P12_S_Bates_Multiblock Polymers - Panacea or Pandora's Box

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DOI: 10.1126/science.1215368, 434 (2012);336Science

et al.Frank S. BatesMultiblock Polymers: Panacea or Pandora's Box?

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Multiblock Polymers:Panacea or Pandoras Box?Frank S. Bates,1* Marc A. Hillmyer,2 Timothy P. Lodge,1,2 Christopher M. Bates,3

Kris T. Delaney,

4

Glenn H. Fredrickson

4,5

Advances in synthetic polymer chemistry have unleashed seemingly unlimited strategies for producingblock polymers with arbitrary numbers (n) and types (k) of unique sequences of repeating units.Increasing (k,n) leads to a geometric expansion of possible molecular architectures, beyondconventional ABA-type triblock copolymers (k= 2,n = 3), offering alluring opportunities to generateexquisitely tailored materials with unparalleled control over nanoscale-domain geometry, packingsymmetry, and chemical composition. Transforming this potential into targeted structures endowedwith useful properties hinges on imaginative molecular designs guided by predictive theory andcomputer simulation. Here, we review recent developments in the field of block polymers.

Block polymers, hybrid macromolecules

constructed by linking together discrete

linear chains comprising dozens to hun-

dreds of chemically identical repeating units,spontaneously assemble into exquisitely ordered

soft materials (1). Precise synthesis of these self-

assembling compounds offers extraordinary con-

trol over the resulting morphology, spanning length

scales from less than a few nanometers to sev-

eral micrometers, enabling a diverse and expand-

ing range of practical applications in, for example,

drug delivery (2), microelectronic materials (3),

and advanced plastics (4). Yet, only a small sub-

set of the vast array of feasible molecular ar-

chitectures has been explored, and just a handful

of these compounds have been developed into

commercial products. Combining more chemi-

cally distinct blocks and block types, beyond theestablished AB and ABA diblock and triblock

copolymers, offers unparalleled opportunities for

designing new nanostructured materials with en-

hanced functionality and properties, often with-

out adding substantially to the cost of production.

But how do we choose among the myriad mo-

lecular design possibilities?

Modern synthetic methods afford access to

a broad portfolio of multiblock molecular archi-

tectures, as illustrated schematically in Fig. 1.

(Although there is no universal definition of the

minimum molar mass that constitutes a polymer

block, generally 10 to 20 monomer repeat units,

and frequently more than 100, are used). Linear

AB diblock copolymers have been investigated

most extensively, leading to a comprehensive ex-

perimental and theoretical understanding of their

bulk- and solution-phase behavior (1). Exten-

sion to linear alternating multiblock copolymers

(ABA, ABABA, etc.) can lead to profound con-sequences on the physical properties (for exam-

ple, enhanced elasticity and fracture toughness)

(5) without drastically influencing the associated

phase behavior. Introduction of a third block

type, C, dramatically expands the spectrum of

accessible nanostructured morphologies ormicro-

phases.Whereas AB and ABA copolymers typ-

ically adopt four familiar microphase structures

(lamellae, double gyroid, cylinders, and spheres),

many more ordered phases have been documented

with ABC triblock terpolymers (1,69). Adding

additional numbers of blocks (n) and chemi-

cally distinct block types (k) rapidly expands the

number of unique sequences (see Box 1), eachcapable of producing a host of nanostructures.

Cyclic and branched architectures further ex-

pand the possibilities (10), leading to dizzyi

phase complexity.

Block polymer phase behavior is determin

by a suite of molecular variables, in additi

to the molecular topology and specific block

quences, as summarized in Table 1. Primary f

tors include the degree of polymerization of ea

block, Ni, and the associated binary segme

segment interaction parameters, cij (where i a

j refer to chemically distinct repeat units) (Box 2). (Alternatively, the composition fi = N

and overall molecular size N= SNican be usin place of defining each block length, wher

common segment volume defines Ni). Secon

ary factors (not discussed further) include blo

flexibility (for instance, stiff versus flexible chain

the distribution in block lengths (dispersity), a

any additional sub-block structure such as

ternating, random, or tapered sequences of

peat units. Obviously, increasing n and/o

even slightly, compounded by scores of prac

cal options in choosing the block chemistr

results in an expansive parameter space an

REVIEW

1Department of Chemical Engineering and Materials Science,University of Minnesota, Minneapolis, MN 55455, USA. 2De-partment of Chemistry, University of Minnesota, Minneapolis,MN 55455, USA. 3Department of Chemistry, The University ofTexas at Austin, Austin, TX 78712, USA. 4Materials ResearchLaboratory, University of California, Santa Barbara, CA 93106,USA. 5Department of Chemical Engineering, University of Cal-ifornia, Santa Barbara, CA 93106, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

Table 1. Molecular variables that influenblock polymer self-assembly.

Primary

Topology; linear, branched, etc.

Number of blocks, n

Number of block types, k

Block degree of polymerization, N iInteraction parameters, c ij

Secondary

Block flexibility, b iMolar mass distribution, i

Sub-block structure (e.g., tapered)Composition heterogeneity

1

11

2

4

23

3

1

Fig. 1.A subset of the vast structural complexity with two (k= 2) or three (k= 3) block types producby varying the number of blocks (n) and the functionality of the connector at each block-block junctu(difunctional, circles; trifunctional, triangles). The number and type of connectors needed are given each structure.

27 APRIL 2012 VOL 336 SCIENCE www.sciencemag.org34


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boundless array of possible structures and che

ical functionalities.

Multiblock polymers, such as those sketch

in Fig. 1, offer unlimited potential for designi

soft materials with precisely specified structur

subject to two critical limitations: (i) The stru

tures can be accessed synthetically, and (ii) p

dictive theoretical tools to guide molecular des

are available. A parallel can be drawn with t

complexity and diverse functions associated w

proteins, another well-known class of multifutional macromolecules. Theorists strive to pred

polypeptide structure and ultimate function ba

on models that employ quantum chemical a

statistical mechanical tools capable of account

for both short-range (~0.1 nm) and long-ran

interactions mediated by an aqueous environm

and constrained by the macromolecular arc

tecture (11). Essentially any sequence of less th

100 residues of the 20 naturally occurring ami

acids (and a bevy of unnatural variants) can

produced using commercially available so

phase peptide synthesizers. The challenge is

anticipate which linear combinations of am

acids, drawn from 20n

possibilities, will leadthe desired secondary, tertiary, and even quat

nary structures and corresponding functions.

In some respects, block polymers appear

pose a less daunting theoretical challenge, p

ticularly for flexible and noncrystalline polyme

because the chain structure at the monomer le

does not play a primary role in determining

domain geometry or packing symmetry. Mo

over, with no direct analog to solid-phase pept

synthesis, contemporary research focuses on p

ducing block polymers with a minimum deg

of complexity (that is, the smallestn andk) n

essary to achieve a desired morphology and co

plement of properties. However, whereas protfolding is intrinsically intramolecular, supram

lecular self-assembly of block polymers lea

to ordered structures with unit cells that may co

tain many thousands of polymer chains (m

lions of atoms), resulting in unique theoreti

and computational challenges. Hence, even

seemingly primitive case of ABC triblocks (k=

n = 3) is only marginally understood in ter

of the resulting phase structure and associat

properties.

Synthesis

Block polymer synthesis has evolved consid

ably since the introduction and development

living anionic polymerization by Szwarc m

than 50 years ago (12). Numerous strategies t

enable the covalent connection of two or m

polymer blocks have been developed, includ

sequential addition of distinct monomers to

active polymer chain, chain-coupling strategi

and transformations of polymer end groups

accommodate diverse and often incompati

polymerization mechanisms. Chemical conn

tors that link together more than two chains

increasingly available as evidenced by the her

syntheses of a 31-arm starblock pentapolym

Box 1. Mnages en blocs.The challenge of enumerating possible block polymer structures is already evident in Fig. 1,

where it is seen that a substantial number of molecules can be constructed by linking multipleblock species using difunctional or trifunctional linkers. An appreciation for the numbers involvedcan be obtained by considering just the set of linear molecules composed of n blocks, selectedfrom kdifferent species, and connected into a chain by means of n 1 difunctional linkers. Theenumeration of all such linear molecules, subject to the restrictions (i) that adjacent blocks alongeach chain are of different species, (ii) only topologically distinct sequences are counted (i.e., ABis not distinguishable from BA), and (iii) allkblock species appear, is a problem in combinatorialanalysismnages en blocsthat is taken up in the supplementary material.

Mnages en blocs has close connections to classic problems in permutations with restrictions,including the 19th century problme des mnages(47,48), which asks for the number of waysof seatingn married couples at a circular table, men and women in alternate positions, no wifenext to her husband. The blocs problem is also related to problems involving enumeration ofprotein sequences (49, 50), although in the latter there are normally no species restrictions onadjacent amino acid residues. The restriction (i) for multiblock polymers amounts to the as-sumption that blocks do not acquire size dispersity by virtue of sequence. Though this assumptionis violated for some classes of block polymers formed by statistical linking of monomers and/orpreformed blocks by various (e.g., condensation) chemistries, such nonideal systems are beyondthe scope of the present discussion.

The object of interest in mnages en blocs isZ(k,n), the number of linear block polymers thatcan be formed from n blocks andkdifferent block species, and is subject to the restrictions (i) to(iii) stated above. This object can be constructed by exclusion from a simpler function Q(k,n) thatis defined as the number of linear block polymers that can be formed fromnblocks selected from

kspecies and subject only to the restrictions (i) and (ii). An expression for Q is derived in thesupplementary material

Q(k, n) k2(k 1)

(n1) n evenk2(k 1)

(n1)/2[1+ (k 1)(n1)/2] n odd

Imposing the final restriction (iii) that all kblock species must appear in each molecule is trivialfor the case ofk= 2, because restriction (i) implies restriction (iii); hence, Z(2,n) =Q(2,n). Fork> 2,the principle of exclusion is used to recursively generate Z(k,n) fromQ(k,n)

Z(k, n) = Q(k,n) k1

j=2

kj

Z(j, n)

In words, block polymers containingk 1 or fewer block species are removed from the setcomprising Q(k,n) to yield Z(k,n), the subset of molecules in Q(k,n) in which all k blockspecies appear.

Application of the above formulas leads to notable growth in block polymer diversity withincreasing kand n. The table below summarizes the Z(k,n) function for block enumerationswith k,n 10. One observes that there are no molecules with k > n, as is required byrestriction (iii). For k= 2, there is an odd-even effect with increasing n (e.g., the mnage trois ABA and BAB at n = 3, ABAB at n = 4). Along the diagonal, block polymer complexityexplodes in a factorial manner with Z(n,n) = n!/2.

Mnages en blocs function Z(k,n)

Curiously, the maximum ofZ(k,n) for n > 4 is above the diagonal, located atZ(n 1,n)for5 n 8,and atZ(n 2,n) forn= 9, 10. Synthesis of all these distinct linear block polymer molecules willsurely keep synthetic polymer chemists busy for some time!

k/n 2 3 4 5 6 7 8 9 10

2 1 2 1 2 1 2 1 2 1

3 0 3 9 24 45 102 189 402 765

4 0 0 12 72 300 1092 3612 11,664 36,3005 0 0 0 60 600 3900 21,000 102,120 466,200

6 0 0 0 0 360 5400 50,400 378,000 2,502,360

7 0 0 0 0 0 2520 52,920 670,320 6,667,920

8 0 0 0 0 0 0 20,160 564,480 9,313,920

9 0 0 0 0 0 0 0 181,440 6,531,840

10 0 0 0 0 0 0 0 0 1,814,400

www.sciencemag.org SCIENCE VOL 336 27 APRIL 2012

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(k = 5, n = 31) (13). In principle, all of t

structures depicted in Fig. 1 could be prepa

with arbitrary choice of block constituents usi

modern polymerization methods and polym

functionalization strategies, including numero

controlled radical, ring-opening, metal-catalyz

and ionic polymerizations, often in combinat

with highly selective and efficient functional-gro

transformation techniques (14, 15).

Modern block polymer molecular design

driven by the intended applications. For exaple, a thermally stable and mechanically rob

membrane endowed with a dense array of sp

cifically sized nanopores necessitates a combin

tion of glassy, ductile, and chemically etchab

(or cleavable) blocks (16). Biocompatibility f

ther limits the possible ingredients. The asso

ated segment-segment interaction parameters a

optimal block architecture demand a carefu

targeted synthetic strategy, possibly includ

multiple routes to the desired structure, draw

from an ever-expanding synthetic tool kit. A

though high degrees of molecular uniformity

generally desirable, in most circumstances so

level of imperfection (e.g., less than n blocand modest chain-length distribution can be t

erated [in fact, tailored dispersity represent

strategic design parameter (17, 18)]; the goa

to build the appropriate structure in the m

synthetically economical way possible.

Figure 2 illustrates routes to all possible

quences of a linear ABC terpolymer (k= 3,n=

designed to incorporate three complement

physical and chemical properties: (i) glassy p

(styrene) (PS), (ii) rubbery poly(isoprene) (P

and (iii) chemically etchable and biorenewa

poly(lactide) (PLA). Although only two of th

three structures have been reported, PI-PLA-

is certainly tractable on the basis of a relatprecedent (19).

The synthetic strategies depicted in Fig. 2

quire various combinations of ring-opening,

ionic, and controlled radical polymerizations w

concomitant end-group manipulations. In fa

the drive to largern and k will almost certain

necessitate multiple polymerization mechanis

and functional-group manipulations, the or

of which will depend on the desired block

quence. Complexity in ABC terpolymer syste

is further increased by introducing the pos

bility of cyclic motifs and three-point junctio

(miktoarm stars or brushes). Putting all the pie

together requires the continued development

efficient processes and realizable retrosynthe

analyses.

Introduction of a fourth block with only th

monomers (k= 3,n = 4) provides an added le

of complexity not realized in ABC terpolyme

Asymmetric ABCA tetrablock terpolym

(NA NA) offer unique opportunities for d

signing bulk materials (highlighted in the Str

ture section) and tailored solution structures (2

However, independent control over the length

all four blocks requires synthetic strategies t

depend on the specific block chemistries. F

10-4 10-3 10-2 10-1 100 101

D D

D

D

D

DDD

OO

O

O

O

N

SO3-

Estimated with respect to polystyrene

Box 2. Complexity with c.Discussions of polymer phase behaviorwhether block polymers, blends, or solutionsare

usually couched in terms of the binary interaction parameter cAB, or simply c. Conceptually, c isa dimensionless measure of the energetic cost of exchanging a repeat unit of polymer A for anequal volume (Vref) unit of polymer B: c = zDw/kT, where kT is the thermal energy (k is theBoltzmann constant, and T is temperature), Dw is the difference between the A/B interactionenergywABand the average of the self-interaction energieswAAandwBB, andzis the number ofnearest neighbors. On the assumption that there is entirely random mixing of A and B units, withentropy arising only from the ideal combinatorial of mixingDSm

id = kS(fi/Ni)lnfi(wherefiand

Niare the volume fraction and degree of polymerization of species i, respectively), the Flory-Huggins (or Bragg-Williams or mean-field or regular-solution) free energy of mixing for A and Bpolymers becomes

DGm = TDSmid + ckTfAfB

However, in experimental practice, c is imbued with whatever attributes are needed todescribe the data; as such, it actually represents an excess free energy of mixing DGm

ex thatexhibits both enthalpic and entropic contributions. Therefore, in practice c is recast as c = cH+cS= a/T+ b (wherea andb are empirical parameters), and either a or b may also depend onfand N. This complexity can lead to much confusion when, for example, comparing differentmeasurements on the same system. Furthermore, either a orb may be positive or negative insign. The case where a > 0 andb < 0 leads to the classicalupper critical solution temperaturephase diagram, whereas a < 0 and b > 0 leads to a lower critical solution temperature (i.e.,

phase separation on heating); both situations are common. Because DSmid

varies inversely withN,it is negligible for typical molecular weights; consequently, the excess free energy is not just acorrection, but the main contribution.

The characteristic magnitude of cABspans several orders of magnitude, as illustrated below.From a predictive point of view, the situation is quite unsatisfactory. For systems in which shortrange, isotropic, dispersive interactions dominate, the solubility parameter approach might beexpected to yield reasonable estimates [i.e., c cH = Vref/kT(dA dB)

2, where dA and dB arethe associated segment solubility parameters]; however, in practice, agreement with experi-ment is often remarkably poor. Additionally, although solubility parameters can be used toanticipate qualitatively where a particular blend might lie on this continuum, there is no generaltheoretical approach for predicting the relative contributions ofcHandcSor for computing eitherterm to within, say, a factor of 2. As phase boundaries in polymer systems typically depend on thedimensionless product cN, it is therefore not possible to even anticipate the molecular weightsneeded to produce an experimentally accessible phase boundary.

Blends of hydrogenous and deuterated isotopomers represent the simplest case. Here, it is

usually found thatc 103

, andcHcan be understood in terms of differences in zero point energyand the polarizabilities of CH and CD bonds. Nevertheless, cSis not negligible, for reasons thatremain elusive. For example, for (H/D) poly(styrene) blends, c(373 K) 0.00014, but cSis ofcomparable magnitude to cHand is negative (51). For poly(styrene)/poly(methylmethacrylate), apair of prevalent thermoplastics, c (373 K) 0.038, a relatively modest value; however, cS ispositive and, surprisingly, represents ~70% of the total, so that c itself has almost no Tdepen-dence (52). When polar or ionic polymers are mixed with nonpolar partners, c can exceed unity.Here, the fundamental interactions are relatively strong, long-ranged, anisotropic, and/or non-pairwise additive, yet entropic contributions still play a substantial role. For example, forpoly(styrene)/poly(lactide), c(373 K) 0.15, but the magnitude of cSat this temperature isabout half that of cH (53).


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example, the addition of water-soluble and bio-compatible poly(ethylene oxide) (PEO) to both

ends of a PS-PI core should be feasible. Se-

quential anionic polymerization of styrene and

isoprene initiated with an alkyllithium reagent

containing a protected hydroxyl group (21) fol-

lowed by end-capping with one unit of ethylene

oxide generates a heterotelechelic diblock co-

polymer. Activation of the free alcohol end en-

ables the ring-opening polymerization of ethylene

oxide, without compromising the protecting group

at the other chain end, leading to a linear BCA

triblock; the terminal end of the A block must

be capped to prohibit further growth during sub-

sequent polymerization of ethylene oxide fromthe B segment in this scheme. Unmasking and

activation of the hydroxyl group at the B ter-

minus allows for independent control of NArelative to NA. Such asymmetric ABCA tetra-

block polymers have not been reported, but this

approach could enable remarkable tunability of

ordered microstructures (see below).

New organocatalytic polymerization methods

(22), controlled synthesis of conducting poly-

mers (23), and various clickstrategies (14), all

developed over the past decade, exemplify how

advances in chemistry have allowed access to

hybrid structures that simply could not be made

before. However, deciding what macromolecular

masterpieceto synthesize using the expanding

palette of monomer paints and polymer syn-

thesis brushesrepresents a growing conundrum.

Targeting new block structures for purely aesthetic

reasons is impractical. Today, synthetic polymer

chemists can prepare nearly any architecture

with any set of desired chemistries, for com-

modity and value-added applications, paralleling

the synthesis of small molecules for therapeutic

markets. Organic chemists can build complicated

small-molecule structures with an amazing, al-

most arbitrary array of chemical functionalities;

to know which molecule is efficacious requiresa better and more complete understanding of

biological action. Analogously, given the heavy

investment required for creating even a single

new multiblock structure (24), advances in the

block polymer/soft materials arena require (i) a

deeper understanding of how block architecture

influences structure and, thus, properties and (ii)

advances in predictive theories that can guide

synthetic chemists.

Structure

The fundamental principles governing block

copolymer self-assembly were established in

the 1970s, culminating in Helfands strong seg-regation (25) and Leiblers weak segregation

(26) analyses of AB diblock copolymers. These

theories have since been subsumed into a com-

prehensive mean-field framework known as self-

consistent field theory (SCFT) (see the Theory

section). Equilibrium-phase behavior represents

a compromise between minimizing unfavorable

segment-segment contacts, mediated bycAB, andmaximizing configurational entropy, which is

inversely proportional toNand quadratically de-

pendent on the extent of chain stretching relative

to the unperturbed state. Simplifying assump-

tions, including Gaussian chain (random walk)

statistics and constant density, enables tractable

computational schemes that have accounted for

all of the experimentally observed diblock mor-

phologies (27). Increasing n and k introduces

additional complexity, greatly compounded by

the choice of block sequences along with the

other molecular parameters listed in Table 1.

In the limit of strong segregation, multiblock

polymers segregate into relatively pure domains

separated from neighboring domains by as many

ask(k 1)/2 distinct and narrow interfaces char-

acterized by interfacial tensiongij~ (cij)1/2; strong

segregation impliesNi Nj>> 10/cij. Reducing

the individual values ofcij leads to broader terfaces and, ultimately, mixing ofi andjbloc

Thus, an ABC triblock terpolymer (k= 3,n =

may contain one, two, or three types of interfa

(or none if entirely disordered). For example,

condition cAB = cBC > 10/N, t

x= 1 tetrablock terpolymer will exhibit a tw

domain lamellar morphology with equival

interfaces separating A and A domains fro

mixed B/C domains (Fig. 3A). Increasing t

segregation strength to cAB= cAC= cBC/2 wresult in a three-domain lamellar configurat

Fig. 2. Synthetic routes to produce the three distinct linear triblockterpolymer structures composed of PS, PI, and PLA blocks, using com-binations of living anionic, metal-catalyzed ring-opening, and revers-

ible addition-fragmentation chain transfercontrolled radical polymerizatioR and R, alkyl groups; Ph, phenyl; p, q, and r, degrees of polymerizatioEt, ethyl.


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with energetically equivalent A/B and C/A in-

terfaces that exactly balance the internal B/C

interface (30). Further elevating cBCwill induce

changes in the domain geometry that minimize

the overall interfacial free energy (P

ij

Aijgij,

where Aij is the i-j interfacial area) subject to

entropic penalties associated with packing the

blocks into the resulting structures. The sequence

of domains sketched in Fig. 3A, radial and axial

segregated cylinders followed by a Janus sphere,illustrates phase transitions that capture this ef-

fect by reducing ABC/(AAC+ AAB), the ratio of

interfacial surface areas. (We note that this hy-

pothetical sequence of morphologies may be su-

perseded by other morphologies and never realized

in practice). Clearly, addingan Ablock to the ABC

sequence offers specific control over the shape

and subdivision of the self-assembled domains.

Interdomain packing can be considerably in-

fluenced by the symmetry parameterx, as shownin Fig. 3B. We have selected the axially segre-

gated cylindrical domain structure to illustrate

this point. Tetrablock terpolymers (k= 3,n = 4)

can produce ordered cylinders containing B andC subdomains, even with NA+ NA = N/2 (31);

cylindrical structures at such low matrix com-

positions are inaccessible with diblock copoly-

mers. For x = 1, there will be no interdomain

axial order because, under the stated conditions,

the intradomain structure does not influence

the corona chains that control cylinder packing;

two-dimensional (2D) hexagonal packing would

be anticipated. However, for x > 1, the C do-mains will be decorated with A blocks, which

are longer than the A blocks emanating from

the B portions of the cylinder. This arrangement

introduces an effective anisotropic interdomain

potential that should induce 3D order. The tend-

ency to pair short and long corona chains rep-

resents a purely entropic effect, one that minimizes

unfavorable chain stretching and compression

at constant density. A different cylinder-packing

symmetry (for instance, tetragonal) would be re-quired to permit uniform chain packing; neither

three- nor sixfold symmetry supports complete

in-plane alternation of C and B domains. Such

symmetry-breaking also should apply to the Janus

sphere geometry (and radial segregated cylinders)

creating dipole-dipole interactions, thus offering

the fascinating possibility of tailoring domain

orientation within specific ordered lattices anal-

ogous to spin alignment in magnetic materials (32)

[e.g., noncentrosymmetric ferromagnetic order-

ing (33) of spheres on a body-centered cubic

(bcc) crystal as shown in Fig. 3B] and pairing of

colloidal Janus particles (34).

Simply changing the sequence from ABCA

to ABAC while holding all other parameters

constant leads to qualitatively different phase be-

havior. Fixing cAB = cAC 1, shorchains (emanating from the red B

mains) will show a preference to pnear long A chains (emanating frthe blue C domains) to maintain costant density while minimizing unfavable chain stretching and compressiThis induces 3D cylindrical order aa ferromagnetic-like arrangementJanus-type spheres.

BC

/AB

A A'

BC

AC

AB

1

21

1

4

R

2d=

+> > >

0

AB=

AC

A B C A'B/C

Noorder

1>1=

Order

2R

d

A

B


REVIEW


7/8

characterized by effective Hamiltonians (or ac-

tions) H that are highly nonlinear and nonlocal

functionals of the field variables and have ex-

plicit dependence on segmental interaction and

chain architecture parameters, such as cijandNi.

The Hamiltonian is generally also a complex-

valued functional, which can have important con-

sequences for generating numerical solutions.

Two major classes of field-based simulations

can be built on top of the Edwards framework:

(i) SCFT (39), where mean-field solutions aresought corresponding to saddle points of H,

and (ii) field-theoretic simulations (FTS), refer-

ring to stochastic numerical sampling of the full

(complex-valued) field theory (40). SCFT sim-

ulations are considerably less expensive than FTS,

because they seek only a single saddle-point field

configuration, but SCFT neglects field fluctu-

ations that are especially important in solvated

polymer systems. In the case of highmolar mass

undiluted block polymers, such fluctuation ef-

fects are substantial only near order-disorder

phase boundaries and associated critical points,

so SCFT can be applied to polymer melts with

relative impunity.Even within the confines of SCFT, however,

establishing the phase map for a given block

polymer [for example, from the (k,n) linear mul-

tiblock family] can be a daunting challenge. First,

the parameter space is large: Even in the case of

an ABC triblock, the phase map is defined by

a complicated surface within the 5D parameter

spacecABN, cACN, cBCN, fA (= NA/N), and fBthat has yet to be delineated in detail and would

entail a Herculean effort to complete. Second,

the mean-field free-energy landscape explored

in SCFT simulations is rough, so simulations

launched in large cells from random initial field

configurations and subject only to periodic bound-

ary conditions tend to yield defective morphol-

ogies that are metastable, rather than stable

(lowest in free energy) for the specified param-

eters. An example is shown in Fig. 4, where such

a large-cell SCFT simulation is conducted for

an ABC triblock melt using parameters (28)that coincide with a PI-PS-PEO system known

to have a stable orthorhombicFddd(O70) phase

(7). The simulation launched from random ini-

tial conditions settles into a metastable, highly

defective Fdddstructure (Fig. 4A), whereas an

analogous simulation initialized from a determi-

nistic seed with the proper symmetry falls quickly

into the stable O70 phase (Fig. 4B). Occasionally,

large-cell simulations will also settle into defect-

free, metastable phases of a different symmetry

than the stable phase.

Such challenges of metastability are familiar

to researchers in a variety of fields that rely on

global optimization; indeed, practitioners of com-putational protein folding struggle with rough

energy landscapes (11). Researchers in that field

have the advantage that nature has apparently

engineered protein sequences that are resistant

to misfolding; nonetheless, they lack a quantita-

tive mean-field theory, and their folding results

do not enjoy the universality across broad fam-

ilies of monomers as do SCFT predictions for

multiblock polymers.

Large-cell SCFT simulations are the tool

choice when prospecting for candidate order

phases in a new polymer system for which

perimental results are not available (38). T

challenges ofcijestimation notwithstanding (

Box 2), one can specify interaction parame

values, block sequences, and block lengths a

then proceed to use large-cell SCFT simulatio

to predict microphase structure candidates. If si

ulations launched from a variety of random ini

conditions consistently lead to a single defefree or defective, but still identifiable morpho

gy, one can be reasonably confident that the stab

structure has been identified. Defective morph

ogies can also be converted to defect-free ones

the deterministic seeding method illustrated

Fig. 4. Conversely, if repeated large-cell SC

simulations from uncorrelated random seeds tu

up multiple structures differing in symmetry, th

it is necessary to compare the free energies

the competitive structures to establish which

them is stable. For this purpose, a second ty

of SCFT simulation is most efficienta unit-c

simulation that attempts to converge a single u

cell of a candidate structure within the confinof the symmetry constraints of that phase (4

Such a simulation seeks a saddle-point condit

on the Hamiltonian with respect to the comp

field variables while simultaneously minimiz

H(the mean-field free energy) with respect to t

lattice parameters of the unit cell.

State-of-the-art numerical methods for lar

cell simulations are based on spectral collocat

(or pseudospectral) techniques with plane wa

bases (38, 42). Fast Fourier transforms are us

to switch between real-space and reciproc

space representations of the fields, allowing

efficient evaluation of the operators, forces, a

energies embodied in SCFT. Large 3D calcutions using up to 5123 = 1.34 108 plane wa

or grid points, such as those shown in Fig

require the use of parallel algorithms imp

mented either on clusters of single or multico

central processing units or, more recently, on

single graphics processing unit containing

to 500 lightcores. The method of choice

unit-cell SCFT simulations is a Galerkin spec

technique developed by Matsen and Schick t

uses Fourier basis functions possessing the sy

metry of the phase being considered (41). A

though the method is not well suited to paral

computing, a smaller number of symmetry-adap

basis functions are generally required for ac

rate unit-cell simulations than the plane wav

used for spectral collocation in parallelepip

cells (43).

In spite of these advances in numerical SCF

the tool is primarily used to predict self-assemb

given a block polymer design. More useful fro

the standpoint of applications is the inve

problemnamely, the identification of polym

designs that will self-assemble into a specif

morphology. Though little progress has be

made to date on this problem, techniques su

as inverse Monte Carlo, in principle, could

Fig. 4.Large-cell SCFT morphol-ogies obtained for an incompres-sible ABC triblock terpolymer melt

using parameters (fA= 0.275,fB=0.550,fC= 0.175,cABN= 13, cACN= 35, cBCN = 13) and a periodicsimulation cell size commensuratewith 3-by-3-by-3 unit cells of theFddd(O70) phase. (A) A highly de-fective structure is obtained froma simulation initialized using anunstructured, random initial fieldconfiguration. (B) A defect-free mor-phology comprising 27 unit cellsof the O70 phase emerges from asimulation initialized using a smallnumber of field harmonics consist-ent with the Fddd space group.

Both simulations were conducted onan NVIDEA Tesla C2070 graphicsprocessing unit.


RE


8/8

mated to SCFT toward this end. It is also no-

table that marked advances have transpired in

algorithms for FTS simulations, which do not

rely on the mean-field approximation (38, 44).

These techniques can be used to explore a wide

range of fluctuation-mediated phenomena and,

when combined with thermodynamic integration

and Gibbs ensemble methods, can yield accurate

order-disorder phase boundaries for complex

block polymers.

Outlook

Do more blocks presage a panacea or Pandoras

box? At a minimum, thermoplastic elastomers

requirek= 2 and n = 3 as evidenced by PS-PI-

PS, introduced in the 1960s and still the most

successful block copolymer product in the world-

wide marketplace. Increasing n at fixed k = 2

generates a host of new opportunities without

conceptual difficulty (see Box 1) and at marginal

or no added cost, provided that the appropriate

synthetic tools are available. Recent advances

in olefin polymerization chemistry, such as the

dual-catalystmediated chain-shuttling mecha-

nism (45), underscore this point. Increasing kquickly complicates the practical design of new

materials yet challenges our imagination with

unbounded opportunities. It is not unreason-

able to claim that virtually any structure (domain

morphologies and packing symmetry) can be

created at length scales between roughly 5 nm

and 1 mm using block polymers, while main-taining robust flexibility over individual block

chemical and physical properties. Every (k,n)

enumeration has the potential to produce a

unique material. The resulting compounds may

address engineering goals directly (e.g., multi-

functional plastics) or enable a host of other

products [for example, as intermediates in drugdelivery, as scaffolds that guide the assembly of

inorganic hard materials (46), or for pattern for-

mation in the manufacturing of microelectronics

(3)]. Realizing these tantalizing prospects requires

overcoming challenges posed by complexity, a

dilemma that throttles many modern technolog-

ical ambitions ranging from biological systems

to global climate prediction. Here, the approach

seems clear: integration of engineering goals, cre-

ative chemistry, and predictive theoretical tools,

augmented by continued advances in structural

characterization methods. Fortunately, even mi-

nor extrapolation in multiblock complexity (e.g.,

k= 3,n = 4) offers a glimpse of what is possible

in the future.

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Acknowledgments:This work was supported by the Mater

Research Science and Engineering Centers Program of the

NSF under award numbers DMR-0819885 (F.S.B., M.A.H., T.

and DMR-1121053 (G.H.F., K.T.D.). C.M.B. thanks C. G. Willso

for his support.

Supplementary Materialswww.sciencemag.org/cgi/content/full/336/6080/434/DC1

Supplementary Text

10.1126/science.1215368

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P12_S_Bates_Multiblock Polymers - Panacea or Pandora's Box