Letter Cite This: ACS Macro Lett. 2019, 8, 1075−1079
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Theory of Microphase Segregation in the Melts of Copolymers with Dendritically Branched, Bottlebrush, or Cycled Blocks Ekaterina B. Zhulina,† Sergei S. Sheiko,‡ and Oleg V. Borisov*,†,§ †
Institute of Macromolecular Compounds of the Russian Academy of Sciences, 199004 St. Petersburg, Russia Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA § Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254 CNRS UPPA, 64053 Pau, France Downloaded via NOTTINGHAM TRENT UNIV on August 14, 2019 at 21:58:41 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
‡
S Supporting Information *
ABSTRACT: Theory of microphase segregation in the melt of diblock copolymers comprising two strongly incompatible blocks of similar or different topologies is developed. The spectrum of considered architectures include copolymers with arbitrary combinations of bottlebrush-like, dendritic, cycled blocks, and so on. Our theory provides quantitative predictions of how the morphology of the microphase segregated structures can be controlled not only by the volume fractions of the incompatible blocks, but also by their architecture. These predictions open perspectives for developing new materials, for example, photonic crystals, with independently adjustable volume fractions and morphology of the domains.
M
Consider a melt of diblock copolymers each comprising blocks A and B with degrees of polymerization NA and NB, respectively. Blocks may have different topologies, for example, linear chain, comb-like, cyclic, or dendritic, see Figure 1. Monomeric units in the blocks have sizes aj and volumes a3j (j = A and B). All linear segments in the blocks are assumed to be flexible with a Khun segment lj = pjaj. Incompatibility between the blocks leads to formation of domains A in a continuous matrix B (or vice versa). In the following we ascribe index i = 1, 2, and 3 to lamellar, cylindrical, and spherical morphologies of domains, respectively. The interfacial free energy γ̃ at A/B boundaries is
elts of block copolymers have been intensively investigated for a number of decades.1−4 The analytical and numerical theoretical methods were applied to study conformations of block copolymers in microphase segregated states (see comprehensive reviews5−7). Microphase segregation of linear diblock copolymers in both weak and strong segregation limits was thoroughly investigated starting from seminal works of Leibler8 and Semenov,9 in remarkable agreement with experimental data.5,6 Recently, significant attention was attracted by copolymers composed of branched blocks.5 Miktoarm stars,10,11 bottlebrush block copolymers,12−16 dendronized copolymers,17−19 and barbwire molecular brushes20−22 invoke novel material properties compared to linear counterparts with the same chemical composition. Graft-copolymers (combs, bottlebrushes, and barbwires) remain in the focus of research due to unique mechanical, optical, and biomedical properties.23−25 Bottlebrushes and barbwire macromolecules in melts and solutions and at interfaces were studied using computer simulations,26−31 scaling,32 and self-consistent field numerical33 approaches. Recent studies34,35 provided the details of lamellar organization of diblock and triblock bottlebrush copolymer melts. However, we are still missing fundamental understanding of the effect of branching on polymorphism of superstructures in microphase segregated melts of diblock copolymers. In this Letter, we present a theoretical framework to describe microphase segregation in melts of block copolymers consisting of two chemically dissimilar blocks with branched (tree-like) or cyclic architectures. © XXXX American Chemical Society
Figure 1. Schematics of selected block architectures: bottelbrush (a), barbwire bottlebrush (b), starlike (c), dendron (d), and macrocyclecontaining block (e). Received: June 29, 2019 Accepted: August 9, 2019
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DOI: 10.1021/acsmacrolett.9b00498 ACS Macro Lett. 2019, 8, 1075−1079
Letter
ACS Macro Letters
distance D and promotes transition of aggregates in the regime with D/R ≤ 1. In this regime, the width of dead zone rapidly decreases as a function of ratio D/R, and therefore the parabolic potential serves as a reasonable approximation to address convex dry brushes formed by branched polymers. The elastic free energy of block j in morphology i can be expressed as
sufficiently large to ensure narrow interfaces between domains A and matrix B (strong segregation limit). Dense packing of blocks A relates the interface area s per copolymer to the domain size Ri as s=
iNAaA3 , Ri
i = 1, 2, 3
(1)
Here Ri is the radius of spherical (i = 3) or cylindrical (i = 2) domain, and half-thickness of lamellar domain A (i = 1). A similar packing condition for blocks B in the surrounding matrix (modeled as Wigner-Seitz cell with radius Ri + Di) leads to an interdomain half-distance
F (j i) =
F
=
F(i) A
+
FB(i)
+
Fs(i)
iR ± sj(z) = sjjj k R
(j , i) f elastic (z)
=
kBT =
(3)
i−1
, i = 1, 2, 3
(6)
T (j , i)(z) 2kBT
3κj2 2sj(z)aj5pj
∫z
R i , Di
z′sj(z′)dz′, i = 1, 2, 3; j = A , B
(7)
with T (z) as the tension flux. By substituting eqs 6 and 7 in eq 5 and performing an integration, one finds (j,i)
Here, and are conformational entropy losses of the blocks, and F(i) s = γ̃s is the free energy of A/B interface. The (i) free energies F(i) A and FB can be obtained by assimilating inner (A) and outer (B) domains to dry concave and convex polymer brushes, respectively. In these calculations we use a strong stretching self-consistent field (SS-SCF) approximation proposed for superstructures of linear−linear block copolymers by Semenov9 and generalized later for dendron brushes36,37 and brushes formed by macromolecules with tree-like or cycled topology.38−40 In this model, the Lagrange potentials U(z) arising due to incompressibility condition, and acting at monomer units of each block at distance z from A/B interface are parabolic, and
z zy zz {
with “+” and “−” signed to matrix (j = B) and domain (j = A) blocks, respectively. As long as the blocks exhibit the Gaussian elasticity, the density of the elastic free energy can be expressed as
F(i) B
UA(z) 3 = 2 κA2(R i2 − z 2) kBT 2aApA
(j , i) f elastic (z)sj(z)dz , i = 1, 2, 3; j = A , B
(5)
where x = VB/VA and Vj = Nja3j is the volume of block j = A and B. The ratio Di/Ri ≥ 1 if x ≥ x c = 2i − 1. Dimensions of the inverted structures are given by eqs 1 and 2 with permuted notations aB ⇄ aA and NB ⇄ NA. In the strong segregation limit, the free energy per block copolymer molecule in morphology i = 1, 2, and 3 is presented as F A(i)
R i , Di
Here, f(j,i) elastic(z), is the density of elastic free energy, the upper limit of integral is Ri for blocks A and Di for block B, z is the distance from A/B interface, and
ÄÅ ÉÑ ÅÅi ÑÑ 3 y1/ i ÅÅjj N a z B B zz − 1ÑÑÑÑ = R [(1 + x)1/ i − 1] , i = 1, 2, 3 Di = R iÅÅÅjjj1 + z i ÑÑ ÅÅ NAaA3 z{ ÑÑ ÅÅÇk ÑÖ (2)
(i)
∫0
bR 2 F A(i) = i 2i ηA2 kBT NAaApA
(8)
with morphology-dependent numerical prefactors b1 = π /8, b2 = π2/16, and b3 = 3π2/80, coinciding with those obtained earlier9 for linear core-forming block. The free energy of matrix block B yields 2
Di2 R i2 FB(i) x = ηB2 η 2g (x)β g (x) = 2 1/ i 2 i kBT NBaBpB [(1 + x) − 1] NAaA2 pA B i
(9)
with β = (aBpA)/(aApB) ≃ 1 for typical flexible polymers, and
UB(z) 3 = 2 κB2(Di2 − z 2) kBT 2aBpB
g1(x) =
(4)
Here, kBT is thermal energy, and κA and κB are the topological coefficients of the blocks established for various architectures elsewhere.38−40 The topological ratio ηj = κj/κj,lin = 2Njκj/π quantifies the increase in the elastic free energy of a branched block compared to linear chain with the same number Nj of monomer units. A scheme to calculate topological ratio η and the analytical expressions for some typical branched architectures are presented in SI. In contrast to the model of microphase segregated copolymer with linear blocks,9 we do not fix free end-points of blocks B on the outer surface of Wigner-Seitz cell, but allow them to fluctuate freely in matrix B. It is known that the parabolic potential approximation applies in planar or concave brushes of linear chains but leads to dead zones depleted of free ends near the grafting surface in convex brushes. However, branching of the brush-forming macromolecules makes the free-end distribution more uniform and could decrease or even eliminate dead zones in convex brushes.36,37,41 Moreover, branching of outer block leads to the decrease in interdomain
π2 x, 8
6π 2 [(1 + x)1/2 − 1]4 · 8x 2 ÉÑ ÄÅ Ñ ÅÅ 1 ÅÅ + 1 [(1 + x)1/2 − 1]−1 ÑÑÑ, ÑÑ ÅÅ 4 3 ÑÖ ÅÇ g2(x) =
g3(x) =
9π 2 [(1 + x)1/3 − 1]5 · 8x 2
ÉÑ ÄÅ Ñ ÅÅ 1 ÅÅ + 1 [(1 + x)1/3 − 1]−1 + 1 [(1 + x)1/3 − 1]−2 ÑÑÑ ÑÑ ÅÅ 5 3 2 ÑÖ ÅÇ
(10)
The free energy per copolymer molecule is presented as F (i) + F A(i) + Fs(i) R i2 iγNAaA F (i) = B = Φi(x) + 2 Ri kBT kBT NAaApA (11)
γ̃a2A/kBT
Here γ = is the surface free energy per area units of thermal energy), and 1076
a2A
(in
DOI: 10.1021/acsmacrolett.9b00498 ACS Macro Lett. 2019, 8, 1075−1079
Letter
ACS Macro Letters Φi(x) = ηB2gi(x)β + ηA2bi
(12)
Minimization of F(i) in eq 11 with respect to Ri gives ÄÅ É ÅÅ iγN 2 ÑÑÑ1/3 A Ñ Å Å ÑÑ R i = aAÅÅ ÅÅÇ 2Φi(x) ÑÑÑÖ (13) and Di is given by eq 2. The equilibrium free energy per molecule in the superstructure of morphology i yields F (i)(x) 3 1/3 = 2/3 (iγ )2/3 N1/3 A Φ i (x ) kBT 2
(14)
with all the architectural details of constituent blocks incorporated in Φi (eq 12) via topological ratios ηA and ηB. The half period of the lamella is specified as ÄÅ É ÅÅ 4γ(1 + x) ÑÑÑ1/3 ÅÅ ÑÑ H1 = R1 + D1 = R1(1 + x) = aAÅÅ 2 2 Ñ N 2/3 ÅÅ π (βη x + η2) ÑÑÑ Ñ B A Ö ÇÅ
Figure 2. Unified morphological diagram of microphase segregated melt of diblock copolymers with branched blocks in generalized coordinates (f, ηA/ηB). Three generic morphologies, lamellae (L), cylinders (C), and spheres (S), are illustrated by cartoons; inverted cylindrical and spherical morphologies are marked as C′ and S′. Black dots indicate transition values of f for linear diblock copolymers calculated by Semenov.9 f C* (red dashed line) and f S* (blue dashed line) indicate compositions, corresponding to Di = Ri for cylinders (i = 2) and spheres (i = 3), respectively. To the left of the corresponding dashed lines, the interdomain distance Di in cylindrical (C) and in spherical (S) domains is smaller than the domain size, Di ≤ Ri.
(15)
with N = NA + NB. Because ηB and ηA are independent of the molecular weights of the blocks, microsegregated block copolymer with chemical composition x = VB/VA exhibits universal power law dependence H1 ∼ N2/3. If topological ratios of two blocks are equal and β ≈ 1, then the lamellar period is independent of the block copolymer composition x. Equality of topological ratios ηA = ηB does not necessarily mean architectural similarity of blocks A and B.39 For example, if β ≃ 1 and one block has starlike architecture with q1 ≫ 1 free branches, while the second block is a dendron of the second generation with branching activity q2 ≈ q1 , their topological ratios is approximately equal,39 and therefore, the lamella period is almost independent of the block copolymer composition x. Notably, the power law dependence for the size of the Wigner-Seitz cell, Ri + Di ∼ N2/3 holds also for spherical and cylindrical morphologies, provided that copolymer chemical composition x is fixed. The condition F(i)(x) = F(i+1)(x) provides equations for binodals separating L (lamella), C (cylindrical), and S (spherical) morphologies of domains A, i 2 Φi(x) = (i + 1)2 Φi + 1(x),
i = 1, 2
The diagram of states in Figure 2 is applicable to a variety of branched and cycled architectures of the blocks. As it follows from Figure 2, an increase in the degree of branching of block B at fixed architecture of block A (i.e., an increase in ηB at fixed ηA in the right part of the diagram) leads to stabilization of spherical morphology of domains A embedded in matrix B, while the effect is opposite for inverted structures with blocks B forming the domains in the matrix of blocks A. As it is also seen from Figure 2, binodals separating S, C, and L morphologies are located in the regimes with D/R ≪ 1, in which dead zones have a minor effect. Because the topological ratio ηj depends only on how the chain segments are connected in a branched polymer, the equilibrium dimensions of the domains obey the same molecular weight dependences, Ri ∼ Di ∼ N2/3, as for linear diblock copolymers. For equal topological ratios of the branched blocks, ηA = ηB, the boundaries between different morphologies do not shift compared to linear block copolymers and remain symmetrically positioned with respect to f = VB/(VA + VB) = 0.5. This result is intuitive for copolymers comprising two topologically identical blocks. However, the degeneracy of topological ratios39 allows constructing topologically asymmetric block copolymers (e.g., star-bottlebrush copolymer) for which the boundaries between different morphologies remain symmetric with respect to the copolymer composition VB/(VA + VB), just as for linear−linear block copolymers. We emphasize that the diagram in Figure 2 was constructed presuming that blocks in the domains are elongated with respect to their Gaussian dimensions due to crowding of junctions between blocks at A/B interfaces. That is, spacers in unconfined individual blocks in the melt would obey the Gaussian statistics. This assumption could be violated in the case of short spacers that exhibit intrinsic tension. For example, the backbone in a bottlebrush comprising side chains with n
(16)
By substituting Φi(x) in eq 16 with account of eq 10, one finds L−C and C−S binodal lines, ηB ηA
β
1/2
l o (4b2 − b1)/[g1(x) − 4g2(x)] L−C o o o =m o o o o (9b3 − 4b2)/[4g2(x) − 9g3(x)] C−S n
(17)
For inverted morphologies C′ and S′ binodals are specified by eq 17, with x → 1/x and ηA ⇄ ηB. By using volume fraction f = VB/(VA + VB) = x/(1 + x) as variable, we present the morphological diagram in the f, ηB/ηA coordinates in Figure 2. The horizontal line ηB/ηA = 1 corresponds to copolymers with identical architectures of the blocks including linear−linear block copolymers. Vertical dashed lines indicate copolymer compositions, f C* and f S*, corresponding to onsets of D ≤ R regime for cylindrical (C) and spherical (S) aggregates. The boundaries between microphase segregated states and uniform melt are not indicated. 1077
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(3) Lazzari, M.; Lin, G.; Lecommandoux, S. Block Copolymers in Nanoscience; Wiley-VCH: Weinheim, 2006. (4) Schacher, F. H.; Rupar, P. A.; Manners, I. Functional block copolymers: nanostructured materials with emerging applications. Angew. Chem., Int. Ed. 2012, 51, 7898−7921. (5) Bates, C. M.; Bates, F. S. 50th Anniversary Perspective: Block Polymers Pure Potential. Macromolecules 2017, 50, 3−22. (6) Liu, J. V.; García-Cervera, C. J.; Delaney, K. T.; Fredrickson, G. H. Optimized Phase Field Model for Diblock Copolymer Melts. Macromolecules 2019, 52, 2878−2888. (7) Matsen, M. W.; Bates, F. S. Unifying Weak- and StrongSegregation Block Copolymer Theories. Macromolecules 1996, 29 (4), 1091−1098. (8) Leibler, L. Theory of Microphase Separation in Block Copolymers. Macromolecules 1980, 13 (6), 1602−1617. (9) Semenov, A. N. Contribution to the Theory of Microphase Layering in Block-Copolymer Melts. Sov. Phys. JETP 1985, 61, 733− 742. (10) Shibuya, Y.; Nguyen, H. V.-T.; Johnson, J. A. Mikto-Brush-Arm Star Polymers via Cross-Linking of Dissimilar Bottlebrushes: Synthesis and Solution Morphologies. ACS Macro Lett. 2017, 6 (9), 963−968. (11) Steinschulte, A. A.; Gelissen, A. P. H.; Jung, A.; Brugnoni, M.; Caumanns, T.; Lotze, G.; Mayer, J.; Pergushov, D. V.; Plamper, F. A. Facile Screening of Various Micellar Morphologies by Blending Miktoarm Stars and Diblock Copolymers. ACS Macro Lett. 2017, 6, 711−715. (12) Rzayev, J. Synthethis of polystyrene-polylactide bottlebrush block copolymers and their melt self-assembly into large domain nanostructures. Macromolecules 2009, 42, 2135−2141. (13) Bolton, J.; Bailey, T. S.; Rzayev, J. Large pore size nanoporous materials from self-assembly of asymmetric bottlebrush block copolymers. Nano Lett. 2011, 11, 998−1001. (14) Bolton, J.; Rzayev, J. Synthesis and Melt Self-Assembly of PSPMMA-PLA Triblock Bottlebrush Copolymers. Macromolecules 2014, 47, 2864−2874. (15) Chang, A. B.; Lin, T.-P.; Thompson, N. B.; Luo, S.-X.; Liberman-Martin, A. L.; Chen, H.-Y.; Lee, B.; Grubbs, R. H. Design, Synthesis, and Self-Assembly of Polymers with Tailored Graft Distributions. J. Am. Chem. Soc. 2017, 139 (48), 17683−17693. (16) Vatankhah-Varnosfaderani, M.; Keith, A. N.; Cong, Y.; Liang, H.; Rosenthal, M.; Sztucki, M.; Clair, C.; Magonov, S.; Ivanov, D. A.; Dobrynin, A. V.; Sheiko, S. S. Chameleon-like elastomers with molecularly encodded strain-adaptive stiffening and coloration. Science 2018, 359, 1509−1513. (17) Grason, G. M.; Kamien, R. D. Self-consistent field theory of multiply branched block copolymer melts. Phys. Rev. E 2005, 71, 051801−0518011. (18) Costanzo, S.; Scherz, L. F.; Schweizer, T.; Kröger, M.; Floudas, G.; Schlüter, A. D.; Vlassopoulos, D. Rheology and Packing of Dendronized Polymers. Macromolecules 2016, 49 (18), 7054−7068. (19) Wurm, F.; Frey, H. Linear-dendritic block copolymers: the state of the art and exciting perspectives. Prog. Polym. Sci. 2011, 36, 1−52. (20) Liang, X.; Liu, Y.; Huang, J.; Wei, L.; Wang, G. Synthesis and characterization of novel barbwire-like graft polymers poly(ethylene oxide)-g-poly(-caprolactone)4 by the ‘grafting from’ strategy. Polym. Chem. 2015, 6, 466−475. (21) Uhrig, D.; Mays, J. W. Synthesis of Combs, Centipedes, and Barbwires:thinspace Poly(isoprene-graft-styrene) Regular Multigraft Copolymers with Trifunctional, Tetrafunctional, and Hexafunctional Branch Points. Macromolecules 2002, 35 (19), 7182−7190. (22) Pelras, T.; Mahon, C. S.; Nonappa, x; Ikkala, O.; Gröschel, A. H.; Müllner, M Polymer Nanowires with Highly Precise Internal Morphology and Topography. J. Am. Chem. Soc. 2018, 140 (40), 12736−12740. (23) Yuan, J.; Müller, A. H. E.; Matyjaszewski, K.; Sheiko, S. In Polymer Science: A Comprehensive Reference; Matyjaszewski, K., Möller, M., Eds.-in-Chief; Elsevier: Amsterdam, 2012.
monomers each, and spacers with the number of monomers m < n1/2 between neighboring grafts, exhibits the Gaussian statistics on length scales ζ ≲ am, is elongated on intermediate length scales am ≲ ζ ≲ an1/2 and constitutes a Gaussian chain of impermeable superblobs with size ∼an1/2 on length scales ≳an1/2.32 Notably, side chains retain the Gaussian size ∼an1/2 at any values of m ≳ 1. In this scenario, implementation of the parabolic potential requires prior renormalization of the bottlebrush parameters. We will address this issue in the forthcoming full paper. To conclude, in this Letter we presented for the first time an analytical approach to investigate microphase segregation in melts of block copolymers with branched or macrocyclecontaining monodisperse blocks in the strong segregation limit. The architectural details of the blocks are accounted through the topological ratios ηj (j = A, B). Knowledge of ηj for specific architectures of both blocks allows for localization of copolymer with composition f on the diagram in Figure 2 and prediction of its morphology in microphase segregated state. The predicted possibility to tune morphology of the microsegregated domains by exploiting topological diversity of copolymer blocks allows obtaining superstructures that are unattainable with conventional linear block copolymers. For example, branching of matrix blocks B stabilizes superstructures with spherical domains A, even if A is a majority block. These theoretical predictions are in good qualitative agreement with MD studies.42,43 The morphological diagram of states for macromolecules with equal topological ratios of the blocks coincides with that for copolymers with linear blocks, and the lamella period H ∼ N2/3 does not depend on copolymer composition. These predictions open novel perspectives in design of nanostructured materials.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.9b00498. Principles of SS-SCF theory of solvent-free brushes formed by branched macromolecules and an example of derivation of the topological coefficient (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Sergei S. Sheiko: 0000-0003-3672-1611 Oleg V. Borisov: 0000-0002-9281-9093 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by Ministry of Research and Education of the Russian Federation within State Contract No. 14.W03.31.0022 and by the National Science Foundation (DMR 1407645, DMR 1436201).
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REFERENCES
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