diblock copolymer melts [33]. Leibler
constructed a Landau theory for weakly ordered diblock copolymer melts, which is similar in
structure to the AM theory, and also calculated the coefﬁcients in the Landau expansion from
an underlying selfconsistent ﬁeld theory of polymer liquids. Leibler considered candidate BCC
19
crystal and hexagonal (H) columnar phases, which were also considered by AM, and a lamellar (L)
phase, which was not explicitly considered by AM. Leibler predicted that cooling of an disordered
(D) liquid phase of a melt of slightly asymmetric diblock copolymers should generally lead to a
phase sequence D → BCC → H → L, consistent with AM’s conclusion that a BCC phase should
form ﬁrst.
Subsequent numerical SCFT calculations by Matsen and Schick [51] established a predicted
phase diagram for diblock copolymers that is consistent with the behavior obtained by Leibler in
the weak segregation regime, and that also includes a gyroid cubic network phase. The predicted
region of stability of the gyroid phase lies between those of the hexagonal and lamellar phases in
moderately segregated systems, but does extend into the weak segregation region in which Leibler’s
theory is rigorously valid. The gyroid phase has also been shown to be stable above a critical value
of χN in an extension of Leibler’s singlewavenumber theory that includes Fourier components of
the order parameter ﬁeld arising from two families of reciprocal lattice wavevectors with slightly
different wavenumbers [? ? ].
Bates and coworkers [8, 18] have mapped the phase behavior of poly(isoprenebstyreneb
ethylene oxide) (ISO) and poly(cyclohexylethylenebethylethylenebethylene) (CE E E) triblocks
copolymers. In both of these systems, they have found a signiﬁcant region in the composition
phase triangle where a orthorhombic network phase having an Fddd space group symmetry is
stable. Tyler and Morse [74] calculated the phase diagram of the ISO triblocks using numerical
SCFT method and obtained results in qualitative agreement with the experimental results.
Tyler and Morse also reanalyzed the phase diagram of diblock copolymers, while allowing for
an F ddd orthorhombic network as one of the candidate phases. They found the F ddd network to
be a stable phase of very weakly segregated diblocks in a narrow window that lies between the Hex
and lamellar phases. The F ddd phase window also borders the gyroid phase, which lies between
the Hex and lamellar phases in more strongly segregated systems. The fact that the F ddd phase
was found to be stable in numerical SCFT only for very weakly segregated diblock melts suggests
20
that the reasons for its stability might be understood within the framework of a Landau theory.
Organization of this Chapter is as follows. In Section 1.2 the basic formalism of Landau theory
of weak crystallization is presented. In Section 1.3, the special F ddd structure that has been found
experimentaly and predicted theoretically is described. In Section 1.4, a simpliﬁed vsrsion of
the Landau theory is discussed and it is shown that the concerned F ddd structure is a candidate
equilibrium structure in any weakly segregated material. In Section 1.5, a more rigorous version
of the theory is applied to the case of diblock copolymer melts. The results of the analytical theory
and the numerical SCFT are compared.
The free energy functional for the F ddd phase constructed under two wavenumber approxi
mation is presented in Appendix. A phase diagram under twowavenumber approximation is not
calculated.
3.2 Landau theory of crystallization
In the Landau theory, we deﬁne an order parameter ﬁeld ψ(r) ≡ δρ(r) that is associated with the
deviation of the ensemble average of a particle concentration ρ(r) from its spatial average value.
This order parameter ﬁeld may be associated with either the concentration of atoms in atomic
liquid, or with a deviation in the local concentration of either of the two types of monomer in a
diblock copolymer melt. In either case, the order parameter ψ(r) in a periodically structure may
be expanded in plane waves as a sum
ψ(r) = ψG exp (iG · r) (3.1)
G
in which G denotes a wavevector that belongs to the reciprocal lattice of the crystal, and the sum is
over all nonzero reciprocal lattice wavevectors. The Fourier coefﬁcients ψ Gi ’s are a measure of or
dering and are chosen as order parameters to describe the phenomenon of weak crystallization. The
ψGi ’s are zero in the disordered phase and are small in a weakly crystallized system. The differ
ence between
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