## Abstract

Advances in synthetic polymer chemistry have unleashed seemingly unlimited strategies for producing block 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, beyond conventional ABA-type triblock copolymers (*k* = 2, *n* = 3), offering alluring opportunities to generate exquisitely tailored materials with unparalleled control over nanoscale-domain geometry, packing symmetry, and chemical composition. Transforming this potential into targeted structures endowed with useful properties hinges on imaginative molecular designs guided by predictive theory and computer 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 hundreds of chemically identical repeating units, spontaneously assemble into exquisitely ordered soft materials (*1*). Precise synthesis of these self-assembling compounds offers extraordinary control over the resulting morphology, spanning length scales from less than a few nanometers to several micrometers, enabling a diverse and expanding range of practical applications in, for example, drug delivery (*2*), microelectronic materials (*3*), and advanced plastics (*4*). Yet, only a small subset of the vast array of feasible molecular architectures has been explored, and just a handful of these compounds have been developed into commercial products. Combining more chemically distinct blocks and block types, beyond the established AB and ABA diblock and triblock copolymers, offers unparalleled opportunities for designing new nanostructured materials with enhanced functionality and properties, often without adding substantially to the cost of production. But how do we choose among the myriad molecular design possibilities?

Modern synthetic methods afford access to a broad portfolio of multiblock molecular architectures, 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 experimental and theoretical understanding of their bulk- and solution-phase behavior (*1*). Extension to linear alternating multiblock copolymers (ABA, ABABA, etc.) can lead to profound consequences on the physical properties (for example, 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 or “microphases.” Whereas AB and ABA copolymers typically adopt four familiar microphase structures (lamellae, double gyroid, cylinders, and spheres), many more ordered phases have been documented with ABC triblock terpolymers (*1*, *6*–*9*). Adding additional numbers of blocks (*n*) and chemically distinct block types (*k*) rapidly expands the number of unique sequences (see Box 1), each capable of producing a host of nanostructures. Cyclic and branched architectures further expand the possibilities (*10*), leading to dizzying phase complexity.

### Ménages 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 multiple block species using difunctional or trifunctional linkers. An appreciation for the numbers involved can be obtained by considering just the set of linear molecules composed of *n* blocks, selected from *k* different species, and connected into a chain by means of *n* – 1 difunctional linkers. The enumeration of all such linear molecules, subject to the restrictions (i) that adjacent blocks along each chain are of different species, (ii) only topologically distinct sequences are counted (i.e., AB is not distinguishable from BA), and (iii) all *k* block species appear, is a problem in combinatorial analysis—ménages en blocs—that is taken up in the supplementary material.

Ménages en blocs has close connections to classic problems in permutations with restrictions, including the 19th century “problème des ménages” (*47*, *48*), which asks for the number of ways of seating *n* married couples at a circular table, men and women in alternate positions, no wife next to her husband. The blocs problem is also related to problems involving enumeration of protein sequences (*49*, *50*), although in the latter there are normally no species restrictions on adjacent amino acid residues. The restriction (i) for multiblock polymers amounts to the assumption that blocks do not acquire size dispersity by virtue of sequence. Though this assumption is violated for some classes of block polymers formed by statistical linking of monomers and/or preformed blocks by various (e.g., condensation) chemistries, such nonideal systems are beyond the scope of the present discussion.

The object of interest in ménages en blocs is *Z*(*k*,*n*), the number of linear block polymers that can be formed from *n* blocks and *k* different 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*) that is defined as the number of linear block polymers that can be formed from *n* blocks selected from *k* species and subject only to the restrictions (i) and (ii). An expression for *Q* is derived in the supplementary material

Imposing the final restriction (iii) that all *k* block species must appear in each molecule is trivial for the case of *k* = 2, because restriction (i) implies restriction (iii); hence, *Z*(2,*n*) = *Q*(2,*n*). For *k* > 2, the principle of exclusion is used to recursively generate *Z*(*k*,*n*) from *Q*(*k*,*n*)

In words, block polymers containing *k* – 1 or fewer block species are removed from the set comprising *Q*(*k*,*n*) to yield *Z*(*k*,*n*), the subset of molecules in *Q*(*k*,*n*) in which all *k* block species appear.

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

**Ménages en blocs function Z(k,n)**

Curiously, the maximum of *Z*(*k*,*n*) for *n* > 4 is above the diagonal, located at *Z*(*n* – 1,*n*) for 5 ≤ *n* ≤ 8, and at *Z*(*n* – 2, *n*) for *n* = 9, 10. Synthesis of all these distinct linear block polymer molecules will surely keep synthetic polymer chemists busy for some time!

Block polymer phase behavior is determined by a suite of molecular variables, in addition to the molecular topology and specific block sequences, as summarized in Table 1. Primary factors include the degree of polymerization of each block, *N _{i}*, and the associated binary segment-segment interaction parameters,

*χ*(where

_{ij}*i*and

*j*refer to chemically distinct repeat units) (see Box 2). (Alternatively, the composition

*f*/

_{i}= N_{i}*N*and overall molecular size

*N*= Σ

*N*can be used in place of defining each block length, where a common segment volume defines

_{i}*N*). Secondary factors (not discussed further) include block flexibility (for instance, stiff versus flexible chains), the distribution in block lengths (dispersity), and any additional sub-block structure such as alternating, random, or tapered sequences of repeat units. Obviously, increasing

_{i}*n*and/or

*k*even slightly, compounded by scores of practical options in choosing the block chemistries, results in an expansive parameter space and a boundless array of possible structures and chemical functionalities.

### Complexity with χ.

Discussions of polymer phase behavior—whether block polymers, blends, or solutions—are usually couched in terms of the binary interaction parameter χ_{AB}, or simply χ. Conceptually, χ is a dimensionless measure of the energetic cost of exchanging a repeat unit of polymer A for an equal volume (*V*_{ref}) unit of polymer B: χ = *z*Δ*w*/*kT*, where *kT* is the thermal energy (*k* is the Boltzmann constant, and *T* is temperature), Δ*w* is the difference between the A/B interaction energy *w*_{AB} and the average of the self-interaction energies *w*_{AA} and *w*_{BB}, and *z* is the number of nearest neighbors. On the assumption that there is entirely random mixing of A and B units, with entropy arising only from the ideal combinatorial of mixing Δ*S*_{m}^{id} = –*k*Σ(ϕ* _{i}*/

*N*)lnϕ

_{i}*(where ϕ*

_{i}*and*

_{i}*N*are the volume fraction and degree of polymerization of species

_{i}*i*, respectively), the Flory-Huggins (or Bragg-Williams or mean-field or regular-solution) free energy of mixing for A and B polymers becomes

Δ*G*_{m} = –*T*Δ*S*_{m}^{id} + χ*kT*ϕ_{A}ϕ_{B}

However, in experimental practice, χ is imbued with whatever attributes are needed to describe the data; as such, it actually represents an excess free energy of mixing Δ*G*_{m}^{ex} that exhibits both enthalpic and entropic contributions. Therefore, in practice χ is recast as χ = χ_{H} + χ_{S} = α/*T* + β (where α and β are empirical parameters), and either α or β may also depend on ϕ and *N*. This complexity can lead to much confusion when, for example, comparing different measurements on the same system. Furthermore, either α or β may be positive or negative in sign. The case where α > 0 and β *<* 0 leads to the “classical” upper critical solution temperature phase diagram, whereas α < 0 and β *>* 0 leads to a lower critical solution temperature (i.e., phase separation on heating); both situations are common. Because Δ*S*_{m}^{id} varies inversely with *N*, it is negligible for typical molecular weights; consequently, the excess free energy is not just a correction, but the main contribution.

The characteristic magnitude of χ_{AB} spans several orders of magnitude, as illustrated below. From a predictive point of view, the situation is quite unsatisfactory. For systems in which short range, isotropic, dispersive interactions dominate, the solubility parameter approach might be expected to yield reasonable estimates [i.e., χ ≈ χ_{H} = *V*_{ref}/*kT*(δ_{A} – δ_{B})^{2}, where δ_{A} and δ_{B} are the associated segment solubility parameters]; however, in practice, agreement with experiment is often remarkably poor. Additionally, although solubility parameters can be used to anticipate qualitatively where a particular blend might lie on this continuum, there is no general theoretical approach for predicting the relative contributions of χ_{H} and χ_{S} or for computing either term to within, say, a factor of 2. As phase boundaries in polymer systems typically depend on the dimensionless product χ*N*, it is therefore not possible to even anticipate the molecular weights needed to produce an experimentally accessible phase boundary.

Blends of hydrogenous and deuterated isotopomers represent the simplest case. Here, it is usually found that χ ≤ 10^{−3}, and χ_{H} can be understood in terms of differences in zero point energy and the polarizabilities of C–H and C–D bonds. Nevertheless, χ_{S} is not negligible, for reasons that remain elusive. For example, for (H/D) poly(styrene) blends, χ (373 K) ≈ 0.00014, but χ_{S} is of comparable magnitude to χ_{H} and is negative (*51*). For poly(styrene)/poly(methylmethacrylate), a pair of prevalent thermoplastics, χ (373 K) ≈ 0.038, a relatively modest value; however, χ_{S} is positive and, surprisingly, represents ~70% of the total, so that χ itself has almost no *T* dependence (*52*). When polar or ionic polymers are mixed with nonpolar partners, χ 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, for poly(styrene)/poly(lactide), χ (373 K) ≈ 0.15, but the magnitude of χ_{S} at this temperature is about half that of χ* _{H}* (

*53*).

Multiblock polymers, such as those sketched in Fig. 1, offer unlimited potential for designing soft materials with precisely specified structures, subject to two critical limitations: (i) The structures can be accessed synthetically, and (ii) predictive theoretical tools to guide molecular design are available. A parallel can be drawn with the complexity and diverse functions associated with proteins, another well-known class of multifunctional macromolecules. Theorists strive to predict polypeptide structure and ultimate function based on models that employ quantum chemical and statistical mechanical tools capable of accounting for both short-range (~0.1 nm) and long-range interactions mediated by an aqueous environment and constrained by the macromolecular architecture (*11*). Essentially any sequence of less than 100 residues of the 20 naturally occurring amino acids (and a bevy of unnatural variants) can be produced using commercially available solid-phase peptide synthesizers. The challenge is to anticipate which linear combinations of amino acids, drawn from 20* ^{n}* possibilities, will lead to the desired secondary, tertiary, and even quaternary structures and corresponding functions.

In some respects, block polymers appear to pose a less daunting theoretical challenge, particularly for flexible and noncrystalline polymers, because the chain structure at the monomer level does not play a primary role in determining the domain geometry or packing symmetry. Moreover, with no direct analog to solid-phase peptide synthesis, contemporary research focuses on producing block polymers with a minimum degree of complexity (that is, the smallest *n* and *k*) necessary to achieve a desired morphology and complement of properties. However, whereas protein folding is intrinsically intramolecular, supramolecular self-assembly of block polymers leads to ordered structures with unit cells that may contain many thousands of polymer chains (millions of atoms), resulting in unique theoretical and computational challenges. Hence, even the seemingly primitive case of ABC triblocks (*k* = 3, *n* = 3) is only marginally understood in terms of the resulting phase structure and associated properties.

## Synthesis

Block polymer synthesis has evolved considerably since the introduction and development of living anionic polymerization by Szwarc more than 50 years ago (*12*). Numerous strategies that enable the covalent connection of two or more polymer blocks have been developed, including sequential addition of distinct monomers to an active polymer chain, chain-coupling strategies, and transformations of polymer end groups to accommodate diverse and often incompatible polymerization mechanisms. Chemical connectors that link together more than two chains are increasingly available as evidenced by the heroic syntheses of a 31-arm starblock pentapolymer (*k* = 5, *n* = 31) (*13*). In principle, all of the structures depicted in Fig. 1 could be prepared with arbitrary choice of block constituents using modern polymerization methods and polymer functionalization strategies, including numerous controlled radical, ring-opening, metal-catalyzed, and ionic polymerizations, often in combination with highly selective and efficient functional-group transformation techniques (*14*, *15*).

Modern block polymer molecular design is driven by the intended applications. For example, a thermally stable and mechanically robust membrane endowed with a dense array of specifically sized nanopores necessitates a combination of glassy, ductile, and chemically etchable (or cleavable) blocks (*16*). Biocompatibility further limits the possible ingredients. The associated segment-segment interaction parameters and optimal block architecture demand a carefully targeted synthetic strategy, possibly including multiple routes to the desired structure, drawn from an ever-expanding synthetic tool kit. Although high degrees of molecular uniformity are generally desirable, in most circumstances some level of imperfection (e.g., less than *n* blocks) and modest chain-length distribution can be tolerated [in fact, tailored dispersity represents a strategic design parameter (*17*, *18*)]; the goal is to build the appropriate structure in the most synthetically economical way possible.

Figure 2 illustrates routes to all possible sequences of a linear ABC terpolymer (*k* = 3, *n* = 3) designed to incorporate three complementary physical and chemical properties: (i) glassy poly(styrene) (PS), (ii) rubbery poly(isoprene) (PI), and (iii) chemically etchable and biorenewable poly(lactide) (PLA). Although only two of these three structures have been reported, PI-PLA-PS is certainly tractable on the basis of a related precedent (*19*).

The synthetic strategies depicted in Fig. 2 require various combinations of ring-opening, anionic, and controlled radical polymerizations with concomitant end-group manipulations. In fact, the drive to larger *n* and *k* will almost certainly necessitate multiple polymerization mechanisms and functional-group manipulations, the order of which will depend on the desired block sequence. Complexity in ABC terpolymer systems is further increased by introducing the possibility of cyclic motifs and three-point junctions (miktoarm stars or brushes). Putting all the pieces together requires the continued development of efficient processes and realizable retrosynthetic analyses.

Introduction of a fourth block with only three monomers (*k* = 3, *n* = 4) provides an added level of complexity not realized in ABC terpolymers. Asymmetric ABCA′ tetrablock terpolymers (*N*_{A} ≠ *N*_{A}′) offer unique opportunities for designing bulk materials (highlighted in the Structure section) and tailored solution structures (*20*). However, independent control over the length of all four blocks requires synthetic strategies that depend on the specific block chemistries. For example, the addition of water-soluble and biocompatible poly(ethylene oxide) (PEO) to both ends of a PS-PI core should be feasible. Sequential anionic polymerization of styrene and isoprene initiated with an alkyllithium reagent containing a protected hydroxyl group (*21*) followed by end-capping with one unit of ethylene oxide generates a heterotelechelic diblock copolymer. Activation of the free alcohol end enables 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 subsequent polymerization of ethylene oxide from the B segment in this scheme. Unmasking and activation of the hydroxyl group at the B terminus allows for independent control of *N*_{A} relative to *N*_{A}′. Such asymmetric ABCA′ tetrablock 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 polymers (*23*), and various “click” strategies (*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 “masterpiece” to synthesize using the expanding palette of monomer “paints” and polymer synthesis “brushes” represents 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 commodity and value-added applications, paralleling the synthesis of small molecules for therapeutic markets. Organic chemists can build complicated small-molecule structures with an amazing, almost arbitrary array of chemical functionalities; to know which molecule is efficacious requires a 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 Helfand’s strong segregation (*25*) and Leibler’s weak segregation (*26*) analyses of AB diblock copolymers. These theories have since been subsumed into a comprehensive 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 by χ_{AB}, and maximizing configurational entropy, which is inversely proportional to *N* and quadratically dependent on the extent of chain stretching relative to the unperturbed state. Simplifying assumptions, including Gaussian chain (random walk) statistics and constant density, enables tractable computational schemes that have accounted for all of the experimentally observed diblock morphologies (*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 as *k*(*k* – 1)/2 distinct and narrow interfaces characterized by interfacial tension γ* _{ij}* ~ (χ

*)*

_{ij}^{1/2}; strong segregation implies

*N*≈

_{i}*N*>> 10/χ

_{j}*. Reducing the individual values of χ*

_{ij}*leads to broader interfaces and, ultimately, mixing of*

_{ij}*i*and

*j*blocks. Thus, an ABC triblock terpolymer (

*k*= 3,

*n*= 3) may contain one, two, or three types of interfaces (or none if entirely disordered). For example, the condition χ

_{AB}= χ

_{BC}<< χ

_{AC}favors two interfaces (A/B and B/C), whereas χ

_{AB}= χ

_{BC}>> χ

_{AC}often produces the maximum three. The relative magnitudes of χ

_{AB}, χ

_{BC}, and χ

_{AC}, along with

*N*

_{i}, are the primary determinants of the interfacial topology and overall surface areas. Simple extension beyond diblocks to an ABC architecture has yielded more than 30 distinct microphases (

*7*), many of which have been captured by SCFT (

*28*). Branched (e.g.,

*k*= 3,

*n*= 3 miktoarm) molecular architectures can enforce interfaces between otherwise incompatible blocks, leading to additional structural diversity (

*29*).

Adding just one additional block (*n* = 4) while keeping the number of block types fixed (*k* = 3) provides a powerful tool for decoupling domain geometry and ordering symmetry, as illustrated in Fig. 3 for the sequence ABCA′, one of nine possible tetrablock terpolymer enumerations, where A and A′ represent chemically identical blocks containing *N*_{A} and *N*_{A′} repeat units. Here, we focus on a restricted subset of available molecular parameters defined by χ_{AB} = χ_{AC} and (*N*_{A} + *N*_{A′})/2 = *N*_{B} *= N*_{C} *= N*/4 (i.e., constant composition) with two adjustable variables: (i) the magnitude of the interaction parameter χ_{BC} and (ii) the molecular symmetry factor ξ = *N*_{A′}/*N*_{A}. First, we consider the symmetric case ξ = 1.

With χ_{BC} = 0 and χ_{AB} = χ_{AC} >> 10/*N*, the ξ = 1 tetrablock terpolymer will exhibit a two-domain lamellar morphology with equivalent interfaces separating A and A′ domains from mixed B/C domains (Fig. 3A). Increasing the segregation strength to χ_{AB} = χ_{AC} = χ_{BC}/2 will result in a three-domain lamellar configuration with energetically equivalent A/B and C/A′ interfaces that exactly balance the internal B/C interface (*30*). Further elevating χ_{BC} will induce changes in the domain geometry that minimize the overall interfacial free energy (*A _{ij}* 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 effect by reducing

*A*

_{BC}/(

*A*

_{AC}+

*A*

_{AB}), the ratio of interfacial surface areas. (We note that this hypothetical sequence of morphologies may be superseded by other morphologies and never realized in practice). Clearly, adding an A′ block to the ABC sequence offers specific control over the shape and subdivision of the self-assembled domains.

Interdomain packing can be considerably influenced by the symmetry parameter ξ, as shown in Fig. 3B. We have selected the axially segregated cylindrical domain structure to illustrate this point. Tetrablock terpolymers (*k* = 3, *n* = 4) can produce ordered cylinders containing B and C subdomains, even with *N*_{A} + *N*_{A′} = *N*/2 (*31*); cylindrical structures at such low matrix compositions are inaccessible with diblock copolymers. For ξ = 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 ξ > 1, the C domains 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 tendency to pair short and long corona chains represents 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 required 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 analogous to spin alignment in magnetic materials (*32*) [e.g., noncentrosymmetric ferromagnetic ordering (*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 ABA′C while holding all other parameters constant leads to qualitatively different phase behavior. Fixing χ_{AB} = χ_{AC} << χ_{BC} leads to spherical (perhaps cylindrical) domains made up of C cores surrounded by shells of A and A′ embedded in a matrix of B; hence, A_{BC} = 0. In this case, interdomain packing can be influenced by moderating χ_{AB} and ξ. Recent investigations with poly(styrene-*b*-isoprene-*b*-styrene-*b*-ethylene oxide) (SISO) tetrablock terpolymers (χ_{SI} ≈ χ_{SO} ≈ χ_{IO}/4) have revealed sphere-packing geometries beyond conventional bcc order, including a σ phase (*35*) and simple hexagonal symmetry (*36*).

Clearly, tetrablock terpolymers represent the tip of the iceberg in considering opportunities for designing multiblock polymers with tailored domain geometry, connectivity, and packing symmetry. Characterizing the resulting structures requires a suite of complementary techniques—most importantly, real space imaging by transmission electron microscopy (TEM) and reciprocal space assessment using small-angle x-ray scattering and small-angle neutron scattering (SAXS and SANS) (*35*).

## Theory

Marked advances have been made over the past two decades in the development of theoretical and computational tools for exploring complex block polymer assembly. Based on a methodology developed by Edwards (*37*), a powerful field-theoretic framework has emerged in which coarse-grained, particle-based models of polymer solutions or melts are converted to statistical field theories by the introduction of auxiliary potential fields (*38*). By this approach, field theory models can be constructed for virtually any type of interacting polymer system, including the multispecies block polymers that are the present focus, and in any statistical mechanical ensemble of interest. The resulting field theories are characterized by effective Hamiltonians (or actions) *H* that are highly nonlinear and nonlocal functionals of the field variables and have explicit dependence on segmental interaction and chain architecture parameters, such as *χ _{ij}* and

*N*. The Hamiltonian is generally also a complex-valued functional, which can have important consequences for generating numerical solutions.

_{i}Two major classes of field-based simulations can be built on top of the Edwards framework: (i) SCFT (*39*), where mean-field solutions are sought corresponding to saddle points of *H*, and (ii) field-theoretic simulations (FTS), referring to stochastic numerical sampling of the full (complex-valued) field theory (*40*). SCFT simulations are considerably less expensive than FTS, because they seek only a single saddle-point field configuration, but SCFT neglects field fluctuations that are especially important in solvated polymer systems. In the case of high–molar mass undiluted block polymers, such fluctuation effects 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 multiblock 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 space χ_{AB}*N*, χ_{AC}*N*, χ_{BC}*N*, *f*_{A} (= *N*_{A}/*N*), and *f*_{B} that 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 boundary conditions tend to yield defective morphologies that are metastable, rather than stable (lowest in free energy) for the specified parameters. 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 orthorhombic *Fddd* (O^{70}) phase (*7*). The simulation launched from random initial conditions settles into a metastable, highly defective *Fddd* structure (Fig. 4A), whereas an analogous simulation initialized from a deterministic seed with the proper symmetry falls quickly into the stable O^{70} 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 computational 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 quantitative mean-field theory, and their folding results do not enjoy the universality across broad families of monomers as do SCFT predictions for multiblock polymers.

Large-cell SCFT simulations are the tool of choice when prospecting for candidate ordered phases in a new polymer system for which experimental results are not available (*38*). The challenges of *χ _{ij}* estimation notwithstanding (see Box 2), one can specify interaction parameter values, block sequences, and block lengths and then proceed to use large-cell SCFT simulations to predict microphase structure candidates. If simulations launched from a variety of random initial conditions consistently lead to a single defect-free or defective, but still identifiable morphology, one can be reasonably confident that the stable structure has been identified. Defective morphologies can also be converted to defect-free ones by the deterministic seeding method illustrated in Fig. 4. Conversely, if repeated large-cell SCFT simulations from uncorrelated random seeds turn up multiple structures differing in symmetry, then it is necessary to compare the free energies of the competitive structures to establish which of them is stable. For this purpose, a second type of SCFT simulation is most efficient—a unit-cell simulation that attempts to converge a single unit cell of a candidate structure within the confines of the symmetry constraints of that phase (

*41*). Such a simulation seeks a saddle-point condition on the Hamiltonian with respect to the complex field variables while simultaneously minimizing

*H*(the mean-field free energy) with respect to the lattice parameters of the unit cell.

State-of-the-art numerical methods for large-cell simulations are based on spectral collocation (or pseudospectral) techniques with plane wave bases (*38*, *42*). Fast Fourier transforms are used to switch between real-space and reciprocal-space representations of the fields, allowing for efficient evaluation of the operators, forces, and energies embodied in SCFT. Large 3D calculations using up to 512^{3} = 1.34 × 10^{8} plane waves or grid points, such as those shown in Fig. 4, require the use of parallel algorithms implemented either on clusters of single or multicore central processing units or, more recently, on a single graphics processing unit containing up to 500 “light” cores. The method of choice for unit-cell SCFT simulations is a Galerkin spectral technique developed by Matsen and Schick that uses Fourier basis functions possessing the symmetry of the phase being considered (*41*). Although the method is not well suited to parallel computing, a smaller number of symmetry-adapted basis functions are generally required for accurate unit-cell simulations than the plane waves used for spectral collocation in parallelepiped cells (*43*).

In spite of these advances in numerical SCFT, the tool is primarily used to predict self-assembly given a block polymer design. More useful from the standpoint of applications is the inverse problem—namely, the identification of polymer designs that will self-assemble into a specified morphology. Though little progress has been made to date on this problem, techniques such as inverse Monte Carlo, in principle, could be mated to SCFT toward this end. It is also notable 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 Pandora’s box? At a minimum, thermoplastic elastomers require *k* = 2 and *n* = 3 as evidenced by PS-PI-PS, introduced in the 1960s and still the most successful block copolymer product in the worldwide 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-catalyst–mediated chain-shuttling mechanism (*45*), underscore this point. Increasing *k* quickly complicates the practical design of new materials yet challenges our imagination with unbounded opportunities. It is not unreasonable to claim that virtually any structure (domain morphologies and packing symmetry) can be created at length scales between roughly 5 nm and 1 μm using block polymers, while maintaining 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., multifunctional plastics) or enable a host of other products [for example, as intermediates in drug delivery, as scaffolds that guide the assembly of inorganic hard materials (*46*), or for pattern formation in the manufacturing of microelectronics (*3*)]. Realizing these tantalizing prospects requires overcoming challenges posed by complexity, a dilemma that throttles many modern technological ambitions ranging from biological systems to global climate prediction. Here, the approach seems clear: integration of engineering goals, creative chemistry, and predictive theoretical tools, augmented by continued advances in structural characterization methods. Fortunately, even minor extrapolation in multiblock complexity (e.g., *k* = 3, *n* = 4) offers a glimpse of what is possible in the future.

## References and Notes

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- ↵ SCFT calculations support the two-domain and three-domain lamellar morphologies illustrated in Fig. 3A.
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**Acknowledgments:**This work was supported by the Materials Research Science and Engineering Centers Program of the NSF under award numbers DMR-0819885 (F.S.B., M.A.H., T.P.L.) and DMR-1121053 (G.H.F., K.T.D.). C.M.B. thanks C. G. Willson for his support.