Abstract:
Miktoarm star shaped copolymers have attracted much attention due to their unique shape and intriguing properties compared to the linear block copolymers, including compact structure, higher critical micelle concentration, lower viscosity, efficient synthetic routes and wide range of morphologies. The different synthetic routes such as anionic polymerization and controlled radical polymerization have made it possible to synthesis diverse molecular architecture of copolymer and these diverse architectured copolymers give numerous morphologies. For example, Archimedean tiling patterns and cylindrical microdomains at symmetric volume fraction for miktoarm star copolymers, which have not been reported for linear block copolymers. This paper summarizes the morphology and microphase separation of miktoarm star copolymers with nonlinear architecture.
Introduction:
Block copolymers have attracted considerable attention because of their morphologies and nanophase structures such as spheres, cylinders, bicontinuous, and lamellae. These morphologies show due to the interacting repulsive force between the components, which particularly affected by the phase separation, which strongly depends on volume fraction of the blocks, the degree of polymerization, entropy variation with molecular weight, the Flory-Huggins interaction parameter, and the molecular architecture of the block copolymers.1–4 All of these nanostructure have been widely used in various field such as optoelectronics, microelectronics, and nanotechnology for various applications such as templates, nanoreactors, membranes, optical materials, and data storage media.5–15 6–8 In particular, in the field of pharmaceutical, vesicles of miktoarm star shaped copolymer have been used as drug delivery vehicles.

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In comparison to any other linear block copolymers, star shaped or miktoarm star shaped copolymers show diverse morphology and physical properties due to their different molecular architecture. For instance, unimolecular micelles of star copolymers displayed much higher stablility than the micelles of linear block copolymers because in the star shaped copolymer the arms are covalently connected to the central core. These highly stable micelles of star shaped copolymer have been using to synthesis monodisperse colloidal nanocrystal. 19-22
In the linear diblock copolymers (AB) and linear triblock terpolymers (ABC), the morphologies or microphase structure are mostly governed by the volume fraction of one of the blocks (fA, fB = 1- fA) and one interaction parameter (χAB), and two volume fraction parameters (fA, fB, fC = 1- fA – fB) and three interaction parameters (χAB, χBC, χCA), respectively. For example, spherical or cylindrical microdomains are only observed at asymmetric volume fractions, while lamellar microdomains are shown at symmetric volume fractions in diblock copolymers. However, nonlinear or mitoarm star shaped copolymers showed cylindrical microdomains even at symmetric volume fraction due to the molecular architecture.
Miktoarm star copolymers (sometimes called asymmetric star copolymers, heteroarm star copolymer or simply miktoarm copolymer) are star shaped copolymer, consist of heteroarms covalently joined to a central core with different chemical compositions or molecular weights For example, AmBn miktoarm star copolymer where, m arms of A homopolymer and n arms of B homopolymer are linked to a central core, while in the star-shaped copolymers homoarms with identical chemical compositions are covalently joined to a central core. For instance, (A-b-B)n star-shaped copolymer where, n arms of A-b-B diblock copolymer are connected to a central core. Here the first written A block represents the inner block (core) and B block is the outer block (shell) of star shaped copolymer, as shown in Fig: 1.

Fig: 1 Schematic architectures of (a) miktoarm star copolymers (AmBn) and (b) star-shaped copolymers ((A-b-B)n).
Miktoarm star shaped copolymers morphologies and their characterization:
The effect of molecular architecture on miktoarm star shaped copolymers morphologies has been extensively investigated theoretically and experimentally.
Theoretical investigation:
In 1996, Milner 36 first reported theoretical phase diagram of AnBn miktoarm star shaped copolymers at the strong segregation limit. The morphology and microphase separation are determined by the competition between reduction of interfacial tension and the increase in stretching free energy as the copolymer blocks stretch away from the interface.

Fig 2: Phase diagram of AnBn miktoarm star shaped copolymers at the strong segregation limit as a function of volume fraction of the B monomer (φB), with increasing asymmetric parameter ε = (nA/nB)(lA/lB)1/2, where nA, nB are the numbers of A and B blocks, and lA, lB are characteristic lengths of A and B, respectively.
In 1997, Floudas 37 calculated spinodal curves for the series of ABn miktoarm star shaped copolymers based on mean field theory. The results of the lower number of the series are plotted in Fig 3. The plot indicates that the critical value of the χNt (Nt = Na + nNb) of ABn miktoarm star copolymers is higher than that of diblock copolymers. Therefore, the microphase separation for ABn miktoarm copolymers becomes more difficult. It also indicates that the maximum critical value of χNt appears at n=3 (for AB3 miktoarm copolymers).

Fig 3: (a) The spinodal curves (χNt vs. fA) for diblock and ABn miktoarm star copolymers with three different values of n (2, 3, and 4). (b) Critical values of χNt plotted as a function of the number of arms of the B block.
In 2004, Grason and Kamien38 have calculated phase diagrams of AmBn miktoarm star copolymers for m = 1 with n = 2, 3, 4, and 5 using self consistent field theory (SCFT), but they did not consider the perforated lamellar (PL) and Fddd (O70, orthorhombic and single-network structure) phases. Later, in 2012, Matsen39 calculated the phase diagram for AB2 miktoarm star copolymer and found perforated lamellae (PL) and Fddd (O70), phases near gyroid phase (Fig 4).

Fig 4: Theoretical phase diagram of AB2 miktoarm star copolymers with PL and Fddd phases.
Experimental investigation:
ABC Miktoarm Star Terpolymer:
Matsushita and coworkers74–76 have investigated microphase separation of AxByCz miktoarm star terpolymers. For that they classified the molecular architecture into different series like I1.0S1.0Px1, I1.0SyP2.0, and I1.0S1.8Px2 where I = polyisoprene, S = polystyrene and P = poly (2-vinylpyridine) and 0.2≤ x ≤ 10, 1.1≤ x ≤ 2.7 and 3.2≤ x2 ≤ 53. In all the TEM images and morphologies, I domain represented by black, S domain by white and P domain by gray color.
Fig. 5 compares TEM images for the series, I1.0S1.0Px1. In figure 2(a) for the sample, I1.0S1.0P0.2, spheres of the highly minor component P are sandwiched with lamellae of two major components, I and S, which is called spheres sandwiched with lamellae. Figure 2(b) is a tiling structure as a cross-sectional view of a cylindrical structure from the sample, I1.0S1.0P0.7. This is one of the 12 Archimedean tiling structures. Figure 2(c) is a lamellar structure for the sample I1.0S1.0P3.0, where one of the lamellae is composed of other lamellae, which is called lamellae-in-lamella structure. Figure 2(d) for the sample I1.0S1.0P10 shows cylinders composed of alternating columnar I and S discs, the cylinders being packed hexagonally in a P matrix: this pattern is called a lamellae-in-cylinder structure.

Fig 5: Various morphologies of the type I1.0S1.0Px1. X1 values are (a) 0.2, (b) 0.7, (c) 3 and (d) 10.
Fig. 6 compares the TEM images of structures series, I1.0SyP2.0, where two Archimedean tilings,
(4.6.12) and (4.8.8) can be recognized easily in figure 6(a) for I1.0S1.3P2.0 and in figure 6(c) for I1.0S2.3P2.0 while another (3.3.4.3.4) tiling is seen in figure 6(b) for (I1.0S2.7P2.0) where the I (dark) and S (bright) domains are opposite to Fig 5(a) because of the composition difference.

Fig 6: Tiling structures for I1.0SyP2.0. (a) I1.0S1.3P2.0 (b) I1.0S2.3P2.0 and (c) I1.0S2.7P2.0
Fig 7(a) is the SAXS diffraction image for I1.0S2.3P2.0, in this pattern there are 12 diffraction spots in the lower q region, four of which belong to {20} and the other eight to {21}. From careful data analyses, it shows that this pattern is corresponded to the Archimedean tiling (3.3.4.3.4) (Fig 7(b)).

Fig 7: (a) SAXS diffraction image for I1.0S2.3P2.0. and (b) the corresponding real-space image.
The TEM images for the series, I1.0S1.8Px2 are reported in Fig 8, where Fig 8(a) for the sample, I1.0S1.8P3.2, shows I and S domains form gyroid membrane in the P domain. Figure 8(b) for I1.0S1.8P6.4 and 8(c) for I1.0S1.8P53 show cylinder-in-lamella and hierarchical structure, respectively.

Fig 8: TEM images for (a) I1.0S1.8P3.2 (b) I1.0S1.8P3.2 and (c) I1.0S1.8P3.2
Fig 9 summarizes microphase separation observed for IxSyPz miktoarm star terpolymers with different volume ratios between the arms.

Fig 9: Kaleidoscopic morphologies from the IxSyPz miktoarm star shaped block terpolymer system. (a) Lamellae-in-sphere, (b) lamellae-in-cylinder, (c) cylinder-in-lamella, (d) hyperbolic tiling, (e) zinc blende, (f) sphere-sandwiched-with-lamella, (g) Archimedean tiling and (h) lamellae-in-lamella.