Full Paper Confinement Effect on the Body-Centered-Cubic Phase of Diblock Copolymers in Film Hongge Tan,* Qinggong Song, Shuang Yang, Dadong Yan,* An-Chang Shi In this paper, we study the morphology of diblock copolymers with the body-centered-cubic (bcc) phase confined between two flat surfaces. Employing the Landau–Brazovskii mean field theory and the single mode approximation, we obtain three amplitude parameters as functions of temperature, surface field strength, and film thickness. Because of the effect of confinement size and the surface inducement, the morphology of confined diblock copolymers is different from the bulk structure. By controlling confinement size and surface field strength, lamella, undulated lamella, cylinder, and distorted cylinder can be observed in the bcc bulk phase of diblock copolymers. Also, we construct a ‘‘phase diagram’’ of confinementinduced structures at different surface field strengths. We compare the present theoretical results with the other relevant theoretical results. The predictions about these interesting confinement-induced structures should be observable in the experiments under suitable conditions. Introduction The self-assembly of diblock copolymers has drawn more and more attention because of the rich ordered microstructures.[1–6] The ordered microstructures can be applied H. Tan, Q. Song College of Science, Civil Aviation University of China, Tianjin 300300, China E-mail: thg@iccas.ac.cn S. Yang, D. Yan State Key Laboratory of Polymer Physics and Chemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, China E-mail: yandd@iccas.ac.cn A.-C. Shi Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada Macromol. Theory Simul. 2008, 17, 45–51 ß 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim in nanotechnology.[7,8] For bulk diblock copolymers, lamellae, hexagonal arrays of cylinders, body-centeredcubic (bcc) arrays of spheres, and the bicontinuous gyroid have been obtained in turn by changing the interactions between the distinct blocks and architectures of those blocks.[1–6] A good understanding of the bulk phase behavior has been obtained. For diblock copolymers confined by the environment of a system, the self-assembly process is more complicated and interesting due to the confinement geometry and the interactions between the block copolymer and the surface. The self-assembly in confined environment is a new approach toward preparing ordered periodic nanostructured materials and has attracted considerable interest in recent years.[7–31] Much attention has been focused on the self-assembly of symmetric diblock copolymers in a confined environment.[9–18] Their phase behavior has been understood DOI: 10.1002/mats.200700064 45 H. Tan, Q. Song, S. Yang, D. Yan, A.-C. Shi fairly well. For confined asymmetric block copolymers with cylindrical bulk morphology, some studies have also been carried out theoretically[19–23] and experimentally.[24–27] Under confinement, diblock copolymers with cylindrical bulk phase cannot only change the orientation of cylinders, but also alternate between different morphologies. In contrast to the lamellar bulk phase and the cylindrical bulk phase, confined diblock copolymers with the spherical bulk phase receive less attention. Although a few papers appeared which deal with some aspects of the bcc phase in thin film, the emphases of these papers are different from that of our work.[28–31] Recently, we studied theoretically the surface effect on the bcc spherical bulk phase.[32] As a continuation of the previous study,[32] in this paper, we devote to the confinement effect on the bcc forming asymmetric diblock copolymers under the interactions between surfaces and diblock copolymers. This paper is organized as follows. In the next section, in weak segregation limit we induce the model in which the diblock copolymer melt with bcc bulk phase is confined between two flat surfaces with surface interaction. In the subsequent section, we study the structure evolution under different film thicknesses and different surface field strengths. The corresponding discussions are also given in this section. In the last section, we give the conclusions. Theory As a continuation of the previous study, the model we describe here is the same as that in ref.[32] We consider an incompressible melt of n AB diblock copolymers in a volume V0 at a temperature T. The total degree of polymerization of the diblock copolymer is N. The degree of polymerization of the A block is fAN, where fA is the fraction of segments on each chain that are of type A. The effective interaction between statistical segments of type A and type B is characterized by the dimensionless Flory–Huggins parameter x in kBT units. The order parameter of diblock copolymers is defined as fðrÞ
fA ðrÞ  fA , the deviation of the local A monomer concentration from its average value. In the weak segregation regime, the Landau–Brazovskii model can be used. Specifically, the Landau–Brazovskii free energy per block copolymer f ½fðrÞ is given by f ½fðrÞ ¼ 1 V Z dr
2  j t g 1 ½ðr2 þ 1Þf2 þ f2  f3 þ f4 2 2 3! 4! (1) Here, the details of the parameters in Equation (1) can be found in ref.[1–3] and[32–34]. j and g are functions of fA; t is a 46 Macromol. Theory Simul. 2008, 17, 45–51 ß 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim function of xN and fA. When xN ¼ xNs (the disorder to order spindal point), t ¼ 0. In the weak segregation limit, fA and xN are chosen close to the order–disorder critical temperature. For a bcc structure in the weak segregation regime, it is sufficient to restrict ourselves to the first mode in the reciprocal space with the reciprocal lattice vectors pffiffiffi ~ i j ¼ q0 ¼ 2p=ðb= 2Þ (b is the lattice parameter and jG ~ i j ¼ 1 in scaled units), instead of using the complete set jG ~ i . In the single mode approximation, the bulk of vectors G bcc spherical phase can be described by the order parameter (see Equation (16) in ref.[32]) ~ 1  rÞ þ 2a2 ðrÞ½cosðG ~ 2  rÞ fðrÞ ¼ 2a1 ðrÞ cosðG ~ ~ þ cosðG3  rÞ þ 2a3 ðrÞ½cosðG4  rÞ ~ 5  rÞ þ cosðG ~ 6  rÞ þ cosðG (2) with ~1 G ~2 G ~3 G ~ G4 ~5 G ¼ z^; pffiffiffi ¼ 12 ð 3x^ þ z^Þ; pffiffiffi ¼ 12 ð 3x^ þ z^Þ; pffiffiffi ¼ p1ffiffi3 ðx^ þ 2y^Þ; pffiffiffi pffiffiffi ¼ 2p1 ffiffi3 ðx^  2 2y^ þ 3z^Þ; pffiffiffi pffiffiffi ~ 6 ¼ p1 ffiffi ðx^ þ 2 2y^ þ 3z^Þ G 2 3 where a1 (r), a2 (r), and a3 (r) are space-dependent amplitude functions describing the variation of microstructures. In the bcc bulk melt, a1 ¼ a2 ¼ a3 ¼ abcc, with which the order parameter describes bcc spheres. Furthermore, when a1 ¼ a2 ¼ ahex and a3 ¼ 0, the order parameter describes the hexagonal arrays of cylinders. When a1 ¼ alam and a2 ¼ a3 ¼ 0, the order parameter describes the lamellae. When the diblock copolymer melt is put in contact with two flat surfaces, the translational symmetry is broken. In order to study the confinement-induced structures in the bcc spherical phase explicitly under interactions between polymers and the surfaces, we choose surface plane as the z ¼ 0 and z ¼ l planes and assume that the surface is presented by a surface potential. Thus, the free energy of the system is given by V¼ 1 V Z drðffðrÞDu½dðzÞ þ dðz  lÞg þ f ½fðrÞÞ (3) Du is the differential affinity of surface tensions with respect to one block of the copolymer.[21] If uAS and uBS are the surface tensions of polymers A and B with respect to the surface, one has Du ¼ uBS  uAS . By uAS and uBS, Du can be related to any available material. d(z) and d(z  l) are delta functions, which mean that the interactions are short range and only exist on the surfaces (z ¼ 0 plane and z ¼ l plane). We assume that the amplitudes a1, a2, and a3 are only z-dependent, and vary slowly on the scale of l0, which DOI: 10.1002/mats.200700064 Confinement Effect on the BCC Phase of Diblock Copolymers . . . is the characteristic period of the lamellar component in the bulk melt. Also, we neglect the effect of the incommensurability between l and l0, although this incommensurability affects the orientation on the microstructure, which is the result of the competition between the surface interaction and the incommensurability. Under the above two assumptions, we can separate the length scale for variations in the amplitude from the length scale of the microstructure. Since we have assumed the amplitudes vary slowly on the scale of l0, we only retain gradient terms up to quadratic order in the Landau–Brazovskii free energy. Thus the free energy per unit surface can be written as (see Equation (26) in ref.[32]) ~ ¼ sXð0Þ  sXðLÞ V !2     Z L ( dX 1 dY 2 1 dZ 2 dt þ þ þ dt 2 dt 2 dt 0 ) þ f ½XðtÞ; YðtÞ; ZðtÞ values of s ¼ 0.9 and 2 corresponding to weaker and stronger surface fields, respectively, as we did in ref.[32] Results and Discussion In this section, we show that how the film thickness affects the bulk structure under different surface field strengths. As the first step, we linearize the free energy of bulk phase f(X, Y, Z) around the bcc structure. Minimizing the total free energy given by Equation (4), we obtain three linear Euler–Lagrange equations. Under the boundary conditions, we obtain the general solutions of these linear equations, DXðtÞ ¼ XðtÞ  Xbulk ¼s 3 h i X CXi ða; LÞeli ðaÞt þ CXiþ3 ða; LÞeli ðaÞt i¼1 (5) ð4Þ DYðtÞ ¼ YðtÞ  Ybulk with X ¼ a1 =g; t ¼ zg=2j; s ¼ Du=jg 2 ; Y ¼ a2 =g; L ¼ lg=2j; ~ ¼ V=2jg 3 V Z ¼ a3 =g; Here, the parameters g and j can be determined by f(X, Y, Z) is the free energy of the bulk diblock fA. copolymers and the expression can be found in ref.[32] Thus X(t), Y(t), and Z(t) are related to the variation of the structures since they are proportional to a1(z), a2(z), and a3(z), respectively. The reduced distance t, the effective film thickness L and the reduced surface field strength s are used instead of z, the film thickness l and Du which can be obtained from the experiments, respectively. Minimizing the free energy in Equation 4, we obtain three-coupled Euler–Lagrange equations for the amplitudes of X(t), Y(t), and Z(t). The formation of these three Euler–Lagrange equations are the same as Equation (27)–(29) in ref.[32] From Equation 4, we can obtain the boundary conditions   dX  dX  ¼   ¼ s dt t¼0 dt t¼L ¼s In the following section, we can solve these threecoupled Euler–Lagrange equations and discuss the results with the parameters L, s and effective temperature a, which appears in f (X, Y, Z) and is defined as t/g 2 in ref.[32] Although we cannot relate these values to any available materials at the moment, we can still select two typical Macromol. Theory Simul. 2008, 17, 45–51 ß 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (6) i¼1 DZðtÞ ¼ ZðtÞ  Zbulk [1–3,32,34]     dY  dY  dZ  dZ  ¼ ¼ ¼ ¼0 dt t¼0 dt t¼L dt t¼0 dt t¼L 3 h i X CYi ða; LÞeli ðaÞt þ CYiþ3 ða; LÞeli ðaÞt ¼s 3 h i X CZi ða; LÞeli ðaÞt þ CZiþ3 ða; LÞeli ðaÞt (7) i¼1 where CXi(a, L), etc. are the coefficients of the solutions of the linear equations and are determined by a and L. In order to study the confinement effect, first we fix the effective interaction s ¼ 0.9 and the effective temperature a ¼ 0.02 (far from the effective transition temperature a ¼ 0:07345, as mentioned in ref.[32]). We obtain Xbulk ¼ Ybulk ¼ Zbulk ¼ 0.27613,[32] l1 ¼ 1.7046, l2 ¼ 1.1034, and l3 ¼ 0.5997. We change the film thickness and consider three different cases as follows. The first case is for a thick film in which we choose L ¼ 12. In this case, by solving the linear Euler–Lagrange equations, the corresponding coefficients are CX1 ¼ 2.783  103, CX2 ¼ 0.1104, CX3 ¼ 1.457, CX4 ¼ 3.638  1012, CX5 ¼ 1.962  107, CX6 ¼ 1.091  103, CY1 ¼ 3.268   102, CY2 ¼ 0.3496, CY3 ¼ 0.5507, CY4 ¼ 4.271  1011, CY5 ¼ 6.212  107, CY6 ¼ 4.124  104, CZ1 ¼ 4.671  102, CZ2 ¼ 0.2314, CZ3 ¼ 0.5589, CZ4 ¼ 6.105  1011, CZ5 ¼ 4.112  107, and CZ6 ¼ 4.186  104. These asymptotic solutions can be used as the initial conditions to obtain the exact numerical solutions of three Euler–Lagrange equations and then we obtain the amplitudes X(t), Y(t), and Z(t) varying with t. The results are shown in Figure 1. On the surfaces, Y ¼ 0.208 and Z ¼ 0.005, which indicate the www.mts-journal.de 47 H. Tan, Q. Song, S. Yang, D. Yan, A.-C. Shi Figure 1. Reduced amplitudes X(t), Y(t), and Z(t) as functions of t for a ¼ 0.02, s ¼ 0.9, and L ¼ 12. Figure 3. Reduced amplitudes X(t), Y(t), and Z(t) as functions of t for a ¼ 0.02, s ¼ 0.9, and L ¼ 1. cylindrical component exists and the spherical component is negligible. The second case is for the medium thickness film, in which we choose L ¼ 6. In this case, the corresponding coefficients are CX1 ¼ 2.783  103, CX2 ¼ 0.1106, CX3 ¼ 1.497, CX4 ¼ 1.006  107, CX5 ¼ 1.474  104, CX6 ¼ 0.0410, CY1 ¼ 3.268  102, CY2 ¼ 0.3501, CY3 ¼ 0.5658, CY4 ¼ 1.181  106, CY5 ¼ 4.667  104, CY6 ¼ 1.548  102, CZ1 ¼ 4.671  102, CZ2 ¼ 0.2317, CZ3 ¼ 0.5742, CZ4 ¼ 1.689  106, CZ5 ¼ 3.089  104, and CZ6 ¼ 1.571  102. Following the same process as above, we obtain the amplitudes X(t), Y(t), and Z(t) varying with t. The results are shown in Figure 2. On the surfaces, Y ¼ 0.191 and Z ¼ 0.0008, which are smaller than those in the first case and indicate the cylindrical component still exists while the spherical component is nearly negligible. Furthermore, all the values of Z(t) are almost zero, which means that the spherical component hardly exists throughout the whole film. The third case is for a thin film, in which we choose L ¼ 1. In this case, the corresponding coefficients are CX1 ¼ 3.402  103, CX2 ¼ 0.1652, CX3 ¼ 3.229, CX4 ¼ 6.186  104, CX5 ¼ 5.482  102, CX6 ¼ 1.772, CY1 ¼ 3.994  102, CY2 ¼ 0.5232, CY3 ¼ 1.220, CY4 ¼ 7.263  103, CY5 ¼ 1.736, CY6 ¼ 0.6698, CZ1 ¼ 0.0571, CZ2 ¼ 0.3462, CZ3 ¼ 1.238, CZ4 ¼ 1.038  102, CZ5 ¼ 0.1149, and CZ6 ¼ 0.6797. We also obtain the amplitudes X(t), Y(t), and Z(t) varying with t shown in Figure 3. Note that all the values of Y(t) and Z(t) become negligible, which indicate that the cylindrical component and the spherical component disappear in the film. To demonstrate the evolution of the confinementinduced structures explicitly, the real space profiles of the system are presented in Figure 4–6 for L ¼ 12, 6, and 1, respectively. From Figure 4, we find that when L is large, the structures in the film change from undulated lamellae near the surfaces to cylindrical structures, then to the distorted cylindrical structures, and finally to the spherical structures near the center of the thin film. From Figure 5, we can find that the spherical structures disappear, and cylindrical structures appear near the center of the film and then convert to undulated lamellar structures near the surfaces. Figure 6 shows that when L is small enough, lamellar phases extend throughout the film. The results for s ¼ 2 are only quantitatively different from those for s ¼ 0.9. To save space, we do not present the results here. In order to illustrate the structure variation with film thickness, we construct a phase diagram for a ¼ 0.02 with s ¼ 0.9 and 2, respectively, as shown in Figure 7. Lines L1 and L2 describe the positions where the amplitudes Y(t) reach 10% of X(t) from the surfaces (t ¼ 0 and L) for s ¼ 0.9 and 2, respectively. To the right of Figure 2. Reduced amplitudes X(t), Y(t), and Z(t) as functions of t for a ¼ 0.02, s ¼ 0.9, and L ¼ 6. 48 Macromol. Theory Simul. 2008, 17, 45–51 ß 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/mats.200700064 Confinement Effect on the BCC Phase of Diblock Copolymers . . . Figure 4. Three-dimensional contour plot of the order parameter fðx; y; zÞ in the region with amplitudes from the solid lines in Figure 1. z is taken as t (in length unit of q1 0 ¼ 4p). these lines, the cylindrical components begin to be appreciable. Lines C1 and C2 describe the positions where the amplitudes Z(t) reach 10% of X(t) for s ¼ 0.9 and 2, respectively. To the right of these lines, the spherical components begin to appear. From Figure 7 one can find that the effect of confinement for weaker and stronger surface field are different. The region where the lamellar phase exists for s ¼ 0.9 is smaller than that for s ¼ 2. The critical thickness Lc1 for s ¼ 0.9, below which the lamellar phases extend throughout the film, is smaller than Lc2 for s ¼ 2. It is because the stronger surface field strength can induce lamellar phases easily. The structures in the confinement film are different from those in ref.[32] First, the main difference comes from the cusp of lines L1, L2, C1, and C2. According to ref.,[32] these lines should always keep the same distances from the surfaces (t ¼ 0 and L) and intersect directly at a point which is in front of the cusp of the lines in Figure 7. This can be considered as the effect of confinement. Secondly, the values of Y(t) and Z(t) near the surface decrease with a decrease in the film thickness (see Figure 1–3), which indicates that the cylindrical component exists for thicker films and then disappears little by little near the boundary with the decrease in the film thickness. Finally, the lamellar phase appears in a thin enough film even for a weaker surface field. This means that the confinement makes the lamellar phase appear for a weaker surface field. In other studies, Pereira also proved that the cubic to cylindrical transition was possible in diblock copolymers confined in the thin film.[28] His theoretical analysis was carried out in the strong segregation limit and confining Figure 5. Three-dimensional contour plot of the order parameter fðx; y; zÞ in the region with amplitudes from the solid lines in Figure 2. Macromol. Theory Simul. 2008, 17, 45–51 ß 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mts-journal.de 49 H. Tan, Q. Song, S. Yang, D. Yan, A.-C. Shi Figure 6. Three-dimensional contour plot of the order parameter fðx; y; zÞ in the region with amplitudes from the solid lines in Figure 3. plates were neutral with respect to each block, which disagreed with our theoretical work. He predicted that the cylindrical phase should appear in thin films and the bcc phase should be observed in thicker film. The conclusion there is in agreement with some of the present results. Therefore, our work and Pereira’s work should be seen as complementary to each other. Tsori reported that the external electric fields could be used to induce a phase transition from the bcc spheres to the hexagonal array of cylinders, which also proves indirectly the present results are reasonable.[29] Till date, a systematic experimental study of the evolution of confinement-induced structures in bcc spherical phase of diblock copolymers was lacking. We expect that the present prediction should be observable under appropriate experimental conditions. Conclusion Figure 7. Phase diagram, i.e., reduced film thickness L versus reduced distance t, for a ¼ 0.02 with s ¼ 0.9 (solid lines) and 2 (dashed lines), respectively. Lines L1 and L2 describe the positions where the amplitudes Y(t) reach 10% of X(t) from the surfaces (t ¼ 0 and L) for s ¼ 0.9 and 2, respectively. Lines C1 and C2 describe the positions where the amplitudes Z(t) reach 10% of X(t) for s ¼ 0.9 and 2, respectively. 50 Macromol. Theory Simul. 2008, 17, 45–51 ß 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim In summary, the evolution of confinement-induced structures is studied based on the Landau–Brazovskii theory in the weak segregation limit. For a weaker surface field strength, in a thick film, undulated lamellae, cylinders, and distorted cylinders appear in sequence near the surfaces and then convert to the spherical structure at the center of the film. With the decrease in film thickness, cylindrical phases appear near the center of the film and then convert to undulated lamellae near the surface. If we continue to decrease the film thickness, lamellar phases appear near the surfaces at first and then extend throughout the film. For a stronger surface field, the results are only quantitatively different from those for a weaker surface field. 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