Stable Frank-Kasper phases of self-assembled, soft matter spheres
Abhiram Reddy, Michael B. Buckley, Akash Arora, Frank S. Bates, Kevin D. Dorfman, Gregory M. Grason
SStable Frank-Kasper phases of self-assembled, soft matter spheres
Abhiram Reddy, Michael B. Buckley, Akash Arora, FrankS. Bates, Kevin D. Dorfman, and Gregory M. Grason Department of Polymer Science and Engineering,University of Massachusetts, Amherst, MA 01003 Department of Physics, University of Massachusetts, Amherst, MA 01003 Department of Chemical Engineering and Materials Science,University of Minnesota, Minneapolis, MN 55455
Abstract
Single molecular species can self-assemble into Frank Kasper (FK) phases, finite approximants ofdodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are max-imally symmetric. FK phases are speculated to emerge as the minimal-distortional packings ofspace-filling spherical domains, but a precise quantitation of this distortion and how it affectsassembly thermodynamics remains ambiguous. We use two complementary approaches to demon-strate that the principles driving FK lattice formation in diblock copolymers emerge directly fromthe strong-stretching theory of spherical domains, in which minimal inter-block area competes withminimal stretching of space-filling chains. The relative stability of FK lattices is studied first usinga diblock foam model with unconstrained particle volumes and shapes, which correctly predictsnot only the equilibrium σ lattice, but also the unequal volumes of the equilibrium domains. Wethen provide a molecular interpretation for these results via self-consistent field theory, illuminat-ing how molecular stiffness regulates the coupling between intra-domain chain configurations andthe asymmetry of local packing. These findings shed new light on the role of volume exchange onthe formation of distinct FK phases in copolymers, and suggest a paradigm for formation of FKphases in soft matter systems in which unequal domain volumes are selected by the thermodynamiccompetition between distinct measures of shape asymmetry. Keywords: self-assembly; Frank-Kasper phases; optimal lattices; block copolymers a r X i v : . [ c ond - m a t . s o f t ] J un . INTRODUCTION Spherical assemblies occur in nearly every class of supramolecular soft matter, from ly-otropic liquid crystals and surfactants, to amphiphillic copolymers [1]. In concentratedor neat systems, self-assembled spherical domains behave as giant “mesoatoms,” adoptingperiodically-ordered crystalline arrangements. While superficially similar to lattices formedin atomic or colloidal systems – which are stabilized largely by bonding or translationalentropy – the periodic order in soft materials is governed by distinctly different principlesbecause lattice formation occurs in thermodynamic equilibrium with the formation of the“mesoatoms” from the constituent molecules themselves. Thus, the equilibrium sizes andshapes of “mesoatoms” are inextricably coupled to the lattice symmetry, and vice versa.In this article, we address the emergence of non-canonical, Frank Kasper (FK) latticesin soft materials, characterized by complex and large-unit cells yet formed by assembly ofa single molecular component. Initially constructed as models of metallic alloys [2, 3], FKlattices are a family of periodic packings [4, 5] whose sites are tetrahedrally-close packed andcan be decomposed into polyhedral (e.g. Voronoi or Wigner Seitz) cells surrounding eachsite containing 12, 14, 15 or 16 faces. Known as the FK polyhedra, these cells (Z12, Z14, Z15and Z16) possess variable volume and in-radii. Hence, FK lattices are natural candidatesto describe ordered, locally-dense packings of spherical elements of different radii such asatomic alloys [3, 5] or binary nanoparticle superlattices [6]. Once considered anomalous insoft matter systems, the past decade has seen an explosion in the observation of FK latticesin a diverse range of sphere-forming assemblies. These include (A15, σ ) liquid-crystallinedendrimers [7, 8], linear ( σ ,A15) tetrablock [9, 10], ( σ , A15, C14, C15) diblock [11–13] and(A15) linear-dendron [14] block copolymer melts, (A15) amphiphilic nanotetrahedra [15, 16],(A15, σ , C14, C15) concentrated ionic surfactants [17, 18] and (C14) monodisperse, func-tionalized nanoparticles [19]. The central puzzle surrounding the formation of FK latticesin these diverse systems is understanding why single-components assemble into phases com-posed of highly heterogeneous molecular environments.A common element distinct to FK formation in soft systems is the thermodynamic costof asymmetry imposed by incompatibility between uniform density and packing of perfectlyspherical objects (Fig. 1). In soft assemblies, the ideally spherically symmetric domainsare warped into lower-symmetry, polyhedral shapes (i.e. topologically equivalent to the2oronoi cells) which fill space without gaps. The minimal free energy state is the one forwhich the quasi-spherical domains (qSD) remain “most spherical.” The most commonlyinvoked notion of sphericity in this context is the dimensionless cell area A to volume V ratio, A ≡ A/ (36 πV ) / , which has a lower bound of 1 achieved by perfect spheres. Thecellular partitions of FK lattices play a key role in the mathematical modeling of dry foams,known as the Kelvin problem [20–23], which seeks minimal area of partitions of space intoequal volume cells. Based on the fact that the lowest-area, equal-volume cellular partitionknown to date, the Weaire-Phelan foam [20], derives from the FK lattice A15, Ziherl andKamien proposed that this lattice is generically favored thermodynamically in so-called“fuzzy colloid” models [24, 25], an argument subsequently adapted to sphere phases of blockcopolymers [26, 27]. Recently, Lee, Leighton and Bates reasoned that average “sphericity”could be increased (i.e. decreased mean A ) below the Weaire-Phelan structure if the equal-volume constraint for distinct cells is relaxed, as would occur for molecular exchange betweendistinct qSD [11]. Based on the Voronoi partitions, which have unequal volumes for FKlattices, σ was argued to have lower mean dimensionless area than A15, and thus should bestable over that lattice according to the sphericity argument, consistent with observations ofa σ lattice in diblock copolymer melts [13] and self-consistent field theory of conformationallyand architecturally asymmetric diblocks [28].While the role of volume asymmetry has been implicated previously in the formationFK lattices by soft qSD assemblies [29], critical questions remain unanswered. First, for agiven lattice, precisely which cell geometries and volumes accurately model qSD formation?Second, what are the relevant measures of sphericity selected by the assembly thermodynam-ics? Finally, how do these in concert select the optimal balance between shape asymmetry(non-spherical domains) and volume asymmetry (molecular partitioning among domains)for a given qSD lattice, and in turn, select the equilibrium lattice and determine the scaleof thermodynamic separation between the many competing FK lattices? We address thesequestions in the context of what we call the diblock foam model (DFM), which quantifies the The tetrahedral coordination of FK lattice implies that their partitions closely approximate the geometricconstraints of Plateau borders, and are therefore near to minimal-area partitions. In addition to A15, atleast two more partitions of FK lattices, σ and H, have also been shown previously [21, 23] to beat thearea of optimum originally conjectured by Kelvin, the BCC partition. In Appendix B 2, we report thatthe FK lattice P also belongs to this rarified category. IG. 1: Chain packing of spherical diblock copolymer domains of the BCC lattice (top), withcorresponding limits of weakly-coupled (bottom left) and strongly-coupled (bottom right) of coredomain shape of polyhedral (truncated-octohedron) cell symmetry. thermodynamic cost of asphericity in terms of a geometric mean of reduced cell area anddimensionless radius of gyration of the cells, and thus, integrates elements of both the
Kelvin and lattice Quantizer problems [30]. These geometric proxies for inter-block repulsion andintra-molecular stretching in qSD exhibit qualitatively different dependencies on cell shape,a factor that we show, based on this model and self-consistent field theory (SCFT) analysis,to be critical to the volume partitioning among distinct qSD and optimal lattice selection.Amongst the various classes of FK-forming soft matter [7–9, 11–15, 17, 19], we positthat diblock copolymers represent the optimal starting point for investigating the selectionof low symmetry FK phases by soft matter spheres. Diblock copolymers are a relativelysimple chemical system, consisting of two flexible chains bonded together at their ends, andthere exist robust theoretical methods for studying their phase behavior in the context ofuniversal physical models [31, 32]. The fundamental mechanisms underlying assembly ofdiblock copolymers that we elucidate here furnish the foundation for subsequent investiga-tions of other soft matter systems, where these basic principles are conflated with additionalphenomena emerging from electrostatics, hydrophobic interactions, and detailed packing ofthe complicated (non-Gaussian) configurations of their constituents.4
I. DIBLOCK FOAM MODEL OF FK LATTICE SELECTION
We adopt what we call the diblock foam model (DFM), first developed by Milner andOlmsted, in which the free energy of competing arrangements is reduced to purely geometricmeasures of the cellular volumes enclosing the qSD [39, 40]. To a first approximation, thesecells are the polyhedral Voronoi cells for a given point packing, whose faces represent coronalbrushes flattened by contact with neighboring qSD coronae. The model is based on strong-stretching theory (SST) of diblock copolymer melts, in which inter-block repulsions driveseparation into sharply divided core and coronal domains and the chains are well-extended.We also consider the case of large elastic asymmetry between core and coronal blocks,which itself derives from asymmetry of the block architecture or the segment sizes. Thiscorresponds to the polyhedral-interface limit [33], in which the core/coronal interface ineach qSD adopts a perfect, affinely shrunk copy of the cell shape (see Fig. 1,bottom right).Polyhedral warping of the interface is favored when the stiffness of the coronal blocks, whichfavors a more uniform extension from the interface to the outer cell wall, dominates overentropic stiffness of core blocks and inter-block surface energy, which both favor roundinterfaces.In this limit, the free energy per chain [26, 40], F ( X ), of a given lattice packing X derivesfrom two contributions, F ( X ) = γ A ( X ) R + κ I ( X ) R , (1)where γ and κ are coefficients fixed by the chain properties (i.e., block lengths, segmentlengths, inter-block repulsion), and R is the radius of a sphere of equal volume to themean volume of cells, or (4 π/ R = n − X (cid:80) n X i =1 V i , where V i is i th cell volume of n X to-tal cells in X (see Appendix A 1 for details). The first term represents the enthalpy ofcore-corona contact, and hence is proportional to the (per volume) interfacial area, whichitself is proportional to the cell area A i , measured by the dimensionless (mean) cell area, A ( X ) = ( n − X (cid:80) n X i =1 A i ) / (4 πR ). The second term represents the entropic costs of extendingpolymeric blocks (here modeled as Gaussian chains) in radial trajectories within qSD. Thiscost grows with the square of domain size and depends on qSD shape through the dimen-sionless square radius of gyration, or stretching moment I ( X ) = ( n − X (cid:80) n X i =1 I i ) / (4 πR / I i = (cid:82) V i d x | x − x i | is the second-moment volume of the i th cell, whose center liesat x i . Optimizing mean cell size ( R ) yields the minimal free energy of lattice X , relative5 IG. 2: A DFM structure for the cubic repeat of C15 is shown in (A), qSD centers shownwithin the Z12 and Z16 cells, red and blue, respectively. In (B), equilibrium shapes for three cellgeometries studied, in which slight curvature of cell faces and edges is visible for the relaxed shapecases. Results of the DFM are shown for 11 competing FK phases (labeled above), plotted asfunction of mean coordination, or average number of cell faces (cid:104) Z (cid:105) : mean dimensionless area (C);mean dimensionless stretching (D); mean free energy (E); and rms volume variation among cellsrelative mean volume (cid:104) V ( X ) (cid:105) (F). In (C)-(E) points are labeled according to the legend in (D)and the dashed and solid lines shows unconstrained and Voronoi results respectively for BCC. In(F), variable volume cell results are compared to qSD volumes extracted from SCFT at χN = 40, f = 0 .
25 and (cid:15) = 2 as described in the text. to the perfect sphere free energy F = ( γ κ ) / , F ( X ) ≡ min R (cid:2) F ( X ) (cid:3) /F = (cid:2) A ( X ) I ( X ) (cid:3) / . (2)This geometric mean favors simultaneously low values of dimensionless area and stretching .While minimal area partitions (at constant volume) are associated Kelvin’s foam problem,lattice partitions that optimize I i (at fixed density) are the object of the Quantizer prob-lem [30], which has applications in computer science and signal processing [34].The Milner and Olmsted model has been studied for flat-faced Voronoi cells of FCC, BCC Assembly thermodynamics depends on the dimensionless ratios of structure-averaged area and stretchingof cells and volume, as opposed to averages of dimensionless cell area and stretching. F ( X ) overarbitrary volumes and shapes of constituent cells in the DFM structure (see Appendix B fordetailed method and tabulated results).To assess the importance of relaxing volume and shape, consider the three distinct ensem-bles of qSD cells, shown for C15 in Fig. 2A,B. We have computed results for equal-volume,relaxed-shape cells, which cannot exchange mass, and centroidal Voronoi cells , which havefixed flat-face shapes but unequal volumes. The former ensemble neglects the possibility ofmass exchange between micelles, while the second optimizes stretching [34] but is suboptimalin terms of cell area . Neither model is realistic but they provide useful points of comparisonfor the unconstrained, relaxed-volume and shape cell model, which strictly minimize F ( X )for given X . Fig. 2C shows that allowing both volume and shape to relax leads to a completeinversion of the trend of A ( X ) with (cid:104) Z (cid:105) . Importantly, there is also a near degeneracy for thefree energy of FK structures in Fig. 2E, which all lie within 0.08% in F ( X ) (as compared tothe relatively large ≈
1% spread for equal-volume qSD). These results confirm the criticalrole of volume exchange among asymmetric qSD in the thermodynamics of lattice forma-tion [11, 12]. Among these nearly degenerate, fully unconstrained DFM structures, the σ phase overtakes A15 (minimal for fixed, equal volume) as the minimal energy phase (withnext lowest energy for P), consistent with its observation upon in annealling [11, 13] as wellas recent SCFT studies of conformationally asymmetric diblocks [28]. Notably, however, inthe relaxed-volume and shape DFM, σ possesses neither the minimal area (C14), nor mini-mal stretching (BCC). Rather, its predicted stability results from the optimal compromisebetween these competing measures of domain asphericity.The interplay between area and stretching underlies the emergent asymmetry in equilib-rium qSD volumes. Comparing the equal-volume to unconstrained DFM results in Fig. 2Cand D shows that volume relaxation has a far more significant effect on relaxation of A ( X )than I ( X ), which changes little by comparison. Relaxation proceeds for all structures by Centroidal Voronoi cells have generating points at the centers of volume of the cell, and hence, for a given X minimize the mean-square distance of all points to their corresponding central point (see Appendix B 1for additional details) IG. 3: The polyhedral warping of the A/B interface, measured by α i from SCFT profiles of χN = 40 and f = 0 .
29 diblocks, of BCC qSD is plotted vs. conformational asymmetry (cid:15) = a A /a B in (A). Corresponding 2D cross sections (normal to [100] through center of primitive cell) of qSDwithin the truncated-octahedral cells of BCC are shown in (B), with composition varying fromred in A-rich regions to blue in B-rich regions (A/B interface is white). AAlso shown in vectorsare the mean orientation of A-block segments (polar order parameter) [36]. In (C), the arealdistortion of Z14 and Z12 qSD from SCFT predictions of A15 are shown (same composition andsegregation strength as BCC), with corresponding section of the Z14 (cut normal to [100] throughface of primitive cell, see Fig. 22) qSD shown in (D) as in (B). Additionally, spatial distributionof the A-block (core forming) chain ends are shown in (E), varying from zero density (blue) tomaximal density (red) within the cores. Schematics illustrating respective discoidal and polyhedral qSD packing are shown in (F). In (G), the volume dispersion (normalized by the DFM prediction)is plotted vs. conformational asymmetry. inflating cells with relatively larger area, and shrinking smaller-area cells, restrained only bystretching cost creating highly unequal domain sizes (Fig. 5). Volume exchange for latticeswith large proportions of lower area Z12 cells (e.g., C14 and C15) achieve relatively large( ≈ A ( X ) when compared to the high- (cid:104) Z (cid:105) end of the spectrum (e.g., ≈ . ≈
19% variance for C14 and C15) is drivenby dramatic reduction in inter-block contact area, a drive that is ultimately limited by thethermodynamic balance with the entropic (stretching) costs of filling space with qSD ofunequal size. These results imply that structures with a larger equilibrium volume disper-sion (such as the lower- (cid:104) Z (cid:105) C14 and C15) structures are more susceptible to the effects ofthermal processing that selectively promote or inhibit chain exchange among equilibratingspheres [12] than phases such as A15, which relax free energy relatively little through volumeequilibration.Previous SCFT studies [12, 28] have shown that the canonical BCC sphere phase isovertaken by a stable σ lattice when the elastic asymmetry, embodied by ratios of statisticalsegment lengths (cid:15) ≡ a A /a B (cid:38) .
5. DFM not only correctly predicts σ as the dominantlystable sphere phase, but also does a remarkable job of predicting the relative hierarchyamong metastable FK competitors. This is evident in Fig. 20A-C, where we compare thefree energies, scaled enthalpies and entropies for σ , Z, C14, C15 and A15 predicted bythe unconstrained cell DFM to AB diblock SCFT calculations using methods describedin ref. [32] at somewhat strong segregation conditions χN = 40, where χ is the Flory-Huggins parameter for A/B contact and N is the degree of polymerization. DFM correctlypredicts the narrow 0 .
01% scale of free-energy splitting between these competitors for (cid:15) = 2diblocks in the composition range f ≤ .
25, where f is the volume fraction of the minorityblock. Moreover, DFM predicts their ranking relative to σ with the exception of Z, whichDFM predicts to be nearly degenerate with C15. The accuracy of DFM extends beyondthermodynamics to structure, most notably the volume asymmetry in Fig. 2F. III. MOLECULAR MECHANISM OF ASPHERICAL DOMAIN FORMATION
To probe the molecular mechanism that underlies the selection of FK lattices in blockcopolymers, we analyze two order parameters that quantify the respective asymmetric shapesand volumes of qSD, computed from the volumes enclosing A-rich cores in SCFT compositionprofiles of diblocks at χN = 40, f = 0 .
29 and for variable conformational asymmetry (see9ppendix C 2). The first parameter, α i = A A / B i − A poly − , (3)measures the degree of polyhedral warping of the core in terms of the dimensionless area A A / B i of the A/B interface of the i th domain relative to a sphere, where A poly i is the dimen-sionless area predicted for the perfectly polyhedral interface of the corresponding cell fromthe unconstrained DFM: α i = 0 for spherical interfaces; and α i = 1 for interfaces that adoptthe polyhedral shapes of the DFM cells. We define a second parameter, ν ( X ), that measuresasymmetry of unequal volumes enclosed within A/B interfaces predicted by SCFT, relativeto the volume asymmetry predicted by polyhedral cells of DFM for the same structure Xν ( X ) = (cid:10)(cid:12)(cid:12) ∆ V i ( X ) (cid:104) V ( X ) (cid:105) (cid:12)(cid:12) (cid:11) / / B (cid:10)(cid:12)(cid:12) ∆ V i ( X ) (cid:104) V ( X ) (cid:105) (cid:12)(cid:12) (cid:11) / (4)where ∆ V i ( X ) = V i − (cid:104) V ( X ) (cid:105) is the volume deviation of the i th domain relative to theaverage in X , and values of ν ( X ) greater (less) than 1 indicate that qSD in SCFT are more(less) polydisperse predicted by relaxed DFM cells.It has been argued previously [27] that the polyhedral warping, or faceting, of core-coronainterfaces should increase with (cid:15) , which controls the ratio of corona- to core-block stiffness,due to the relatively lower entropic cost of more uniformly stretched coronae achieved bypolyhedral interfaces. This expectation is consistent with the observed monotonic increaseof α from 0 at (cid:15) = 1 to the saturated value of α ≈ .
05 for (cid:15) (cid:38) − . As shown in Fig. 21, the polyhedral warp of the interface growsalso with increasing f , due to the increased proximity of the qSD cell boundary to theinterface and relatively shorter coronal blocks at larger core fractions. While clearly farfrom a sharply faceted shape, the increase in core shape anisotropy is obvious from 2D cutsthrough of the qSD shown in Fig. 3B, showing a visible warp of A/B interface towards thetruncated-octahedral shape of the BCC cell at (cid:15) = 3.For the FK phases, which are composed of distinct-symmetry qSD, areal distortion ex-hibits a markedly different dependence on increased coronal/core stiffness, as illustrated bythe plots of α and α vs. (cid:15) for A15 in Fig. 3C. Z12 domains exhibit a monotonic, albeit While this extends beyond what is realized with most flexible linear diblocks, bulky side chains includingbottlebrush configurations and miktoarm polymers would make the upper limit accessible. (cid:15) . Surprisingly, for the Z14 domains, the excess areadrops from its maximal value of α (cid:39) . (cid:15) = 1down to a lower, yet significant plateau value of α (cid:39) .
2, roughly twice the areal distortionfor BCC.The origin of this counterintuitive drop in dimensionless area of the Z14 cells with in-creased outer block stiffness is illustrated in Fig. 3D, which compares 2D sections of the Z14qSD of A15 at (cid:15) = 1 and (cid:15) = 3. While the shape for larger outer-block stiffness ( (cid:15) = 3)is consistent with a quasi-faceted interface that copies the polyhedral cell (with roundededges) of the Z14 domain, the conformationally symmetric case ( (cid:15) = 1) is neither facetednor spherical. It instead adopts oblate, or discoidal shape. The contrast in core shape is fur-ther reflected in the sub-interface (vector) orientational order parameter of A-segments [36]and the spatial distribution of A-block chain ends, also shown in Fig. 3D,E. For larger (cid:15) ,the preference for more uniform coronal block stretching drives the quasi-polyhedral domainshape, with radial chain trajectories extending from the center of the domain, a point atwhich core block ends are concentrated. In contrast, for the case of (cid:15) = 1, the stiffness ofthe core blocks is sufficient to resist deformations away from uniform core thickness. Oc-cupying the somewhat flattened Z14 cell with a qSD of uniform core thickness then leadsto the discoidal shape, in which chain ends spread laterally in a quasi-lamellar core rimmedby a quasi-toroidal packing at its circumference. The preference for uniform core thicknesswithin the relatively oblate Z14 cell, which gives rise to a larger area discoidal interface for (cid:15) = 1, ultimately gives way to the quasi-polyhedral qSD shape, and corresponding radialchain stretching, with increased outer block stiffness for (cid:15) (cid:38) discoidal → polyhedral transition qSD within the mostoblate cells of other FK phases, C15 and Z, leading to a corresponding drop in excess area α i from (cid:15) = 1 to (cid:15) ≈ (cid:15) = 1) realize a volumedispersion that is strongly divergent from the polyhedral geometry in the DFM, includingboth greater ( ν ( X ) >
1, for A15) and lesser ( ν ( X ) < (cid:15) (cid:38)
2, relatively stiffer coronal blocks pull the cores into radial-stretching, quasi-polyhedralshapes. This transition to more compact cores, in turn, results into volume redistributingamong equilibrium qSD tending to the ν ( X ) → IG. 4: Correlation between polyhedral warping of core shapes ( α i ) within symmetry-distinctqSD extract from SCFT at χN = 40, f = 0 .
25 and (cid:15) = 2 and the degree of frustration of chainstretching in the correspond cell, quantified by the (cell-wise) dimensionless stretching moment, I i . asymmetric volumes of DFM and SCFT shown in Fig. 2F.Notwithstanding the broad agreement between SCFT and DFM predictions, the degreeof polyhedral warping of qSD shape is both arguably modest (i.e., α (cid:46) . (cid:15) (cid:29) χN and f ) and highly variable in the FK structures, suggesting a heterogeneous degree ofshape frustration among cells. Moreover, the discoidal → polyhedral transition occurs onlyin high- α qSD, whereas low- α cells (e.g., Z12 cells of A15) maintain radial stretching and amonotonic dependence on (cid:15) . What controls the variability of coupling between cell geometryof polyhedral distortion? Fig. 4 shows the correlation between α i for qSD extracted fromSCFT at χN = 40, f = 0 .
25 and (cid:15) = 2 (i.e. in the quasi-polyhedral shape regime) plottedas a function the dimensionless stretching I i for the corresponding cells from the DFM.The generically increasing trend of α i with I i for cell geometries across competing phasesargues that the variable degree of shape frustration within distinct qSD, and its consequentimpact on qSD core shape, is regulated by the constraints of asymmetric chain-stretchingin polyhedral cells. In other words, the ultimate degree of asphericity of core distortionof qSD (measured by dimensionless area), is in fact, controlled by the local asphericity inradial stretching required by space-filling chain packing (measured by dimensionless radiusof gyration). 12 V. CONCLUDING REMARKS
We anticipate that the emergence of optimal FK lattice structure and thermodynamicsvia a balance of competing measures of domain asymmetry highlighted here for high molec-ular weight diblock copolymers will extend to other copolymer systems where these phaseshave been observed, including architecturally asymmetric copolymers, linear multiblocks,low molecular weight/high- χ systems and blends. In particular, lower molecular weightpolymers drive the system closer towards the strong segregation limit and away from themean-field limit. Each of these materials exhibit different molecular mechanisms throughwhich the relative stiffness of the coronal domain transmits the asymmetry of the local qSDpacking into the core shape. For example, the observation of polygonal/polyhedral warpingof outer zones of core-shell domains of linear mulitblock polymers [37] provides a plausiblemechanism to stabilize the σ phase observed in linear tetrablocks [9]. On the other hand,accurately modeling the formation of σ by low molecular weight conformationally asym-metric diblocks [11, 13], likely requires a non-Gaussian (finite extensibility) model of chainstretching, but one which nevertheless, like the dimensionless radius of gyration I ( X ), favorscompact domains and competes against the minimal area preference for unequal domain vol-umes. Beyond copolymers, we speculate further that additional intra- and inter-molecularmechanisms play the role of balancing the drive for minimal domain area in the formationFK phases, from giant nanotetrahedra [15, 16] to ionic surfactants [17, 18].The present results for the DFM also shed new light on the non-equilibrium pathwaysfor stabilizing metastable FK competitors, as has been demonstrated of conformationalasymmetric linear diblocks quenched from high-temperature disorder sphere phases to lowtemperature metastable A15, C14 and C15 phases [12, 38]. The low temperature quench issuspected to freeze out the inter-domain chain exchange needed to achieve the equilibrium σ state, thus the kinetically-trapped quenched state inherits the volume distribution of hightemperature micelle liquid state. The DFM suggests a new way to analyze the stabilityof FK states when domain volumes are out of equilibrium, suggesting the observation ofC14 and C15 may be selected among the low-temperature kinetically trapped arrangementsbecause it inherits a volume distribution that is both smaller in average cell size and possiblymore polydisperse than the equilibrium state at the low temperature, and hence, a betterfit to the “aggregation fingerprint” of low- (cid:104) Z (cid:105) packings.13 cknowledgments R.Gabbrielli, J.-F. Sadoc, R. Mosseri and G. Schr¨oder-Turk are acknowledged for valu-able input on geometric models of cellular packings. This research was supported by theAFOSR under AOARD award
Appendix A: Diblock Foam Model1. Polyhedral Interface Limit of Strongly-Segregated Diblock Sphere Lattices
We briefly overview the strong-segregation theory (SST) calculation of Milner and Olm-sted [39, 40] for spherical domains in polyhedral interface limit (PIL), also known as the straight path ansatz . We further show how the free energy of competing sphere packingsis computed from purely geometric measures of the cellular volumes that enclose distinctspheres [26, 27], which forms the basis for the Diblock Foam Model (DFM).Here, we focus on the case of AB linear diblocks with conformational asymmetry, butthe theory can be generalized to other architectures like miktoarm stars [27]. We considera chain with total segment number N = N A + N B , with f = N A /N the fraction of theA-block. Segments are taken to have equal volumes ρ − and potentially unequal statisticalsegment lengths, a A and a B , for the respective blocks. The ratio of segment lengths definesthe conformational asymmetry (cid:15) ≡ a A /a B . Within SST, the total free energy F ( X ) (in unitsof k B T ) of a periodic repeat spherical assembly of lattice packing X decomposes into twoterms F ( X ) = F int + F st , (A1)which represents the respective costs of inter-block repulsions at a core/coronal interfacesand the entropic cost of stretching of (Gaussian) chains from random walk configurations.The first term F int = Σ A int simply derives from the product of total area of core/coronal14ontact, A int , times Σ to give the surface area energy between phase separated A and Bdomains [41], Σ = ρ a (cid:114) χ (cid:16) (cid:15) / − (cid:15) − / (cid:15) − (cid:15) − (cid:17) , (A2)where χ is the Flory-Huggins parameter for AB repulsion and a ≡ √ a A a B is the geometricmean of segment lengths . For the i th cell of X , the core/corona interface is an affinelyshrunk copy of the outer cell that encloses a fraction f of the total cell volume. Hence,the area of the core interface of i th domain is f / A i , where A i is the cell area, and A int = f / (cid:80) n X i =1 A i , where n X is the number of domains (and cells) per periodic repeat.The entropic contribution from chain stretching for domain α in cell i (denoted as volume V α,i ) can be evaluated using the SST entropy derived from the “parabolic brush” theory [42],which can be expressed as [31], F ( α ) st,i = 3 π ρ N α a α (cid:90) V α,i d x z (A3)where z is the distance from the AB interface, where junction points are localized, fromwhich chain trajectories are assumed to extend along the radial lines extending from the cellcenter x i to the outer wall of cell i , and (cid:82) V α,i d x is the integral over volume. For sphericaldomains, these integrals are evaluated by describing the cell shape as a function of theradial directions ˆΩ extending from the cell center at x i : R i ( ˆΩ) and R (cid:48) i ( ˆΩ) are the respectivedistances to the interface and outer wall of the cell in direction ˆΩ. Because the core chainsoccupy a fixed fraction f of each “wedge” in the PIL, we have R (cid:48) i ( ˆΩ) = f / R i ( ˆΩ), and thestretching contributions from each block are proportional to the same geometric stretchingmoment, F ( A ) st,i = 3 π ρ N A a A (cid:90) d ˆΩ (cid:90) R (cid:48) i (ˆΩ)0 dr r (cid:2) R (cid:48) i ( ˆΩ) − r (cid:3) = π ρ f / N a A S i , (A4)and F ( B ) st,i = 3 π ρ N B a B (cid:90) d ˆΩ (cid:90) R i (ˆΩ) R (cid:48) i (ˆΩ) dr r (cid:2) R (cid:48) i ( ˆΩ) − r (cid:3) = π ρ (1 − f / ) (6 + 3 f / + f / )80(1 − f ) N a B S i , (A5)where S i ≡ (cid:90) d ˆΩ R i ( ˆΩ) , (A6) Note that Σ is varies with conformational asymmetry (cid:15) , and that it reduces to the standard result forinterfacial tension between immiscible polymer melts in the symmetric limit Σ( (cid:15) →
1) = ρ a ( χ/ / . I i = (cid:90) V i d x | x − x i | = (cid:90) d ˆΩ (cid:90) R i (ˆΩ)0 dr r = S i / . (A7)Combining these together and summing over the cells in the periodic repeat we have thetotal stretching free energy F st = π ρ N a (cid:104) f / (cid:15) + (cid:15) (1 − f / ) (6 + 3 f / + f / )(1 − f ) (cid:105) n X (cid:88) i =1 I i . (A8)Since melt assembly occurs at fixed total density, equilibrium states correspond to statesof minimal free energy per chain. Defining the mean volume of the cells in X as V = n − X (cid:80) n X i =1 V i , the total number of chains per periodic repeat is n X V ρ /N . The mean volumeper cell also defines a measure of the mean cell dimension R = (3 V / π ) / , the radius of asphere of equal volume to V . Using this definition we can rewrite the area per volume as (cid:80) n X i =1 A i n X V = 3 A ( X ) R (A9)and the stretching per volume as (cid:80) n X i =1 I i n X V = 35 I ( X ) R (A10)where the dimensionless quantities A ( X ) = n − X (cid:80) n X i =1 A i / (4 πR ) and I ( X ) = n − X (cid:80) n X i =1 I i / (4 πR /
5) depend only on cell shapes and are independent of R , or mean do-main size. Using these quantities and dividing F int + F st by the total chain number we arriveat 1, where the coefficients are given by, γ = N a (cid:112) χ/ (cid:16) (cid:15) / − (cid:15) − / (cid:15) − (cid:15) − (cid:17) , (A11)and κ = 3 π N a (cid:104) f / (cid:15) + (cid:15) (1 − f / ) (6 + 3 f / + f / )(1 − f ) (cid:105) , (A12)which are independent of structure X and are fixed for a given set chain properties. Opti-mizing F ( X ) with respect to R , we find a equilibrium mean domain size( R ) eq = R s (cid:16) A ( X ) I ( X ) (cid:17) / , (A13)where R s = ( γ/κ ) / ∝ ( χN ) / N / a is the thermodynamically selected radius of domainsif cells were equal volume spheres (i.e. A = I = 1).16e note that this model relies on the so-called parabolic brush theory [42] in the expres-sions for Gaussian chain entropy in eq. (A3), which are known to fail for brush-like domainswith convex curvature due to the presence end-exclusions zones missing from the parabolicmodel. Notwithstanding, the failure to properly account for these exclusion zones in thecoronal blocks of this calculation [43], this approximation only modifies the coefficient κ andits dependence of f . The proportionality of the stretching free energy with (cid:82) d ˆΩ R ( ˆΩ)follows on the general grounds that each “wedge” of the domain includes a number of chainsproportional to d ˆΩ R ( ˆΩ), each of which is stretched a distance proportional to R ( ˆΩ) andhence acquires a free energy penalty proportional to R ( ˆΩ).
2. Unequal Domain Volumes in A15 Lattices: Weighted Voronoi Partitions
Here we illustrate the dependence of the DFM energy F ( X ) = (cid:2) A ( X ) I ( X ) (cid:3) / onthe volume difference between symmetry-distinct cells of FK lattices. For this purposewe consider the A15 lattice, which can be decomposed into two Z12 cells (at the centerand corners of the primitive, cubic cell) and six Z14 cells (two positions decorating eachface of the primitive, cubic cell). An analytical relation for F (A15) can be obtained usingexpressions for cellular area, volume and second-moments of volume for the weighted Voronoicells of A15. Standard Voronoi partitions derive from the polyhedra constructed by planesthat bisect the center-to-center neighbor separation vectors normally. Here, we use theweighted partitions of A15 derived by Kusner and Sullivan [22], correspond to the (flat-face)polyhedra constructed from planes at variable separation between the Z12 and Z14 sites(i.e., non-bisecting).Fixing the length of the primitive cubic cell to 2, and the mean volume per cell is fixed to V = 1 (non-dimensional lengths), the total area of the cells can be expressed as a functionof c , which parameterizes the size of the dodecahedral Z12 cells: the volumes of these cells Specifically, it can be shown that the coronal brush free energy in a spherical geometry is proportionalto h , where h is the brush thickness, times a function of h/R s , where R s is the spherical radius. In thisgeometry h ( ˆΩ) = R ( ˆΩ) − R (cid:48) ( ˆΩ) = (1 − f / ) R ( ˆΩ) and R s = R (cid:48) ( ˆΩ) so that h/R s = ( f − / −
1) for eachwedge, independent of ˆΩ V Z12 = c /
2; which implies V Z14 = (4 − V Z12 ) /
3. The mean cell area [22] is18 (cid:88) i =1 A i ( c ) = 32 + 32 √ √ − √ − c (A14)while the total second moments per cubic repeat was calculated by Kashyap and Neuhoff [45]as 18 (cid:88) i =1 I i ( c ) = 132 (3 c − c + 10) . (A15)Normalizing these by the area and second moment of spheres of V = 1 gives the dimension-less free energy of A15 for flat faced cells, F c (A15) = (cid:16) (cid:17) / (cid:16)(cid:2)
12 + 12 √ √ − √ − c ] (3 c − c + 10) (cid:17) / (A16)The dimensionless area, stretching and free energy are plotted as function of the volumedifference ∆ VV = 23 ( c − , (A17)where ∆ V = V Z12 − V Z14 between Z12 and Z14 cells in Fig. 5. This shows that the dimen-sionless area is minimal for vanishing Z12 volume ( c = 0), while dimensionless stretching isin fact minimized by the standard (centroidal) Voronoi partition ( c = 5 / V /V = − .
03. The competition between these drives for unequalcell volumes which results in optimal free energy ( F c (A15) = 1 . c min = 1 .
22 and ∆
V /V = − . Appendix B: Numerical optimization of cell geometry
Beyond the case of the flat-faced A15 cell results shown in Fig. 5 and discussed in the pre-vious section, the relaxation of the cellular partitions from competing SD phases is performedusing the
Surface Evolver (SE) [35]. While most commonly used for area optimization prob-lems (e.g. dry foam models, minimal surfaces), SE generically optimizes a target function(e.g., energy, area) defined on the facets of triangulated surface mesh subject to variousgeometric constraints, for example periodic boundary conditions or volumes within bodiesenclosed bodies (e.g., cells or bubbles).For the DFM (and for Kelvin and Quantizer problem results below) we construct initialconfigurations that are input into SE by generating the Voronoi partitions from the point18
IG. 5: (A) Plot of A , I and F for A15 lattice as a function of varying volumes of Z12 and Z14cells; in (B) zoom in of minimal for free energy and stretching. In (C), schematics of A15 primitivecell with unequal cell volumes: Z12 (blue) and Z14 (green) polyhedra. lattice positions of competitor structures within triply periodic, rectilinear box. The as-pect ratio of the periodic cell dimensions and the initial coordinates of the cell centers areextracted from references listed for each FK lattice in Sec. B 3 below. Voronoi cells arecomputed using Voro + + [44] and then converted to SE input files via a custom pythonscript. In addition to the initial topology of the “foam” structure, the SE input file alsodefines a discrete approximation of the dimensionless stretching, I , area A , and DFM freeenergy, F .Dimensionless area derives directly from computed total facet area and enclosed volumesof cells, while the stretching is computed as follows. For cell i in structure X , an initialcenter x (cid:48) i is chosen as fixed reference point within the body. Since x (cid:48) i is, in general, not thecentroid of i , the stretching integral x i splits into two contributions, I i = I (cid:48) i ( x (cid:48) i ) − V i | x i − x (cid:48) i | (B1)where I (cid:48) i ( x (cid:48) i ) ≡ (cid:82) V i d x | x − x (cid:48) i | is the second moment of V i with respect to the referencepoint x (cid:48) i . For a triangulated mesh composed of triangular facets with center X f , area ∆ A f ,19outward) normal N f , these quantities can be approximated using the discrete sums, I (cid:48) i ( x (cid:48) i ) = 15 (cid:88) f ∈ i ∆ A f N f · ( X f − x (cid:48) i ) | X f − x (cid:48) i | , (B2)and x i − x (cid:48) i = 14 V i (cid:88) f ∈ i ∆ A f (cid:2) N f · ( X f − x (cid:48) i ) (cid:3) ( X f − x (cid:48) i ) . (B3)These quantities are evaluated by use of facet general integral in SE. In the limit of∆ A f → F ), area ( A ) or stretching ( I ) for a given set of constraints, fixed periodicity, number of cellsand with or without enforcing equal volume among distinct cells. Numerical optimizationproceeds by successive interaction of vertex relaxation followed by mesh refinement steps. Foreach mesh refinement, vertices are relaxed until the optimized quantity ( F , A or I ) changesby less than 10 − . Mesh refinements proceed until the total change of post-relaxation valueof the target quantity falls below 10 − upon successive mesh refinements. FIG. 6: (A) Computed minimal dimensionless stretching moments ( I ( X )) computed from: equalvolume constraint (blue) and Centroidal Voronoi Partitions (green). (B) Computed minimal di-mensionless area ( A ( X )) for equal volume cells (i.e. Kelvin problem results). Dashed lines showresults for BCC. . Minimal stretching cells: Quantizer problem For comparison to DFM result, we compute the cell geometries that optimize I for thecompeting FK lattices. This optimization is directly related to the Quantizer problem [45–47] that seeks the optimal decomposition of lattice into cells, at a given cell density, whoseaverage square distance to the generating points x (cid:48) i is minimal (i.e., minimal sum of thesecond moments (cid:82) V i d x | x − x (cid:48) i | ) ). For a fixed set of generators , x (cid:48) i , the cells that minimize I are given by the (unweighted) Voronoi partition (VP) derived from x (cid:48) i . Additionally, fora fixed set of cells , V i , the generators that minimize I are given by the centroids x i = V − i (cid:82) d x x of V i . Hence, VP whose generating points are cell centroids, so-called centroidalVoronoi partitions (CVP), are local minima of I , for a given cell topology [34, 48]. Therefore,in the context of the DFM, the CVP correspond to cell geometries that rigorously minimizethe entropic cost of chain stretching.We compute the CVP for competing lattices by minimizing I ( X ) within SE startingwith generating points corresponding to reported lattice site positions of FK lattices (assummarized in Sec. B 3). CVP results from the minimization of I at fixed mean volume ofcells V (fixed dimensions of the periodic repeat) but without constraints on the individualcell volumes V i are shown in Fig.6A. For comparison, we also compute the minimal stretchingcells for fixed, equal cell volumes , V i = V , a constraint which accounts for increased values of I ( X ) that decrease with mean coordination of cells (cid:104) Z (cid:105) . This trend is consistent with Fig.2where the asymmetry in volume among cells for CVP is smallest for large- (cid:104) Z (cid:105) structures(A15) and largest at for small- (cid:104) Z (cid:105) structures (C14, C15). This trend implies that imposingthe equal cell volume constraints requires a smaller distortion from the optimal stretching cellgeometry for structures whose CVP are closest to equal volume (i.e. at large- (cid:104) Z (cid:105) ). Notably,no FK structure beats the minimal stretching value of BCC partition, I (BCC) = 1 .
2. Minimal area cells: Kelvin problem
For comparison to DFM result, we also compute the cell geometries that optimize A forthe competing FK lattices. As shown earlier in Sec.A 2 for flat faced cells (weighted VP) ofA15, optimizing A in the absence of the volume constraints is unstable due to shrinking of21ells to zero volume, and hence is not well-defined with respect to comparison of partitionsof different lattices X . A well-defined comparison is possible with constraints on the relativevolumes among cells, such as in the case of the Kelvin problem , which seeks partitions into equal volume cells ( V i = V ) that minimize mean cell area, or equivalently, surface areaenergy. The results from the minimization of A at fixed, equal volume of cells is shown inFig. 6B. Notably, the minimal area partition deriving from A15, the Weaire Phelan foam, isminimal [20], beating Kelvin’s conjectured optimal foam, deriving from BCC. Partitions ofFK structures σ and H also been previously reported [21, 23] to achieve lower areas than theKelvin foam (yet still larger than A15). Here, we report a third counterexample of Kelvin’sconjecture, deriving from the FK lattice P ( A (BCC) = 1 . > A (P) = 1 . > A (A15) = 1 . σ , δ and M, all with dimensionless area exceeding the Kelvin foam.
3. Competing lattices: Results
Here, we summarize the SE results for competing structures in terms of dimensionlessfree energy F , dimensionless surface area energy A , dimensionless stretching moment I forvarious optimizations (optimal area, stretching or DFM free energy) and under different con-straints (with and without equal cell volume constraints). For comparison, we also includedimensionless area, stretching and DFM free energy for Voronoi partitions correspondingto the initial generating points (not necessarily centroidal). Additionally, we compare vol-ume histograms for both centroidal Voronoi partitions (upper histogram) and unconstrainedDFM cells (lower histogram), plotted in terms of the volume fraction φ i occupied by cell type i , in each structure. For each structure, cell types are classified and color coded in terms ofnumber of faces: Z12 (blue); Z14 (green); Z15 (orange); and Z16 (red). Additionally, in thehistograms, cell types are annotated according to the Wyckcoff positions corresponding totheir generating points. We note that reported values for dimensionless area by Phelan, which from Fig. 2 of ref. [21] gives A ( σ ) = 1 . > A (H) = 1 . IG. 7: BCC; space group: Im ¯3 m ; periodic cell: (cubic) a:b:c=1:1:1; n X = 2, (cid:104) Z (cid:105) = 14FIG. 8: A
15; space group:
P m ¯3 n ; periodic cell: (cubic) a:b:c=1:1:1; n X = 8; (cid:104) Z (cid:105) = 13.5, init.coords.: Cr Si from ref. [49] IG. 9: H ; space group: Cmmm ; periodic cell: (orthorhombic) a:b:c=1:3.88:1; n X = 30; (cid:104) Z (cid:105) =13.466; init. coords.: ref. [51] (note that reference indicates Z12 spheres are situated at Wyckoffpositions 2(a) and 4(e) whereas we have used positions 2(a) and 4(f)) IG. 10: σ ; space group: P /mnm ; periodic cell: (tetragonal) a:b:c=1.9:1.9:1; n X = 30; (cid:104) Z (cid:105) =13.466; init. coords.: ref. [50]FIG. 11: Z ; space group: P /mmm ; periodic cell: (hexagonal*) a:b:c=1:1:0.993; n X = 7; (cid:104) Z (cid:105) =13.428; init. coords.: ref. [52] (*For SE calculations, we have used an equivalent orthorhombic unitcell with twice the number of cells than that of hexagonal unit cell) IG. 12: P ; space group: P bnm , periodic cell: (orthorhombic) a:b:c=1.91:3.57:1; n X = 56; (cid:104) Z (cid:105) =13.428; init. coords.: ref. [53] IG. 13: δ ; space proup: P ; periodic cell: (orthorhombic) a:b:c=1.03:1.03:1; n X = 56; (cid:104) Z (cid:105) = 13.428; init. coords.: ref. [54]FIG. 14: pσ ; space group: P bam ; periodic cell: (orthorhombic) a:b:c=1.95:1.64:1; n X = 26; (cid:104) Z (cid:105) =13.385; init. coords.: ref. [55] IG. 15: M ; space group: P nam ; periodic cell: (orthorhombic) a:b:c=1.89:3.3:1; n X = 52; (cid:104) Z (cid:105) =13.385; init. coords.: ref. [56] IG. 16: µ ; space group: R ¯3 m ; periodic cell: (hexagonal*) a:b:c=1:1:5.2; n X = 39; (cid:104) Z (cid:105) = 13.385;init. coords.: W F e from ref. [49] (*For SE calculations, we have used an equivalent orthorhombicunit cell with twice the number of cells than that of hexagonal unit cell) IG. 17: C
14; space group: P /mmc ; periodic cell: (hexagonal*) a:b:c=1:1:1.63; n X = 12; (cid:104) Z (cid:105) =13.333; init. coords.: M gZn from ref. [49] (*We have used an equivalent orthorhombic unit cellwith twice the number of cells than that of hexagonal unit cell)FIG. 18: C
15; space group:
F d ¯3 m , periodic cell: (cubic) a:b:c=1:1:1; n X = 24; (cid:104) Z (cid:105) = 13.333; init.coords.: Cu M g from ref. [49] ppendix C: Self-consistent field theory of conformationally asymmetric diblocks We use self-consistent field theory (SCFT) of a Gaussian chain model of diblock copoly-mer melts [31] to predict structure and thermodynamics of a multi-chain qSD formation.In particular, we consider a model where chains possess N A = f N and N B = (1 − f ) N segments of A and B type monomers each having statistical segment lengths as a A and a B = (cid:15) − a A respectively but having the same segment volume ρ − , with the Flory-Hugginsinteraction parameter χ describing the enthalpic repulsion between A and B blocks. InSCFT, the key statistical quantities are the chain distribution functions q ( x , n ) and q † ( x , n )which capture the statistical weights (constrained partial partition functions) of chains“diffusing” from their respective A and B ends to the n th segment located at position x . Following methods described in ref. [32] and elsewhere, these are determined self-consistently according to inter-segment interactions deriving from the mean compositionsprofiles φ A,B ( x ) = VN Q (cid:82) A,B dn q ( x , n ) q † ( x , n ), where Q = (cid:82) d x q ( x , n ) q † ( x , n ) is the singlechain partition function and (cid:82) A,B dn corresponds to the integration over the A or B blocksegments.
1. Thermodynamics of SD phases
Here, we summarize results for thermodynamics of qSD phases, in comparison to DFMpredictions, for conformational asymmetries (cid:15) >
1. For modest conformational asymmetry,i.e (cid:15) =1.5 and 2, Xie et al. [28] and Kim et al. [12] have shown that σ is the equilibrium forAB diblock copolymers in melt over a range of compositions between a stable BCC (low- f )and hexagonally ordered cylinders (high- f ). Kim et al. have additionally reported resultsfor FK candidates, σ , A15, Z, C14 and C15, for χ N = 40, which we analyze in more detail.In Fig.19 we also report two new metastable structures H and pσ for AB diblocks at χ N =25. Although metastable, these phases all beat BCC over a range of f , and H is shown tobe competitive with σ and A15 over the entire range of metastable compositions studied,0 . ≤ f ≤ .
33. 31
IG. 19: Relative Free energy per chain of FK lattices w.r.t BCC as a function of volume fractionfor χ N=25
To compare the DFM predictions to SCFT results, for σ , A15, Z, C14 and C15, at thehighest segregation strength computed ( χN = 40) we normalize the free energy per chain bythe value of A15, as plotted in Fig. 20A, since according to DFM, the free energy per chainfor each structure is proportional to the same quantities (a function of χN , (cid:15) and f ) thatvary with cell geometry. While DFM models are strictly constant with f and SCFT resultsshow at least slight variation of relative free energy with f , we note that the relatively closefree energies of SCFT are remarkably consistent with scale of separation predicted by DFMpredictions.We also compare the relative ranking of σ , A15, Z, C14 and C15 in terms of the enthalpicand entropic contributions to the free energy per chain, ˜ F (cid:48) enthalpy = V − (cid:82) d x χφ A ( x ) φ B ( x ),and ˜ F (cid:48) entropy = ˜ F (cid:48) tot − ˜ F (cid:48) enthalpy , which are computed from SCFT solutions as described in ref.[31] and elsewhere (here, primed quantities refer to values derived from SCF and unprimedquantities refer to their values from DFM). To extract strictly the geometric dependence ofthese thermodynamic quantities, we note from the DFM model (predicated on the strong-segregation and the polyhedral interface limits) that˜ F enthalpy = γ A R ; ˜ F entropy = κ I R (DFM) (C1)which motivates the definition of scaled-enthalpy A (cid:48) and scaled-entropy I (cid:48) computed from32CF results for ˜ F (cid:48) enthalpy and ˜ F (cid:48) entropy , appropriately scaled by the mean sphere radius R (cid:48) (theradius of a sphere of equal mean volume to equilibrium SD for a given structure) accordingto A (cid:48) ≡ γ − ˜ F (cid:48) enthalpy R (cid:48) ; I (cid:48) ≡ F (cid:48) entropy κR (cid:48) (SCFT) . (C2)This definition scales out the variation of enthalpic and entropic contributions due to thedifference in mean domain sizes from structure to structure. Fig. 20 B and C plots therespective SCFT results for scaled enthalpy ( A (cid:48) ( X )) and scaled entropy ( I (cid:48) ( X )) normalizedby the value for A15, and compared to DFM predictions. Additionally, Fig. 20 D plots themean domain sizes R (cid:48) ( X ) (relative to A15) computed from SCFT, which largely confirmsthat generic prediction of DFM, in eq. (A13) that structures corresponding to relativelysmall stretching costs favor relatively larger domain sizes (aggregation numbers per sphere). FIG. 20: (A) Relative free energy (B) Scaled enthalpy (C) Scaled entropy of FK lattices relativeto A15 from SCF (circles) and DFM (dashed lines) (D) Relative mean domain sizes of FK lattices(relative to A15). SCF results are for χN = 40 and (cid:15) = 2.
2. Geometric analysis of spherical domains
The geometry of SD core shapes predicted by SCFT are analyzed in terms of the ABinterface, which can be extracted from the equilibrium composition profiles, specificallythe isosurfaces where A and B have equal volume fractions, φ A ( x ) = φ B ( x ) = 0 .
5. Fromthe isosurfaces, numerically extracted using MatLab, the total areas and enclosed volumeswithin each SD in the predicted SCFT structure can be directly computed. Because thecore blocks constitute a fixed fraction f of the entire chain, the core volume accounts for thesame fixed fraction of the entire qSD. As shown in Fig.3 for A15 and BCC at f = 0 .
29, the33real distortion parameter, α i , of the core interface varies with conformational asymmetry.In Fig. 21 we also show results at higher core composition, f = 0 .
34. These indicate thatdegree of polyhedral warping of interface increases with f , due to the enhanced proximity ofthe interface to the outer boundary (or cell “wall”) between neighbor domains. Fig. 21 alsoshows the variation of α i with f for C15, highlighting the presence of two populations of SDin the structure: relatively spherical Z16 domains (low α i ), and less spherical Z12 domains(higher α i ). FIG. 21: Measure of areal distortion of AB interface of distinct domains for competing FK phasescomputed from SCF predictions at χN = 40, f = 0 .
29 (open circles) and f = 0 .
34 (filled circles).
Like the case for the Z14 cells of A15 (shown in Fig.3), these higher- α i cells of C15and Z also undergo a discoidal-to-radial transition as (cid:15) is varied from 1 (conformationallysymmetric) to ≈ polar orientational order parameter, t A ( x ),of A block segments using methods described in ref. [36], t A ( x ) = V N Q (cid:90) A dn [ q ∇ q † − q † ∇ q ] , (C3)where the vector orientation of the segments is defined to point from the free A end towardsthe junction point along the chains. These orientational profiles are shown in 2D sectionsthrough spherical domains of BCC and A15 in Fig.3. In Fig. 22 we also show 2D cutsthrough Z12 domains and Z15 domains of C15 (B,C) and Z (E,F), respectively. Note thatonly streamlines of t A ( x ) are shown in these figure, and thus only the local orientation of Asegments, but not the degree of alignment, is visible. Both of these domains show a trendconsistent with the sub-domain morphology of the Z14 cell in A15. For conformationallysymmetric chains ( (cid:15) = 1), the interface shape is more oblate than the polyhedral cell en-34losing the domain (which itself can be observed from the flow lines about the orientationalorder parameter), and the core regions are composed of a quasi-lamellar “puck” encircledby quasi-toroidal rim. In contrast, when conformational asymmetry imposes a sufficientlylarger cost on coronal stretching ( (cid:15) (cid:38) q † ( x , n = 0) and shown in Fig. 22 D and G.35 IG. 22: (A) 3D density plot of core block forming A15 from SCFT data at f =0.29, (cid:15) =1, χN = 40in primitive cell (on left) with the Z14 cell surrounding a corresponding qSD on the [100] face shownin green (on right) corresponding to the sections shown in Fig.3. (B) shows the same but for theC15 structure, and a 2D section through a < > plane through the center of a Z12 cell shownin blue (on right). The composition and segment orientation for the Z12 domain of C15 are shownin (C), with the end distribution of the core A-block shown in (D) for conformationally symmetricand asymmetric cases. (E) shows a hexagonal cell of Z phase from SCFT results at the sameconditions at (A) and (B) (on left), with a cut through the center of the Z15 cell shown in orange(on right). The composition and segment orientation for the Z15 domain of Z are shown in (F),with the end distribution of the core A-block shown in (G) for conformationally symmetric andasymmetric cases.
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