- Nano Express
- Open Access
Disorder to Order Transition and Ordered Morphology of Coil-Comb Block Copolymer by Self-Consistent Field Theory
© Jiang et al. 2015
- Received: 11 May 2015
- Accepted: 1 August 2015
- Published: 18 August 2015
The disorder to order transition and the ordered patterns near the disordered state of coil-comb copolymer A-b-(B m + 1-g-C m ) are investigated by the self-consistent field theory. The phase diagrams of coil-comb copolymer are obtained by varying the composition of the copolymer with the side chain number m = 1, 2, and 3. The disorder to order transition is far more complex compared with the comb copolymer or linear block copolymer. As the side chain number m increases, the Flory-Huggins interaction parameter of disorder to order transition (DOT) increases and the lowest DOT occurs when the volume fractions of blocks A, B, and C are approximately equal. When one component is the minority, the disorder to order transition curve is similar with binary copolymer, but the curve shows the asymmetric property. The comb copolymer is more stable with larger side chain number m and shorter side chain. The ordered patterns from the disordered state are discussed. The results are helpful for designing coil-comb copolymers and obtaining the ordered morphology.
- Coil-comb block copolymer
- Disorder to order transition
- Phase separation
- Self-consistent field theory
The ability of block copolymers to spontaneously form microphase-separated structures with length scales of 10–100 nm indeed makes them extensive applications in nanoscience, such as pore materials , drug delivery , and templates [3, 4]. Recently, researchers are devoted to design block copolymers with different architectures to engineer functional materials. Even the crystal structures of block copolymers [5–9] were intensively studied. For example, the Frank-Kasper σ phase can form in two different single-component, sphere-forming, block copolymer melts near the order-disorder transition temperature (T ODT) [6, 7]. Xie et al.  studied asymmetric miktoarm block copolymers which can self-assemble into fcc, bcc, A15, and the complex σ phase. They even designed multiblock terpolymers which can self-assemble into various binary mesocrystals with space group symmetries of a large number of binary ionic crystals, including NaCl, CsCl, ZnS, α-BN, AlB2, CaF2, TiO2, ReO3, Li3Bi, Nb3Sn(A15), α-Al2O3, etc. . This study showed that block copolymer with different architectures can be used to obtain the complex nanostructures or even construct crystal phase by macromolecular metallurgy in a mesoscale.
Coil-comb copolymers, which self-assemble at a length scale of a few nanometers, can be constructed by attaching oligomers or small molecules to polymers via covalent bond and noncovalent-bonding interactions, for example, hydrogen bonding, electrostatic interactions, or metal coordination [1, 3, 4, 10–27]. They have potential applications such as in electrical, biological, and other functional materials. Feng et al. [12–14] focused on double hydrophilic coil-comb copolymer, in which the side chain grafted to the backbone through covalent bond. And they found its potential applications in templates and biological materials. Ten Brinke and co-workers [3, 4, 16–21, 26] reported a series of A-b-(B m + 1-g-C m ) coil-comb copolymers, where the small molecules C are weakly connected to backbone B through hydrogen bonds, and the coil-comb copolymers can self-assemble into hierarchically ordered structures. In these structures, there were two different length scales: the larger length scale period is mainly driven by the separation between the comb part and the coil part, whereas the small length period is produced by the segregation within the comb part.
Besides, some researchers also found these hierarchically ordered structures of coil-comb copolymer based on theoretical simulation [28, 29]. For instance, by using the real-space implemented self-consistent field theory, Wang et al.  reported that the coil-comb block copolymers exhibit hierarchically ordered microstructures, including parallel and perpendicular lamella-within-lamella, cylinder-within-lamella, lamella-within-cylinder, and cylinder-within-cylinder.
The ordered phase can form when the repulsive interaction parameter χN is larger than the interaction parameter of the disorder to order transition. Therefore, it is very important to determine the disorder to order transition of block copolymers. It can help us to find ordered microstructures easily. Therefore, more and more researchers pay attention to the disorder to order phase transition of supramolecular polymers [30–32]. At the meantime, self-consistent field theory (SCFT) method has been largely used to study the phase behavior of block copolymers [33–37]. It is also used to study the disorder to order transition of block copolymers [38, 39]. Even the SCFT method has been used to successfully study the polydispersity-induced disorder to order transition of block copolymers [33, 40–42]. In this work, we consider the disorder to order transition of coil-comb copolymer A-b-(B m + 1-g-C m ) with different side chain numbers by using the SCFT. Due to the complexity of this copolymer, we only consider the stability of the homogeneous phase relative to the microseparated state. The related parameters are the side chain number m, the volume fraction of blocks A, B, and C, and the Flory-Huggins parameter χN. We construct the phase diagram of disorder to order transition of coil-comb copolymer by continuously varying the composition of the block copolymer.
Here, we solve Eqs.(4)–(10) directly in real space by using a combinatorial screening algorithm proposed by Drolet and Fredrickson [43, 44]. Note that one must solve the diffusion equation first for q C(r, s) and q A(r, s) with initial condition q C(r, 0) = 1 and q A(r, 0) = 1, then for q B(r, s, t) with q B(r, 0, 1) = q A(r, N A), q B(r, 0, t) = q C(r, N B, t − 1)q C(r, N C) for t > 1 and for q B +(r, s, t) with q B +(r, N B, m + 1) = 1, q B +(r, N B, t) = q B +(r, 0, t + 1)q C(r, N C) for t < m + 1, and last for q C +(r, s, t) and q A +(r, s) with q C +(r, N C, t) = q B(r, N B, t)q B +(r, 0, t + 1) and q A +(r, N A) = q B +(r, 0, 1). Each iteration continues until the phases are stable. Several times are repeated by using different initial conditions to avoid the trapping in a metastable state. In addition, we also minimize the free energy with respect to the system size because it has been pointed out that the box size can influence the morphology . The implementation of the self-consistent field theory is carried out in a two-dimensional L x × L y lattice with periodic boundary conditions.
We consider the disorder to order transition with different side chain numbers m and the ordered pattern near the disorder to order transition. In our calculation, we consider the condition: symmetrical interaction parameters χ AB N = χ AC N = χ BC N = χN. Thus, we calculate the crossover curve of the disordered phase and an (unspecified) ordered phase in terms of the normalized Flory-Huggins parameter χN, the relative composition, and the number of teeth in the comb m. By systematically changing the volume fractions of A, B, and C blocks, we can construct the component triangle phase diagrams in the entire range of the copolymer composition.
Phase diagrams of disorder to order transition for A-b-(B m + 1-g-C m ) (m = 1–3)
Disorder to order transition for A-b-(B m + 1-g-C m ) when fixing blocks A, B, and C
Ordered pattern near the disorder to order transition
Based on the phase diagrams of the disorder to order transition, we also predicted the ordered pattern near the disorder to order transition. Several times are repeated by using different initial conditions to avoid the trapping in a metastable state. In addition, we also minimize the free energy with respect to the system size because it has been pointed out that the box size can influence the morphology. From the ordered pattern, we can clearly see the self-assembly structure of the coil-comb block copolymer. Here, we only present the phase diagrams for symmetrical interaction parameters, i.e., χ AB N = χ AC N = χ BC N = χN. We only consider the side chain number m = 1–3.
Phase Diagram Based on the Ordered Pattern Near DOT
For m = 2, the phase diagram is similar with that for m = 1. But the region of hexagonal phase (HEX and CSH) enlarges. Another new phase (CSH2) occurs, and the phase TET2 for m =1 does not appear here. For m = 3, there are four stable phases. The complex phase CSH2 does not occur here, and the phase region of CSH phase enlarges. Comparing the three phase diagrams, we can see that the effect of the comb part becomes strong with the increase of the side chain number of the comb part. When the coil part block A is the minority, the phase separation of the coil-comb block copolymer is complex. The separation between the coil and comb part is natural. The separation between the main chain and the side chain will compete with the separation between the coil and comb part; therefore, the complex phases (TET2, CSH2, HEX2) form.
The disorder to order transition and the ordered patterns from the disordered state of coil-comb copolymer A-b-(B m + 1-g-C m ) are investigated by the self-consistent field theory. The phase diagrams of coil-comb copolymer are obtained by varying the composition of the copolymer with the side chain number m = 1, 2, and 3. The disorder to order transition is far more complex compared with the comb copolymer or linear block copolymer. As the side chain number m increases, the Flory-Huggins interaction parameter of disorder to order transition (DOT) increases and the lowest DOT occurs when the volume fractions of blocks A, B, and C are approximately equal. When one component is the minority, the disorder to order transition curve is similar with binary copolymer, but the curve shows the asymmetric property. The comb copolymer is more stable with larger side chain number m and shorter side chain. There are six ordered phases near the DOT: two-color lamellar phase (LAM2), hexagonal phase (HEX), core-shell hexagonal phase (CSH), two interpenetrating tetragonal phase (TET2), hexagonal lattice outside core-shell hexagonal phase (CSH2), and hexagonal lattice outside hexagonal phase (HEX2). The results are helpful for designing coil-comb copolymers and obtaining the ordered morphology.
This work was financially supported by the National Natural Science Foundation of China (grant nos. 21474051, 21074053 and 51133002) and Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT). The numerical calculations in this paper have been done on the IBM Blade cluster system in the High Performance Computing Center (HPCC) of Nanjing University.
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