Microphase Separation of Mixed Polymer Brushes: Dependence of the

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Microphase Separation of Mixed Polymer Brushes: Dependence of the Morphology on Grafting Density, Composition, Chain-Length Asymmetry, Solvent Quality, and Selectivity Jiafang Wang*,†,‡ and Marcus Mu¨ller† Institut fu¨r Theoretische Physik, Georg-August UniVersita¨t, D-37077 Go¨ttingen, 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: April 6, 2009; ReVised Manuscript ReceiVed: June 22, 2009

Microphase separation of binary mixed A/B polymer brushes exposed to different solvents is studied using Single-Chain-in-Mean-Field simulations. Effects of solvent quality and selectivity, grafting density, composition, and chain-length asymmetry are systematically investigated, and diagrams of morphologies in various solvents are constructed as a function of grafting density and composition or chain-length asymmetry. The structure of the microphase segregated morphologies lacks long-range periodic order, and it is analyzed quantitatively using Minkowski measures. I. Introduction Polymer brushes refer to a collection of macromolecules grafted onto a substrate, and they have been extensively used in applications such as colloidal stabilization, drug delivery, surface coating, wetting, and adhesion.1-3 Recently, multicomponent polymer brushes consisting of two or more incompatible polymers have attracted continuing interest as responsive surface coatings due to their ability to switch surface properties and morphologies in response to changes in their environment. These smart materials can consist of block copolymers or a mixture of homopolymers. Immiscible, multicomponent polymer systems exhibit phase separation to decrease unfavorable contacts among immiscible components. Because of the immobility of grafting points, however, macroscopic separation is impossible in multicomponent brushes, and instead, microphase separation leads to a domain structure on the nanometer scale. The structure can be modulated by changing molecular parameters of the brush (e.g., chemical nature of monomers, chain architecture, molecular weight, and grafting density) and external stimuli (temperature, substrate interaction, solvent, etc). Among all these factors, solvent selectivity is an important and experimentally accessible manner to reversibly control surface properties. Experiments have demonstrated that by changing solvent selectivity, the surface of polymer brushes can be switched from hydrophobic to hydrophilic and vice versa.4-6 Such a switching of the surface morphology and properties can be repeated in a cyclic way by exchanging the solvent.7 For some applications, for example, wetting and adhesion, only the gross composition of the uppermost layer is important, but for applications such as templates for nanoscale structures, long-range order of the microphase-separated structures matters. Unlike thin films of block copolymers, however, the microphaseseparated morphologies of multicomponent polymer brushes lack long-range order, an effect that has been attributed to the pinning of domains arising from fluctuations of the density and composition of the grafting points.8,9 These fluctuations in the grafting points are strongly amplified by the structure formation * Corresponding author. E-mail: [email protected]. † Georg-August Universität. ‡ Chinese Academy of Sciences.

in the multicomponent polymer materials. Thus, during cyclic exposure of the brush to different solvents, microphase-separated domains exhibit a strong tendency to form at the same location, an effect termed domain memory measure. Recently, experiments demonstrated that this domain memory effect strongly (anti)correlates to the ability of multicomponent brushes to transport nano-objects.9-13 In mixed polymer brushes, there are two limiting scenarios for microphase separation: (i) One component may segregate to the top of the brush while the other component is preferentially located at the grafting surface. This perpendicular segregation yields a sandwichlike structure. (ii) Alternatively, lateral segregation can occur, in which the size of the domains is dictated by the lateral coil size of grafted chains, but it is also influenced by the quenched fluctuations of the grafting points. Theoretical studies and computer simulations have been performed to investigate the interplay between lateral and perpendicular segregation in binary polymer brushes under various conditions. The first theoretical approach based on the selfconsistent field theory has been devised by Marko and Witten.14,15 They studied the onset of instability from a disordered phase in a symmetric mixed polymer brush, employing a strongstretching but weak-immiscibility approximation. In this limit, density and composition of the brush decouple and the influence of segregation on the classical conformations of grafted chains can be neglected. They predicted that an increase in incompatibility leads to a second-order phase transition from a disordered to a laterally segregated, ripple phase. Early simulations indicated that perpendicular layering and lateral segregation may occur in mixed polymer brushes in contact with nonselective solvents.16-21 These simulation studies, however, were limited to rather small system sizes and the equilibration of the morphology in the simulations is difficult. Using scaling arguments, Zhulina and Balazs considered the lateral microphase segregation of tethered Y-shaped A/B copolymer brushes in a nonselective poor solvent and constructed a phase diagram for symmetric conditions as a function of grafting density and incompatibility between the species.22 At high grafting densities, there is a transition from a homogeneous, mixed structure at low incompatibility to a laterally segregated, periodic structure at higher incompatibility. At low

10.1021/jp903161j CCC: $40.75  2009 American Chemical Society Published on Web 07/28/2009

Microphase Separation of Mixed Polymer Brushes grafting densities, the brush loses its lateral homogeneity and splits into separate coils in good or θ solvents, or it aggregates into “pinned micelles” under poor solvent conditions. A phase diagram of binary polymer brush in nonselective good or marginal solvents has been obtained by three-dimensional, selfconsistent field calculations.23 A ripple phase is stable in mixed brushes with nearly symmetric composition at small incompatibility, whereas at higher incompatibility or for brushes with asymmetric composition, dimple structures become stable instead. In dimple phases, one species forms clusters arranged on a quadratic or hexagonal lattice and embedded into a matrix of the other component. The square arrangement has a narrow region of stability for nearly symmetric composition and intermediate incompatibilities; hexagonal packing of the dimples occurs in most of the parameter region. The approach has been extended to asymmetric molecular weights24,25 where lateral and perpendicular segregation occur simultaneously. Both scaling considerations and self-consistent field theory assume spatially periodic structures with long-range order, in contrast to what is observed in experiments. Most theoretical studies focus on mixed brushes in nonselective solvents. The role of solvent selectivity is only incompletely understood. The latter property, however, is crucial for applications. The dependence of the morphology, switching, and reorganization of mixed brushes in selective solvents has attracted vast experimental interest. The first experiments demonstrated that a mixed brush of polystyrene/polyvinylpyridine can reversibly switch from a hydrophobic surface to a hydrophilic one when the solvent changes from toluene to acid water.4-6 The change of the morphology goes along with a change in surface properties. In mixed brush poly(methyl acrylate)/poly(styrene-co-2,3, 4,5,6-pentafluorostyrene), experiments7 observed sandwichlike morphologies or a combination of vertical and lateral microphase segregation depending on solvent selectivity. In selective solvents, the preferred polymer occupies the top of the brush, whereas in nonselective solvents, both polymers are present in the top layer.26 The asymmetry in chain length also affects morphologies.24 For small chain-length asymmetry, brushes exhibit lateral or perpendicular segregation, depending on solvent quality. Upon increasing the chain-length asymmetry, a transition from lateral segregation to a layered, sandwichlike structure occurs. The location of this transition can be tuned by changing the solvent selectivity. Feng and co-workers27 observed that very densely grafted mixed brush polystyrene/poly(methyl methacrylate) (PS/PMMA) exhibits dimple structures in cyclohexane and isobutyl alcohol but less microphase separation in tetrahydrofuran. A class of well-studied mixed brushes is Y-shaped grafted mixed brushes, in which two chains of different chains are grafted on a single point. This technique eliminates composition fluctuations of the grafting points. Julthongpiput et al.28,29 found Y-shaped polystyrene/poly(butyl acrylate) mixed brushes form segregated pinned micelles or craterlike micelles upon treatment with selective solvents. Zhao and co-workers30-32 studied the effect of molecular-weight asymmetry of Y-shaped PMMA/PS in selective solvents and observed that mixed PMMA/PS brushes with the chain length of PS that is slightly less than that of PMMA exhibit a relatively ordered nanopattern after treatment with a selective solvent for PMMA. Recently, new techniques were developed to investigate in situ the switching of morphology of Y-shaped polystyrene/ poly(acrylic acid) (PS/PAA) polymer brushes in various solvents33 and detect structural features below the top layer.34

J. Phys. Chem. B, Vol. 113, No. 33, 2009 11385 In view of these rich experimental observations, there are comparably few theoretical studies of mixed polymer brushes in selective solvents. Singh and co-workers studied the binary brushes in selective solvents in the limit of low grafting densities and found that if the solvent is bad for one component and good for the other, pinned micelles will form with the solvophobic component collapsing to a dense core shielded from the unfavorable solvent by the solvophilic one.35 Self-consistent field theory predicted that for a symmetric mixed brush, introducing solvent selectivity can induce a morphological transition from a ripple to a dimple phase, and experimental observations have confirmed this prediction.25 A systematic investigation of the role of solvent selectivity on the structure and thermodynamics of mixed polymer brushes and its interplay with the other characteristics (e.g., grafting density or molecular weight) is required to rationalize the wealth of experimental observations and provide guidance for the design of multicomponent brushes for specific applications. In this study, we perform Single-Chain-in-Mean-Field (SCMF) simulations to study microphase separation in mixed brushes exposed to various solvents. We systematically investigate the dependence of the morphology of mixed polymer brushes on solvent quality and selectivity, composition, f, of the A/B brush, chain-length asymmetry, and grafting density. The simulation results are related to previous experiments and theoretical predictions. The morphologies without long-range order are analyzed quantitatively using Minkowski measures. II. Model and Methods We study microphase separation of a mixed binary A/B brush exposed to solvents, S, using Single-Chain-in-Mean-Field simulations.36-38 This coarse-grained simulation technique allows us to investigate the structure formation in large, threedimensional systems with an experimentally relevant invariant degree of polymerization. Within our coarse-grained model, the molecular chain of each homopolymer, A or B, is discretized into NA or NB effective interaction centers (segments), respectively. The connectivity along the molecular backbone gives rise to Gaussian chain statistics on large length scales, and it is modeled by a discretized Edwards Hamiltonian for a conformation {rR,i(s)}, where R ) A or B denotes the chain species, and the indices i and s run over all molecules and segments along a molecule, respectively.

Hb[{rR,i(s)}] ) kBT

nR NR-1

∑ ∑∑

R)A,B i)1 s)1

3(NR - 1) 2RR,eo2

[rR,i(s + 1) rR,i(s)]2

(1)

If not stated otherwise, we will use the discretization NR ) 32 with R ) A, B. RR,eo2 is the mean square end-to-end distance of polymer species R in a dense melt. It is the only characteristic length scale of a Gaussian chain, and we set Reo for a polymer with N ) 32 as the unit length scale in the following. nR denotes the number of polymers of species R and n ) nA + nB is the total number of polymers. We consider a simulation cell of lateral extension L ) 16Reo, and periodic boundary conditions are applied along the lateral directions, x and y. 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 at random positions onto the substrate with their first segment. The height of simulation box Lz is much

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TABLE 1: Interaction Parameters of the Implicit-Solvent Modela solvent

SA

SB

SC

SD

VAA VBB VAB wAAA wBBB wAAB wABB

0.325 275 0.258 649 2.078 56 0.212 006 0.150 328 0.189 051 0.168 581

-2.797 21 0.173 237 0.638 477 1.041 8 0.141 527 0.535 554 0.275 31

4.628 45 -11.041 2 -2.338 79 0 3.666 96 0 0

-9.256 9 -11.041 2 -8.529 73 3.878 62 3.666 96 3.806 74 3.736 2

a

Compare to eq 2.

larger than the brush height, and the upper surface is modeled as an impenetrable hard wall. An equation-of-state approach is adopted to account for the nonbonded interactions:39

Hnb[{FA, FB}] ) kBT

∫V Rdr3 eo

(

∑

1 V F (r) Fβ(r) + 2 R,β)A,B Rβ R

∑

)

1 w F (r) Fβ(r) Fγ(r) 3 R,β,γ)A,B Rβγ R

(2)

where FR(r) is the local number density of polymer species, R, in a volume Reo3 defined as

Reo3 nR FR(r) ) N i)1

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). The partition functional in the canonical ensemble can be expressed as

Z∼

(

∫ D[{rR,i(s)}]exp -

)

Hb[{rR,i(s)}] + Hnb[{FA, FB}] kBT (4)

nR D[rR,i(s)] integrates over all where D[{rR,i(s)}] ) ΠR)A,B Πi)1 molecular conformations. In SCMF simulations, we simulate a large ensemble of polymer conformations in a fluctuating external field.36-38 This fluctuating external field mimics the interaction of a polymer with its instantaneous local environment. The molecular conformations of the noninteracting molecules subjected to the external fluctuating field are generated by Smart-Monte Carlo moves,40 where the local segment displacement is proposed according to the forces that act on the segment. After all segments have, on average, the chance to be moved once, that is, one Monte Carlo step (MCS), the instantaneous density is calculated from the explicit molecular conformations. To this end, eq 3 has to be regularized. In our numerical implementation, we use a fine collocation grid with spacing ∆L to calculate the densities at a grid vertex, c, according to

NR

∑ ∑ δ[r - rR,i(s)]

(3)

s)1

VRβ and wRβγ are the second-order virial coefficient between species R and β, and the third-order virial coefficient among species R, β, and γ, respectively. Here, only the second- and third-order interactions are included, and the higher-order terms are neglected. Thus, the expression should not be interpreted as a low-density expansion, but the coefficients are effective parameters that characterize the interactions over an extended range of densities. This local free-energy functional, which describes the nonbonded interactions, is the simplest one that allows for the interplay between liquid-vapor (solvent) phase coexistence and liquid-liquid demixing of the different species, A and B. If more experimental information about the systems becomes available, generalizations to include a more realistic equation of states or packing effects can be envisioned. In the present equation-of-state approach, the solvent degrees of freedom are integrated out. The quality of solvent is implicitly reflected in the solvent-mediated, effective virial coefficients. If the solvent is bad for one species, the second-order coefficient is negative. The density inside the collapsed domains and the compressibility are dictated by combinations of attractive second- and repulsive third-order coefficients. Mixed secondorder coefficients are related to the Flory-Huggins parameter between the species. The values of the virial coefficients are compiled in Table 1. We use the same interaction parameters as in our previous study of block-copolymer brushes39 to allow for a quantitative comparison between the two systems. The parameters of the equation of state are chosen to mimic poly(acrylic acid)/ polystyrene (PAA/PS) in various solvents, including DMF (solvent SA), MEK (solvent SB), and water. 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.

Fˆ R(c) )

nR

NR

N∆L3 i)1

s)1

Reo3

∑ ∑ Π(ri(s), c)

(5)

where Π(r, c) is a linear assignment of the segment at the position r onto the point c of the collocation grid;41-43 i.e.,

Π(r, c) )

∏

j)x,y,z

(

1-

|rj - cj | ∆L

)

(6)

if the distance between the grid point and the segment position along each Cartesian direction is less than ∆L, and Π(r, c) ) 0, otherwise. ∆L characterizes the spatial range of interactions, and it is chosen to be of the same order as the typical distance, Reo/N, between bonded segments. Similar schemes are used for particle-in-cell techniques in plasma physics44 or particlemesh methods in electrostatics45,46 to assign a particle-based density or charge distribution onto a lattice. In dense systems, this technique is computationally advantageous because the interaction of a particle can be calculated without explicitly evaluating all interactions with its neighbors. The nonbonded interactions, eq 2, take the form

Hnb[{Fˆ A, Fˆ B}] ∆L3 ) kBT Reo3

∑ c

(

∑

1 V Fˆ (c) Fˆ β(c) + 2 R,β)A,B Rβ R

∑

)

1 w Fˆ (c) Fˆ β(c) Fˆ γ(c) 3 R,β,γ)A,B Rβγ R

(7)

The local instantaneous density of the finite but large ensemble of explicit chain conformations fluctuates, and we obtain the external, fluctuating field, which acts on the R-species, according to37

Microphase Separation of Mixed Polymer Brushes

wˆR(c) )

Reo3

∂ Hnb ∂F ˆ ∆L R(c) kBT 3

J. Phys. Chem. B, Vol. 113, No. 33, 2009 11387

(8)

It can be shown37 that the simulation method will capture fluctuations if the external, fluctuating fields are frequently recalculated from the instantaneous densities. To achieve the equilibrium at a low temperature, we apply a slowly quenching scheme with Tj ) kTj-1, where Tj is the temperature at the jth quenching step and k is the scaling factor reflecting the rate of temperature decrease. In our simulations, we set a quenching step to be 7000 MCS and k ) 0.9, and it takes 60 quenching steps to reach the final temperature, after which an additional 70 000 MCS are performed to obtain the final morphologies at 490 000 MCS. Longer annealing steps are used to confirm equilibration. Due to the soft potentials and the crossability of the macromolecules in the course of their motion, the mobility of the molecules is much larger than that in a coarse-grained model with harsh repulsive interactions, such as a Lennard-Jones potential. III. Simulation Results The morphology of the microphase-separated structure of a mixed polymer brush depends on the surrounding solvent, and the ability of the brush to adopt to its environment has been exploited in many applications. The response of the brush to different solvents depends not only on the quality and selectivity of the solvent but additionally on the composition of the mixed brush, its grafting density, and asymmetry of molecular weight between the two grafted homopolymers. In the following, we first present diagrams of morphologies in terms of composition and grafting density for mixed brushes with symmetric chain lengths, NA ) NB, and then describe the dependence of morphology on chain-length asymmetry for mixed brushes of symmetric composition (f ) 0.5). Hereafter, the composition f is defined to be the number fraction of A-polymers (i.e., f ) nA/(nA + nB)), and the grafting density is σ ) n/L2, which usually is expressed in the dimensionless form σReo2 ) n/(L/Reo)2. A. Mixed Polymer Brushes with Symmetric Chain Length. Solvent SA is marginally good for both A and B polymers, with a slight preference for the A species. Therefore, both A and B polymer chains swell slightly in solvent SA. Figure 1A presents the diagram of morphologies for mixed A/B brushes in solvent SA as a function of composition and grafting density. Due to a small selectivity of the solvent, the diagram of morphologies is nearly symmetric around f ) 0.5. Laterally microphase-separated structures form; that is, both polymer species A and B are present at the top of the brush. The composition has critical influence on the lateral morphology. For brushes of very asymmetric composition, the majority component forms a continuous phase (matrix) into which clusters of the minority component (dimples) are dispersed. For brushes with nearly symmetric composition, bicontinuous structures (ripple) are formed. At intermediate composition, a rather gradual crossover between ripple and dimple structures is observed. Figure 1B illustrates some typical surface patterns of mixed polymer brushes with different compositions exposed to solvent SA at σReo2 ) 25. As mentioned above, both A and B homopolymers, are swollen by solvent SA, and therefore, the local polymer density deceases as we decrease the grafting density, σ. Laterally averaged density profiles of the two components as a function of the distance, z from the grafting substrate are shown in Figure

Figure 1. (A) Diagram of morphologies for mixed A/B polymer brushes exposed to solvent SA, in terms of grafting density, σReo2, and composition, f. Both grafted homopolymers have identical chain lengths, NA ) NB ) 32. Triangles and squares denote dimple and ripple morphologies, respectively. Circles mark the broad and gradual crossover between the two laterally segregated structures. Lines are only a guide to the eye. (B) Some typical morphologies at grafting density σReo2 ) 25 are illustrated by configuration snapshots. f is indicated in the key. The A component is yellow; the B species is represented in blue.

2. In panel A, we present density profiles of a symmetric brush, f ) 0.5, for different grafting densities. Because the solvent SA slightly prefers the A component, there is a small (average) surface enrichment of the A species at the top of the brush. Since there is a pronounced lateral phase separation between the two components, the data indicate that the A-rich domains are slightly more extended into the solvent than the B-rich domains; that is, the slight solvent selectivity also induces a small topographical contrast between domains. As the grafting density decreases, the local polymer density of the brush decreases, as well. Qualitatively, the grafting density has only a minor influence on the shape of the density profiles and the lateral morphologies. The data in Figure 1 indicate that the boundaries between different morphologies shift to more symmetric composition upon decreasing σ. At low grafting densities, σReo2 ≈ 10, it may become possible to alter the morphology from a dimplelike structure to a ripplelike one by increasing the grafting density. At even lower grafting densities than the ones we have studied (or larger swelling due to a better solvent quality) but still in the brush regime, σReo2 > 1; however, we anticipate that the density inside the brush may become too small for lateral phase separation to occur. In panel B of Figure 2, we investigate the laterally averaged density profiles as a function of composition. The majority component exhibits the typical parabolic density profile of a dense brush in a good solvent predicted by self-consistent field calculations. The density profile of the minority component, however, strongly differs from the parabolic shape. If the solvophilic component, A, is the minority component, the density profile is rather flat in order to expose more segments to the favorable solvent. In the opposite case of a solvophobic

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Figure 3. (A) Diagram of morphologies of mixed A/B brushes with symmetric chain length NA ) NB ) 32 exposed to solvent SB as a function of grafting density and composition. Triangles and squares denote dimple and ripple morphologies, respectively. Circles mark the broad and gradual crossover between the two laterally segregated structures. Lines are only a guide to the eye. (B) Some typical morphologies at grafting density σReo2 ) 25 are displayed in the bottom row. Figure 2. Density profiles along the perpendicular direction for mixed A/B brushes exposed to solvent SA. (A) Symmetric composition, f ) 0.5, and varying grafting densities as indicated in the key. (B) Constant grafting density, σReo2 ) 25, and varying composition as indicated in the figure legend.

minority component, the B species is enriched at the grafting substrate to reduce the unfavorable interactions with the A component. The laterally averaged density profiles also provide information about the height of the microphase-separated domains. The height of A minority domains (dimples) will be larger than that of the matrix formed by B homopolymers because the solvent slightly prefers the A species. For dimples of the solvophobic B component, the opposite behavior is observed. Solvent SB is marginally good for polymer B and moderately poor for the A component. Figure 3A presents the diagram of morphologies of a mixed A/B homopolymer brush in this solvent, SB, as observed in SCMF simulations. Since in solvent SB, B polymers are swollen and A chains are collapsed, the effective volume fraction of the B component is larger than the composition, which quantifies the relative number density of chains, regardless of the effective volume of a molecule. The increased effective composition of the B polymer shifts the boundaries between different morphologies to a larger value of f. This rationalizes the amount of skew in the gradual transition between different morphologies as compared with the nonpreferential solvent, SA (cf. Figure 1A). The asymmetry becomes particularly pronounced at lower grafting densities. Despite the solvent selectivity, lateral phase separation is dominant. Qualitatively similar to the behavior in solvent SA, dimple and ripple structures are formed, and there is a broad, gradual crossover between different morphologies as a function of grafting density or composition.

Figure 3B presents some typical surface patterns of mixed A/B brushes exposed to solvent SB. Since this solvent prefers polymer B to A, the height of a B-rich domain is always higher than that of an A-rich one. Nevertheless, the solvent selectivity is not sufficient to make polymer B cover the top layer of the brush completely for most compositions. Only if the B content is extremely large, 1 - f > 0.9375, will the solvophobic minority component A collapse into dimples close to the grafting substrate. The collapsed A dimples are covered with the swollen B component. These dimples resemble the craterlike structures observed by Julthongpiput and co-workers in Y-shaped brushes of PS/PAA in PAA-selective water, in which the solvophilic polymer, PAA, is short, and therefore, the swollen layers will not be able to cover the solvophobic PS core completely.28 If the concentration of the solvophilic B component is very low, dimples of B species form because the surrounding majority component A acts like a bad solvent. The dimple structure differs from the A dimples formed at the opposite extreme of composition (i.e., small f) in that the B domains do not collapse at the grafting surface, but B dimples grow out of the A matrix like sprouts to benefit from the solvent selectivity. Solvent SC is very good for the A component and very poor for the B species. In solvent SC, A polymers are highly swollen, and B polymers are collapsed. Therefore, perpendicular segregation is dominant, with collapsed B domains always pinned to the substrate and shielded from the solvent by a swollen layer of A polymers. Localizing the collapsed B domains at the substrate mitigates the effect of unfavorable SC/B-contacts by replacing them with contacts with the nonpreferential grafting substrate. Figure 4A summarizes the morphologies that mixed A/B brushes exhibits when exposed to solvent SC, and part B


Figure 4. (A) Diagram of morphologies of a mixed A/B brush with equal chain lengths exposed to solvent SC in terms of grafting density and composition. Triangles, squares and diamonds denote dimple, ripple, and complete layer, respectively. Lines are only a guide to the eye. (B) Configurational snapshots at grafting density σReo2 ) 12.5 are depicted in the bottom row.

displays some typical surface patterns at grafting density, σReo2 ) 12.5. The lateral morphology of collapsed B domains strongly depends on the fraction of B polymer, 1 - f. If its fraction is large, the collapsed B polymers completely cover the substrate and form a continuous layer. Since the repulsion between solvent SC and B polymers is so strong, we find that when the fraction of A polymers is small, A chains will be retained in collapsed B domains to avoid very unfavorable B/SC-contacts. The swelling of the B domains by a small amount of A species increases the effective volume fraction of B. If the A content becomes larger, around f > 0.35, A polymers will separate from the collapsed B domains. Decreasing the relative amount of the B species, we observe that this layer becomes perforated and develops a network structure, and for very small values of 1 f, it forms patches that shrink in size. Such a sequence is qualitatively similar to the behavior of one-component brushes in a poor solvent.8,47,48 Solvent SD is bad for both polymers, A and B, but slightly worse for B than for A. Figure 5A shows the morphology diagram of mixed brushes exposed to solvent SD. Characteristic configurational snapshots of surface patterns are compiled in panel B. Due to the bad solvent quality, both polymers, A and B, are collapsed. From laterally averaged density profiles perpendicular to the grafting substrate (not shown here), we deduce that the grafting density affects mainly the brush height and has only minor effects on the local polymer density, which is dictated by the coexistence density of the corresponding polymer melt in contact with the solvent. Therefore, grafting density has only a minor effect on microphase separation. Lateral segregation is dominant, and both polymers tend to be present at the top surface except when the fraction of A is much larger than that of B and the grafting density is high, σReo2 ) 25. In this limit, the top layer will be completely covered by A polymers. Lateral structures sensitively depend on


Figure 5. (A) Diagram of morphologies of a mixed brush (NA ) NB) exposed to solvent SD in terms of grafting density and composition. Triangles and squares denote dimple and ripple morphologies, respectively. Circles mark the broad and gradual crossover between the two laterally segregated structures. Lines are only a guide to the eye. (B) Typical surface patterns and their side views of configuration at grafting density, σReo2 ) 25, are illustrated in the lower tableau.

composition. Increasing f, we observe the following sequence of morphologies: dimples of A polymers, bicontinuous ripples, and dimples of B polymers. As in the other systems, there are no abrupt transitions between morphologies. The small preference of solvent SD for polymer species A makes the diagram of morphologies slightly asymmetric with respect to composition and the boundaries between different morphologies shift to lower values of f. Ripple structures form when the composition is slightly smaller than f ) 0.5, and the stability region of dimples of B polymers is larger than that of A polymers. B. Mixed Polymer Brushes of Asymmetric Chain Lengths. If the two components have equal chain extension, lateral structure formation dominates, and both components are simultaneously present at the top surface. Only in a solvent of very strong selectivity, solvent SC, is a pronounced perpendicular reorganization observed. In this section, we will study the effect of the asymmetry in chain lengths between the two components of the mixed brush. For simplicity, we restrict ourselves to symmetric composition (f ) 0.5); that is, the number of grafted polymers of both species is identical. We vary the chain length of one polymer while keeping the discretization of the other species constant, N ) 32. For one-component brushes, the variation of chain length changes the brush height and the lateral size of grafted coils. Therefore, introducing asymmetry in chain length will also affect microphase separation in mixed polymer brushes in two aspects: the lateral domain size and the height of each domain. Asymmetric chain lengths may give rise to topographic features, perpendicular segregation, or both. The different morphologies of such an asymmetric system with a fixed chain length of polymer B, NB ) 32, exposed to solvent, SA, are summarized in Figure 6 A). Snapshots of the different structures that form at grafting density σReo2 ) 25 are

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Figure 6. (A) Diagram of morphologies of mixed brushes with an equal number of A and B polymers exposed to solvent SA as a function of grafting density and chain length, NA of the A polymers. The chain length of the B polymers is kept constant at NB ) 32. Triangles and squares denote perpendicularly segregated and laterally segregated morphologies, respectively. Circles mark the broad and gradual crossover between the two structures. Lines are only a guide to the eye. (B) Some typical morphologies at σReo2 ) 25 are illustrated by snapshots of configurations. The bottom row is the topographic maps. The topographic maps are the height contours of the iso surfaces of the 3D density profiles, which are averaged over 25 configurations obtained every 500 MCS.

depicted in panel B. The brushes with NA ) 32 and varying NB exhibit similar behavior. In the symmetric case, NA ) NB ) 32, a bicontinuous lateral ripple pattern is observed with little height contrast between domains of different species. Chainlength asymmetry gives rise to a nonuniform surface topography. Usually, the polymer of longer chain length has the larger height, considering the very small preference of solvent, SA. At a modest chain-length asymmetry, the lateral pattern is retained on the top surface despite the topographic difference. At a large chain-length asymmetry, two layers can be distinguished perpendicular to the substrate: a laterally structured bottom layer that consists of A and B domains and a top layer that contains only the longer polymer species. At an extreme chain-length asymmetry, the surface becomes smooth. Moreover, chain-length asymmetry also affects the lateral structure of the bottom layer because increasing the chain length also increases the volume fraction of this component. Although for symmetric chain lengths (and f ) 0.5) we observe a bicontinuous ripple structure, in the asymmetric case, the shorter component tends to form a networklike structure (perforated

layer), and the longer component tries to form a dispersed phase, especially at a large chain-length asymmetry, when two-layer structures form. In this case, in the bottom layer, the shorter polymers are squeezed to the substrate, and the longer polymers are stretched, and therefore, this kind of arrangement helps to decrease unfavorable A/B contacts. Such a tendency is corroborated by the height-dependent Minkowski measures shown in Figure 17, which will be discussed later. The behavior in solvent SB is presented in Figure 7. It is similar to that observed in solvent SA. The solvent selectivity is reflected in the asymmetry of the morphology diagram in A and B with respect to exchanging NA and NB. By virtue of the solvent’s preference for the B component, a layered structure, where the B species is on top of a laterally structured bottom layer, will readily form if the B polymers are longer than the A polymers. Even in the opposite case, NA > NB, B polymers may cover A-rich domains. This is observed, for instance, at NA/NB ) 32/28 ≈ 1.14 (and σReo2 ) 25). If we decrease the length of the B polymers to NB ) 20 , NA ) 32 (i.e., NA/NB ≈ 1.6), however, both species will be present at the brush’s top surface.



Figure 7. (A, B) Diagram of morphologies of symmetric brushes exposed to solvent SB in terms of grafting density and chain length of one species, keeping the chain length the other polymer fixed at 32 throughout. Triangles denote perpendicularly segregated morphologies with the top occupied by the longer chain completely, and circles mark the broad and gradual crossover between the laterally segregated and the perpendicularly segregated structures. (C, D) Typical morphologies at σReo2 ) 25 are indicated by snapshots. The last row in part C is the topographic maps.

Figure 8 shows the morphology diagram of a mixed brush in solvent SC as a function of σReo2 and NA in panel A with fixed NB ) 32, and NB in tableau B with fixed NA ) 32. In solvent SC, A polymers are highly swollen, and the B component is collapsed. To a first approximation, we can assume the density inside B domains is independent of NB, then the lateral structure of B domains is determined mainly by nBNB ) NBσL2(1 - f). So changing NA but keeping NB ) 32 has little effect on the lateral structure as shown in panel A. The A

polymers only help to avoid the unfavorable B/SC contacts, and the strong SC/B repulsion gives rise to an effective attraction between A and B. If the A polymers are short, the height of the B domains will be small, and they will cover a larger substrate area. For example, at σReo2 ) 12.5, a perforated layer forms. Increasing NA from 12 to 20 and then to 32, the number of pores increases, as shown in Figure 8C. If we change NB but keep NA unaltered, the behavior will be different, with the boundary roughly obeying σ ∼ 1/NB. Increasing NB, we observe

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Figure 8. (A, B) Diagram of morphologies of symmetric brushes exposed to solvent SC in terms of grafting density and chain length of one species, keeping the chain length of the other polymer fixed at 32 throughout. Diamonds, triangles, and squares denote complete layer, dimples, and ripples, respectively. The dotted line in part A separates the region of polymer A being adsorbed in the collpased B layer from that of polymer A forming a swollen layer. (C, D) Snapshots indicate the different morphologies at σReo2 ) 12.5.

that B domains change their structure from dispersed patches, bicontinuous layer, and perforate layer to a continuous layer, as shown in Figure 8D. Figure 9 presents the morphology diagram of mixed A/B brushes exposed to solvent SD, which is bad for both A and B species, with a slight preference for the A component. The basic effect of chain-length asymmetry on the surface patterns is similar to those in solvent SB. Due to selectivity of solvent SD, there is an asymmetry between varying NA (shown in panel A) and NB (depicted in tableau B). If polymer A is longer than polymer B, it is easy to completely cover the top layer with A polymers. If the longer component is polymer B, a large asymmetry in chain lengths is necessary for polymer B to occupy the top layer completely. Since the surface of A domains is nonwettable for the B component, polymer B will rather shrink instead of spread on the surface of polymer A.

C. Comparison to Experiments and Other Theoretical Approaches. In the previous subsections, we have systematically investigated the dependence of the microphase-separated structure of a mixed homopolymer brush as a function of solvent quality and selectivity, chain-length ratio, and composition. Numerous structures have been observed, and they are separated by broad, gradual crossovers as the external control parameters of the system are altered. For specific parameter regions, our results corroborate previous predictions and experimental findings. Marko and Witten used analytical strong-stretching theory to predict that laterally segregated ripple phase forms in symmetric A/B mixed brushes under melt conditions and the perpendicularly segregated, layered phase is preempted.14 Monte Carlo simulations by Lai also provided evidence that the ripple



Figure 9. (A, B) Diagram of morphologies of symmetric A/B brushes exposed to solvent SD in terms of grafting density and chain length of one species, keeping the chain length the other polymer fixed at 32 throughout. Triangles and squares denotes perpendicularly and laterally segregated morphologies, respectively, and circles mark the broad and gradual crossover between them. (C, D) The configurational snapshots below display typical morphologies at σReo2 ) 25. The last row in part C is the topographic map.

phase is stable for symmetric mixed brushes in nonselective good solvents.17 Using numerical self-consistent field theory, lateral segregated phases (ripple and dimple structure) were found to be stable against perpendicularly layered phases in mixed brushes exposed to a nonselective marginal solvent, and a phase diagram has been obtained as a function of composition and grafting density.23 Our simulation results complement these theoretical predications. For mixed polymer brushes (f ) 0.5) with symmetric chain length, only lateral segregation is observed

in solvents SA, SB, and SD. This selection of solvents encompasses good and bad solvents, which exhibit only a weak selectivity. The lateral patterns, which form in these solvents, consist of bicontinuous ripple structures and dimple morphologies without long-range order. Even for brushes with asymmetric composition, f * 0.5, lateral segregation will dominate if the solvent selectivity is weak (solvents SA, SB, and SD) and no pronounced perpendicular segregation is observed in our simulations.

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Experiments demonstrate that the surface properties of mixed brushes (e.g., wettability or adhesion) can be controlled by solvent selectivity or chain-length asymmetry.24 These physical properties are dictated by the average chemical composition in the uppermost layer of the brushes. A variety of experiments have studied the switching of surface wettability by changing solvent selectivity.24,30 For instance, in a nonselective solvent, CHCl3, PS/PMMA brushes have a flat but microscopically heterogeneous surface, with both PS and PMMA being present on the top surface, but in glacial acetic acid, which is a good solvent for PMMA and a bad one for PS, dimple phases form.30,31 These dimples have flower-micelle structures, where PS aggregates into dense cores shielded by swollen PMMA chains. Similarly, in our simulations, when exposed to cosolvent SA, a flat and chemically heterogeneous surface forms, but when exposed to a selective solvent SC, pinned micelles covered by polymer A appear when the grafting density of polymer B is sufficiently low (see Figure 4, f ) 0.75). For mixed brushes with symmetric composition, one observes a gradual crossover from a laterally segregated ripple structure to a dimple structure with a pronounced perpendicular segregation by replacing a nonselective solvent with a selective one.25 In our simulation, mixed brushes with f ) 0.5 exposed to solvent SA have laterally segregated ripple structures but in solvent SC, perpendicular segregation occurs, and aggregated B polymers form discrete patches (i.e., dimples). Unlike the predictions of the selfconsistent field theory, these dimples do not arrange on a hexagonal (or square) lattice, but their long-range structure is disordered. Moreover, the shape of each dimple is not regular (i.e., spherical), but rather, rough and jagged. The irregular shape is dictated by fluctuations in the grafting density of the minority component, which has a pronounced influence on the location (domain memory measure) and the shape of an aggregate. An alternative to controlling the surface properties by exposing the mixed brush to different solvents (nonselective or selective) consists of modulating the properties by changing the chain-length asymmetry. In PS/PMMA mixed brushes with fixed chain length of PMMA treated by nonselective solvent CHCl3, it has been found that the contact angle of water on the brush surface changes from the value obtained for a pure PMMA surface to that of a PS surface when increasing the chain length of PS.31 This can be understood from the morphology diagram in Figure 6 showing the dependence of morphology on the chain-length asymmetry for mixed brushes in solvent SA. Although lateral segregation always occurs near the substrate, if one polymer is much longer than the other, the longer one will completely occupy the uppermost layer and, thus, dictate the wettability of the coated surface. If the chain-length asymmetry is not sufficiently strong, the surface will be heterogeneous, and both polymers will be present on the surface. Therefore, changing the chain-length asymmetry, one can achieve continuous control over the surface wettability. An additional mechanism for changing the surface wettability consists of varying the composition of the mixed brush. According to the morphology diagram as a function of composition and grafting density (cf. Figures 1, 3, and 5 for brushes exposed to weakly selective solvents SA, SB, and SD, respectively) both A and B polymers are simultaneously present at the uppermost layer and the composition of the top layer approximately coincides with the average composition of the brush. Therefore, surface wettability can be adjusted by changing the composition of the brushes. The diagrams of morphologies explore how the different control parameters (solvent quality and selectivity, chain-length

Wang and Mu¨ller asymmetry, and composition) affect the morphology of the brush and its surface properties in a combined way. If the solvent is nonselective or good for the longer polymer, chain-length asymmetry will amplify perpendicular segregation. If the solvent prefers the shorter polymer, solvent quality and chain-length asymmetry will act antagonistically. When chain-length asymmetry is small, lateral segregation will be dominant. In the limit of large chain-length asymmetry, however, mixed brushes will exhibit both lateral and perpendicular segregation, where lateral segregation occurs near the substrate, above which a layer of the longer polymer is formed. Our simulations also investigate the practically important case of a strongly selective solvent. In solvent SC, the chain length of the soluble polymer A has little effect on lateral structure, which consists of pinned aggregates of polymer B. Santer and co-workers investigated the effect of chain-length asymmetry on PS/PMMA mixed brushes with fixed length of the PS molecules in acetone, which is a good solvent for PMMA.13 Increasing the chain length of PMMA, they found all brushes have dimple structures with pinned PS domains covered by swollen PMMA. Zhao studied the effect of grafting density on PMMA/PS mixed brushes exposed to glacial acetic acid. He also found pinned micelles, which formed when the ratio of PS to PMMA grafting density was in the range from 0.67 to 2.17 and the overall grafting density was about 0.85 nm2. At a smaller grafting density of PS, the topographical feature was negligible, but at a higher grafting density of PS, the pinned micelles become connected and finally build a flat surface.32 This observation is in accord with our results in solvent SC. IV. Integral Geometry Analysis Although the surface patterns do not exhibit long-range order and there are no thermodynamic transitions between different structures, nevertheless, these crossover regions are practically important because the mixed brush is very susceptible to external stimuli in these regions; that is, small changes of the external control parameters are amplified by the structure formation of the multicomponent system. It is difficult to quantitatively characterize the different morphologies and describe the gradual crossover between them. Minkowski measures from the integral geometry are efficient to describe such segregated morphology lacking long-range order.49-53 According to integral geometry, two-dimensional black-andwhite patterns can be characterized by three Minkowski functionals: the covered area, S; the boundary length, U, of the domains; and the Euler characteristic, χE. The latter quantity, χE, reflects the connectivity of domains and is defined as the difference between the numbers of domains of the two kinds. A positive χE means that the white domains are dispersed and the black domains forms the continuous matrix, and vice versa. Small values of χE are typically for a ripple structure or a bicontinuous morphology. The information extracted from the Minkowski measures complements the characterization of the more routinely utilized structure factor, which provides an accurate estimate for the characteristic in-plane length scale but is rather insensitive to the details of the structure of the disordered morphology. In this section, we apply Minkowski measures to analyze the microscopically segregated surface patterns using three methods. A. Binary Maps Generated by Different Thresholds. To describe the distinct morphologies of microphase-separated polymer brushes exposed to different solvents by twodimensional Minkowski measures, the continuous, threedimensional spatial distribution of species has to be converted



Figure 10. Illustration of binary level contours at different threshold values for mixed A/B brushes. ΣzFt-∆L/(σReo3) is used as the thresheld variable, and its valued is labeled under each figure. Polymer B is enriched in the white areas, and the black areas correspond to A-rich domains. The thin dashed line denotes the boundary between domains at the representative threshold value. (A) Ripple morphology for mixed brushes exposed to solvent SA, with NA ) NB ) 32, σReo2 ) 25, and f ) 0.5. (B) Dimple morphology for mixed brushes exposed to solvent SA, with NA ) NB ) 32, σReo2 ) 25 and f ) 0.75.

into a two-dimensional, black-and-white map. In our model with implicit solvent, there are two independent local densities, FA and FB, and the microphase-separation can be characterized by the local density, F+ ≡ FA + FB, or the composition contrast, F- ≡ FB - FA. Depending on solvent selectivity, F+ or F- better reflects the local structure of the mixed polymer brush. For instance, in a very selective solvent, SC, the lateral structure is mainly dictated by the nonsoluble component B, which collapses into dense aggregates. In this case, the morphology resembles that of a one-component B brush in a bad solvent. Thus, the local density F+ is a suitable measure characterizing the morphology. Moreover, ΣzF+ can detect lateral structures of dense B aggregates in solvent SC irrespective of chain-length asymmetry. In nonselective or weakly selective solvents, SA, SB, and SD, and at sufficiently large grafting density, however, the total density, F+, of the brush is rather uniform and resembles that of a one-component brush.14 In this case, the composition contrast, F-, is more suitable for characterizing the lateral structure. If the solvent is only weakly selective (i.e., solvents SA, SB, and SD) and if the chain lengths of the two components are nearly symmetric, lateral segregation will be dominant and the perpendicularly averaged composition, ΣzF-, yields a faithful representation of the morphology. We emphasize that the local density and composition variations are also measurable by scanning force microscop. These experimental data can be analyzed in a way similar to our simulation results and provide a quantitative characterization of the observed surface morphology. Increasing the chain-length asymmetry, we observe that lateral and perpendicular segregation occur simultaneously and the lateral structure depends on the distance, z, from the grafting substrate. In this case, we analyze the Minkowski measures for the lateral composition contrast, F-(z), as a function of z. This technique provides information about the variation of lateral structure with height. To calculate Minkowski measures from the 2-dimensional variations of the density or composition, we transform the continuous contour-level plots into binary, black-and-white patterns by a threshold procedure. First, the images are discretized into a collection of pixels. Pixels with local variable ξ (ξ being F-(z), ΣzF+ or ΣzF-) higher or lower than a given threshold value ξt are colored white or black, respectively. Then, Minkowski measures are calculated as a function of the threshold, ξt.54,55

Figure 10 illustrates typical binary level contours at different thresholds and compares them with the original, continuous composition contrast. A ripplelike structure of a symmetric brush in solvent SA and a dimple structure at f ) 0.75 are shown. The lighter shaded areas correspond to B-rich domains, whereas the darker regions mark A-rich domains. The binary level contours strongly depend on the threshold. For a range of intermediate threshold values, ΣzF- ≈ 0, the black-and-white images provide a faithful description of the domain structure, and they do not strongly depend on the threshold value. For more extreme values of the threshold, the minority domains break up into many smaller spots, indicating the location of extreme local composition fluctuations. The threshold value, at which this qualitative change occurs, provides information about the magnitude of the composition contrast inside a domain. For instance, if the lateral structure formation extends throughout the film in a perpendicular direction, the composition contrast between the two domains will be large and the intermediate range of threshold values, where the black-and-white image faithfully describes the morphology, will extend to larger, absolute values of the threshold. If there were a top layer consisting of the longer polymer species, then the composition contrast between the domains would be significantly reduced and also, the intermediate range of threshold values would be much narrower. Figure 11 illustrates the dependence of the Minkowski measures on the threshold value for different morphologies, which form in different solvents and at different compositions. Typically, there is an intermediate range of threshold values, where the Minkowski measures characterize the laterally microphase-separated morphology and the Minkowski measures vary slowly with the threshold value. Such a well-defined plateau of the Minkowski measures as a function of the threshold is observed in all weakly selective solvents, SA, SB, and SD, where pronounced lateral segregation is observed. The center of this plateau defines the representative threshold, ξc. For mixed brushes in weakly selective solvents, we use ξc ) 0; that is, ΣzF- ) 0 for equal chain lengths, NA ) NB, and F- (z) ) 0 for mixed brushes with asymmetric chain lengths (cf. Figure 17). The limits of this intermediate plateau regime are marked by rapid variation of the Minkowski measures with threshold, for example, a pronounced decrease in S, a peak in U, or a switch of χE from negative to positive values. These rapid variations

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Figure 11. Minkowski measures as a function of threshold values for mixed A/B brushes of symmetric chain length NA ) NB ) 32 exposed to solvent (A) SA with σReo2 ) 25, (B) SB with σReo2 ) 25, (C) SC with σReo2 ) 12.5, and (D) SD with σReo2 ) 25. The composition is shown in the figure legends.

occur when the threshold value exceeds the average density or composition contrast of a domain. Outside the intermediate regime of threshold values, the Minkowski measures no longer characterize the domain structure but, rather, the random, extreme fluctuations of density or composition inside the physical domains. The behavior in the strongly selective solvent, SC, with f ) 0.25 is qualitatively different. There is no plateau in the dependence of the Minkowski measures on the threshold value, and only the rapid variations, which mark the limits of the intermediate range of threshold values in the weakly selective solvents, can be observed. In this case, the configurational snapshots in Figure 4 confirm the absence of lateral domains. Thus, the width of the intermediate plateau quantifies the strength of the lateral structure formation. In accord with this interpretation, we observe an indication of a plateau for f ) 0.5 and f ) 0.75, where the nonsoluble, minority component forms clusters, as illustrated in Figure 4. For solvent SC, the representative threshold is specified in the middle of the plateau, although the plateau is narrow and masked by fluctuations inside domains. Figure 12 shows the Minkowski measures for the representative threshold value for mixed brushes of different compositions. S refers to the area covered by B polymers, and it increases with the composition, 1 - f. In weakly selective solvents, SA, SB, and SD, the fractional area, S, of white (B-rich) domains in the lateral map of ΣzF- ) 0 is very close to the fraction, 1 f, of grafted B polymers. If the solvent prefers the B species (e.g., solvent SB), S will be slightly larger than 1 - f, and in the opposite case (i.e., solvent SD, which prefers the A component), the normalized area, S, is slightly smaller than 1 - f. These deviations are the stronger, the smaller is the grafting

density, and they become notable only for the lowest grafting density, σReo2 ) 6.25. With an increase in f, the length of the domain boundary, U, first increases, passes through a maximum, and then decreases. Small values of U indicate a low density of domains or domains of small boundary length. Thus, at the extreme compositions, f , 0.5 and f . 0.5, there are fewer and smaller domains. The maximum of the boundary length approximately corresponds to bicontinuous or ripple phases. The Euler characteristics increase from negative to positive values as we increase the composition. Positive values of χE characterize dispersed white (B-rich) domains, which are embedded in an A-rich matrix. Negative values indicate that the A-rich phase forms clusters embedded in a B-rich matrix. In both cases, the absolute value, χE, counts the number of domains in the area L2. Ripple morphologies are characterized by χE ≈ 0, for there are about equal amounts of A and B clusters. Thus, the variation of χE characterizes the gradual change from A dimples via a ripple structure to B dimples upon increasing f. Since there are no abrupt changes in χE, the change in the morphology is gradual, and it does not give rise to a thermodynamic singularity. The composition, where the ripple structure occurs, depends on the solvent selectivity. The transition is around f = 0.5 in nonselective solvent SA; it occurs at A-rich composition, f > 0.5, in the solvent SB, favoring polymer B; and it happens at f < 0.5 in solvent SD, which prefers polymer A. The effects of solvent selectivity are more pronounced at lower grafting density, where the density difference between swollen and collapsed conformations is larger. At a lower grafting density, we also observe that the f dependence of U is stronger. The effective immiscibility between



Figure 12. Minkowski measures calculated at the representative threshold as a function of composition for mixed A/B brushes exposed to solvent (A) SA, (B) SB, (C) SC, and (D) SD. The grafting density is indicated in the key.

components is weaker at a smaller grafting density, and therefore, the interface is more diffuse and the concomitant interface tension is lower. Thus, the interfaces are more jagged due to thermal fluctuations and the quenched fluctuations due to the random, immobile grafting points. Both effects tend to increase the boundary length. The behavior of the Minkowski measures is very different for mixed brushes in a strongly selective solvent, SC. The nonsoluble component B forms a collapsed layer. This layer will be continuous if the grafting density is large or if the composition of the A component is large. For very A-rich compositions, however, this collapsed layer will break up. Such a perforated layer gives rise to larger values of the boundary length and negative values of the Euler characteristics. For very low grafting densities and extremely A-rich composition, the B component forms small clusters instead of a perforated layer. This morphology is characterized by positive Euler characteristics, and the boundary length decreases as the clusters become smaller and more sparse. In addition to the overall characterization of the lateral structures, the Minkowski measures extracted at the representative threshold value provide quantitative information about the domains of dimple structures; that is, well-defined clusters of the minority component embedded in a matrix consisting of the majority species. Then the absolute value counts the number of dimples, and properties of a single domain can be extracted.51 The average lateral area of a dimple is given by

{

S/ (1 + χE) χE > 0 s) ( 2 ) L - S / (1 - χE) χE < 0

(9)

and, assuming the domains are circular, we estimate the average domain size according to

r ) U/2π(1 + |χE | )

(10)

Therefore, we define R ≡ r/(s/π)1/2 to reflect the circular shape of dispersed domains. For monodisperse, disklike domains, R ) 1. Noncircular domain shapes or significant poly dispersity in the disk size lead to deviation of R from unity. Furthermore, we can define a measure of the dominant lateral length, Λ.

Λ ) [L2 / (1 + |χE | )]1/2

(11)

The domain size, r, and the dominant lateral length, Λ, are shown in Figure 16. The average domain area, s, and shape parameter, R, are not shown here. In solvent SA, the average domain area, domain size, and shape parameter decrease as we approach extreme compositions, f , 0.5 or f . 0.5. This behavior coincides with the decreases in the size of dimples observed for very asymmetric systems. Around f ) 0.5, all characteristics develop a singular behavior because the morphology consists of ripples. In this region, |χE| does not quantify the number of domains, and the values do not have a physical meaning. Mixed brushes exposed to solvents SB, SC, and SD exhibit qualitatively similar behavior, the difference being only the location of the transition between dimple and ripple structures, which will depend on the grafting density if the solvent is selective. Interestingly, the size of dimples is remarkably insensitive to the solvent quality.

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Figure 13. Illustration of binary level contours during domain expanding for A/B mixed brushes. White phase is polymer B and black phase is polymer A. Red dashed line denotes the unmoved interface. (A) Ripple morphology for mixed brushes exposed to solvent SA, with NA ) NB ) 32, σReo2 ) 25, and f ) 0.5; (B) Dimple morphology for mixed brushes exposed to solvent SA, with NA ) NB ) 32, σReo2 ) 25, and f ) 0.75.

B. Binary Maps Generated by Parallel Displacements of Domain Boundaries. As illustrated in Figure 11, it is useful to generate a family of Minkowski measures for each configuration. In the previous section, we utilized different threshold values to obtain a family of morphology maps. An alternative consists of defining one binary contour map via the representative threshold value such that the binary map faithfully characterizes the lateral morphology and then expanding the domains by a signed distance, δ.53 Negative values of δ expand black domains, whereas positive values are used to indicate that the white domains have been enlarged by a distance, δ, normal to the boundary. If δ is positive (negative) and a pixel p has a local variable higher (lower) than the representative threshold value in the original map, then all pixels of the expanded map at a distance away from pixel p smaller than |δ| are also colored white (black). This method of parallel boundaries is a complementary method for generating a family of maps and corresponding Minkowski measures from a configurational snapshot. It has the advantage that the parameter δ has an immediate interpretation as a length, but information about the strength of the segregation or the contrast between the domains is lost. In Figure 13, this technique is illustrated for a ripple structure and a dimple morphology, which are formed by a mixed brush with equal chain lengths, grafting density, σReo2 ) 25, and composition, f ) 0.5 and f ) 0.75, respectively. Increasing δ, one erodes the black domains, and if δ becomes on the order of the domain width (i.e., the widths of the ripples for f ) 0.5 or the distance between two boundaries of dimples for f ) 0.75), the black domains break up into many unconnected clusters and the white domains join and form a single cluster (matrix). Decreasing δ to negative values, one decreases with widths of white domains. In the ripple structure at f ) 0.5, this leads to a breaking up of the white domains. In the dimple structure at f ) 0.75, the white disklike domains shrink and eventually disappear when |δ| becomes comparable to the dimple radius. Figure 14 presents the Minkowski measures as a function of δ for the two cases. Increasing δ, we observe that S increases monotonically from S ) 0 at large negative values of δ to S ) L2 at large positive δ because we expand the area of white domains. For the symmetric ripple structure at f ) 0.5, the dependence is symmetric with respect to δ f -δ and S f L2 - S. At around |δ| > 0.6Reo, the value of S adopts its limiting values, S ) 0 or L2. This indicates that the width of a ripple is 1.2Reo and the dominant characteristic length scale is 2.4Reo.

This number is slightly larger than the prediction of selfconsistent field calculations, which yield Λ ≈ 1.8Reo for a lamellar structure with long-range periodicity.14,23 This value, however, agrees with experimental observations,25 and we speculate that the difference arises from the random grafting and the concomitant, quenched fluctuations that are ignored in the self-consistent field calculations. For asymmetric compositions, f ) 0.25 and f ) 0.75, the S-vs-δ curves are shifted. For f ) 0.75, S starts to deviate from S ) 0 at around δ ) -0.4Reo. This value characterizes the radius of the B dimples. The limiting value S ) L2 is approached for δ > 0.9Reo. At this point, the B domains have been expanded to such an extent that no matrix phase remains; that is, the (maximal) width of the A-rich matrix phase in the original, representative map is about 1.6Reo, and the dominant lateral length scale is about 2.4Reo. For f ) 0.25, the behavior is analog, except that the dimples are formed by the A polymers. We also note that the S-vs-δ curves have an s shape for the dimple structures, and they are more linear for the symmetric ripple structures. This behavior indicates the shape of the domains. For the ripple structure, the area change is a linear function of δ. For the disklike domains of the dimple structure, however, the change is large for δ ≈ 0 because the dimples have a large radius, and it becomes smaller as the size of the disklike domains decreases. The behavior of the boundary length, U, parallels the behavior of S. When S approaches its limiting values, there is no domain structure, and U vanishes. Therefore, in a symmetric ripple structure, U has a broad plateau around δ ) 0. The width of the plateau characterizes the thickness of a ripple, and the interval over which U decays from its plateau value to zero quantifies the fluctuations of the ripple thickness. The curves for the asymmetric dimple structures at f ) 0.25 and f ) 0.75 are different. First, there is no plateau of U as a function of δ because, in contrast to ripples, changing the width of disklike domains, we also change their boundary length. Second, the curves are not symmetric: the maximum of U is not located at δ ) 0, and the two distinct slopes at small and large values of δ are not similar. For f ) 0.75, increasing δ B-rich dimples emerge and increase in radius (cf. Figure 13). The radius of a dimple in the binary map with value δ is given by r + δ, where r characterizes the radius of a dimple in the original map. Since the number of dimples remains constant, the boundary length increases approximately linearly with δ for



Figure 14. Variation in Minkowski measures when expanding the domains by a length δ for mixed brushes exposed to selective solvent (A) SA with σReo2 ) 25, (B) SB with σReo2 ) 25, (C) SC with σReo2 ) 25, and (D) SD with σReo2 ) 25.

small values of δ. At large δ, the boundary length decreases because the dimples are expanded and start to overlap. This gives rise to a sudden decrease in U. The structure of the morphology is also clearly visible in the dependence of the Euler characteristics on the parallel distance, δ. For the symmetric ripple structure, χE is a symmetric function of δ. At δ ≈ 0, there is a broad plateau, χE ≈ 0, indicating that there are similar numbers of A and B domains. As we increase (decrease) δ, we erode dark, A-rich (white, B-rich) domains. If |δ| reaches half of the characteristic width of a ripple, they break up into many domains, and the absolute value of χE increases. Increasing |δ| further, the broken-up domains erode further and vanish. Thus, |χE| passes through a maximum and then decreases to zero. The dependence of χE for a dimple structure is markedly different. Around δ ) 0, the absolute value |χE| adopts large values counting the number of disklike domains. The sign identifies which of the two components forms the dimples and which constitutes the matrix. Changing δ, we observe that the disklike domains expand or shrink, but their number and, hence, |χE| remain unaltered. If δ is changed such that the size of the disklike domains further decreases (i.e., increasing δ for f ) 0.25 or decreasing δ to negative values for f ) 0.75), the dimples will disappear at δvanish, and the Euler characteristics approaches 0. In the opposite limit, the disklike domains merge, and the matrix breaks up into many small domains. In this case, χE approaches 0 and changes sign, where δ is marked by δswitch. Varying δ further, the domains of the matrix disappear, and χE approaches 0. From the variation of χE with δ, we can extract characteristic lengths of lateral morphology. As shown in Figure 15, the

Figure 15. A typical example illustrating determination of domain ˜ , using the method of parallel size, r˜, and dominant lateral length, Λ displacement of interface. NA ) NB ) 32, solvent SA, σReo2 ) 25, and f ) 0.75.

domain size r ) 2|δvanish| is related to the values of δvanish, where domains disappear (χE ) 0), and the dominant lateral length Λ can be determined from 2|δswitch - δvanish|. Figure 16 shows the sizes of dimple structures, r, and the dominant lateral length, Λ, as function of f, and the results are compared with those determined from eqs 10 and 11. In the composition range where the dimple structure is stable, both methods agree. At symmetric composition, where the ripple structure is formed, the method of parallel surface gives physically meaningful results, whereas using the previous method with different thresholds fails (because χE ≈ 0 does not quantify the number of domains). Overall, the dominant lateral length scale depends only weakly on the composition. The dominant lateral length slightly increases when the composition becomes more symmetric and the morphology changes from dimplelike to ripplelike. Note that at the same characteristic length scale, bicontinuous ripple structures have a larger boundary length. It also increases when

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Wang and Mu¨ller

Figure 16. Domain size and lateral dominant length as a function of composition for mixed A/B brushes exposed to solvent (A) SA, (B) SB, (C) SC, and (D) SD. The grafting density is indicated in the key. The hollow symbols are obtained from eqs 10 and 11, and the solid symbols are calculted using the method of parallel dispacement of interface.

systems become extremely asymmetric because the number of domains decreases. Both ripple structures and dimple morphologies are characterized by 2.2Reo e Λ e 3Reo. The lateral length scale increases with grafting density. This can be ascribed to weak effective immiscibility at smaller grafting density and the entropic penalty (i.e., chain stretching) associated with forming domains. By the same token, brushes exposed to a bad solvent, SD, form denser aggregates and are characterized by a large incompatibility. They have larger dominant lateral lengths than those structures formed by exposing the brush to marginal solvents, SA and SB. C. Minkowski Measures As a Function of z. Introducing solvent selectivity or asymmetry in the chain length, perpendicular segregation additionally occurs. In this case, the lateral structure varies with the distance, z, from the grafting surface. To quantitatively describe the variation in lateral structure with z, we calculate Minkowski measures of maps, F-, in planes parallel to the grafting surface at different distances. In Figure 17, we present the Euler characteristics as a function of position, z, and parallel interface displacement, δ, for brushes with different NA, keeping NB ) 32 constant. For brushes exposed to solvent SA, and NA ) NB ) 32 (i.e., symmetric chain length), a ripple structure forms. In the region very close to the grafting substrate, there is no “plateau” at χE ≈ 0, indicating that in the ultimate vicinity of the grafting substrate, no lateral phase separation occurs due to the grafting constraint and stretching of grafted chains. Further inside the brush, 0.25Reo < z, the Euler characteristics (and the morphology) does not vary with the position, and χE(z) exhibits the typical structure of the ripple structure as a function of δ. This indicates the ripple structure persists in the perpendicular direction, z. Near the top surface, the value of |χE| become very

large, which indicates the thermal fluctuation of the composition in the region where the density is already very small (cf. Figure 2). Moreover, above them, there is a change of the sign of χE, signaling the crossover between brushes and the pure solvent. These features give information about the brush height. From the sign of χE at the top of brushes, we can judge which polymer reaches out farther away from the substrate. We identify a chain-length ratio (NA/NB)) where both polymer have the same height. (NA/NB)) depends on the solvent selectivity. The results show that for the weakly selective solvent, SA, (NA/NB)) ) 1; for solvent SB, preferring polymer B, 44/32 > (NA/NB)) > 36/ 32; and for solvent SD, 20/32 < (NA/NB)) < 28/32. Changing the chain-length asymmetry away from (NA/NB)), a difference in height appears, and the topographic morphology is obtained. Increasing the chain-length asymmetry even further, we observe the gradual formation of a two-layer structure. The lower layer exhibits lateral structure formation, which is clearly detectable in the dependence of the Euler characteristics on the parallel boundary displacement. The upper layer consists of only the longer polymer species, and it is either laterally homogeneous (χE ) 0, NA ) 44) or undulating (χE * 0; e.g., NA ) 36), depending on chain length asymmetry. Moreover, introducing chain-length asymmetry, the lateral structure of the bottom layer tends to become a perforated layer: the component of shorter chain length form the continuous phase, and the longer chains form the dispersed phase, as indicated by χE(δ ) 0) and the bending direction of bands of χE (which changes at (NA/NB))). At (NA/NB)), the lateral structure persists throughout the entire brush. With chain-length asymmetry different from (NA/NB)), the lateral structure varies gradually in the perpendicular direction. By the method illustrated in Figure 15, we can deduce that the pore (dimple)



Figure 17. Euler characteristics of the two-dimensional composition as a function of the distance from the grafting surface and the parallel bondary distance. Data for mixed brushes with composition f ) 0.5 exposed to different solvents with constant NB ) 32 and varying NA are analyzed. (A) Mixed brushes with σReo2 ) 25 in solvent SA, (B) brushes with σReo2 ) 25 in solvent SB, (C) brushes with σReo2 ) 12.5 in solvent SC, and (D) brushes with σReo2 ) 25 in solvent SD.

size gradually varies along the perpendicular direction. The dominant lateral length decreases with the chain length of the shorter polymer. Brushes exposed to solvents SB and SD have behavior similar to those exposed to solvent SA. Only the value of (NA/NB)) differs, reflecting the solvent selectivity. Brushes in solvent SC are quite different from those in solvents SA, SB, and SD. In solvent SC, polymer B, collapses and forms a layer pinned to the substrate, which is mixed with some stretched A polymers. So χE(δ ) 0) shows a large negative value, which does not yield information about the pore number but, rather, reflects the distribution of individual polymer A in the polymer B matrix. When the A polymers are short, NA ) 20, most of them are adsorbed in collapsed B domains, and the A species also covers the surface of B domains. Increasing the chain length of polymer A, we observe a swollen layer of polymer A forms above the collapsed layer of polymer B. The χE(z, δ) clearly shows the height of the collapsed layer and the swollen layer. V. Conclusions Using SCMF simulations, we systematically explore the morphologies of mixed polymer brushes as a function of grafting density, composition, chain-length asymmetry, solvent quality, and solvent selectivity. The solvent is implicitly described by a third-order equation of state, which is the simplest form capable of simultaneously accounting for liquid-liquid separation

between the polymer species and liquid-vapor (solvent) phase coexistence. The parameters of our model are inspired by the experimental system PAA/PS exposed to various solventssDMF, MEK, and watersat high pH and water at low pH. The general features of the variation of the structure of the microphaseseparated brushes, however, apply to a wider class of materials. Our findings have been compared to experimental results and theoretical predictions. Various isolated portions of the large parameter space have been previously explored, and generally good agreement is found. However, in contrast to the predictions of the self-consistent field theory but in agreement with experiments, we observe that the morphology lacks long-range order and that the dominant characteristic length scale of the ripple structures is larger than predicted by the mean field theory. Both effects can be traced back to quenched fluctuations of the grafting points, which are strongly amplified by the microphase separation in the mixed brush. The lack of longe-range order results in an absence of thermodynamic phase transitions between different morphologies. We rather observe a gradual change in the microphaseseparated structure in response to changes in the various control parameters. Since the structure factor provides information only about the dominant lateral length scale of the morphology but is rather insensitive to the geometrical features of the domains, we use two-dimensional Minkowski measures (i.e., area covered by one species, boundary length of the domains, and Euler characteristics of the domain pattern) to characterize the

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disordered structures. Two sets of Minkowski measures are utilized: one that is generated by using different thresholds of the density or composition differences and one that is generated by parallel displacement of the interfaces. A wealth of information can be extracted from these integral measures, including the average size and shape of the domains and the dominant lateral length scale. Since this analysis can be applied to experimental results, as well, we hope that this method will find its application in analyzing experiments of disordered but structured morphologies so that experiments can be quantitatively compared to our simulation results. Acknowledgment. It is a great pleasure to thank 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 is gratefully acknowledged. Ample computing time has been provided by the GWDG Go¨ttingen, the HLRN Hannover, and the JSC Ju¨lich. References and Notes (1) Zhao, B.; Brittain, W. J. Prog. Polym. Sci. 2000, 25, 677. (2) Polymer Brushes; Advincula, R. C., Brittain, W. J., Caster, K. V., Ru¨he, J. Eds.; Wiley-VCH: Weinheim, 2004. (3) Luzinov, I.; Minko, S.; Tsukruk, V. V. Prog. Polym. Sci. 2004, 29, 635. (4) Sidorenko, A.; Minko, S.; Schenk-Meuser, K.; Duschner, H.; Stamm, M. Langmuir 1999, 15, 8349. (5) Minko, S.; Usov, D.; Goreshnik, E.; Stamm, M. Macromol. Rapid Commum. 2001, 22, 206. (6) Minko, S.; Patil, S.; Datsyuk, V.; Simon, F.; Eichhorn, K.-J.; Motornov, M.; Usov, D.; Tokarev, I.; Stamm, M. Langmuir 2002, 18, 289. (7) LeMieux, M.; Usov, D.; Minko, S.; Stamm, M.; Shulha, H.; Tsukruk, V. V. Macromolecules 2003, 36, 7244. (8) Wenning, L.; Mu¨ller, M.; Binder, K. Europhys. Lett. 2005, 71, 639. (9) Santer, S.; Kopyshev, A.; Donges, J.; Ru¨he, J.; Jiang, X.; Zhao, B.; Mu¨ller, M. Langmuir 2007, 23, 279. (10) Prokhorova, S. A.; Kopyshev, A.; Ramakrishnan, A.; Zhang, H.; Ru¨he, J. Nanotechnology 2003, 14, 1098. (11) Santer, S.; Ru¨he, J. Polymer 2004, 45, 8279. (12) Santer, S.; Kopyshev, A.; Donges, J.; Yang, H.-K.; Ru¨he, J. AdV. Mater. 2006, 18, 2359. (13) Santer, S.; Kopyshev, A.; Yang, H.-K.; Ru¨he, J. Macromolecules 2006, 39, 3056. (14) Marko, J. F.; Witten, T. A. Phys. ReV. Lett. 1991, 66, 1541. (15) Marko, J. F.; Witten, T. A. Macromolecules 1992, 25, 296. (16) Brown, G.; Chakrabarti, A.; Marko, J. F. Europhys. Lett. 1994, 25, 239. (17) Lai, P.-Y. J. Chem. Phys. 1994, 100, 3351. (18) Soga, K. G.; Guo, H.; Zuckermann, M. J. Eurphys. Lett. 1995, 29, 531. (19) Soga, K. G.; Zuckermann, M. J.; Guo, H. Macromolecules 1996, 29, 1998.

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Microphase Separation of Mixed Polymer Brushes: Dependence of the

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