11112
Langmuir 2007, 23, 11112-11119
Assembly
of
Copolymer Blend on Nanopatterned A Molecular Simulation Study
Surfaces:
Houyang Chen, Changjun Peng, Lei Sun, Honglai Liu,* and Ying Hu Lab for AdVanced Material and Department of Chemistry, East China UniVersity of Science and Technology, Shanghai 200237, China
Jianwen Jiang* Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, 4 Engineering DriVe 4, Singapore 117576 ReceiVed June 14, 2007. In Final Form: August 15, 2007 We report a molecular simulation study on the assembly of an (A7B5)5/A7B5 copolymer blend on nanopatterned surfaces. The density distributions, anisotropic radii of gyration, and conformations of both copolymers are quantitatively characterized. As the width of stripes on the surface decreases, the shape and thickness of the assembled film are found to be in qualitative agreement with those from experiments. The simulation results indicate that the shape and conformation of ordered film can be modulated by tuning the adsorption energy between the surface and the polymer or by adjusting the width of the stripes on the surface. We can regulate the width of the stripes to obtain a desired polymer conformation without altering the assembled film. In remarkable contrast to the pure copolymer, the radii of gyration of the blend in three directions are consistently smaller. The simulation reveals that the addition of a short chain during assembly is of central importance in restructuring the conformations of the long chain.
1. Introduction Molecular assembly of polymers can be induced by localized structures, such as pattered surfaces, buried dislocation arrays, and functionalized nanoparticles.1 The shape and microstructure of an assembled film or layer are predominantly controlled by the conformation of the adsorbed polymer. One key challenge in nanotechnology is to manipulate structures from the bottomup, which could lead to novel materials of unique properties. Assembly of polymers or biopolymers on well-defined surfaces has turned out to be a promising alternative for the fabrication of nanodevices, microelectronics, and specialized materials with nanoscale dimensions.2,3 Nanopatterned surfaces, as one type of well-defined surfaces, have received increasing interest over the past decade. A large number of studies, including experiments,4-8 theories, and simulations, have been conducted to investigate the assembly of polymers on nanopatterned surfaces. More details can be found in recent reviews.2,9,10 Here we briefly summarize previous simulations and theoretical studies in this area, which are closely relevant to our study. Muthukumar11 and Zheligovskaya et al.12,13 proposed a twostage pattern recognition mechanism (adsorption and rearrangement) for a random heteropolymer and a copolymer adsorbed * Corresponding author. Tel.: 86-21-64252921; fax: 86-21-64252921; e-mail:
[email protected] (H.L.). Tel: 65-65165083; fax: 65-67791936; e-mail:
[email protected] (J.J.). (1) Leroy, F.; Eymery, J.; Gentile, P.; Fournel, F. Surf. Sci. 2003, 545, 211. (2) Barth, J. V.; Costantini, G.; Kern, K. Nature 2005, 437, 671. (3) Joachim, G.; Gimzewski, J. K.; Aviram, A. Nature 2000, 408, 541. (4) Pallandre, A.; Meersman, B.; Blondeau, F.; Nysten, B.; Jonas, A. M. J. Am. Chem. Soc. 2005, 127, 4320. (5) Agheli, H.; Malmstrom, J.; Hanarp, P.; Sutherland, D. S. Mater. Sci. Eng. C 2006, 26, 911. (6) Lei, S.; Wang, C.; Wan, L.; Bai, C. J. Phys. Chem. B 2004, 108, 1173. (7) Shi, F.; Wang, Z.; Zhao, N.; Zhang, X. Langmuir 2005, 21, 1599. (8) Kim, S. O.; Solak, H. H.; Stoykovich, M. P.; Ferrier, N. J.; de Pablo, J. J.; Nealey, P. F. Nature 2003, 424, 411. (9) Cheng, J. Y.; Ross, C. A.; Smith, H. I;. Thomas, E. L. AdV. Mater. 2006, 18, 2505. (10) Park, C.; Yoon, J.; Thomas, E. L. Polymer 2003, 44, 6725.
on a pattered surface. Muthukumar et al.,14,15 Balazs et al.,16 and Genzer17,18 employed the self-consistent field theory to calculate the density profile distributions of copolymer near a chemically heterogeneous patterned surface. Integrating the mean-field theory and Monte Carlo (MC) simulation, Chakraborty et al.19,20 found a sharp transition from weak to strong adsorption regime. Using a “minimal” statistically mechanical model, Kriksin et al.21,22 investigated copolymer adsorption on selectively functionalized surfaces, and suggested that a marked pattern recognition is possible for copolymers with relatively short blocks at high polymer/surface attraction. Nath et al.23 proposed a density functional theory to analyze the ordering of block copolymers near patterned surfaces. Huber and Vilgis24,25 found that the localization behavior of copolymers is qualitatively affected by surface penetrability. Semler and Genzer26,27 used MC simulation to study the effects of domain size and interaction potential on copolymer assembly on chemically patterned planar surfaces. They revealed that the commensurability between the monomer (11) Muthukumar, M. J. Chem. Phys. 1995, 103, 4723. (12) Zheligovskaya, E. A.; Khalatur, P. G.; Khokhlov, A. R. J. Chem. Phys. 1997, 106, 8598. (13) Zheligovskaya, E. A.; Khalatur, P. G.; Khokhlov, A. R. Phys. ReV. E 1998, 59, 3071. (14) Petera, D.; Muthukumar, M. J. Chem. Phys. 1997, 107, 9640. (15) Petera, D.; Muthukumar, M. J. Chem. Phys. 1998, 109, 5101. (16) Balazs, A.; Singh, C.; Zhulina, E. Macromolecules 1998, 31, 6369. (17) Genzer, J. AdV. Colloid Interface Sci. 2001, 94, 105. (18) Genzer, J. Macromol. Theory Simul. 2002, 11, 481. (19) Chakraborty, A. K.; Bratko, D. J. Chem. Phys. 1998, 108, 1676. (20) Chakraborty, A. K.; Golumbfskie, A. J. Annu. ReV. Phys. Chem. 2001, 52, 537. (21) Kriksin, Y. A.; Khalatur, P. G.; Khokhlov, A. R. J. Chem. Phys. 2005, 122, 114703. (22) Kriksin, Y. A.; Khalatur, P. G.; Khokhlov, A. R. J. Chem. Phys. 2006, 124, 174904. (23) Nath, S. K.; Nealey, P. F.; de Pablo, J. J. J. Chem. Phys. 1999, 110, 7483. (24) Huber, G.; Vilgis, T. A. Eur. Phys. J. B: Condens. Matter Phys. 1998, 3, 217. (25) Semenov, A.; Engel, A.; Hiibers, H. W.; Il’in, K.; Siegel, M. Eur. Phys. J. B: Condens. Matter Phys. 2005, 47, 495. (26) Semler, J. J.; Genzer, J. J. Chem. Phys. 2003, 119, 5274. (27) Semler, J. J.; Genzer, J. Macromol. Theory Simul. 2004, 13, 219.
10.1021/la701773a CCC: $37.00 © 2007 American Chemical Society Published on Web 09/29/2007
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Figure 1. Segment-density profiles of an (A7B5)5/A7B5 blend along the z-axis at λ ) 1.0 and w ) 30 (i.e., Ns ) 2) (the adsorbing domain is located in the range x/σ ) 30 to 60): (a) x/σ ) 45, the center of the adsorbing domain; (b) x/σ ) 31, 1σ away from the interface between the adsorbing and non-adsorbing domains.
sequence distribution of copolymers and the size and spatial distribution of adsorbing surface sites plays a critical role in copolymer adsorption. Patra and Linse28 investigated the structural properties of polymer brushes adsorbed on nanopatterned surfaces. The results showed that the central part of a patterned brush remains nearly unchanged as long as the pattern is several times wider than the height of the brush, which agrees qualitatively with the recent experimental observation from atomic force microscopy. Striolo29 explained how the presence of a solid hard mask, used to mimic nanoscale patterns on an underlying hydrophobic surface, affects surfactant adsorption. Jayaraman et al.30,31 successfully developed a simulation technique to generate optimal surfaces that can recognize and preferentially adsorb a certain sequence in copolymer. We recently reported the recognition of multiblock polymers on nanopatterned surfaces.32 Our results showed that the recognition affinity becomes stronger with increasing the stripe width, the adsorption strength, and the number of adsorbing segments in the copolymer chain. In practical applications, the assembly of copolymers on patterned surface may involve a polymer mixture. Currently, it is not well understood how the existence of the second component would influence the microstructures of an assembled layer. As an extension of our pervious study, this work aims to elucidate the pattern-specific assembly of a copolymer blend near nanopatterned surfaces through MC molecular simulation. Emphasis is on the effects of short diblock copolymers on the assembly of long multiblock copolymers. Specifically, the assembled shape and conformation are examined with the existence of short copolymers at various stripe widths and polymer-surface interactions, which are two primary factors affecting the functionality of assembled materials.33,34 Our simulation finding is compared with available experimental (28) Patra, M.; Linse, P. Nano Lett. 2006, 6, 133. (29) Striolo, A. J. Chem. Phys. 2006, 125, 094709. (30) Jayaraman, A.; Hall, C. K.; Genzer, J. Phys. ReV. Lett. 2005, 94, 078103. (31) Jayaraman, A.; Hall, C. K.; Genzer, J. J. Chem. Phys. 2005, 123, 124702. (32) Chen, H.; Peng, C.; Ye, Z.; Liu, H.; Hu, Y.; Jiang, J. W. Langmuir 2007, 23, 2430.
observations.35 Following this introduction, the model and methodology are described in section 2. Simulation results are presented and discussed in section 3, including density distributions, radii of gyration, conformations of adsorbed polymer, and so forth. Section 4 gives concluding remarks.
2. Model and Methodology The copolymer blend under study is composed of a binary mixture of a multiblock (A7B5)5 and a diblock A7B5. (A7B5)5 and A7B5 are represented as freely jointed chains of 60 segments and 12 segments, respectively, with an identical diameter σ for A and B segments. Coarse-grained models with attractive interactions such as square-well potential32 and FENE potential36 are commonly used for chemically bonded atoms of self-assembly molecules. Here, we adopt the square-well potential. The interaction between segments is
{
rij > 1.5σ 0 u(rij) ) -ij σ < rij < 1.5σ ∞ rij < σ
(1)
where rij is the center-to-center distance between segments i and j; ij is the cross well-depth calculated by the geometric combining rule (iijj)1/2; ii is the well-depth between segments of the same type, and for simplification we have AA ) BB. To better distinguish the copolymer assembly, A-A and B-B interactions are assumed to be attractive; however, A-B interaction is repulsive. Although the coarse-grained model for copolymers (A7B5)5 and A7B5 cannot account for the atomistic detail, the essential characteristics of a copolymer chain are captured, (33) Scho¨n, J. H.; Kloc, C.; Batlogg, B. Nature 2000, 406, 702. (34) Scho¨n, J. H.; Kloc, C.; Dodabalapur, A.; Batlogg, B. Science 2000, 289, 599. (35) deGans, B. J.; Sanchez, C.; Kozodaev, D.; Wouters, D.; Alexeev, A.; Escuti, M. J.; Bastiaansen, C. W. M.; Broer, D. J.; Schubert, U. S. J. Comb. Chem. 2006, 8, 228. (36) Bosko, J. T.; Todd, B. D.; Sadus, R. J. J. Chem. Phys. 2004, 121, 1091.
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Figure 2. Segment-density contours (a-e) and snapshot (f) of an (A7B5)5/A7B5 blend in the xz plane at λ ) 1.0 and w ) 30 (i.e., Ns ) 2): (a) total density; (b) A in (A7B5)5; (c) B in (A7B5)5; (d) A in A7B5; (e) B in A7B5; (f) snapshot (A segments: gray, B segments: black).
including the excluded volume, chain connectivity, internal flexibility, and segment difference. The nanopatterned surface is mimicked by a plain surface with alternating adsorbing and nonadsorbing stripes along the x direction. Each stripe is equally sized with a width of w. The adsorbing and nonadsorbing stripes have different interactions with segments A and B:
{
0 ri-surface g σ βi-wall ) λ 0 e ri-surface < σ ∞ ri-surface < 0
(2)
where ri-surface is the distance between segment i and the surface. We assume that the adsorbing stripe has an attractive interaction only with segment B, while the nonadsorbing stripe has no preference for either segment A or B. MC simulations were performed in the canonical ensemble with a box of Lx × Ly × Lz ) 60σ × 60σ × 30σ. The number of stripes along the x direction is Ns ) Lx/w. The periodic boundary conditions were exerted in the x and y directions, while, in the z direction, two identical impenetrable nanopatterned surfaces were located at Lz ) 0 and Lz ) 30σ, respectively. The system consisted of two (A7B5)5 chains and 10 (A7B5) chains in the semi-infinite three-dimensional (3D) space. Each chain was subject to two types of trial moves, namely, translation and
jiggling. In translation, a randomly selected chain is translated with a random displacement in the x, y, or z direction. In jiggling, a bond between three successive segments is subject to a random rotation. The polar angle θ and the azimuthal angle φ of the new bond are supposed to uniformly distribute between [0,θmax] and [0,2π], respectively. Here, θmax is a maximum angle governed by the acceptance ratio. More details can be found in the literature.37-39 Specifically, three steps are involved to generate discrete points of (θ,φ) ∈ [0,θmax] × [0,2π]. First, a random number κ ∈ [cos θmax,1] is generated according to
κ ) cos θmax + (1 - cos θmax)$
(3)
where ω is a random number uniformly distributed within (0,1). Second, a random number ξ is generated between 0 and 1, and the two angles (θ,φ) are given by
θ ) arccos κ, φ ) 2πξ
(4)
Third, the Cartesian coordinates are calculated by
x ) sin θ cos φ, y ) sin θ sin φ, z ) cos θ
(5)
(37) Dickman, R.; Hall, C. K. J. Chem. Phys. 1988, 89, 3168. (38) Cai, J.; Liu, H. L.; Hu., Y. Fluid Phase Equilib. 2002, 194-197, 281. (39) Ye, Z. C.; Chen, H. Y.; Cai, J.; Liu, H. L.; Hu, Y. J. Chem. Phys. 2006, 125, 1.
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Figure 3. Total segment-density profiles of an (A7B5)5/A7B5 blend along the x-axis at λ ) 1.0 and (a) w ) 30, (b) w ) 10, (c) w ) 3, and (d) w ) 1.5.
Finally, random variables are obtained that uniformly distribute in a certain angle on a spherical surface. In our simulations, the reduced temperature was set as T* ) kBT/AA ) 6. From our previous work40 and the work of Hehmeyer et al.,41 this temperature is above the critical temperature of the copolymer under confinement. Typically, 107 trial moves were employed to reach equilibration, and the ensemble averages were obtained from the subsequent 4 × 107 trial moves. The statistical error of our simulation was typically less than 1%.
3. Results and Discussion 3.1. Segment-Density Distributions. Figure 1 shows the segment-density profiles of (A7B5)5 and A7B5 in their blend along the z-axis at λ ) 1.0 and w ) 30 (i.e., Ns ) 2). The adsorbing domain is located at x/σ ) 30-60. As characterized by the density profiles at the center of the adsorbing domain x/σ ) 45 in Figure 1(a), a thin film is formed near the surface. The film contains two layers, including one layer of B segments closer to the surface and another layer of A segments. In either layer, the density of A or B segments in the multiblock (A7B5)5 is larger than that of the diblock A7B5. Similar behavior was observed in the adsorption of multicomponent polymers at the solid-liquid interface.42 Generally, a longer chain tends to be excluded from the surface because of a greater loss of entropy upon adsorption.43 However, here the longer (A7B5)5 is preferentially adsorbed on the patterned surface because of the selective affinity to B segments. In other words, there is a competition in adsorption between the energetic and entropic effects. The second layer is formed as a result of the chain connectivity. The tail of the first layer diminishes rapidly beyond the range of the square-well potential, while the tail of the second (40) Chen, T.; Liu, H. L.; Hu, Y. Macromolecules 2000, 33, 1904. (41) Hehmeyer, O. J.; Panagiotopoulos, A. Z. J. Phys. Chem. B 2004, 108, 6809. (42) Jiang, J. W.; Liu, H.; Hu, Y. Macromol. Theory Simul. 1998, 7, 113. (43) Chandler, D.; McCoy, J. D.; Singer, S. J. J. Chem. Phys. 1986, 85, 5977.
layer extends further away from the surface. A closer look at Figure 1a suggests that A segments stay within the range of square-well potential because of the chain connectivity, whereas B segments reside out of the potential range because of the excluded-volume effect. As a result of both the chain connectivity effect and the excluded-volume effect, the long chain (A7B5)5 has a stronger affinity to the patterned surface than the short chain A7B5. The density profiles at x/σ ) 31, which is 1σ away from the interface between the adsorbing and non-adsorbing domains, are shown in Figure 1b. Assembly of A and B segments in the copolymer blend primarily resembles the behavior observed in Figure 1a at x/σ ) 45. However, the density of B segments in the multiblock (A7B5)5 is nearly equal to that in the diblock (A7B5)5, which is not the case in Figure 1a. This implies that each component in the local ordered film at x/σ ) 31 differs from that at x/σ ) 45. A similar result was previously found on a hemispherical gradient surface.44 Figure 2 shows the ensemble averaged segment-density contours of an (A7B5)5/A7B5 blend in the xz plane at λ ) 1.0 and w ) 30 (i.e., Ns ) 2). A thin gradient film is observed in the adsorbing domain of the surface. The thickness decreases slightly upon departure from x/σ ) 45, that is, the middle of the adsorbing domain. While B segments in both (A7B5)5 and A7B5 locate primarily near the surface, A segments can extend away from the surface. In addition, the contours of B segments, both in (A7B5)5 and A7B5, are more compact than those formed by A segments because the adsorbing domain near the wall attracts B segments. A simulation snapshot is shown in Figure 2f. We can see more chains are in the adsorbing region due to the attraction with B segments. From Figures 1 and 2, it is obvious that only microscopic phase separation occurs between A and B segments; nevertheless, there is no macroscopic phase separation between multiblock and diblock copolymers. (44) Choi, S. H.; Newby, B. Z. Langmuir 2003, 19, 1419.
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Figure 4. Segment-density contours of an (A7B5)5/A7B5 blend in the xz plane at λ ) 1.0 and (a) w ) 30, (b) w ) 10, (c) w ) 3, and (d) w ) 1.5.
Figure 3 shows the total segment-density profiles of an (A7B5)5/A7B5 blend along the x-axis at w ) 30, 10, 3, and 1.5 (i.e., Ns ) 2, 6, 20, and 40, respectively). The copolymers can be recognized on the patterned surface at the four w values but with different shapes of the film. These are consistent with previous studies, for example, the optimizing photoembossed experimental data,35 the nonlinear 3D simulation based on the Navier-Stokes equation,45 and the laser irradiation measurement.46 In addition, our results qualitatively match the experimentally determined thickness and shape of the films.35 As the width of the stripe decreases, the thickness of the ordered films decreases. The reason is more copolymer molecules can cross the interface between adsorbing and non-adsorbing stripes as w decreases, partially because of the role of chain connectivity. To further illustrate, the segment-density contours for the same systems are shown in Figure 4. In the density profiles between adsorbing and non-adsorbing domains, the difference is clearly seen to diminish with decreasing w. It is evident that the topographic features in the assembled films of the (A7B5)5/A7B5 blend on the patterned surface are highly extended in the direction perpendicular to the underlying patterned surface.47 This behavior differs from the behavior of copolymers adsorbed on uniform substrates. As shown in Figure S1 of the Supporting Information,32 in the same conditions, pure copolymer (A7B5)5 behaves nearly the same as the (A7B5)5/A7B5 blend in the characteristic density profiles along the z direction at x/σ ) 31 and 45, as well as the ordered film. The interaction between a surface and a polymer is of central importance in surface modification and in tuning surface properties, such as wetting/adhesion and friction/lubrication. To investigate the effect of attractive interaction between the surface (45) Verma, R.; Sharma, A.; Kargupta, K.; Bhaumik, J. Langmuir 2005, 21, 3710. (46) Barrett, C. J.; Natansohn, A. L.; Rochon, P. L. J. Phys. Chem. 1996, 100, 8836. (47) Edwards, E. W.; Stoykovich, M. P.; Solak, H. H.; Nealey, P. F. Macromolecules 2006, 39, 3598.
Figure 5. Segment-density profiles of an (A7B5)5/A7B5 blend along the z-axis at λ ) 2.0, w ) 30, and x/σ ) 45.
and B segments, we show the segment-density profiles of the (A7B5)5/A7B5 blend at λ ) 2.0 in Figure 5, instead of λ ) 1.0 as in Figure 1. At a larger λ, film becomes more compact, and the tail of the density profile is shorter. This implies that the surface interaction with B segments governs the density distribution, and the excluded-volume effect plays only a minor role. As interestingly demonstrated from a recent experimental study, it is possible to increase the density of film by exerting an external field.45 3.2. Radii of Gyration. Polymer conformation is a crucial ingredient to elucidate the microscopic structure of ordered film. In this section, the radii of gyration are presented for all polymer molecules in the system, and in the next section, the detailed
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Figure 6. Radii of gyration of (a) (A7B5)5 and (b) A7B5 in an (A7B5)5/A7B5 blend at λ ) 1.0 and w ) 30.
conformations including tail, loop, and train are shown for polymer molecules with direct contact with the surface. The anisotropic radii of gyration Rg,x2, Rg,y2, and Rg,z2 in the x, y, and z directions are defined separately as
Rg,p2 )
1
∑i (pi - pcom)2
N
(6)
where p ) x, y, or z, pi is the coordinate of the ith segment, pcom is the center of mass of a polymer chain, and N is the number of the segments in the chain.48 Figure 6a shows the Rg2 of (A7B5)5 in an (A7B5)5/A7B5 blend at λ ) 1.0 and w ) 30 (i.e., Ns ) 2). To clearly identify the stripe effect, the Rg,p2 values (p ) x, y, z) in both non-adsorbing and adsorbing domains are separately plotted. In the non-adsorbing domain, Rg,p2 is approximately zero near the surface and increases moving away from the surface. This reveals a strong depletion to (A7B5)5 by the surface. The value of Rg,x2 is not the same as that of Rg,y2, indicating a weak stripe effect. From surface to bulk, (A7B5)5 exhibits an ellipsoidal shape, but the lengths of principal axes in the ellipsoid vary with the distance z from surface. In the adsorbing domain and the full space, all three Rg,p2 values are negligible but rise rapidly with increasing z. Rg,x2 is nearly equal to Rg,y2, indicating a slight stripe effect, and neither of them is larger than Rg,z2, which suggests the confinement effect of the external surface. Near the surface (region 1), (A7B5)5 first adopts a coiled conformation. Moving away from the surface, the conformation changes to elongated (region 2) and finally to globular in bulk phase (region 3). The stripe effect is enhanced gradually as w decreases. In principle, the stripe becomes structureless when w approaches zero. As shown in Figure S2 for w ) 10 and 1.5 (Ns ) 6 and (48) Eisenriegler, E.; Kremer, K.; Binder, K. J. Chem. Phys. 1982, 77, 6296.
40) in the Supporting Information, it is clear that the width of the stripe is an important factor in controlling the conformation of adsorbed polymer. In this sense, one can monitor the width of the stripe to obtain a certain assembled polymer film without altering the shape. It is instructive to distinguish the assembly behavior of (A7B5)5 in an (A7B5)5/A7B5 blend from that of pure (A7B5)5. First, the stripe effect is less distinct in the blend. The reason is that, in the blend, the short A7B5 is more likely to reside at the interface of adsorbing and non-adsorbing domains, and the long (A7B5)5 tends to move into the center of the adsorbing domain because of the excluded-volume effect. Overall, (A7B5)5 in an (A7B5)5/ A7B5 blend prefers to stay near the surface with a coiled conformation. Second, in the blend, the values of Rg2 in the adsorbing domain are much larger than those in the non-adsorbing domain. On the contrary, the values of Rg2 in pure (A7B5) are approximately the same in both adsorbing and non-adsorbing domains. From the assembled shape and conformation, we can conclude that the Rg2 values of (A7B5)5 in an (A7B5)5/A7B5 blend in all three directions become smaller than those in pure (A7B5)5, although the assembled shape is not appreciably changed. These reveal that the addition of the short chain plays an important role in restructuring the conformation of the long chain. Figure 6b shows the Rg2 value of A7B5 in an (A7B5)5/A7B5 blend at λ ) 1.0 and w ) 30 (i.e., Ns ) 2). In the non-adsorbing domain, the stripe effect is clearly observed with Rg,x2 > Rg,y2. It is interesting to find adsorption from Rg,x2 and Rg,y2 while depletion is found from Rg,z2. This implies that A7B5 exhibits elongation in both the x and y direction, but exhibits coiling in the z direction. In the adsorbing domain and full space, A7B5 adopts an ellipsoid shape predominantly along the x and y directions near the surface (region 1), then turns the direction of the ellipsoid away from the surface (region 2), finally exhibiting
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Figure 7. Size distributions of tail (circles), loop (squares), and train (triangles) versus conformation length for (a) (A7B5)5 and (b) A7B5 in an (A7B5)5/A7B5 blend at λ ) 1.0 and w ) 30.
a globular shape (region 3), and the volume increases slightly in bulk phase (region 4). A sharp contrast exists between the long chain (A7B5)5 and short chain A7B5 in their blend. Because of the short length, A7B5 can be more easily moved closer to the surface. This is because the stripe can accommodate the whole short chain or block, but not the whole long chain. The characteristics of the long chain in full space are primarily determined by the adsorbing domain, while, for the short chain, both the adsorbing and nonadsorbing domains come into play. 3.3. Conformations of Adsorbed Copolymers. The conformations of adsorbed copolymers can be classified into three types: tail, loop, and train. Their definitions can be referred to in our preview work.32 The probability to find a conformation q with length l is
Pq(l) ) Nq(l)/
∑ Nq(l)
(7)
where Nq(l) is the number of conformation q with length l. The average size of conformation q is estimated from
Lq )
∑ lPq(l)/∑ Pq(l)
(8)
The fraction of conformation q is calculated by49
fq )
∑j j × Nq(j)/∑q ∑j j × Nq(j)
(9)
Figure 7 shows the size distributions of tail, loop, and train as a function of the conformation length in an (A7B5)5/A7B5 blend at λ ) 1.0 and w )30 (i.e., Ns ) 2). For the long chain (A7B5)5 in Figure 7a, the length of the tail appears mainly as (9 + 12n)σ with n ) 0, 1, 2, 3, 4. Interestingly, here “12” is the characteristic length of (A7B5)5 with 12 repeated segments. As is well-known, the non-adsorbing block A7 tends to stay in bulk, while the adsorbing block B5 prefers being near the surface. As (49) Chen, H. Y.; Ye, Z. C.; Peng, C. J.; Liu, H. L.; Hu, Y. J. Chem. Phys. 2006, 125, 204708.
Figure 8. Average sizes and fractions of tail (circles), loop (squares), and train (triangles) for (a) (A7B5)5 and (b) A7B5 in an (A7B5)5/A7B5 blend at λ ) 1.0 and different Ns.
a consequence of this factor and chain connectivity, long tails are normally observed. The length of the loop is about 1 or (11 + 12n)σ with n ) 0, 1, 2, 3. Apparently, the probabilities of the tail and loop reduce with increasing length. This is caused by the competition between the entropic penalty and the energy contribution.26 The train mainly exists at very short lengths (
