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Morphology and Phase Diagram of Comb Block Copolymer Am+1(BC)m Zhibin Jiang, Rong Wang,* and Gi Xue Department of Polymer Science and Engineering, State Key Laboratory of Coordination Chemistry, Nanjing National Laboratory of Microstructures, School of Chemistry and Chemical Engineering, Nanjing UniVersity, Nanjing 210093, China ReceiVed: September 14, 2008; ReVised Manuscript ReceiVed: April 21, 2009
The morphologies and the phase diagram of comb copolymer Am+1(BC)m are investigated by the self-consistent field theory. By changing the volume fractions of the blocks, the interaction parameters between the different blocks, and the side chain number, nine phases are found, including the two-colored lamellar phase, threecolored lamellar phase, hexagonal lattice phase, core shell hexagonal lattice phase, two interpenetrating tetragonal lattice, core shell tetragonal lattice, lamellar phase with beads inside, lamellar phase with alternating beads, and disordered phase. The phase diagrams are constructed for Am+1(BC)m with different side chain numbers of m ) 1, 2, 3, and 5. Due to the asymmetric topology of comb copolymer Am+1(BC)m, the phases and the diagrams are very different from linear ABC triblock copolymer or star ABC triblock copolymer. When the volume fraction of one of the blocks is the domination, the (core shell) hexagonal phase or two interpenetrating tetragonal lattice can form, depending on which block dominates and the interaction between the blocks. The (core shell) hexagonal phase easily forms at the corner of the block A (fA g 0.5). The side chain number m affects the phase diagram largely due to the fact that the architecture of a comb copolymer is not invariant under the interchange between the three different monomers. Due to the connectivity of the blocks B and the inner blocks A, Am+1(BC)m comb copolymers with the longer main chain A or longer side chain with short block C, i.e., longer block B, are difficult to phase separate. The results are helpful to design nano- or biomaterials with complex architecture or tailor the phase behavior of comb copolymers. 1. Introduction Self-assembly spontaneously creates structures or patterns with a significant order parameter out of disordered components.1 Of these self-assembly materials, block copolymer is an important one. Block copolymers can self-assembly into ordered patterns in nanoscale. The downsizing of functional units can result in a significant decrease in device energy consumption and more efficient production processes.2 Novel phenomena have been revealed at the nanometer scale. Another important factor is that polymers have relatively low cost and good mechanical properties and are compatible with most patterning techniques. For example, a truly unique approach to surface patterning has been realized through the self-assembly of block copolymers.3 Over the past decades, branched block copolymers with complex architectures have attracted considerable attention not only due to their importance in understanding the relationship of architectures with properties but also due to their potential applications.4-9 Comb copolymer (also brush copolymer), as a typical branched polymer, can be composed of a long, flexible or stiff, hydrophobic backbone and dense, long or short, hydrophilic side chains that branch from this main chain. If the selective solvent is sufficiently poor for the side chains but good for the backbone, then this kind of grafted comblike copolymers is the simplest one in which both chemical and topological complexities are combined, ensuring a wide variety of different regimes of structure formation. Therefore, they are an important and interesting class of macromolecules of nonlinear global architecture. As one particular type of amphiphiles, recent * To whom correspondence should be addressed. E-mail: rong_wang73@ hotmail.com.
developments have demonstrated that comb-shaped polymer, where the side chains are attached to the backbone by physical interactions such as hydrogen bonding, ionic bonding, coordination complexation, etc., offer a unique concept to design functional polymeric materials.9-15 New synthetic techniques allow synthesis of graft block copolymers having well-defined molecular architectures.8,9,16,17 These techniques provide precise control over the backbone molecular weight, the arm molecular weight, the arm polydispersity, the placement of the branch points along the backbone, and the number of arms grafted to each branch point. Molecules with specific, incremental backbone lengths are produced, allowing multiple grafts with well-defined architectures to be isolated and characterized. A number of theoretical studies of chemical disorder effects are on the self-assembly behavior of polymer blends and copolymers.18-29 The primary interest to such investigators has been the phase behavior of AB random copolymers and AB random multiblock copolymers with different statistical models of the sequence distributions of the two types of monomers. There have also been a few investigations of AB graft copolymers where the locations of the grafting points are random variables.30-35 Some common themes arise from these studies. The conformational properties of comblike macromolecules in a dilute solution have been studied by several authors using both analytical approaches36-38 and computer simulations.39-46 Even the structure of poly(methyl methacrylate-r-polyoxyethylene methacrylate) (P(MMA-r-PEOM)) comb polymer films and the blend of polylactide (PLA)/P(MMA-r-PEOM) at the water/polymer surface was predicted by a self-consistent field lattice model.47,48 Almost all of the previous theoretical calculations concentrated on the density profiles or the structures of
10.1021/jp811281t CCC: $40.75 2009 American Chemical Society Published on Web 05/07/2009
Comb Block Copolymer Am+1(BC)m
Figure 1. Schematic representation of a comb copolymer molecule. Thick solid, thin solid, and thin dashed lines represent main chain A and side blocks B and C, respectively.
the comb copolymers. The phase transitions of comb polymer Am+1Bm, such as order-to-disorder transition and order-to-order transition, were also studied theoretically. 32,49,50 In recent years, the comb copolymer Am+1(BC)m was experimentally studied.8,9 The comb-shaped polyethers with endfunctionalized side chains can self-aggregate into cross-linked polymer networks.9 The cross-shaped polyethers can serve as hosts for electrolyte solutions, i.e., a salt dissolved in organic aprotic solvents. The brush copolymers poly(2-hydroxyethyl mathacrylate)-graft-poly(ε-caprolactone)-block-poly(ethylene oxide) (PHEMA-g-(PCL-b-PEO)) have double crystallizable side chains.8 The crystallization temperature (Tc), melting temperature (Tm), and degree of crystallinity (Xc) of poly(2-hydroxyethyl mathacrylate)-graft-poly(ε-caprolactone) (PHEMA-g-PCL) were enhanced with the chain length increase of poly(ε-caprolactone) (PCL). As for PHEMA-g-(PCL-b-PEO) brush copolymer, the Tm and Xc values of PCL block decrease with the chain length increase of PEO block. The above results show that the special architecture of the comb copolymer can influence the phase behaviors. Therefore, it is very important to study the phase behaviors of the comb copolymer Am+1(BC)m with complex architectures. In this work, we consider the stable morphologies of Am+1(BC)m with different side chain numbers at the given interaction parameters between three different blocks by using the self-consistent field theory (SCFT), which is widely used in the prediction of phase separation of block copolymer and other related systems, such as in solution51-54 and in thin film,55-57 besides in melt. Even the phase behaviors of block copolymer under external field (electric field, shear) are studied.58,59 Due to the complexity of this comb block copolymer Am+1(BC)m, we only consider the side chain number m ) 1, 2, 3, and 5. By systematically changing the volume fractions of the A, B, and C blocks, we have found many stable morphologies and constructed the component triangle phase diagrams in the entire range of the copolymer composition. Our results are helpful to design nano- or biomaterials with complex architecture or tailor the phase behavior of comb copolymers to obtain the stable and ordered morphologies. 2. Calculation Algorithm We consider n comb copolymer Am+1(BC)m with polymerization N in a volume V, and there are m branching points (or the side chain number) along the main chain A, which divide the main chain as m + 1 equal parts (we called it “divided sections”) with polymerization NA and each side chain BC has NB and NC segments for blocks B and C, respectively. So, N ) (m + 1)NA + m(NB + NC). The schematic diagram of a comb copolymer molecule is presented in Figure 1. The monomers of the main chain and the side ones are assumed to be flexible with a statistical length a. Therefore, the compositions (average volume fractions) are fA ) (m + 1)NA/N, fB ) mNB/N, fC ) 1 - fA - fB for the main chain A and the blocks B and C of the side chains BC, respectively.
J. Phys. Chem. B, Vol. 113, No. 21, 2009 7463 With the different architectures of the blocks A, B, and C, we define four distribution functions, i.e., qBC(r, s), qBC+(r, s, t), qA(r, s, t), and qA+(r, s, t), where s is the contour along the main chain for A and along the side chain for BC and t is the number along the main chain A divided by the side chains BC and it belongs to [1, m + 1]. With these definitions, the polymer segment probability distributions q and q+ for main chain A and side chain BC satisfy the modified diffusion equations:
∂ Na2 2 q) ∇ q - wq ∂s 6
(1)
∂ + Na2 2 + q )∇ q + wq+ ∂s 6
(2)
where w is wA when s belongs to block A, wB when s belongs to block B, and wC when s belongs to block C. The initial conditions are qBC(r, 0) ) 1, qA(r, 0, 1) ) 1, qA(r, 0, t + 1) ) qBC(r, NB + NC) qA(r, NA, t), qA+(r, NA, m + 1) ) 1, qA+(r, NA, t) ) qA+(r, 0, t + 1) qBC(r, NB + NC), and qB+(r, NB + NC, t) ) qA(r, NA, t) qA+(r, 0, t + 1), where t ∈ [1, m]. Accordingly, the partition function of a single chain subject to the mean field wi, where i represents block species A, B, and C, can be written as Q ) ∫drqA(r, NA, m + 1). With the above description, the free energy function (in units of kBT) of the system is given by
F ) -ln(Q/V) + 1/V
∫ dr[ ∑ χijNφiφj - ∑ wiφi i*j
i
ξ(1 -
∑ φi)]
(3)
i
where χij is the Flory-Huggins interaction parameter between different species (i, j ) A, B, C, i * j), φi is the monomer density of each species (i ) A, B, C), and ξ is the Lagrange multiplier (as a pressure). Minimization of the free energy to mean field, density, and pressure, δF/δw ) δF/δφ ) δF/δξ ) 0, leads to the following self-consistent field equations that describe the equilibrium state:
φA(r) )
V QN
m+1
∑ ∫0
∑ ∫N
V QN t)1
m
φC(r) )
dsqA(r, s, t) qA+(r, s, t)
(4)
t)1
m
φB(r) )
NA
NB+NC C
∑ ∫0
V QN t)1
NC
dsqBC(r, s) qBC+(r, s, t)
dsqBC(r, s) qBC+(r, s, t)
(5)
(6)
wA(r) ) χABNφB(r) + χACNφC(r) + ξ(r)
(7)
wB(r) ) χABNφA(r) + χBCNφC(r) + ξ(r)
(8)
wC(r) ) χACNφA(r) + χBCNφB(r) + ξ(r)
(9)
φA(r) + φB(r) + φC(r) ) 1
(10)
Here, we solve eqs 4-10 directly in real space by using a combinatorial screening algorithm proposed by Drolet and Fredrickson.60,61 Note that one must solve the diffusion equation first for qBC(r, s) with initial condition qBC(r, 0) ) 1, then for qA(r, s, t) with qA(r, 0, 1) ) 1, qA(r, 0, t + 1) ) qBC(r, NB + NC) qA(r, NA, t), for qA+(r, s, t) with qA+(r, NA, m + 1) ) 1, qA+(r, NA, t) ) qA+(r, 0, t+1) qBC(r, NB + NC), and last for qBC+(r, s, t) with qBC+(r, NB + NC, t) ) qA(r, NA, t) qA+(r, 0, t + 1). Each iteration continues until the free energy converges to 10-6. Several times are repeated by using different initial conditions to avoid the trapping in a metastable state. In addition,
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Figure 2. Ordered morphologies for Am+1(BC)m comb block copolymer, m ) 1, 2, 3: (a) two-color lamellar phase (LAM2), (b) three-color lamellar phase (LAM3), (c) hexagonal lattice phase (HEX), (d) core shell hexagonal lattice phase (CSH), (e) two interpenetrating tetragonal lattice (TET2), (f) core shell tetragonal lattice (CST), (g) lamellar phase with beads inside (LAM+BD), (h) lamellar phase with alternating beads (LAM+AB). The chain packing information of the eight morphologies is also presented in the right side of the morphology correspondingly by taking A2(BC) as an example.
Figure 3. Phase diagrams for A2(BC) comb block copolymer: (a) χ˜ ) 35; (b) χ˜ ) 55.
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.62 The implementation of the selfconsistent field theory is carried out in a two-dimensional Lx × Ly lattice with periodic boundary conditions. The results presented in this paper are based on the 2D model, and hence may not obtain those intrinsic 3D structures, such as the cubic bcc and complex tricontinuous gyroid structures. However, the 2D model is not completely artificial. The potential applications of complex block copolymers as nanolithographic templates, membranes, and precursors for quantum electronic arrays often
involve thin films with a thickness comparable to the radius of gyration of the block chains. Moreover, even in a 3D system, the microphases with translational invariance along certain directions, such as lamellar and cylindrical phases, can also be investigated by a 2D model.63 Due to the complexity of this comb block copolymer Am+1(BC)m, we only consider the side chain number m ) 1, 2, 3, and 5. The interaction parameters between the different blocks are also considered in this work. In our calculation, the interaction parameters between different blocks are set to be equal: χABN ) χACN ) χBCN ) χ˜ . Due to the χN of the order-
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Figure 4. Phase diagrams for A3(BC)2 comb block copolymer: (a) χ˜ ) 50; (b) χ˜ ) 80.
to-disorder transition (ODT) and the order-to-order transition (OOT) increases largely as the side chain number m increases.29,32,49,50 Thus, we search the morphologies of Am+1(BC)m at the larger χ˜ with the increase of the side chain number m. The selected χ˜ is approximately proportional to the disorder-order transition of the comb copolymer. By systematically changing the volume fractions of the A, B, and C blocks, we can construct the component triangle phase diagrams in the entire range of the copolymer composition. 3. Results and Discussion The nine morphologies are found for comb block copolymer Am+1(BC)m by changing the volume fractions of different blocks and the side chain number. They are two-colored lamellar phase (LAM2), three-colored lamellar phase (LAM3), hexagonal lattice phase (HEX), core shell hexagonal lattice phase (CSH), two interpenetrating tetragonal lattice (TET2), core shell tetragonal lattice (CST), lamellar phase with beads inside (LAM+BD), lamellar phase with alternating beads (LAM+AB), and disordered phase (DIS). Figure 2 shows the eight ordered phase separated morphologies. The microphase patterns, displayed in the form of density, are the red, green, and blue, assigned to blocks A, B, and C, respectively. The final color plotted at each point is the mixture of three colors, in which the concentration of each color is proportional to the local volume fraction of an individual block. For a clear presentation of the final pattern, the linear dimensions of the unit cell are replicated two times in each direction and each dot of the pattern is enlarged by four times. The chain packing formation for each pattern is also shown in Figure 2 by taking A2(BC) as an example. We can clearly understand the chain arrangement when forming the ordered pattern. 3.1. Phase Diagram for Comb Block Copolymer A2(BC) (m ) 1). When m ) 1, A2(BC) comb copolymer resembles linear ABC block copolymer with an additional arm of block A. Figure 3 represents the phase diagrams of comb block copolymer A2(BC). When χ˜ ) 35, see Figure 3a, seven ordered morphologies are found except disordered phase (DIS): LAM2, LAM3, HEX, CSH, TET2, CST, and LAM+AB. Compared to ABC linear block copolymer,63 the morphologies are more complex. An additional three phases occursLAM2, CST, and LAM+ABsdue to the additional connecting point for block A. Due to the additional arm of the block A, the A-C reflection symmetry of the phase diagram disappears, although the interaction parameters between different blocks are equal. When the main chain A has the majority, the CSH is the stable phase. However, when the end block C is the majority, the CST phase
occurs besides the CSH phase, and when the middle block B is the majority, the TET2 and LAM+AB phases occur compared to the linear ABC triblock copolymer. The CSH phase is dominant when fA g 0.5, but the CSH phase occurs when fA g 0.6 for the linear ABC triblock copolymer.63 Thus, the composition of the phase transition is different. Comparing the three corners of the phase diagram, we can see that the disordered phase exists at the B corner, which shows that the comb copolymer with the longer B block is difficult to phase separate. The comb copolymer with the longer C block has the trend to phase separate into the CSH phase. When the middle block B is the minority, the phase transition from CSH to CST to LAM3 to CSH is observed with the increase of the volume fraction of the main chain A (two arms of the block A). When the volume fractions of the three components of the block copolymer are comparable, the stable phase is lamellae, but the lamellar region is smaller and shifts to the BC region, which is due to the additional arm of the block A. When the end block C or A is the minority, it is not easy to phase separate well for the comb copolymer compared with the case that the middle block B is the minority. The phase transitions occur from hexagonal phase (HEX or CSH) to lamellar phase (LAM3 or LAM2) to hexagonal phase (HEX) and finally to disordered phase. When the interaction parameter between different blocks strengthens, such as χ˜ ) 55, the block copolymer is well separated. The phase diagram is shown in Figure 3b. The LAM2 and DIS phases disappear. Another phase LAM+BD appears. Most HEX phases at χ˜ ) 35 separate into TET2 phases at χ˜ ) 55 where the end blocks A and C are the minority. The phase transition resulting from the interaction parameter χ˜ has some similarity with the linear ABC triblock copolymer which also has the free end for blocks A and C. The difference is the disappearance of the A-C reflection symmetry of the phase diagram due to the additional end block A. 3.2. Phase Diagram for Comb Block Copolymer A3(BC)2 (m ) 2). When the side chain number is m ) 2, the comb block copolymer becomes A3(BC)2. Due to the χN of the order-todisorder transition (ODT) and the order-to-order transition (OOT) increases largely as the side chain number increases.29,32,49,50 Thus, we search the morphologies of Am+1(BC)m at the larger χ˜ with the increase of the side chain number m. The phase behaviors of A3(BC)2 are studied at χ˜ ) 50 and 80. When χ˜ ) 50, five stable phases are found: DIS, HEX, CSH, LAM2, and LAM3. The phase diagram of A3(BC)2 is shown in Figure 4. Due to the two ends of part of the block A and the block B being restricted by the architecture, the disordered phase occurs at the corner of the blocks A and B. However, the HEX phase
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Figure 5. Phase diagrams for (a) A4(BC)3 comb block copolymer at χ˜ ) 60 and (b) A6(BC)5 comb block copolymer at χ˜ ) 100.
is stable at the block C corner. When the interaction parameters between the different blocks increase to χ˜ ) 80, the LAM2 phase disappears and phase separates into LAM3 or LAM+BD. The HEX phase region contracts, and the CSH phase region enlarges. The DIS phase separates into the HEX phase at the block A corner. Comparing the two phase diagrams of A3(BC)2, we can see that the DIS phase easily forms when the comb copolymer has the longer main chain A or longer side chain with short block C, i.e., longer block B. The reason is the connectivity of two ends of the block B and the middle block A, resulting in their stability (not easy separation). 3.3. Phase Diagram for Comb Block Copolymer A4(BC)3 (m ) 3) and A6(BC)5 (m ) 5). When the side chain number m ) 3 and 5, the comb block copolymer becomes A4(BC)3 and A6(BC)5. Here, we only consider the phase behaviors at χ˜ ) 60 for A4(BC)3 and at χ˜ ) 100 for A6(BC)5. The phase diagrams are shown in Figure 5a and b, respectively. The seven stable morphologies are DIS, LAM2, LAM3, LAM+BD, HEX, CSH, and TET2. When the block B is the majority, the comb block copolymer cannot phase separate easily due to the two ends of the block B being connected by the blocks C and A. Also, the disordered phase still exists in the block A corner which results from the end connection of part of the block A. The two-colored lamellar phase (LAM2) will easily form at the small volume fraction of block C, such as fC ) 0.1 when 0.3 e fA e 0.5. With the side chain number m increasing, the comb copolymer becomes more stable and it cannot easily phase separate. Comparing the phase diagrams of Am+1(BC)m comb block copolymer with different side chain numbers m, we can clearly see that the phase transitions are similar ((DIS) f HEX or CSH f LAM2 or LAM3 f CSH f (HEX) f (DIS)) when increasing the composition of the main chain (block A) at the very short middle block B (e.g., fB ) 0.1). However, the phase transitions are more complex when increasing the composition of the main chain (block A) at the very short side chain end (block C). This property can be used to tailor the morphology of Am+1(BC)m comb block copolymer to obtain the complex phase separated ordered pattern. 4. Conclusions The phase behaviors of Am+1(BC)m (m ) 1, 2, 3, and 5) comb copolymer are studied by the numerical version of the selfconsistent field theory. Nine stable morphologies are found: LAM2, LAM3, HEX, CSH, TET2, CST, LAM+BD, LAM+AB, and DIS. The phase diagrams are constructed for different side chain numbers m. Due to the spinodals of the microphase separation of comb copolymer,29,32,49,50 the phase diagrams are constructed at different interaction parameters, which is pro-
portional to the spinodals. The phase diagrams are asymmetrical, which is due to the fact that the architecture of a comb copolymer is not invariant under the interchange of A-, B-, and C-monomers. Comparing the phase diagrams of Am+1(BC)m, we can see that the DIS phase easily forms when the comb copolymer has the longer main chain A or longer side chain with short block C, i.e., longer block B. The reason is the connectivity of two ends of the block B and the middle block A, resulting in their stability (not easy separation). Our results are helpful to design nano- or biomaterials with complex architecture or tailor the phase behavior of comb copolymers to obtain the stable and ordered morphologies. Acknowledgment. We gratefully acknowledge financial support by National Natural Science Foundations of China (Nos. 20674035, 20504013, 20874046, and 50533020), Nanjing University Talent Development Foundation (No. 0205004107), and Natural Science Foundation of Nanjing University (No. 0205005216). References and Notes (1) Palmer, L. C.; Stupp, S. I. Acc. Chem. Res. 2008, 41, 1674. (2) Ariga, K.; Hill, J. P.; Lee, M. V.; Vinu, A.; Charvet, R.; Acharya, S. Sci. Technol. AdV. Mater. 2008, 9. (3) Nie, Z. H.; Kumacheva, E. Nat. Mater. 2008, 7, 277. (4) Dziezok, P.; Sheiko, S. S.; Fischer, K.; Schmidt, M.; Moller, M. Angew. Chem., Int. Ed. 1997, 36, 2812. (5) Lin, J. J.; Cheng, I. J.; Chen, C. N.; Kwan, C. C. Ind. Eng. Chem. Res. 2000, 39, 65. (6) Sheiko, S. S.; Moller, M. Chem. ReV. 2001, 101, 4099. (7) Sheiko, S. S.; Sun, F. C.; Randall, A.; Shirvanyants, D.; Rubinstein, M.; Lee, H.; Matyjaszewski, K. Nature 2006, 440, 191. (8) Yuan, W. Z.; Yuan, J. Y.; Zhang, F. B.; Xie, X. M.; Pan, C. Y. Macromolecules 2007, 40, 9094. (9) Jannasch, P.; Loyens, W. Solid State Ionics 2004, 166, 417. (10) Viville, P.; Leclere, P.; Deffieux, A.; Schappacher, M.; Bernard, J.; Borsali, R.; Bredas, J. L.; Lazzaroni, R. Polymer 2004, 45, 1833. (11) Ikkala, O.; ten Brinke, G. Science 2002, 295, 2407. (12) Maki-Ontto, R.; de Moel, K.; de Odorico, W.; Ruokolainen, J.; Stamm, M.; ten Brinke, G.; Ikkala, O. AdV. Mater. 2001, 13, 117. (13) Kosonen, H.; Ruokolainen, J.; Knaapila, M.; Torkkeli, M.; Jokela, K.; Serimaa, R.; ten Brinke, G.; Bras, W.; Monkman, A. P.; Ikkala, O. Macromolecules 2000, 33, 8671. (14) Thunemann, A. F.; Lochhaas, K. H. Langmuir 1998, 14, 4898. (15) Ruokolainen, J.; Makinen, R.; Torkkeli, M.; Makela, T.; Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280, 557. (16) Xenidou, M.; Hadjichristidis, N. Macromolecules 1998, 31, 5690. (17) Iatrou, H.; Mays, J. W.; Hadjichristidis, N. Macromolecules 1998, 31, 6697. (18) Dobrynin, A. V.; Leibler, L. Macromolecules 1997, 30, 4756. (19) Dobrynin, A. V.; Leibler, L. Europhys. Lett. 1996, 36, 283. (20) Fredrickson, G. H.; Milner, S. T.; Leibler, L. Macromolecules 1992, 25, 6341. (21) Fredrickson, G. H.; Milner, S. T. Phys. ReV. Lett. 1991, 67, 835. (22) Semenov, A. N. Eur. Phys. J. B 1999, 10, 497. (23) Semenov, A. N. J. Phys. II 1997, 7, 1489. (24) Semenov, A. N. Phys. ReV. Lett. 1998, 80, 1908.
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