数理解析研究所講究録 1330 巻 2003 年 10-17

10Mathematical Aspects of Microphase Separation of Diblock
Copolymers
Rustum Choksi
Department of Mathematics
Simon Eraser University
Abstract
Diblock copolymer melts, dubbed “designer materials”, have the remarkable ability to self-Bemble
into various ordered structures. These structures are key to the many properties that make diblock
copolymers of great technological interest. The density functional theory of Ohta and Kawasaki leads toanonlocal variational problem, and presents an excellent setting for the analysis of microphases.
In this note we will first discuss the origins and derivation of this theory, presenting it in connection
with the self-consistent mean field theory. Then, focusing on what is know as the strong segregation
regime, we will discuss some analytical techniques which provide insight on the scales of minimizing
structures (phases). These techniques have the advantage that they are ansatz-free, that is they are notbased upon any preassigned bias for the phase geometry. In particular, we will derive ascaling law for theminimum energy in three space dimensions, and will address properties of optimal structures achieving
this scaling law.
This note includes joint work with X. Ren (Utah State University) and work in progress with G.
Alberti (University of Pisa) and F. Otto (University of Bonn).

1The Physical Problem
Adiblock copolymer is alinear-chain molecule consisting of two subchains joined covalently to each other.
One of the subchains is made of monomers of type Aand the other of type B. Below acritical temperature,even aweak repulsion between unlike monomers Aand induces astrong repulsion between the subchains,causing the subchains to segregate. Amacroscopic segregation whereby the subchains detach from one
another can not occur because the chains are chemically bonded. Rather, in asystem of many such macr0molecules, the immisibility of these monomers drives the system to form structures which minimize contacts
between the unlike monomers and this tendency to separate the monomers into Aand -rich domains iscounter balanced by the entropy cost associated with chain stretching. Because of this energetic competition, aphase separation on amesoscopic scale with Aand -rich domains emerges. The mesoscopic domains
which are observed are highly regular periodic structures; for example lamellar, bcc centered spheres, circulartubes, and bicontinuous gyroids (see for example, [4], [11]). These ordered structures are key to the material
properties which make diblock copolymers of great technological importance.
Three dimensionless material parameters are needed for modeling the microphase separation: , theFlory-Huggins interaction parameter measuring the incompatibility of the two monomers and is inversely
proportional to the temperature; $N$ , the index of polymerization measuring the number of monomers per
macromolecule; and , the relative length of the A-monomer chain compared with the length of the wholemacromolecule. In the mean field approximation, where thermal fluctuations are ignored, one finds that the
microphase separation depends only on the two quantities
and . The phase diagram (either theoreticallyor experimentally constructed) indicates several regimes for the phase separation. In particular, for afixedvalue of one finds with increasing;adisordered regime wherein the melt exhibits no observable phaseseparation, the weak segregation regime (WSR) where the size of the and -rich domains are of roughly of$\mathrm{B}$$\mathrm{B}\sim$$\mathrm{B}$$\chi$$a$$\chi N$$a$$a$$\chi N$

$A$

$B$

11

the same order as the interfacial (overlapping) regions around the bonding points, an intermediate segregation
regime, and the strong segregation regime (SSR) wherein the domain size is much larger than the interfacial
where aslength. In the SSR, it has been observed (cf. [12, 13]) that the domain size scales likethe interfacial length scales like.
$\chi^{1/6}N^{2/3}$$\chi^{-1/2}$

2Ohta-Kawasaki Density Functional Theory(DFT) which reduces to the niinimization
In [22], Ohta and Kawasaki derived adensity functionalof aCahn-Hilliard-like free energy. Following $[21, 8]$ , we write the functional in arescaled, nondimensional
form as afunction of the relative (averaged) macroscopic monomer density (i.e. the difference between the
averaged Aand monomer densities):$\mathrm{t}\mathrm{h}\infty \mathrm{r}\mathrm{y}^{1}$$u$$\mathrm{B}$$E_{\epsilon,\sigma}(u)$$:= \frac{\epsilon^{2}}{2}\int_{D}|\nabla u|^{2}d\mathrm{x}+\int_{D}W(u)d\mathrm{x}$$+ \frac{\sigma}{2}\int_{D}|(-\triangle)^{-:}(u(x)-m)|^{2}d\mathrm{x}$

,

(2.1)

with unit volume
where 6is the Laplacian operator with Neumann boundary conditions; $D$ is asubset of(representing the rescaled physical space
upon which the melt exists); $W$ has adouble well structure
preferring pure Aand phases $(u=\pm 1);\epsilon$ represents the interfacial thickness (suitably rescaled) at the. More precisely, the parameters ,and $B$ monomer intersections; and ais inversely proportional toare related to the parameters , $N$, ,via (cf. [8])$R^{3}$$\Omega$$A$$\mathrm{B}$$N^{2}$$\chi$$a$$\epsilon$$\sigma$$|\Omega|$$\sigma=\frac{36|\Omega|^{2/3}}{a^{2}(1-a)^{2}l^{2}\chi N^{2}}$$\epsilon^{2}=\frac{l^{2}}{3a(1-a)\chi|\Omega|^{2/3}}$

,

(2.2)

denotes the Kuhn statistical length which measures the average distance between two adjacentmonomers. Conservation of the order parameter requires we maintain the constraint

where

$l$

$u$

$\int_{D}ud\mathrm{x}$$=m=2a-1$ .

Prom this functional it is easy to see the incentive for pattern formation. The double-well term prefers
pure phases of or $B$ monomers, but for $m\neq\pm 1$ , the conservation constraint dictates amixture. Transitions
between phases are penalized by the gradient term but the nonlocal term prefers oscillations between phases.
The latter is best seen in one space dimension. Indeed, this functional can be regarded as ahigher-dimensional
analogue of afunctional introduced by Miiller in [20] as atoy problem for capturing multiple scales Let$m=0$ , $\sigma/2=1$ . Setting $u=v_{x}$ , gives$A$$\int_{0}^{1}\frac{\epsilon^{2}}{2}|v_{xx}|^{2}+W(v_{x})+v^{2}dx$

.

(2.3)

is itself a
In particular, in one space dimension the nonlocal energy is in fact local: every function inin (2.1),
term
primitivethenonlocalhigherisderivative. In
dimensions the analogue for thenorm of thewhich for periodic functions on the cube is simply thenorm squared. From (2.3), one can easily see
why the third term induces fine structure. If
asaw-tooth function $v(x)$ with slopes 81 lowers its, suchnorm with increasing oscillations. Hence the minimum energy is zero but is not attained. If
oscillations are penalized and one expects the competition to result in oscillations on afine but specific scale.
$L^{2}$$L^{2}$$H^{-1}$$u$$L^{2}$$\epsilon=0$$\epsilon>0$3Derivation of the Ohta-Kawasaki DFT
In this section we give asummary of the main steps in deriving the free energy (2.1). The purpose is to givethe reader some idea of how one derives such afunctional from the statistical physics of Gaussian chains.
lsae related work in [3], [16], [20]

12

We follow [8] but provide few details. The derivation is based on two steps. The first is what is commonlyreferred to as the Self-Consistent Mean Field Theory (SCMFT) which has been developed and applied over
the years by many researchers, see for example [11], [17] and the references therein. The copolymer melt is
modeled with aphase space of of continuous chains which prefer to be randomly coiled. Thus we consider
aphase space$n$$\Gamma=\{r=(r_{1}, \ldots,r_{n}):r:\in C([0,N],\mathrm{R}^{3})\}$

equipped with aproduct measureconsisting essentially of copies of Wiener measure. The $A(B$respectively) monomers “occuPy” the interval $\mathrm{I}_{A}=(0, N_{A})(\mathrm{I}_{B}=(NA, N)$ respectively). Within thisspace one introduces amonomer interaction Hamiltonian to reflect the immisibilty of the different monomer
types. At this point one can write the associated partition function $Z$ , the free
, and Gibbs
canonical distribution $D(r)$ . Defining the microscopic densities as$d\mu$$n$$\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}-\beta^{-1}\log Z$, $k=A$ , ,$B$$\beta k(X, r)=\sum_{\dot{*}=1}^{11}\int_{\mathrm{I}_{k}}\delta(x-r:(\tau))d\tau$the desired macroscopic monomer densities should be given by$( \rho_{k}(x)\rangle=\int_{\Gamma}\rho_{k}(x,r)D(r)$tip. $k=A,B$(3.4)

None of these can actually be calculated because of the nonlocal character of the Hamiltonian. The SelfConsistent Mean Field Theory is based upon avariational principle whereby the true free energy is approximated by aminimization over aclass of distributions generated by asingle external field $U=(U^{A}, U^{B})$acting separately on the and monomers. More specifically, setting
$A$

$B$

$H_{U}(r)= \sum_{\dot{|}=1}^{n}\sum_{k}\int_{\mathrm{I}_{k}}U^{k}(r:(\tau))$dr.

with the resulting partition function and Gibbs canonical distribution,$Z_{U}= \int_{\Gamma}\exp(-\beta H_{U}(r))d\mu$$D_{U}(r)= \frac{1}{Z_{U}}\exp(-\beta H_{U}(r))$,

one approximates the true ffae energy by minimizing$F(U)= \int_{\Omega}[\frac{V^{km}}{2\rho_{0}}\langle\rho_{k}(x))_{U}(\rho_{m}(x)\rangle_{U}-U^{k}(x)\langle\rho_{k}(x))_{U}]d\mathrm{x}$ $- \frac{1}{\beta}\log Z_{U}$over all external fields $U=(U^{A}, U^{B})$ . Here,$\langle\cdot\rangle_{U}$

.

(3.5)

denotes the expectation with respect to $D_{U}(r)d\mu;\beta$ is the(the Boltzmann constant has been

reciprocal of the absolute temperature measured in units ofset to one); $V^{km}$ represents the interaction Parameters with$(\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y})^{-1}$$\chi=\beta V^{AB}-(\beta/2)(V^{AA}+V^{BB})>0$ ;and $\rho 0=nN/|\Omega|$ (the average monomer density number). The explicit nature of the external field allows
one to compute all the variational integrals via Feynman-Kac integration theory.
The second step entails writing the ffee energy entirely in terms of the macroscopic monomer density.
The&st term (i.e. the interaction term) in (3.5) is already written in terms of the monomer density andnaturally gives rise to the double well energy in (2.1). The main step in turning the second and third termin (3.5) to afunctional of
involves the inversion of the relationship between the dependence of (on
via the linearization about $\beta=0$ (i.e. at infinite temperature). This is done via the solutions to thebackward and forward modified heat equations which come ffom the Feynman-Kac integration theory. The
details are too cumbersome to summarize here but very briefly, this linearization entails convolution ofwith acertain tensor whose Fourier transform can be computed explicitly. We keep only the short and longrange expansions. After some calculations and the introduction of the monomer difference order parameter,
at both the squared gradient and nonlocal term in (2.1), with the respective coefficients reflectin$\langle\rho\rangle$$\rho\rangle u$$\beta U$$\beta U$$,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}\cap\cap\backslash$13

4Scaling Laws
Accepting the free energy (2.1), the natural question arises as to what minimizers look like for small orworded slightly differently, what are necessary properties for configurations to be minimizing. One approach
could be via the asociated gradient flow dynamic equations which for the conserved order parameter
would be the Cahn-Hillia dynamics, formally written as$\epsilon j$

$u$

$rd$

t4

$= \triangle\frac{\delta F}{\delta u}$.

norm (see [10]). Here we take adifferent
More precisely, this is gradient flow with respect to theof the minimum energy and the resulting
scaling
approach, namely adirect method, and address the issue ofconsequences on minimizing structures. That is, we ask: In the material parameter regime of interest, howdoes the minimum energy scale with respect to the material parameters, and which structures attain this$H^{-1}$

optimalThe simplest approach to this question is based upon setting an ansatz for possible structures with afew
degrees of freedom, and then minimizing the free energy amongst these structures alone. This approach,often dubbed domain theory is ubiquitous; for example Landau used it in his study of ferromagnetism andtyPe-I superconductivity (cf. [15]). In the present context of copolymers, this has been done to to determine
the optimal period size ([22], [4]) which one can also infer via formal dimensional analysis ([3]). These, or in terms of $N$ , lke
calculations all yield that the domain width (or periodicity) scales like$N^{2/3}$. This scaling law has been experimentally confirmed in $[12, 13]$ . While these calculations provide alot of physical insight they leave open the fundamental question of what exactly sets the optimal scale. Are
periodic structures truly minimizing or could anonperiodic geometry yet to be observed and constructed byan ingenious theorist result in even lower energy?
To address these questions rigorously, Ohnishi et al [23] worked in one space dimension with the extra
assumption that admissible structures were what they called “ -layered”solutions (see [23] for the definition).
, and
They concluded that within this smaller class, the global minimizer had aperiod of order. Ren and Wei ([24]) recently obtained the same result with no assumption
an energy of orderon admissible structures. In higher space dimensions it is unlikely that minimizing structures are exactly
periodic. What then can one prove? One approach, first used in solid-solid phase transformations (cf. [14]),is via ageometry-independent lower bound on the total free energy. To motivate this, let us go back to anansatz driven calculation. We consider an ansatz of lamellar structures with the periodicity as the only
degree of ffaedom. One can then write the free energy entirely in terms of the material parameters and :. Factoring out (frnomalizing) the conjectured scaling gives$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{h}.\mathrm{n}\mathrm{g}^{7}$$(\epsilon/\sigma)^{1/3}$$\mathrm{n}$$(\epsilon/\sigma)^{1/3}$$\epsilon^{2/3}\sigma^{1/3}$

$d$

$d$

$E_{\epsilon,\sigma}(d)$$E_{\epsilon,\sigma}(d)=\epsilon\S\sigma\# F(d, \epsilon, \sigma)$

One then optimizes in$d$.

to find that, in the parameter regime of interest $(0<\epsilon\leq\sigma<C)$ ,and$d_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}1}\sim(\epsilon/\sigma)^{1}3$$F(d_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}1}, \epsilon,\sigma)\sim 1$

.

(4.6)

For the lower bound, we make no assumption on the domain structure (essentially $u\in H^{1}$ ) and after
renormalzation(4.7),$E_{\epsilon,\sigma}(u)=\epsilon\S\sigma:F(u,\epsilon, \sigma)$

we find that, in the relevant parameter regime,$F(u, \epsilon,\sigma)\geq C$

for some constant$C$

independent of , , and$\epsilon$$\sigma$

$u$

,

(4.8)

.

also deduced the dependence on $m(=2a-1)$ . Through out this note, we will fix $m\in$ (-), 1) and do not address
scaling issues pertinent to this parameter.
sFbr adetailed look at more complicated ansatz driven calculations see [7]$2\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{y}$14

To be more specific, we follow [5]. Let D be the unit cube, and since we are interested here in the scaling
with respect to and , let m $=0$ , and $W(u)=1-u^{2}$ . Our approach would give the same scaling in andfor m $\in(-1,$ 1) fixed. We work for convenience within the class of admissible states which satisfy the zero
flux (Neumann) boundary condition. That is, the class of admissible states A is,$\epsilon$$\sigma$$\epsilon$$\sigma$,$A:= \{u\in H^{1}(D)|\frac{\partial u}{\partial\nu}|\partial D=0$

where

$\nu$

denotes the outer normal to$\partial D$$\int_{D}ud\mathrm{x}=0\}$,. Hence,$\int_{\Omega}|(-\triangle)^{-:}u|^{2}d\mathrm{x}=\sum_{\mathrm{n}\in \mathrm{Z}^{\theta}}\frac{|u_{\mathrm{n}}|^{2}}{|\mathrm{n}|^{2}}$

,

where
are the appropriate Fourier coefficients. This is the $H^{-1}$ norm squared on thespace offunctions with zero average. Within this formulation, steps (4.7) and (4.8) require us to boundbelow the sum$u_{\mathrm{n}}(\mathrm{n}\in \mathrm{Z}^{3}, \mathrm{n}\neq 0)$$L^{2}$$\frac{1}{M(\epsilon,\sigma)}$($\int_{\Omega}\epsilon|\nabla u|^{2}$$+$)$\frac{1}{\epsilon}(1-u^{2})d\mathrm{x}$

,

$+$

(4.9)

$M^{2}( \epsilon, \sigma)\sum_{\mathrm{n}\in \mathrm{Z}^{S}}\frac{|u_{\mathrm{n}}|^{2}}{|\mathrm{n}|^{2}}$where in the relevant parameter regime we have
, for some constant $C>0$ . As is well-known
ffom the work of Modica and Mortola (cf. [18]), the sum in the parentheses is bounded below by a $BV$
norm of , and hence we seek an interpolation-like inequality between the spaces $BV$ and. Lemma 2.1
in [5] (following work in [6]) bounds below the sum in (4.9) by thenorm squared of , and allows us toconclude the desired lower bound (4.8). The upper bound is obtained by following (4.6). We arrive at:$M(\epsilon, \sigma)\geq C$$H^{-1}$$u$$L^{2}$$\mathrm{w}$Theorem 4.1 If$0<\epsilon_{\sim}<\sigma_{\sim}<1^{4}$,$\epsilon^{\S}\sigma^{\S}\sim<\min_{u\in A}E_{\epsilon,\sigma}\leq\epsilon^{2}\mathrm{z}\sigma^{1}\mathrm{w}$.

Here we have adopted the notation that for functions and of the parameters and ,means
for some constant $C>0$ independent of and awe have $f<Cg$ . In the present context, the constant may
in general depend on the structure of $W$ and $m$ .
While the approach of matching upper and lower energy bounds may not appear to say anything about
the minimizer’s domain size in the way the ansatz driven calculation (4.6) did, it does yield ansatz-freematching upper and lower bounds for the minimizer’s average length scale -more precisely for the totalinterfacial perimeter per unit volume (see [5, 6, 7]). However, we emphasize that Theorem 4.1 does not
imply that (for small ) minimizers are periodic structures on the scale. It is suggestive that they
possess an inherent scale ofbut certainly one would like astronger result. In the next section, we
present such aresult.
$f$

$g$

$\epsilon$$\sigma$$f_{\sim}<g$$\epsilon$$(\epsilon/\sigma)^{1/3}$$\epsilon$$(\epsilon/\sigma)^{1/3}$

5Uniform distribution of energy in asharp-interface limit
Here we report on some work in progress with Alberti and Otto ([1]). We are interested in obtaining further
rigorous support for the following conjecture: For small, minimizers of (2.1) are nearly periodic structureson the scale. Exactly what one means by nearly periodic has of course to be made clear.
One approach is via asharp interface limit whereby tends to zero. We pause to note that one can easily
obtain asharp interface limiting energy functional by considering (2.1) in terms of the original material
parameters (2.2); fixing ;taking;and letting $N$ tend to infinity: Thus we keep the sample
size of the melt on the same (conjectured) length scale of the domains. One can easily show (cf. [8, 25]) that$\epsilon$$(\epsilon/\sigma)^{1/3}(i.e.

N^{2/3})$$\epsilon$$\chi$$|\Omega|^{1/3}\sim N^{2/3}l$$4\Pi[5]$ this notation was not used but insteadforParticular constants were chosen in the hyPothesis
convenience in Proving the uPPer and lower bounds. This is insignificant as it only effects the constants in the conclusion;
however, we remark that they were in fact incorrectly chosen for their purpose$0<\epsilon\leq \mathrm{a}$$\sim<1$15

as $Narrow\infty$ , the functional (2.1) (suitably rescaled) -converges(in the sense of De Giorgi (cf. [9, 18])) to anongenerate sharp-interface variational problem.
In this section, we follow the idea presented in [2] which will in the end result in studying (essentialy)
andof minimizers of
the same sharp-interface variational problem. Consider asequence
fix aposition in the melt (we will take $s=0$ for simplicity). We blow up at $s=0$ , sending the scale
to 1and removing all finer scales: That is, consider the functions of amicroscopic variable :$\Gamma$$u_{\epsilon}(x)(\epsilonarrow 0)$$E_{\epsilon,\sigma}$$s$$\mathrm{t}$$(\epsilon/\sigma)^{1/3}$$v_{\epsilon}( \mathrm{t}):=u_{\epsilon}((\frac{\epsilon}{\sigma})^{1}\mathrm{z}\mathrm{t})$.

tends to periodic functions
In terms of the blow-ups , the previous conjecture can be rephrased as follows:$values\pm
1$phases).
andpure$O(1)$
tothe(corresponding
taking on only the teoo
with periodisvia the asymptotics
approachOur
We very briefly present apartial result in support of this conjecture.
themselves. The
minimizers
theof
asymptotics
the;thereby capturingof the energy written in terms of
weareforced to deal
Thus
infinite.
becomessize
is defined over domains whoseenergy functional on
notion ofappropriate
anwithconcerned
naturallybewith several issues. The first being that we shouldofoverdomains
definedterm
nonlocaltheaspatially local rninirnizer (see below). The second pertains toetc.).
minimizer,
constraint,ofalocalnotionincreasing size (i.e. boundary conditions, the conservation
These issues are dealt with by considering anatural relaxation of the nonlocal term. Here we will only beconcerned with the limiting sharp interface problem, and hence let us describe this relaxation in that context.
Let be abounded, open set. For $v\in BV(A, \pm 1)$ we introduce asecond dependent variable coupled to(interpreted in the sense of distributions), and replace the $H^{-1}$ norm squaredby the constraint
of by$v_{\epsilon}$$v_{\epsilon}$

$A$

$B$

$v_{\epsilon}(\mathrm{t})$$v_{\epsilon}(\mathrm{t})$$\mathrm{b}$$A$$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{b}=v$

$v$

$v$

$\int_{A}|\mathrm{b}|^{2}$$\mathrm{b}L^{2}\min_{\in}$.

$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{b}=v$

One can then reduce the original variational problem (at least$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{U}\mathrm{y}^{5}$) to the following sharp interface

problem:$\min E(v, \mathrm{b}, A):=\int_{A}|\nabla v|+|\mathrm{b}|^{2}$over$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{b}=v$,$v\in BV(A, \pm 1)$

.

,
,
and
with
Now we say
is alocal minimizer of on if for all open
an
prove
one
can
framework
this. Withinsuch that support(b-b) CC , we haveis alocal minimizer of $E$ on ,uniform distribution of energy for local minimizers of $E$ . That is, if$r\geq1$, we haveandthen for every$E$$\mathrm{d}\mathrm{i}\mathrm{v}\tilde{\mathrm{b}}=\tilde{v}$$(\tilde{v},\tilde{\mathrm{b}})$$A$$A\subset\subset\Omega$$\Omega$$(v, \mathrm{b})$$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{b}=v$$E(\tilde{v},\tilde{\mathrm{b}}, A)\leq \mathrm{E}(\mathrm{v}, \mathrm{b},A)$$\Omega$$(\tilde{v},\tilde{\mathrm{b}})$$B(r)\subset\Omega$$E(\tilde{v},\tilde{\mathrm{b}},$$\mathrm{B}(\mathrm{r})\sim|B(r)|$,the respective constants independent
where $B(r)$ is aball of radius andboth, and . The lower bound for this assertion follows from an interpolation-like argument similar to oneofused in the previous section. The upper bound follows from direct construction of suitable comparison fields.
The details will be presented in [1].
$r$$\sim \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}$$\leq \mathrm{a}\mathrm{n}\mathrm{d}$$\sim>_{\mathrm{W}}\mathrm{i}\mathrm{t}\mathrm{h}$$\mathrm{b}\sim$$\tilde{v},$$r$

6Remarks
We have discussed issues and results pertaining to scales and the distribution of energy for minimizers of the
Ohta Kawasaki energy in the SSR. Whereas these results seem encouraging in terms of building ansatz-fiae
tools for capturing properties of the microphases, one should be alerted to the fact that the derivation ofthis functional was based upon the linearization about $\beta=0$ (i.e. about infinite temperature). Thus theconnection via -convergence is still formal as we do not as yet have acompactness result at the -level(cf. [9]) for
this tyPe of local minimizers$\epsilon$$5\mathrm{T}\mathrm{h}\mathrm{e}$$\Gamma$16

physical validity of the density functional theory in regimes other than the WSR remains unclear. In the
SSR, it does seem to predict the basic scaling features of the domain size; however, one must be skeptical as
to whether or not it retains all the essential physics of the problem-as the pure SCMFT seems to $[4, 17]$ . On
the other hand, some recent simulations of the bicontinuous gyroid phase of Teramoto and Nishiura ([26])indicate that this theory does predict rather nonstandard structures (i.e. other than lamellar, cylindricaland spherical) which have been observed at temperatures placing one in the intermediate segregation regime,
and have been previously predicted by the SGMFT ([17]).

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