Rational design of ABC triblock terpolymer solution nanostructures with controlled patch morphology

Block copolymers self-assemble into a variety of nanostructures that are relevant for science and technology. While the assembly of diblock copolymers is largely understood, predicting the solution assembly of triblock terpolymers remains challenging due to complex interplay of block/block and block/solvent interactions. Here we provide guidelines for the self-assembly of linear ABC triblock terpolymers into a large variety of multicompartment nanostructures with C corona and A/B cores. The ratio of block lengths NC/NA thereby controls micelle geometry to spheres, cylinders, bilayer sheets and vesicles. The insoluble blocks then microphase separate to core A and surface patch B, where NB controls the patch morphology to spherical, cylindrical, bicontinuous and lamellar. The independent control over both parameters allows constructing combinatorial libraries of unprecedented solution nanostructures, including spheres-on-cylinders/sheets/vesicles, cylinders-on-sheets/vesicles, and sheets/vesicles with bicontinuous or lamellar membrane morphology (patchy polymersomes). The derived parameters provide a logical toolbox towards complex self-assemblies for soft matter nanotechnologies.


Supplementary Note 1: Determination of solvent swelling factor, q, of polystyrene
To determine a swelling factor, q, of the PS block in acetone/isopropanol mixtures, dynamic light scattering (DLS) measurements of polystyrene-block-poly(tert-butyl methacrylate) (S 366 T 456 ) diblock copolymer micelles were performed in different acetone/isopropanol ratios. For this purpose, the refractive index and the dynamic viscosity was of the solvent mixtures were determined using a refractometer and an ubbelohde viscosimeter ( Supplementary Fig. 2) and used to evaluate the DLS data.
The hydrodynamic radius linearly increases with increasing acetone content and from the linear fit the swelling factor could be calculated for each solvent composition ( Supplementary Fig. 3a). To exclude a volume change of the corona forming PT block, a PT homopolymer star (3 arm star prepared with ATRP, M n,arm = 50 kg/mol) was investigated giving no change in the hydrodynamic radius ( Supplementary Fig. 3b). From TEM measurements of the diblock copolymer in isopropanol (assuming a non-swollen state of the PS block), a radius of the PS core is determined to be R core,ipa = 14.8±1.0 nm. Subtracting this value from the micellar hydrodynamic radius, R h , in isopropanol a thickness of the PT corona of d corona = 54.7 nm was estimated. The size increase, s, and the volume swelling factor, q, of the PS core are calculated according to equations (1) and (2) with R x being the hydrodynamic radius at certain acetone content. All calculations are based on a constant aggregation number of the micelles, N agg . (1) (2)

Supplementary Note 2: Generalized theoretical description of polymorphism of patchy micelles and morphology of patches
The following theoretical arguments aim at rationalizing the polymorphism of patchy micelles of ABC triblock terpolymers. Our theory is general and based on the scaling approach proposed in refs. [19,28]. Let N A , N B , N C be the degrees of polymerization of the respective blocks;  A ,  B are the volumes of the monomer units of the insoluble A-and B-block. The second virial coefficient (excluded volume) of the C-monomer unit, v C ∼ℓ 3 is positive under good solvent conditions and ℓ is the monomer unit (or the Kuhn segment) length, which is assumed to be the same for all the blocks.
Polymorphism of patchy micelles. The morphological transitions from patchy spherical to patchy cylindrical micelles and further to patchy bilayers (lamellar) are governed by a delicate balance between the gain in the conformational entropy of the core-forming A-blocks and the penalty in curvature-dependent part of repulsive interactions in the solvated coronal C-domains. These transitions occur when the thickness of the corona, H, gets smaller than the radius R A of the central Acore, H corona ≤ R A , that is, the micelles have the crew-cut shape ( Supplementary Fig. 7). The free energy (in k B T units) of patchy crew-cut micelle with arbitrary morphology i (i = 3, 2, 1 corresponds to spherical, cylindrical or lamellar structure of the A-core, respectively) comprises the following contributions: (4) accounts for repulsions between coronal blocks where (5) is the free energy per chain in the aggregate with large, R A >> H corona >> R B , curvature radius of the central A-core, R B and D are the radius of the B-patch and semi-distance between centers of the neighboring patches, respectively; , where the correction term N C is specified below ( Supplementary Fig. 7), and (6) is the morphology-dependent increment to the corona free energy. The term F core accounts for the conformational entropy losses in the core-forming insoluble A-blocks where the numerical coefficients are b 1   2 / 8; patchy micelles or between cylindrical patchy micelles and patchy lamellae (i=1).
and , that is a decrease in the length of the soluble C-blocks (or the solvent quality) leads to successive transitions from patchy spherical to patchy cylindrical micelles and further to patchy A-sheets. The parameters of the B-block enter the correction term, equal to the number of monomer units in a segment of the C-block protruding from the B-domain up to the inter-patch distance .
Morphology of patches. The patchy micelles are thermodynamically stable and the above equations apply as long as R B ≤ D ≤ H corona . This is the case, if For shorter soluble C-blocks or/and longer insoluble B-blocks the size of patches R B becomes comparable to or smaller than the extension of the corona H corona . In this regime ( Supplementary Fig.  8) one could expect transformation of quasi-semi-spherical B-domains to semi-cylindrical ones and further to core-shell structure. This transition is driven by the gain in the conformational entropy of the B-blocks, which is balanced by an increase in the overlap and repulsions between the C-blocks protruding from the surface of the B-domains. The coronal contribution to the free energy can be presented as (12) where (13) and j=3,2,1 correspond to semi-spherical patches, cylindrical stripes or continuous layer formed by Bblocks at the A/S interface. Supplementary Eq. (8) for the interfacial free energy assumes the form where with the B-domain geometry-dependent coefficient  j and we have neglected contribution of the A/S interface assuming distance between neighboring B-domains . The exact numerical factors, which quantify the difference in the conformational entropy of the B-blocks confined in spherical or cylindrical segments adjacent to the surface of the A-core (similar to Supplementary Eq. 7) are not available. However, one could expect that the transitions from B-patches to B-stripes and from B-stripes to layered structures occur at (14) The latter equation does no include any dependence on the length of the A-block, since it pre-assumes that the curvature radius of the A-core exceeds by far that of the B-domains decorating its surface. This is the case if N A v A N B v B . However, at this condition is violated and more refined analysis is required to unravel interference between morphological transitions in the A-and Bcore domains.