pubs.acs.org/Langmuir © 2009 American Chemical Society
Memory Effects of Diblock Copolymer Brushes and Mixed Brushes Jiafang Wang*,†,‡ and Marcus M€uller† †
‡
Institut f€ ur Theoretische Physik, Georg-August Universit€ at, D-37077 G€ ottingen, Germany, and State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, P. R. China Received July 8, 2009. Revised Manuscript Received September 2, 2009
Memory effects of microphase segregation in diblock copolymer brushes and binary mixed homopolymer brushes exposed to solvents of different quality and selectivity are studied using Single-Chain-in-Mean-Field (SCMF) simulations. We gauge these memory effects by a fluctuation memory measure, reflecting the correlation between the quenched fluctuations of grafting points and the microphase-separated morphology, and a domain memory measure, quantifying the correlation between surface morphologies during cyclic exposure to different solvents. The fluctuation and domain memory measures are closely correlated, and both of them have their root in the broken translational symmetry of the distribution of grafting points. They become stronger upon increasing the fluctuations of the grafting points. The effects of solvent quality and selectivity, grafting density, and composition of brushes on the memory measures are discussed.
I. Introduction Due to their efficiency in surface modification, polymer brushes are extensively used in various applications such as colloid stabilization,1 drug delivery,2 and modification of surface wetting and adhesion.3,4 With additional freedom in adjusting surface properties, multicomponent polymer brushes (MCPBs), including block copolymer brushes and mixed polymer brushes, become promising materials for fabricating smart surfaces and open opportunities for a variety of applications.5-13 Usually, there is immiscibility and/or contrast in solvent affinity between components, and therefore, phase-separation and structure transformation in MCPBs can be induced, for example, by changing temperature or solvent. Due to the immobility of the grafting points arising from the grafting constraint, lateral phase separation in MCPBs is limited to the molecular length scale of the (1) Pincus, P. Macromolecules 1991, 24, 2912. (2) Galaev, I. Y.; Mattiasson, B. Trends Biotechnol. 1999, 17, 335. (3) Leger, L.; Raphael, E.; Hervet, H. Polym. Confined Environ., 1999 1999, 138, 185. (4) Mansky, P.; Liu, Y.; Huang, E.; Russel, T. P.; Hawker, C. J. Science 1997, 275, 1458. (5) Zhao, B.; Brittain, W. J. Prog. Polym. Sci. 2000, 25, 677. (6) Luzinov, I.; Minko, S.; Tsukruk, V. V. Prog. Polym. Sci. 2004, 29, 635. (7) Brittain, W. J.; Minko, S. J. Polym. Sci., Part A: Polym. Chem. 2007, 45, 3505. (8) Zhao, B.; Brittain, W. J.; Zhou, W.; Cheng, S. Z. D. J. Am. Chem. Soc. 2000, 122, 2407. (9) Zhao, B.; Brittain, W. J.; Zhou, W.; Cheng, S. Z. D. Macromolecules 2000, 33, 8821. (10) Sidorenko, A.; Minko, S.; Schenk-Meuser, K.; Duschner, H.; Stamm, M. Langmuir 1999, 15, 8349. (11) Minko, S.; Usov, D.; Goreshnik, E.; Stamm, M. Macromol. Rapid Commun. 2001, 22, 206. (12) Julthongpiput, D.; Lin, Y.-H.; Teng, J.; Zubarev, E. R.; Tsukruk, V. V. J. Am. Chem. Soc. 2003, 125, 15912. (13) Julthongpiput, D.; Lin, Y.-H.; Teng, J.; Zubarev, E. R.; Tsukruk, V. V. Langmuir 2003, 19, 7832. (14) Zhulina, E.; Singh, C.; Balazs, A. C. Macromolecules 1996, 29, 6338. (15) Singh, C.; Balazs, A. C. J. Chem. Phys. 1996, 105, 706. (16) Brown, G.; Chakrabarti, A.; Marko, J. F. Macromolecules 1995, 28, 7817. (17) Yin, Y.; Sun, P.; Li, B.; Chen, T.; Jin, Q.; Ding, D.; Shi, A.-C. Macromolecules 2007, 40, 5161. (18) Wang, J.; M€uller, M. Macromolecules 2009, 42, 2251. (19) Matsen, M. W.; Griffiths, G. H. Eur. Phys. J. E 2009, 219. (20) Marko, J. F.; Witten, T. A. Phys. Rev. Lett. 1991, 66, 1541. (21) Marko, J. F.; Witten, T. A. Macromolecules 1992, 25, 296. (22) Lai, P.-Y. J. Chem. Phys. 1994, 100, 3351.
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lateral size of polymer coils.14-27 Experiments and theoretical studies have shown that lateral and perpendicular segregation or their combination can occur in multicomponent brushes, depending on the molecular parameters of the polymer chains (e.g., chemical nature of monomers, chain architecture, degree of polymerization, and grafting density) and the external environment (e.g., temperature, solvent, substrate, etc.). For instance, by exposing a MCPB to solvents of different selectivity, the surface of mixed polymer brushes can be switched from hydrophobic to hydrophilic and vice versa, and such a switching of the surface properties can be repeated in a cyclic way.10,11,28,29 Some surface properties, such as wettability and adhesion, only depend on the overall average composition of the top surface, which is accessible to the solvent, because wetting and adhesion occur on a length scale much larger than the molecular length scale in laterally microphase-separated MCPBs and the fine details of the morphology will be averaged out. However, details of microphase-segregated structures are important for other applications. From the very beginning, it has been suggested that MCPBs be applied in the fabrication of high-capacity datastorage devices, waveguides, and so on, just like block copolymer thin films.30,31 These applications require as large an area of perfectly ordered morphology as possible. Some applications, for example, porous materials for filtration devices or catalysis, do not heavily rely on the long-range order of the structures but just the well-defined microscopic length scale of surface structures. For instance, MCPBs have been devised to adsorb proteins or nanoparticles from solution.32 In this case, the hydrophobic patches on the top of the brush with a lateral dimension (23) Soga, K. G.; Guo, H.; Zuckermann, M. J. Europhys. Lett. 1995, 29, 531. (24) Zhulina, E.; Balazs, A. C. Macromolecules 1996, 29, 2667. (25) M€uller, M. Phys. Rev. E 2002, 65, 030802. (26) Merlitz, H.; He, G.-L.; Sommer, J.-U.; Wu, C.-X. Macromolecules 2009, 42, 445. (27) Wang, J.; M€uller, M. J. Phys. Chem. B 2009, 113, 11384. (28) Minko, S.; Patil, S.; Datsyuk, V.; Simon, F.; Eichhorn, K.-J.; Motornov, M.; Usov, D.; Tokarev, I.; Stamm, M. Langmuir 2002, 18, 289. (29) LeMieux, M.; Usov, D.; Minko, S.; Stamm, M.; Shulha, H.; Tsukruk, V. V. Macromolecules 2003, 36, 7244. (30) Russell, T. P. Curr. Opin. Colloid Interface Sci. 1996, 1, 107. (31) Hamley, I. W. Nanotechnology 2003, 14, R39. (32) Hoy, O. Ph.D. Thesis, Clemson University, 2008.
Published on Web 10/06/2009
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comparable to the size of the solute are the key factor. Considering the ability of MCPBs to switch surface structures and adsorb nanoparticles, a mechanism has been proposed to impart or enhance random motion of adsorbed nanoparticles by reversible changes of the lateral structure of the top layer in MCPBs.33-35 To produce a random, diffusive motion, the correlation between laterally segregated surface patterns upon subsequent solventswitching cycles must be very low. In the context of these requirements, two questions arise: (i) What leads to the loss of long-range order of the morphology in MCPBs? In principle, microphase separation in MCPBs can lead to periodic ordered structures possessing long-range order, just like those in block copolymer films. Such possibilities have been considered by scaling theory,14,24 numerical self-consistent field calculation,19,20,25 and computer simulations.17 All these theoretical investigations and simulations are based upon the common assumption that grafting points are completely uniform or regularly distributed. In experiments, however, only structures with a well-defined dominant length but only very poor longrange order have been observed. Several reasons may contribute to the loss of long-range order, one of which is the quenched fluctuations of the grafting points. Due to the random nature of the grafting reaction, spatial inhomogeneity of the distribution of grafting points is inevitable for brushes prepared by conventional “grafting from” or “grafting to” methods. This inhomogeneous distribution of grafting points includes fluctuations in density and composition. The latter fluctuations only matter for those MCPBs with more than one kind of grafting point, for example, mixed brushes. There are correlations between surface morphologies and fluctuations in grafting points, which we call fluctuation memory. Such a fluctuation memory exists even in one-component brushes. Wenning and co-workers have demonstrated that, for one-component brushes in a bad solvent and immiscible mixed brushes, small fluctuations in grafting points can be amplified in the morphologies of the brush and destroy the long-range order in surface patterns.36 (ii) The other question refers to correlations between laterally segregated surface patterns upon subsequent solvent-switching cycles. In cyclic solvent-switching experiments, surface morphologies alternate between a topographically flat and a patterned state.37 Experiments demonstrated that there are correlations between the patterned surface structures, which is denoted domain memory.38,39 It is believed that the domain memory also has its origin in the inhomogeneous distribution of grafting points. Santer and coworkers have compared domain memory effects of polystyrene/ poly(methyl methacrylate) (PS/PMMA) mixed brushes in the melt state with conventional and Y-shaped grafting manners, which has been corroborated by simulations. Although both brushes exhibit similar microphase-segregated morphologies, Y-shaped mixed brushes have a significantly weaker domain memory than conventional mixed brushes because there are no composition fluctuations of grafting points in Y-shaped mixed brushes.39 Both fluctuation memory and domain memory have their root in fluctuations of the density of grafting points, and therefore, they are closely correlated to each other. The microphase-separated (33) Santer (Prokhorova), S.; R€uhe, J. Polymer 2004, 45, 8279. (34) Santer, S.; Kopyshev, A.; Donges, J.; Yang, H.-K.; R€uhe, J. Adv. Mater. 2006, 18, 2359. (35) Yu, K.; Wang, H. F.; Han, Y. C. Langmuir 2007, 23, 8957. (36) Wenning, L.; M€uller, M.; Binder, K. Europhys. Lett. 2005, 71, 639. (37) It is also possible to switch between two segregated structures. (38) Santer, S.; Kopyshev, A.; Donges, J.; Yang, H.-K.; R€uhe, J. Langmuir 2006, 22, 4660. (39) Santer, S.; Kopyshev, A.; Donges, J.; R€uhe, J.; Jiang, X.; Zhao, B.; M€uller, M. Langmuir 2007, 23, 279.
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morphology of MCPBs is dictated by a balance between the repulsion between the different components, which favors large domain sizes, and the entropic penalty of chain stretching, which favors smaller characteristic length scales in turn. Polymer molecules in areas with denser grafting points can aggregate with less chain stretching and therefore have a propensity of forming a cluster. Such clusters can serve as heterogeneous nucleating centers for microphase segregation. Thus, pronounced fluctuations in the spatial distribution of grafting points act as a template for the morphology, giving rise to fluctuation memory and domain memory. While there are several studies on equilibrium morphology of microphase separation in MCPBs, only recently have memory effects in microphase segregation of MCPBs attracted attention.33,36,39 For instance, experiments demonstrated that mixed homopolymer brushes of poly(methyl methacrylate)/poly(glycidyl methacrylate) (PMMA/PGMA) have much stronger domain memory effects than diblock copolymer brushes of PMMAb-PGMA.33 Due to their importance for applications, a systematic study of memory effects is warranted. Recently, we have employed Single-Chain-in-Mean-Field (SCMF) simulations to investigate the different microphase-separated morphologies of mixed brushes and diblock copolymers as a function of chain architecture, grafting density, and solvent quality and selectivity.18,27 In the present manuscript, we explore whether and to what extent these factors affect memory effects in MCPBs. The SCMF simulation technique allows us to study the three-dimensional equilibrium morphology of large systems, which enable us to investigate memory effects of microphase-separated brushes without finite-size effects. The present work is a significant extension of our previous ones.36,39 Instead of mixed homopolymer brushes in the melt state, we consider the behavior of MCPBs in various solvents. We study how the morphology affects memory effects, and compare block copolymer brushes with mixed homopolymer brushes. In addition to a comprehensive exploration of experimentally accessible factors such as grafting density, composition, and solvent quality and selectivity, our simulation also allows us to control the arrangement of grafting points. Thereby, it provides direct insights which are not easily obtained by experiments. Importantly, memory effects in MCPBs in solvents are more complex than those in the melt, where the domain memory only depends on the grafting density via the strength of the quenched density fluctuations of the grafting points.39 In the present work, we demonstrate that a change in the grafting density additionally gives rise to structural changes, which, in turn, affect the susceptibility of the morphology with respect to the quenched fluctuation of the grafting points. Our manuscript is arranged as follows: In the next section, we briefly summarize the salient features of the coarse-grained model and the SCMF simulations. This section also provides the definition of the two memory measures that we will investigate. The third section presents our results for diblock copolymer brushes, and the subsequent one details our findings for mixed homopolymer brushes. Some strategies for controlling memory effects are discussed in the conclusions.
II. Model and Simulation Methods The microphase separation of diblock copolymer A-B brushes and mixed A/B homopolymer brushes exposed to solvent, S, is studied using Single-Chain-in-Mean-Field simulations.40-42 (40) M€uller, M.; Smith, G. D. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 934. (41) Daoulas, K. Ch.; M€uller J. Chem. Phys. 2006, 125, 184904. (42) Daoulas, K. Ch.; M€uller, M.; de Pablo, J. J.; Nealey, P. F.; Smith, G. D. Soft Matter 2006, 2, 573.
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Table 1. Interaction Parameters of the Implicit-Solvent Model solvent υAA υBB υAB υAAA υBBB υAAB υABB
SA
SB
SC
SD
0.325275 0.258649 2.07856 0.212006 0.150328 0.189051 0.168581
-2.79721 0.173237 0.638477 1.0418 0.141527 0.535554 0.27531
4.62845 -11.0412 -2.33879 0 3.66696 0 0
-9.2569 -11.0412 -8.52973 3.87862 3.66696 3.80674 3.7362
The same model and simulation methods have been used to explore the equilibrium microphase segregation in diblock copolymer brushes18 and mixed brushes,27 and thus, we only describe the necessary model parameters here. For the details on the methodology, an overview of the different morphologies as a function of grafting density, chain architecture, and solvent properties, and an analysis of the gradual crossovers between different morphologies via integral geometry techniques, we refer the reader to refs 18 and 27. The diblock copolymer brush consists of nAB grafted A-B diblock copolymer chains, and the mixed homopolymer brush comprises nA A-homopolymer chains and nB B-chains. n is defined as the total number of all polymer chains. For diblock copolymer brushes, n = nAB, and for mixed brushes n = nA þ nB. For diblock copolymer brushes, f denotes the fraction of A-monomer in a diblock copolymer, and, for mixed homopolymer brushes, f = nA/n is the number fraction of A-polymers. The molecular contour of a homopolymer -A or B- or diblock copolymer -A-B- is discretized into NA, NB, or NAB effective interaction centers (segments), respectively. The connectivity along the molecular backbone is modeled by a discretized Edwards Hamiltonian. If not stated otherwise, we will use the discretization NR = 32 for all molecular species, A, B, and A-B. In the following, we use the mean square end-to-end distance of a polymer with N = 32 in a dense melt, Reo, as the unit length scale. The simulation cell has a lateral extension L = 16Reo, and periodic boundary conditions are applied along the lateral directions, x and y. The grafting density is given by σ = n/L2. The grafting substrate at z = 0 is modeled as a hard, impenetrable wall that does not exhibit any preference for a polymer species. Polymers are irreversibly grafted onto the substrate with their first segment. Diblock copolymers are always grafted with the free end of the A-block. The height of the simulation box, Lz, is much larger than the brush height, and the upper surface is modeled as an impenetrable hard wall. A third-order equation of state approach is employed to account for the effective nonbonded interactions between the polymer segments, where the solvent is considered implicitly.18 In Table 1, four sets of virial coefficients are devised to mimic poly(acrylic acid)/polystyrene (PAA/PS) in DMF (solvent SA), MEK (solvent SB), and water (solvents SC and SD). DMF is a cosolvent for PAA and PS, and MEK is a good solvent for PS but a bad solvent for PAA. Water is a very bad solvent for PS, and the solubility of PAA in water can be adjusted by its pH value. Water is a good solvent for PAA at a high pH value (solvent SC), and it is a bad one at low pH value (solvent SD). More details about the model and method are given in the Supporting Information. The molecular conformations are generated by Smart-MonteCarlo moves in the fluctuating mean fields,44 which are updated from the instantaneous densities in a quasi-instantaneous way.40-42 A slowly quenching scheme is adopted to achieve equilibrium at low temperatures or strong segregation.17,43 In our numerical (43) Yin, Y. H.; Jiang, R.; Li, B. H.; Jin, Q. H.; Ding, D. T.; Shi, A.-C. J. Chem. Phys. 2008, 129, 154903. (44) Rossky, P. J.; Doll, J. D.; Friedman, H. L. J. Chem. Phys. 1978, 69, 4628.
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implementation, the instantaneous densities, FR(c), of species R at grid points c, are calculated from the conformation in the continuous space, {ri(s)}, FR ðcÞ ¼
Ni n X Reo 3 X Πðri ðsÞ, cÞ γR ði, sÞ NΔL3 i ¼1 s ¼1
ð1Þ
where the assignment function Π(r,c) = Πj=x,y,z[1 - (|rj - cj|/ΔL)] if the distance between the grid point c and the segment position r along each Cartesian direction is less than the grid spacing, ΔL, and Π(r, c) = 0 otherwise,45-47 and γR(i,s) = 1 if the sth segment on the ith polymer chain belongs to species R, and otherwise, γR(i,s) = 0. To analyze memory effects of the microphase-segregated morphology, we sum the density FR(x,y,z) = FR(c) along the z-direction to obtain a two-dimensional, lateral density map, FR(x,y). From these data, two memory measures are calculated for each individual component, A or B, or their sum. The fluctuation memory measure (fmm) is defined as the correlation between the density fluctuations of the distribution of grafting points and the morphology36 *
+
P
x, y ½Fi ðx, yÞ -F i ½F0 ðx, yÞ -F 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2P 2 x, y ½Fi ðx, yÞ -F i x, y ½F0 ðx, yÞ -F 0
fmm
ð2Þ i
where the average Æ...æi runs over all independent configurations generated from different random initial states with the identical distribution of grafting points. Fi(x,y) is the lateral profile of configuration i, and F0(x,y) is the grafting density convoluted with a Gaussian smoothing function of width 0.3Reo. F is the average of F over the x - y plane. The domain memory measure (dmm) is defined as the correlation between independent configurations quenched from initially disordered configurations with the same grafting points38,39 * dmm
P
x, y ½Fi ðx, yÞ -F i ½Fj ðx, yÞ -F j
+
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2P 2 ½F ðx, yÞ -F ½F ðx, yÞ -F i j x, y i x, y j i, j
ð3Þ
where the average Æ...æi,j is taken over all pairs (i,j) of independent configurations. Additionally, the structure factor, Sg(q), of composition fluctuations is calculated as the circular average of Sg(q), which is, in turn, defined as 1 Sg ðqÞ ¼ n P Nj
2 + * Nj X n X ½γA ðj, sÞ -γB ðj, sÞexp½iqrj j ¼1 s ¼1
ð4Þ
i
j ¼1
where the sum runs over all segments. In this work, 32 independent runs with the same grafting points are used to calculate memory measures and structure factors.
III. Diblock Copolymer A-B Brushes Since the diblock copolymer A-B brushes are grafted by the A-terminus, there are only density fluctuations in the density of grafting points but no composition fluctuations. To quantify how these density fluctuations are reflected in the microphase-separated (45) M€uller, M.; Pastorino, C. Europhys. Lett. 2008, 81, 28002. (46) M€uller, M.; Daoulas, K. C. J. Chem. Phys. 2008, 128, 024903. (47) Detcheverry, F. A.; Daoulas, K. Ch.; M€uller, M.; de Pablo, J. J. Macromolecules 2008, 41, 4989.
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Figure 1. Three typical grafting point configurations with different quenched density fluctuations, which are controlled by the minimal distance between any two grafting points, dmin. (A) dmin/ dhex = 0.05, (B) dmin/dhex = 0.45, and (C) dmin/dhex = 0.9, where dhex = (2/(31/2)σ)1/2 is the minimal distance in a perfect, hexagonal lattice. These configurations contain n = 1600 grafting points on a substrate area, L2 = (16Reo)2. The configurations of the grafting points have been generated by Monte Carlo movement of hard disks of diameter dmin on a plane.
Figure 3. (A) Typical morphologies of diblock copolymer A-B brushes exposed to solvent SA with σReo2 = 25 and f = 0.75 and different grafting point configurations. In the snapshots of block copolymer brushes shown here and hereafter, the red is for B-block and the yellow for A-block. (B) Structure factor of the microphasesegregated morphologies shown in panel (A).
Figure 2. Fluctuation memory measure, fmm, and domain memory measure, dmm, as a function of dmin for diblock copolymer A-B brushes of σReo2 = 25 and f = 0.75 in solvent SA. Isolated symbols are for brushes with grafting points arranged on a square lattice.
morphology of diblock copolymer brushes, we generate configurations of grafting points with different strengths of fluctuations. To generate a configuration of grafting points, we consider a system of hard disks with diameter, dmin, on a planar substrate. Configurations of grafting points are sampled by a Monte Carlo simulation using local, random displacements of the disks. The density fluctuations are dictated by the minimal distance between any two grafting points, dmin, and the density, σ. The parameter, dmin, can be thought of as the minimal distance between two initiator groups in a “grafting-from” scheme or the lateral spatial extent of an anchoring terminus in a “grafting-to” scheme. In the following, we will use it as a convenient parameter to control fluctuation effects. If the disks arrange on a perfect hexagonal lattice, their mutual distance is given by dmin = dhex (2/(31/2)σ)1/2. This is the highest grafting density that is possible. Since there are no defects, this regular configuration does not exhibit fluctuations in the density distribution on the length scale Reo . dmin. Any decrease in σdmin2, however, allows for density fluctuations. The smaller σdmin2 is, the larger are the density fluctuations in the grafting points. Some typical examples for σReo2 = 6.25 are depicted in Figure 1. The dependence of the memory measures in diblock copolymer A-B brushes in solvent SA on the density fluctuations of the grafting points, dmin, is shown in Figure 2. The fraction of the grafted A block is f = 0.75, and the grafting density is σReo2 = 25. In this figure, we also include data for grafting points that are regularly arranged on a square lattice with dmin| = 1/σ1/2. They 1294 DOI: 10.1021/la902438e
are marked by the unconnected points at dmin/dhex = ((31/2)/2)1/2 ≈ 0.93. In this case, the fluctuation memory is vanishingly small because grafting points do not favor specific positions of domains; that is, all lateral surface patterns are equivalent irrespective of orientation and position. Therefore, no fluctuation memory is expected. Both the fluctuation memory of the A-component and that of the B-component grow as we decrease dmin and thereby increase the density fluctuations of the grafting points. These density fluctuations of the grafting point distribution break the translation symmetry of the substrate and differentiate between microphase-separated morphologies with different domain positions and orientations. Morphologies of lower free energy are selected and observed with a higher probability. The fluctuation memory measures of A-blocks are positive, and those of B-blocks are negative. This observation can be rationalized by the fact that diblock copolymer chains are tethered by the ends of A-blocks. Therefore, A-blocks tend to aggregate in areas with higher grafting density, while B-blocks have a propensity to fill the remaining space. Such an arrangement undoubtedly decreases the entropic penalty of stretching. The fluctuation memory grows with an increase of the fluctuations of the grafting point density. The strong fluctuation memory is also reflected in the domain memory measure, that is, the correlation among multiple surface patterns of a brush generated from independent disordered starting configurations with the same distributionof grafting points. This scheme mimics the limiting case where the morphology is completely erased during a cyclic exchange of solvent. A large fluctuation memory measure indicates that the same grafting point configuration leads to similar surface patterns in independent runs, and therefore, these systems also exhibit strong domain memory measures. Figure 2 demonstrates that the domain memory measure increases with a decrease in dmin/dhex. Both A-domains and B-domains have similar memory effects, but their sum, the total density ρ = FA þ FB, exhibits much weaker memory effects. The latter indicates that these surface patterns exhibit only little topographical heterogeneity. Figure 3A shows top views of the morphologies for block copolymer brushes with f = 0.75 and σReo2 = 25, which are Langmuir 2010, 26(2), 1291–1303
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Figure 4. Fluctuation memory measure and domain memory measure as a function of dmin for diblock copolymer A-B brushes with σReo2 = 6.25 and f = 0.75 in solvents SC and SD. Unconnected symbols mark the properties of brushes in solvent SC with grafting points that are arranged on a square lattice.
Figure 5. Configuration snapshots of morphologies of diblock copolymer A-B brushes exposed to solvents SC and SD, with σReo2 = 6.25, f = 0.75, and different values of dmin.
exposed to solvent SA but assemble on different patterns of grafting point. Panel (B) presents the corresponding two-dimensional, circularly averaged structure factors, Sg(q). The structure factor of copolymer brushes with grafting points that are arranged on a square lattice exhibits a pronounced peak. From its location, qmax, a dominant length can be determined as Λ 2π/qmax ≈ 2Reo. The snapshots of the morphologies on different patterns of grafting points in Figure 3A resemble each other, and they are also characterized by similar structure factors. A closer inspection reveals that a decrease in dmin slightly shifts the location, qmax, of the peak of the structure factor toward lower values and the peak also broadens. Thus, the domains increase in size and their extension becomes less uniform upon increasing fluctuations in the grafting point density. Figure 4 presents the memory measures as a function of dmin for block copolymer brushes in solvents SC and SD. The asymmetry of the diblock is f = 0.75 and σReo2 = 6.25. Some typical morphologies are shown in Figure 5. Although the morphologies that form in solvents SC and SD are very different from those of brushes exposed to SA, the fluctuation memory measure and domain memory measure increase with the fluctuations in the grafting density, and they will nearly vanish if grafting points are arranged on a regular, square lattice. This observation corroborates that the fluctuations in grafting points are the origin of memory effects in morphology independently of the solvent. The fluctuation memory and the domain memory are closely Langmuir 2010, 26(2), 1291–1303
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Figure 6. Comparison of memory measures of diblock copolymer A-B brushes with σReo2 = 6.25 and f = 0.75 in various solvents. All brushes have the same configuration of grafting points with dmin/dhex = 0.05.
correlated, and both arise from the broken translational symmetry of distribution of grafting points. If the grafting point configurations are completely homogeneous (a limit that selfconsistent field calculations often consider), we will not expect memory effects. If there are fluctuations in the distribution of grafting points, solvent quality and selectivity, grafting density, and the fraction of A block copolymer will affect the strength of memory effects of diblock copolymer brushes. A comparison between Figures 2 and 4 shows that the same diblock copolymer brushes exposed to different solvents may have very different memory measures. In the following, we will discuss these effects in detail. In Figure 6, we compare the memory measures for A-B diblock copolymer brushes in various solvents using f = 0.75 and σReo2 = 6.25 throughout. All these brushes have the same configuration of grafting points with dmin/dhex = 0.05. The morphologies and their memory effects are visualized in Figure 7, which presents a comparison between snapshots obtained from independent runs with the same grafting point configurations exposed to different solvents. As previously reported, the morphology of diblock copolymer A-B brushes strongly depends on the solvent.18 Solvent SA is a marginal cosolvent for both blocks. The slight swelling of both blocks mitigates the immiscibility between A and B. Since the grafting density is low, no pronounced structure formation is observed. The lateral inhomogeneities mainly consist of density fluctuations of the grafted block, A. Thus, the grafted A-blocks have small memory measure and the B-blocks exhibit even lower memory effects. Solvent SB is marginally good for B-blocks but bad for A blocks. Therefore, the grafted A-blocks collapse and form dense domains at the substrate. These domains are shielded from the solvent by swollen B-blocks. Due to the low grafting density, the collapsed A-domains do not form a complete layer at the substrate but a laterally heterogeneous structure (perforated layer) emerges similar to a one-component brush in a bad solvent.36 Since the holes in the A-layer preferentially form at locations where the grafting density is low, the A-component has a notable memory measure. Memory effects for the swollen B-component are small. Solvent SC is a very good solvent for the grafted block, A, but bad for B-blocks. Therefore, B-blocks collapse and form small clusters (micelles) floating in a matrix of highly swollen A-blocks. This structure is similar to tethered micelles.14 Since the highly swollen A-blocks have more conformational freedom, both components exhibit negligible memory effects. Solvent SD is a bad solvent for both blocks, A and B. In this solvent, the whole diblock copolymer brush acts like DOI: 10.1021/la902438e
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Figure 7. Snapshots obtained from independent simulations of diblock copolymer A-B brushes with σReo2 = 6.25, f = 0.75, and dmin/dhex = 0.05 in various solvents, visualizing the dependence of memory effects on the solvent.
a one-component brush. It collapses and forms a dense perforated layer at the grafting substrate. Therefore, the morphology shows strong memory effects with voids tending to form at identical positions. In this morphology, the top B-block forms some dispersed micelles on the surface of A-domains because of the immiscibility of the different components. Since their positions are tied to the location of the A-domains, these micellar domains also bear a strong memory albeit it is somewhat smaller than the memory of the grafted component. These examples illustrate the rich behavior that can be found in the rather large parameter space spanned by solvent quality, grafting density, composition, and molecular architectures. While this is only a limited selection, some general trends can be inferred: (a) In all cases, the fluctuation memory measure and the domain memory measure are strongly correlated. (b) The grafted block exhibits stronger memory effects those of the top block. (c) Good solvents give polymers more conformational freedom and decrease the influence of the grafting constraint. Thus, brushes in good solvents show weaker memory effects than brushes in bad solvent. (d) The solvent quality of the grafted block affects the memory measures stronger than the solvent quality of top block. Memory effects not only depend on the solvent quality but they also vary with composition, f, or grafting density, σReo2. Figure 8 illustrates the f-dependence of the memory measures in solvent SA. The copolymer brushes are characterized by σReo2 = 25 and dmin = 0.1dhex. The subsequent Figure 9 shows the morphologies 1296 DOI: 10.1021/la902438e
Figure 8. Memory measures as a function of the fraction of the grafted A-block, f, for block copolymer brushes exposed to solvent SA with σReo2 = 25 and dmin/dhex = 0.1.
of copolymer brushes with different values of composition, f. Memory effects are most pronounced around composition, f = 0.7. If the A-block is short, f = 0.375, the block copolymer brushes will exhibit perpendicular segregation; that is, a sandwichlike structure will form with a uniform layer of the grafted A-block at the substrate and a top layer comprising the B-component. This laterally uniform structure shows very weak domain memory. Langmuir 2010, 26(2), 1291–1303
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Figure 9. Typical morphologies for block copolymer brushes of different fractions of A-block, f, exposed to solvent SA with σReo2 = 25 and dmin/dhex = 0.1.
Figure 10. Memory measures as a function of fraction of A-block, f, for diblock copolymer A-B brushes exposed to solvent SC with σReo2 = 6.25 and dmin/dhex = 0.05.
Increasing f, we observe that the interface between the two layers, which runs parallel to the substrate, becomes undulated. The locations of these undulations correlate with the fluctuation of the grafting point density. Since the total height of the brush remains largely unaffected, the fluctuation memory measures of the grafted A-block is positive, the fmm of the top block, B, is negative, and the total density exhibits a much weaker fmm. As we increase f further toward f ≈ 0.7, the undulations of the interface grow stronger and develop into a lateral pattern with an indication of hexagonal arrangement. This morphology exhibits large memory effects. For even larger compositions, f, the top layer of B-blocks breaks up and the lateral structure washes out. This leads to a sharp decline of memory effects for f J 0.75. Figure 10 presents the behavior of diblock copolymer brushes of different compositions in solvent SC. The grafting density, σReo2 = 6.25, and its fluctuations, dmin = 0.05dhex, are kept constant. Morphologies and memory effects of independent runs with the identical distributions of grafting points are shown by the snapshots in Figure 11. Increasing f, we observe that the strength of memory effects decreases and, around f ≈ 0.6, there is a sharp drop in memory measures. Our previous study has shown that around this composition a gradual crossover occurs between a collapsed, perforated B-layer or pinned B-domains, which are located close to the grafting substrate, and depinned micelles with B-cores, which float in the matrix of highly swollen A-blocks.18 For f = 0.25 and f = 0.5, the grafted A-blocks are short, and therefore, the grafting constraint limits the available space for the collapsed B-domains. Thus, the B-domains collapse onto the substrate to avoid the unfavorable contacts with the solvent, SB. B-domains preferentially form in regions of high grafting density Langmuir 2010, 26(2), 1291–1303
to avoid excessive stretching of the grafted A-blocks. Moreover, the short A-blocks can be incorporated into the B-domains to facilitate the aggregation of B-blocks. Thus, in contrast to the previous cases, the fluctuation memory measure, fmm, for the nongrafted B-component is notably large and positive, indicating the collapsed B-domains are formed in regions of high grafting density. At larger values of f, the tethered A blocks are long and the collapsed B-domains resemble tethered micelles that are free to float in swollen A-blocks. In this case, both components exhibit neglectable memory effects and the fmm of the B-component is negative, quantifying that the B-cores of the micelles have a slight preference for locations of small grafting densities of A-blocks. The grafting density, σ, also affects memory effects. Generally, one expects that higher grafting densities or larger values of dmin reduce the relative amplitude of the quenched density fluctuations of the grafting points on the length scale Reo and, by consequence, lead to smaller domain memory measures. This simple argument, however, assumes that coupling between the quenched fluctuations of the grafting points and the morphology of the microphase-separated brush is not affected by varying σReo2. Our previous study of the morphologies of diblock copolymer brushes has demonstrated, however, that the structure of the brush does depend on the grafting density.18 At very low grafting densities, the substrate is not completely covered by tethered polymers and the lateral heterogeneity does not stem from composition fluctuations of a microphase-separated morphology but merely from density fluctuations. This effect is particularly pronounced in bad solvents. In this case, the memory effects simply arise because tethered polymers tend to be close to their grafting points. This behavior is found, for example, in a diblock copolymer brush with f = 0.75 and σReo2 = 6.25 in solvent SA (cf. Figure 7A). Although there is no overt microphase segregation between A and B, the brush exhibits some memory effects. At higher grafting densities, the lateral heterogeneity mainly arises from microphase segregation. The higher the grafting density is, the larger is the density inside the brush and the larger is the incompatibility. Thus, larger grafting densities in our model of a compressible polymer brush give rise to stronger microphase separation. This microphase segregation couples to the quenched fluctuations of the density distribution of the grafting points, and this coupling, in turn, gives rise to the memory effect. Diblock copolymer brushes of composition f = 0.75 and grafting density σReo2 = 25 (see Figure 2) are strongly microphase-segregated and exhibit larger memory effects than systems with sparsely grafted diblocks, σReo2 = 6.25. This observation illustrates that the morphology of the diblock copolymer brush (the spatial arrangement of the domains as well DOI: 10.1021/la902438e
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Figure 11. Top view on morphologies for block copolymer brushes of different fractions of A-block, f, exposed to solvent SC with σReo2 = 6.25 and dmin/dhex = 0.05.
Figure 12. Memory measures of binary A/B mixed brushes in different solvents, with NA = NB = 32, σReo2 = 6.25, and f = 0.75. All the brushes have the same random configuration of grafting points with dmin/dhex = 0.05.
as the strength of segregation) strongly affects the coupling between the quenched distribution of the grafting points and the morphology and, therefore, memory effects. Thus, the changes of morphologies in response to external stimuli often go along with pronounced changes in memory effects as the example in Figure 10 illustrates.
IV. Mixed A/B Brushes Figure 12 presents the memory measures of mixed A/B brushes with composition f = 0.75 and low grafting density σReo2 = 6.25 exposed to different solvents. The grafting point configuration has been randomly generated with the constraint that the distance between grafting points be larger than dmin/dhex = 0.05 independent of the type, A or B. Typical configurations of the grafting points are presented in Figure 1, and in the case of mixed A/B homopolymer brushes we randomly grow A- or B-polymers from these locations. Therefore, the distribution of grafting points exhibits both density fluctuations as well as composition fluctuations. The grafting point configurations for the data in Figure 12 are identical for all solvents. The microphase-separated morphologies in Figure 13 reveal that the mixed brushes show lateral segregation in solvents SA, SB, and SD. In cosolvent SA, the domains are less segregated, while in the solvent SD, which is bad for both components, the segregation is larger and both components collapse into dense domains. Solvent SC differs from the other solvents due to the 1298 DOI: 10.1021/la902438e
large difference in solvent selectivity between the two components. The A-component is highly swollen, while the B-component collapses onto the substrate in form of dense clusters. This explains the large memory effects of the B-component and the significantly smaller value of the A-component in solvent SC, while in all other solvents, SA, SB, and SD, both components of the mixed brush exhibit comparable fluctuation and domain memory effects. We also note that, due to the very low grafting density σReo2 = 6.25, the brush in the bad solvent SD cannot uniformly cover the substrate but in addition to the lateral microphase separation between A- and B-components there are also pronounced spatial variations of the total density. The four snapshots in the lower row of Figure 13 illustrate that these “holes” tend to form at the same locations, yielding a rather large value of the memory measure of the total density, ρ = FA þ FB. These domain memory effects for the individual components and the total density of the mixed brush of f = 0.75 in solvent SD are studied as a function of the grafting density, σReo2, in Figure 14, and the corresponding morphologies and structure factors are shown in Figure 15. In the range of grafting densities considered, the domain memory measure of the individual components, A and B, increases with σReo2 at small grafting densities and levels off toward a constant value around σReo2 = 50. This behavior differs from previous studies of memory effects in binary melt brushes.39 The latter study utilized a significantly larger grafting density, 64 e σReo2 e 256, and, importantly, the melt has been assumed to be incompressible; that is, the density is independent of the grafting density. In our present model with implicit solvent, the sum of the densities of polymer and solvent is assumed to be incompressible but the polymer density itself varies in space and depends on the state of the system, that is, grafting density, solvent quality, and so on. The increase in the grafting density, σReo2, has two counteracting effects: (a) The relative composition fluctuations, Δjg, in the vicinity of the substrate decrease according to Δjg ∼ 1/(σReo2)1/2.39 This is the only effect present in a melt brush. (b) In a compressible system, the density of the brush, ρ, increases with σReo2, and the segregation, χ ∼ υABρ, increases in turn. These effects make the morphology of the brush more susceptible to fluctuations of the grafting points. A composition fluctuation, Δjg(q), at wavevector, q ≈ 2π/Reo, creates a spatially varying surface field of amplitude χΔjg(q). Within linear response, one expects that the change of the brush morphology is of the order S~g(q)χ Δjg(q). Langmuir 2010, 26(2), 1291–1303
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Figure 13. Snapshots obtained from independent simulations of binary mixed A/B brushes with σReo2 = 6.25, f = 0.75, and dmin/dhex = 0.05 in various solvents, visualizing the memory effects for mixed brushes in different solvents. In the snapshots of mixed brushes shown here and hereafter, the blue is for B-blocks and the yellow for A-blocks.
Figure 14. Dependence of domain memory measure on the grafting density for mixed brushes of f = 0.75 and dmin/d/hex = 0.1 exposed to solvent SD. d/hex/Reo = (2/25(3)1/2)1/2.
S~g(q) describes the amplification of composition fluctuations due to microphase separation in the brush. We expect that S~g(q) qualitatively resembles the structure factor of composition fluctuations, Sg(q). In a melt brush, incompatibility, χ, and morphology, Sg(q), are independent of grafting density. In our compressible system, however, χ ∼ υAB increases with σR2 and the observation that the microphase segregation becomes more pronounced with σReo2 suggests that also Sg(q) increases with σR2. The increase of the susceptibility, Sg(q), is shown in Figure 15B. For the grafting densities considered in Figure 14, Langmuir 2010, 26(2), 1291–1303
Figure 15. (A) Configuration snapshots of morphologies of mixed
A/B brushes exposed to solvents SD, with f = 0.75, dmin/d/hex = 0.1, / and different values of σReo2. dhex /Reo = (2/25(3)1/2)1/2. (B) Structure factor of the microphase-segregated morphologies shown in panel (A). The inset is the structure factors for the bottom layer of the microphase-segregated brushes, with a depth, δL, whose values are indicated in the legend. DOI: 10.1021/la902438e
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Figure 16. (A) Regularly arranged configurations of grafting points of high symmetry for σReo2 = 6.25 and different f. (B) Illustration of dmin-controlled composition fluctuation of the grafting points on a square lattice of constant a for f = 0.75 and σReo2 = 6.25.
the concomitant increase of the susceptibility of the mixed brush with respect to quenched fluctuations of the grafting points outweighs the reduction of the strength of the quenched fluctuations upon increasing the grafting density. In contrast to the behavior of the individual components, when decreasing σReo2, the domain memory measure of the total density first decreases at small values of σReo2 and then increases slowly at large values of σReo2. Density fluctuations of the grafting points on the length scale Reo decrease with σReo2. At a low grafting density, for example, σReo2 = 6.25, the lateral inhomogeneity in the total density of the brush, ρ = FA þ FB, is obvious due to the existence of some voids. Increasing grafting density, we observe that such lateral inhomogeneity of the total density of the brush arising from grafting points decreases. Therefore, the domain memory measure of the total density decreases when we increase σReo2 from 6.25 and 12.5. At higher values of grafting density, effects of enhanced segregation, χ ∼ υΑBρ, and solvent interaction become dominant, which leads to a slow increase in the domain memory measure of the total density. For many parameters, density and composition fluctuations in the grafting points are coupled and, by the same token, both types of quenched fluctuations of the grafting points influence the memory measures. One advantage of computer simulations consists of the opportunity of investigating the role of the two types of quenched fluctuations separately. First, we consider the effect of composition fluctuation in grafting points. In order to suppress density fluctuations of the grafting points, we arrange them on a two-dimensional, square lattice with lattice constant a = 1/σ1/2. Then each grafting point is assigned a type, A or B. By controlling the minimum distance, dmin, between two nearest grafting points of the minority species, we adjust the strength of composition fluctuations. As illustrated in Figure 16A, dmin/a = 21/2 for f = 0.5, dmin/a = 2 for f = 0.75, and dmin/a = 2(2)1/2 for f = 0.875 results in a regular arrangement of grafting points with σReo2 = 6.25. In this case, there are neither density nor composition fluctuations on the scale Reo or larger. Decreasing dmin, we introduce composition fluctuations. An example of a configuration of grafting points for σReo2 = 6.25 and f = 0.875 is depicted in Figure 16B, which clearly shows thatsmall values of dmin result in large composition fluctuation. Using these idealized configurations of grafting points, we investigate the effect of composition fluctuations on the memory 1300 DOI: 10.1021/la902438e
Figure 17. Memory measures as a function of composition fluctuation in grafting points, dmin, for binary mixed brushes A/B exposed to solvent SA, with σR2eo = 25 and different f.
of binary A/B mixed brushes. The results for mixed brushes in solvent SA are shown in Figure 17: Both memory measures, fmm and dmm, monotonically decrease with increasing dmin. For regular arrangements of grafting points with no quenched composition fluctuations, both fmm and dmm nearly vanish. In Figure 17, we also observe that the dependence of memory measures on composition, f, depends on two aspects: On the one hand, mixed brushes with more symmetric f and identical dmin exhibit smaller quenched composition fluctuations of grafting point, which tends to decrease memory effects toward more symmetric f. On the other hand, mixed brushes of more asymmetric composition (but identical dmin and incompatibility) are less segregated, and thus, quenched fluctuations are less amplified by the microphase separation inside the mixed brush. This effect results in a decrease of memory effects toward more asymmetric compositions. The combination of these two counteracting aspects leads to a nonmonotonic dependence of the memory measures on composition. At small dmin, for example, dmin/a = 1, the latter effect is dominant; that is, the memory measures of mixed brushes decrease when the composition f becomes asymmetric (increasing f from 0.5 to 0.875 in Figure 17). At a relative large dmin, for example, dmin/a = 21/2, the former effect dominates resulting in decrease of the memory measures as we change f from 0.875 to 0.75. Langmuir 2010, 26(2), 1291–1303
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Figure 19. Memory measures as a function of composition fluctuation in grafting points, dmin, for mixed brushes of f = 0.75 exposed to solvent SA, with different grafting density.
Figure 18. (A) Independent snapshots of mixed A/B brushes in solvent SA with σReo2 = 25 and f = 0.5, showing domain memory effects for grafting point patterns with different composition fluctuations. (B) Structure factor, Sg(q), of the microphase-segregated morphologies shown in (A), along with those of Y-shaped mixed brushes.
Figure 18 illustrates the difference between mixed brushes with different strengths of quenched composition fluctuation of grafting points. Panel (A) depicts four configuration snapshots with dmin/a = 21/2 and dmin/a = 1. The brushes are symmetric, f = 0.5, and they are exposed to solvent SA. The grafting density is σReo2 = 25. By virtue of the symmetric composition, these brushes form ripple structures. The brushes with the regular arrangement of grafting points (dmin/a = 21/2, checkerboard) not only exhibit much smaller memory effects, that is, there are little similarities in the position and orientation of the domains, but also form more well-defined structures than the brushes with quenched composition fluctuations in the grafting points, dmin/a = 1. Moreover, ripples in the former systems tend to align parallel and form a fingerprint-like pattern similar to what is observed in thin films of diblock copolymers. The corresponding structure factors, Sg(q), are presented in panel (B) of Figure 18. These data quantify the difference due to quenched composition fluctuations. The peak height of Sg(q) is slightly larger, and the peak is more narrow for the brushes with dmin/a = 21/2 than for the system with quenched composition fluctuations, dmin/a = 1. The interplay between quenched composition fluctuations and the grafting density on the memory of mixed brushes in solvent SA is presented in Figure 19. At the same grafting density, memory effects decrease with the strength of quenched composition fluctuations. As discussed before for solvent SD, the domain memory measure of the individual components increases as we increase the grafting density from σReo2 = 6.25 to 25, indicating that the enhanced immiscibility at the higher grafting density increases the susceptibility of the microphase-separated morphology with respect to the quenched composition fluctuations of the grafting points. Since SA is a cosolvent for both components, we expect compressibility effects to be much stronger than those in Langmuir 2010, 26(2), 1291–1303
Figure 20. Memory measures as a function of density fluctuation of grafting points, dmin, for Y-shaped mixed brushes with f = 0.5 and different values of grafting density, exposed to different solvents.
solvent SD, which is investigated in Figure 14. Similar to the case of solvent SD, we also observe that domain memory effects of the total density decrease with σReo2. The fluctuation memory measure exhibits a much weaker dependence on the grafting density, σReo2, and, for small quenched fluctuations of the composition of the grafting points, it even slightly decreases with σReo2. This observation suggests that there is a rather subtle balance between the increase of the susceptibility of the microphase-separated morphology with respect to quenched composition fluctuations and the decrease of the amplitude of those quenched fluctuations as one increases σReo2. The effect of quenched density fluctuations in the grafting points can be isolated by investigating Y-shaped mixed brushes.39,43 These systems have been studied experimentally12,13 by grafting diblock copolymers at the junction point between the two blocks. Moreover, to graft triblock copolymers via their short middle block is another experimentally accessible way to obtain Y-shaped mixed brushes. This technique eliminates composition fluctuations of the grafting points because two chains of different type emerge from each grafting point. Density fluctuations of the grafting points, however, are still present, and, as in the case of diblock copolymer brushes, their strength can be adjusted by the minimal distance between the grafting points. Figure 20 presents the two memory measures as a function of dmin for Y-shaped mixed polymer brushes exposed to different solvents. Compared with conventionally grafted mixed polymer, Y-shaped mixed brushes are characterized by small memory measures, although they have similar surface morphology.43 DOI: 10.1021/la902438e
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Figure 21. Independent snapshots of binary mixed A/B Y-shaped brushes with σReo2 = 12.5 and dmin/dhex = 0.9 and 0.1 in solvents SC, showing the different morphologies and memory effects as a function of the strength of the quenched density fluctuations of the grafting points.
In all cases, the domain memory measure and the fluctuation memory measure decrease with increasing dmin, indicating that memory effects arise from the quenched density fluctuation of the grafting points. Especially in solvents SA and SB, memory effects are particularly small. Panel (B) of Figure 18 also includes data of the structure factor of Y-shaped mixed brushes in solvent SA with f = 0.5 and σReo2 = 25. The structure factors of Y-shaped binary brushes with different minimal distances, dmin, between the joined grafting points nicely agree with previous results for the regular checkerboard arrangement of the grafting points, dmin/a = 21/2. Even the structure factor of Y-shaped brushes with large density fluctuations, dmin/dhex = 0.1, has only a slightly lower peak height at lower qmax and a slightly broader width of the peak. This result indicates that quenched composition fluctuations play a dominant role for the morphology but density fluctuations are only of minor importance for the structure in mixed brushes exposed to solvent SA. Thus, these systems are promising candidates for fabricating ordered structures. Memory effects in the other two solvents, SC and SD, are significantly larger. In the strongly selective solvent SC, density and composition fluctuations are strongly coupled because the A-component is highly swollen and the B-component collapsed into dense domains. The snapshots in the first two rows of Figure 21 depict morphologies of Y-shaped mixed brushes with different quenched density fluctuation. If quenched fluctuations of the density of grafting points are small, dmin/dhex = 0.9, B-polymers will collapse onto the substrate and will form a uniform layer. This morphology is characterized by very small memory effects. In the case of pronounced quenched density fluctuations, dmin/dhex = 0.1, however, a perforated layer forms and the location of the “holes” correlates with the low density regions of the grafting points, and they appear at similar locations in the four snapshots in the row. The behavior is reminiscent of a one-component brush in a bad solvent and gives rise to large memory effects. In the bad solvent, SD, we also observe surprisingly large memory effects due to quenched density fluctuations. The coupling of quenched density fluctuations to the microphase-separated morphology inside the brush may be partially attributed to the density difference inside the segregated domains of the A- and the B-component of our compressible model. The marked contrast with the small memory effects in solvents SA and SB suggests that the morphology in solvent SD is more susceptible to density 1302 DOI: 10.1021/la902438e
fluctuations (or changes of density) than the brush in the other two solvents. This observation is in accord with the observed increase of the memory effects as we increase grafting density because increasing the grafting density slightly increases the density inside the brush and enhances segregation.
V. Conclusions Using Single-Chain-in-Mean-Field simulations, we have studied memory effects of the microphase-separated morphologies of multicomponent polymer brushes (MCPBs). Both diblock copolymer A-B brushes and binary mixed A/B brushes at different compositions and grafting densities and immersed into different solvents have been considered. Correlations between fluctuations of the distribution of grafting points and morphology, the fluctuation memory measure fmm, and correlations between morphologies that independently self-assemble on the same distribution of grafting points, the domain memory measure dmm, have been calculated. Both memory effects, fmm and dmm, stem from the quenched fluctuations in grafting points, which break the translational and orientational symmetry of the system and therefore favor specific arrangements of domains. The strength of the memory effects is dicated by two factors: (a) the type and strength of the quenched fluctuations of the grafting points and (b) the susceptibility of the microphase-separated morphology to amplify those fluctuations at the substrate. Two types of quenched fluctuations can be distinguished: density and composition fluctuations. The latter only occur in mixed homopolymer brushes but not in copolymer brushes if the molecules are all grafted with the same block. In the mixed homopolymer brush, composition fluctuations can be suppressed by using Y-shaped brushes, where two molecules of different types emerge from the same grafting point.12,13,43 Generally, the amplitude of quenched fluctuations decreases with grafting density and with the dominant lateral length scale because a domain comprises a larger number of polymers. The susceptibility of the microphase-separated morphology, which amplifies those quenched fluctuations of the grafting points, can become large because already very small quenched fluctuations may result in significant memory effects.36 This factor, however, strongly depends on the details of the system. For instance, the susceptibility of mixed polymer brushes in a common good solvent with respect to Langmuir 2010, 26(2), 1291–1303
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quenched density fluctuations is very small but it is quite significant with respect to composition fluctuations. In strongly selective solvents, SC, or in bad solvents, SD, where both components differ in their density, however, density and composition fluctuations are coupled and, in this case, quenched density fluctuations will also give rise to strong memory effects. Typically, the susceptibility is larger in mixed brushes than in copolymer brushes and it is larger in bad or selective solvents than it is in good solvents. Since in our compressible system the grafting density affects the morphology in the vicinity of the grafting surface, that is, the density and the incompatibility increase and these effects give rise to stronger segregation. An increase of the grafting density also increases the susceptibility of the system with respect to quenched fluctuations of the grafting points. For rather small grafting densities, we observe that this effect can outweigh the reduction of the amplitude of the quenched fluctuations of the grafting points upon increasing σReo2. This behavior is in contrast to the observation in incompressible mixed brushes at large grafting densities,39 where the near-substrate morphology is rather independent of the grafting density. In this case, the susceptibility does not increase strongly with σReo2 and the memory effects decrease like the amplitude of the quenched fluctuations, 1/(σReo2)1/2. Upon increasing the chain length, we expect that memory effects decrease. According to the parametrization of our soft, coarse-grained model, the number of interaction centers, N, along a chain does not strongly affect our simulation results. If one increases the chain length in an experiment at fixed grafting density, σ, one will increase both the reduced grafting density,
σReo2 ∼ N, and the scaled strength of nonbonded interactions, υReo3/N ∼ N1/2. The repulsion, χ, between chemical monomeric repeat units of different species, however, will remain unaltered. Our simple estimate, fmm or dmm ∼ SgχΔjg ∼ Sg/(σReo2)1/2, suggests that the memory measure will decrease as σReo2 ∼ N increases, provided the susceptibility of the bottom layer, Sg, remains approximately constant. For compressible brushes, this limiting behavior may only be observed for very long chains or high grafting densities. Fluctuation memory is one of reasons for the loss of long-range order in the morphologies. To achieve an improved long-range order, it is necessary to minimize the fluctuation in grafting points. To this end, new synthesis techniques have to be devised. Y-shaped mixed brushes12,13 can decrease the memory effectively by suppressing quenched composition fluctuation of the grafting points. Another promising method is to prepare the brushes with regular grafting points by crystallizing a crystalline end block of a block copolymer. By this way, diblock copolymer brushes with uniform grafting points were prepared by crystallizing linear triblock copolymer.48,49 Even in conventional brushes with larger fluctuations of the distribution of grafting points, shear or confinement could be exploited to improve long-range order.
(48) Xiong, H.; Zheng, J. X.; Van Horn, R. M.; Jeong, K.-U.; Quirk, R. P.; Lotz, B.; Thomas, E. L.; Brittain, W. J.; Cheng, S. Z. D. Polymer 2007, 48, 3732. (49) Huang, W. H.; Luo, C. X.; Zhang, J. L.; Yu, K.; Han, Y. C. Macromolecules 2007, 40, 8022.
Supporting Information Available: Details of the model and simulation methods. This material is available free of charge via the Internet at http://pubs.acs.org.
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Acknowledgment. It is a great pleasure to thank S. Santer, K. Hinrichs, I. Luzinov, S. Minko, and M. Stamm for stimulating discussions. Financial support by the DFG-NSF Materials World Network program under Grant Mu 1674/4 and EU-INFLUS is gratefully acknowledged. Ample computing time has been provided by the GWDG G€ ottingen, the HLRN Hannover, and the JSC J€ulich.
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