Structure of Mixed Brushes Made of Arm-Grafted Polymer Stars and

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Article pubs.acs.org/Macromolecules

Structure of Mixed Brushes Made of Arm-Grafted Polymer Stars and Linear Chains Alexey A. Polotsky,*,†,‡ Frans A. M. Leermakers,§ and Tatiana M. Birshtein†,∥ †

Institute of Macromolecular Compounds, Russian Academy of Sciences, 31 Bolshoy pr., 199004 Saint Petersburg, Russia Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University), Kronverkskiy pr. 49, 197101 Saint Petersburg, Russia § Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Wageningen, The Netherlands ∥ Physics Department, Saint Petersburg State University, 1 Ulyanovskaya ul., 198504 Petrodvorets, Saint Petersburg, Russia ‡

ABSTRACT: The structure of a mixed brush made of armgrafted polymer stars and grafted linear chains is investigated using the Scheutjens−Fleer self-consistent field method. It is shown that the mixing of stars and chains is thermodynamically favorable with respect to their lateral segregation in the brush. On the other hand, a segregation of linear and starlike macromolecules in the direction perpendicular to the grafting surface is observed. Conformations of stars and linear chains in the brush are determined by the overall molecular weight and the longest path length of linear and starlike macromolecules. Short linear chains occupy the space adjacent to the grafting surface and push the stars toward the brush periphery. In the case of long chains the interior of the brush is filled by the stars while the chains pass through the layer of the stars and expose their ends at the brush periphery. The most interesting is the intermediate situation where the linear chains have larger longest path but lower molecular weight than the stars. In this case, conformations of stars and linear chains depend essentially on the brush composition. When the amount of linear chains is small, they behave as effectively long, but larger fractions behave as short ones. The transition between two regimes is characterized by a bimodal distribution of linear chains and large fluctuation in the position of their free ends.

I. INTRODUCTION Polymer brushesmonolayers of polymer chains densely grafted onto impenetrable substratesbecame objects of intensive theoretical and experimental studies more than 30 years ago.1−3 Various synthetic and physical approaches were developed to create polymer brushes. Of no less importance, polymer brushes were recognized as constituent parts of more complex systems,1 in particular, as elements of self-assembled block copolymer micelles and supercrystalline structures,4 neurofilaments,5 endothelial glycocalyx,6 casein micelles,7 etc. Surfaces are decorated by polymer brushes to obtain certain desired properties such as improved biocompatibility, better barrier or antifouling properties of surfaces,8−11 or surfaces from hydrophobic to hydrophilic or vice versa.12 In recent years large interest and great expectations were linked to the synthesis and investigations of a new class of polymer brushes, where not “traditional” linear chains but macromolecules of more complex branched topologies are grafted onto a substrate. Examples of such branched macromolecules are combs, stars, and dendrons. Dendrons, or regularly branched treelike macromolecules, have a high molecular weight, a well-developed branched structure, and a large amount of terminal groups accessible for chemical modification (functionalization). This means that brushes made of dendrons, or dendritic brushes, have a large potential to functionalize or activate surfaces. It is noticeable that such a © 2015 American Chemical Society

massive macromolecule is tethered to the substrate via a single group, namely the end group of the root spacer. Theoretical and computer simulation studies of dendritic brushes have revealed their nontrivial internal structure. Already the analysis of the simplest dendritic brush made of the firstgeneration dendrons, which may be identified as polymer stars and the brush may be referred to as a “star brush”, has shown that at a moderate grafting density, when the overlap between neighboring molecules starts to become important, the brush acquires a “two-population” structure.13−15 The first population consists of stars with an almost completely stretched “stem”, or root spacer, and free arms that are extended toward the solvent (brush periphery) to balance the strong extension of the stem. The stars belonging to the second population are less extended and fill the space between the grafting surface and the imaginary plane where the branching points of the stars of the first population are located. As a result, the overall density profile decreases monotonously from the grafting surface to the periphery, thus resembling a parabolic-like profile of a brush made of linear chains, or “linear brush”. At low grafting density there is effectively only one, nonextended, population of stars. Received: February 18, 2015 Revised: March 16, 2015 Published: March 27, 2015 2263

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half of a domain in the case of a lamellar structure is nothing else but a polymer brush, where the domain interface plays a role of an impenetrable grafting surface).26−34 Another, principally different, way to make composite brushes is to combine chemically different polymers in a single brush. In this case the macromolecules that form the brush may have a different affinity to the solvent or to the grafting surface or may be partly or fully incompatible.12,35 These properties are exploited when a brush acts as a “smart surface” sensitive to variations of the environmental conditions like temperature, solvent quality, solvent composition (if the solvent is mixed), pH, etc.12,36−39 When, for example, a few admixed chains are added to an inert brush which serves as a matrix, the admixed chain can work as a sensitive molecular switch.40 In the present work we will consider the situation akin to a bidisperse brush with chemically identical chains differing only in molecular weight. The key difference is that the brush-forming macromolecules will have a different topology. In particular, in the current case, we will consider a mixed brush made of endgrafted linear chains and arm-grafted stars. We will see that an addition of linear chains to a brush made of arm-grafted star macromolecules, which in itself has a nontrivial internal organization, leads to remarkable new features of the resulting mixed brush. More specifically, we will focus on which chain will have its segments on the solution side and which one is found most predominantly near the substrate. Insight into how the rules are for this competition is important for the design of “smart” surfaces and end-graft modifications as explained already. The remainder of the paper is organized as follows. In section II we formulate the model of mixed star−linear brush and briefly outline the SF-SCF approach used in our study. We present the results in section III. In section IV we summarize our conclusions and discuss perspectives for the future work.

The two-population structure of the star brush was predicted and analyzed in detail by using the numerical solution of the selfconsistent field (SCF) equations obtained with the aid of the Scheutjens−Fleer (SF) approach13 and later observed in molecular dynamics (MD) simulations by Merlitz et al.14 These authors also developed a simple analytical theory (on the basis of a boxlike model for the star brush) that quantitatively described the results of the MD simulations.16 The analytical SCF theory of the planar brush of starlike macromolecules was developed by Zhulina et al.17 In brushes made of higher generation dendrons, a similar “multipopulation” structure is observed.13,18,19 Dendrons’ populations have the following distinctive features: At the extremities there are completely unstretched dendrons and dendrons with strongly stretched spacers; in between there are dendrons with stretched spacers of first, second, ..., jth shell and less stretched or unstretched spacers of (j + 1)th ... gth shell. In a single dendron, spacers in the same shell are all more or less equally stretched on average. For the planar dendritic brush Pickett has suggested a figurative name “dendrimer forest”.20 Indeed, a dendron itself resembles in a way a tree (dendron (δένδρoν) is a Greek word for “tree”), and a densely grafted array of dendrons looks like a forest. By analogy, a star brush resembles a “palm forest” because a single grafted many-arm star resembles a palm tree.15 Moreover, due to the two-population structure of a star brush, the palm forest consists of “palms” (extended population of stars having a stretched stem) and a “bush” composed of stars from the weakly stretched population. In the present work we suggest to extend the notion of a “palm forest” by “planting” some “grass” underneath. The role of the latter will be played by linear chains. Since end groups distribute throughout the brush, a certain fraction is buried deep within the brush and cannot play a functional role (for example, they are inaccessible for the fuctionalization with small ligand molecules). From this point of view, the addition of relatively short linear chains can fill the space adjacent to the grafting surface, and the end groups of the dendron (which one may wish to functionalize) will be more predominantly on the brush periphery, as it is desired. The idea of mixing different macromolecules in a brush is in a certain sense natural. First of all, in general various macromolecules forming a brush can have different molecular weights. Moreover, this situation usually arises by itself because it is not so easy to prepare a strictly monodisperse polymer sample; it always has a certain molecular weight distribution (MWD). This of course also applies to polymer brushes. On the other hand, one can intentionally create a polymer brush with short and long chains (the so-called bidisperse brush; the MWD in this case is bimodal). Analytical SCF theory of planar bidisperse and polydisperse brushes has been developed in pioneering works of Milner et al.21 and Birshtein et al.22 It was shown that a complete stratification of the locations of the free chain ends above the grafting surface takes place in the brush, according to the MWD. For a bidisperse brush this means a complete segregation of short and long chains’ ends (closer to the grafting surface and at the brush periphery, respectively). Another important conclusion concerns a thermodynamic preference of mixing chains with different molecular weight in a brush as compared to their segregation. These predictions were subsequently confirmed by Monte Carlo simulations23 and in experiments.24,25 Mixing of short and long chains in a brush is widely utilized in controlling the morphology of block copolymer supercrystalline structures (recall that a domain of a supercrystalline structure or

II. MODEL AND METHOD A. Model of Mixed Brush. Choice of the System Parameters and the Presentation Scheme. We consider a polymer brush made of starlike and linear macromolecules grafted to a solid planar substrate with grafting density σ (number of macromolecules per unit surface area) (Figure 1). The grafting

Figure 1. Schematic illustration of a mixed brush made of arm-grafted three-arm stars (green) and linear (red) macromolecules.

points of the stars and chains are randomly and homogeneously distributed along the substrate. The stars are composed of f chemically identical flexible arms, each consisting of a linear sequence of n spherically symmetric monomer units, which are freely jointed (arm is fully flexible). The total degree of polymerization of a star is N = f n; the length of the longest path connecting the root of the dendron to one of its terminal points is 5 = 2n. The linear chains are chemically identical to the stars (i.e., made of the same monomer units) and have a degree of polymerization M. The brush is immersed into an athermal solvent; in terms of the Flory−Huggins interaction parameter this corresponds to χ = 0. 2264

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Figure 2. Average position (height) of free ends of stars and linear macromolecules (solid lines) with their variations (as error bars) and the average height of all free ends (dashed line) and star branching point (dotted line) as functions of the fraction of linear chains q in a mixed brush composed of stars with f arms and linear chains with M = 2n = 200 monomer units in brushes with grafting density σ. The values of f and σ are indicated in each panel.

It is clear that the study of such mixed brush is multiparametric. This demands working-out a rational scheme for the presentation of our results. We choose a substitution scheme: We will consider mixed brushes at fixed total grafting area of stars and linear chains and fixed total number of macromolecules in the system. An increase of the fraction of linear chains, which we denote by q, corresponds to replacing starlike macromolecules by linear ones. When the macromolecules are grafted onto the surface of fixed area S, this means that the total number of grafted stars and linear macromolecules, Q = Qstar + Qlinear, is invariant, the grafting density σ = (Qstar + Qlinear) /S = constant, and q = Qlinear/(Qstar + Qlinear) is a control parameter determining the degree of substitution of stars by linear macromolecules (the fraction of stars is equal to 1 − q). The limits q = 0 and 1 correspond to one-component brushes: the pure star brush (q = 0) and the brush made of linear chains only (q = 1). We have performed SCF calculations for brushes composed of stars having f = 3−8 arms. The arm length n was taken equal to n = 100 in all calculations presented throughout the paper. This limitation is not crucial and does not affect the final result and conclusions about the structure of brushes composed of armgrafted stars as long as the star arms are flexible. In the majority of our calculations the chain length M was chosen to be a multiple of the arm length; again, this does not restrict the generality of our consideration. Intermediate chain length aliquant to n are briefly discussed in section IV. The grafting density σ was varied over a wide range. For each set of (f, M, σ) the composition of the mixed brush was varied over the full range 0 ≤ q ≤ 1. B. Scheutjens−Fleer Self-Consistent Field Method. To study the structure and thermodynamics of the mixed brush, we use the numerical lattice SCF approach developed by Scheutjens and Fleer.41 This approach has a long history as it was successfully used to model polymer brushes including star and dendritic brushes13,15,18,19 and polydisperse linear brushes.42

Many details of the SF-SCF approach can be found in the literature, in particular in the book of Fleer and co-workers;41 Its implementation for a brush of starlike macromolecules was recently described in ref 15; the generalization to our case of a mixed star-linear brush is straightforward. According to the symmetry of the problem, we will limit ourselves to use the onegradient version of the SF-SCF approach, and more specifically we will use the planar geometry. This means that the lattice sites are organized in planar layers; each layer is referred to with a coordinate z = 1, 2, ... normal to the grafting plane. Within a layer z the concentrations of all monomeric components are averaged; i.e., fluctuations in the x−y directions are ignored. In each z-coordinate, we apply an incompressibility condition. This implies that the sum of the volume fraction of the polymers plus the solvent adds up to unity. The chain model used in the SF-SCF theory is the freely joined chain (FJC). This means that two segments along the chain can freely choose between the six neighboring sites (unless the segment is next to the surface). Chain backfolding is allowed. This backfolding problem is on average counteracted by the incompressibility constraint. Unlike in the Gaussian chain model, the FJC has the property that the chains have a finite extensibility. This is important for our calculations as it occurs in our system that parts of the polymer stars (the arm that is used to graft the star onto the substrate, or the stem) become fully extended. The calculations were performed by using the “SFBox” program developed at Wageningen University which routinely solves the SCF equations with at least seven significant digits. The primary outcome of the calculations is structural quantities such as the equilibrium density profiles of all components, including (i) overall, (ii) star branching point, (iii) star ends, and (iv) linear chain ends density profiles. Additionally, thermodynamic quantities (e.g., the free energy) of the composite brushes are available. These are used below to test the propensity of the 2265

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Figure 3. Average height of free ends of stars and linear macromolecules with their variations (as error bars) and the average height of all free ends (dashed line) as functions of the fraction of linear chains q in a mixed brush composed of stars with f arms and linear chains with M = 3n = 300 monomer units with grafting density σ = 0.1 (a) and 0.3 (b).

of stars (q = 0) and chains (q = 1), in particular, the average position of the ends of the stars ⟨zse⟩ (q = 0) = ⟨ ztot e ⟩ (q = 0) and that of linear chains ⟨zle⟩ (q = 1) = ⟨ztot e ⟩ (q = 1). At M = 2n = 200 ⟨zse⟩ (q = 0) > ⟨zle⟩ (q = 1); hence, the average height of all free ends decreases when the stars are replaced by linear macromolecules, i.e., with increasing q (Figure 2). Let us call such linear chains “short” (and the stars, “long”). An opposite regime of “long” chains (and “short” stars) with ⟨zse⟩(q = 0) < ⟨zle⟩ (q = 1) and increasing ⟨ztot e ⟩ (q) dependence is shown in Figure 3. Consider now the characteristics of the mixed brush in the regimes of “short” and “long” chains. One can see that in spite of the above-discussed differences, there are obvious similarities between the dependences shown in Figures 2 and 3. Star and linear chain ends are well segregated (even the error bars corresponding to variances in zse and zle do not overlap), and the degree of segregation becomes larger as the grafting density and/ or number of arms in a star increase. With an increase in the amount of chains in the brush, the average height of ends of star and linear chains as well as the branching points of the stars change in the opposite direction than the average height of all end groups. In the short chain regime, zbp, zse, and zle increase with increasing q, whereas ztot e decreases. In the long chain regime, on the contrary, zbp, zse, and zle decrease while ztot e increases with increasing q. In the short chain regime the free ends of the linear chains are located on average below the free ends of the stars and the branching points, while in the long chain regime the free ends of the linear chains are located above those of the stars. In both regimes, with an increase in the content of stars (or chains) in the brush, the corresponding fluctuations of zbp and zse(or zle) increase. Fluctuations of ztot e are approximately the same over the whole q range (data not shown). These data (presented in Figures 2 and 3) show that in both regimes the “short” component of the two-component brush occupies the inner part adjacent to the grafting surface. The “long” component passes through this inner layer and organizes the peripheral part of the brush. In the next sections we will investigate the mixed brush structure in short and long chain regimes. We will also consider an intermediate regime, where the average height of the free ends in one-component brushes (grafted at the same grafting density) approximately coincide: ⟨zse⟩ (q = 0) ≈ ⟨zle⟩ (q = 1). B. Structure of Mixed Brush in the Short Chain Regime. As we have seen, in the short chain regime (Figure 2 presents results for M = 5 = 2n), linear chains get suppressed by the stars

mixed brush to segregation into domains which are rich in one and starved in the other species.

III. RESULTS A. Structure of Mixed Brush: Two Distinct Regimes. Let us first determine the possible modes of self-organization of mixed brushes by considering typical integral characteristics. Figures 2 and 3 show the dependences of the average position of free ends and branching points of the stars on the brush composition q. For the free ends, data for each species and all free ends are presented. They are calculated from the density profiles as follows. For end groups ⟨zes, l ,tot⟩

=

∑z zφes, l ,tot(z) ∑z φes, l ,tot(z)

(1)

where φse(z), and φle(z) are the density profiles of the end groups s l of stars and linear chains, respectively, whereas φtot e = φe + φe is the total density of end groups in the brush. In a similar way the average position of the branching points of the stars is calculated

⟨z bp⟩ =

∑z zφbp(z) ∑z φbp(z)

(2)

where φbp(z) is the density distribution of the branching points in the brush. The dependences for f = 3, M = 2n (that is, the chains are of the same length as the longest path in the star, M = 5 ) and two grafting densities σ = 0.1 and 0.3 are shown in Figure 2a,b; to illustrate the effect of increasing the number of arms in the stars, in Figure 2c we show the data for f = 8 and σ = 0.1. One can see that independently of the grafting density and the number of arms in a star, a similar picture emerges. For the larger chain length, M = 3n and f = 3 (this means that the linear chain has the same molecular mass as the star but is longer than the longest path in the star), we see from Figure 3 that the general picture is also independent of the grafting density (σ = 0.1 and 0.3). However, there is a marked difference between Figures 2 and 3. Increasing the chain length from M = 2n to 3n has a dramatic effect on the structure of mixed brush: The relative positionings of the ends of the stars and linear chains are exchanged, and the character of the compositional dependences changes to the opposite. It should be noted that the molecular parameters of stars and chains define all characteristics of “pure” one-component brushes 2266

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Figure 4. Overall (solid lines), star (dashed lines), and linear chain (dotted lines) density profiles in a mixed brush composed of stars with f = 3 arms and linear chains with M = 2n = 200 monomer units grafted at the density σ = 0.3 for the fraction of linear macromolecules q = 0.2 (a) and 0.8 (b). Long dashed curves in (a) and (b) give the overall polymer density profiles for the star brush and linear brush, respectively.

Figure 5. Distributions of free ends of linear chains (a) and that of stars (b), and branching points (c) in mixed brushes composed of stars with f = 3 arms and linear chains with M = 2n = 200 monomer units grafted with grafting density σ = 0.3 for various fraction of linear macromolecules, q, as indicated.

well depth at q = 0.8 (Figure 4b) is indeed very close to this minimum value, while at small q the well depth exceeds φstar,min, thus pointing to the partial extension of the transit chain and/or to the fact that not only the stem of the stars is present in the inner (“chain”) part of the mixed brush. The overall polymer density profile demonstrates no anomalies and has a typical shape: with increasing distance from the grafting surface the density decays monotonically. The comparison of the overall profiles for different values of q (Figure 4) reveals that the replacement of stars by linear chains with lower molecular weight leads to a decrease in both the brush density and the brush thickness (the upper bound of the profile, i.e., the minimal value of z at which the density becomes zero). Recall that at the same time the average position of a star and the chain ends demonstrate the opposite dependencethey increase with increasing q (Figure 2). Branching point and free ends distributions shown in Figure 5 behave also in full agreement with this picture. For the linear

of a larger molecular weight and hence are located close to the grafting surface. Examination of the individual contributions of stars and linear chains to the polymer density profile in the brush, φ(z) = φstar(z) + φlinear(z) shown in Figure 4 supports this view as well. The density profile of linear chains adjoins the grafting plane, and the density decays monotonously with increasing distance from the grafting plane. In contrast, the density profile for the stars has a characteristic “well” close to the grafting surface. With increasing q the height and the width of the chain profile and also the depth and the width of the star density “well” increase. The depth of the well near the grafting surface indicates that at large fraction of the chains the “transit” parts of the stars are strongly, almost extremely, stretched. Indeed, at maximum possible extension of the stems of the stars the volume fraction must acquire the minimum possible value, which is equal to the number of stars per unit area: φstar,min = σ(1 − q) (that is, φmin = 0.24 and 0.06 at q = 0.2 and 0.8, respectively). We see that the 2267

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Figure 6. Overall (solid lines), star (dashed lines), and linear chain (dotted lines) density profiles in a mixed brush composed of stars with f = 3 arms and linear chains with M = 3n = 300 monomer units with grafting density σ = 0.3 for the fraction of linear macromolecules q = 0.2 (a) and 0.8 (b). Long dashed lines in (a) and (b) correspond to the overall polymer density profiles for a one-component brush of star macromolecules and linear chains, respectively.

Figure 7. Distributions of the free ends of linear chains (a) and that of stars (b), and branching points (c) in mixed brushes composed of stars with f = 3 arms and linear chains with M = 3n = 300 monomer units with grafting density σ = 0.3 for various fraction of linear macromolecules, q, as indicated.

Therefore, the fact that short chains replace the stars of the lower, weakly stretched, population explains the observed increase of the height of the ends for both stars and linear chains with an increase in q. For chains this is a consequence of increasing “living space” for chains (best seen in Figure 4) as q increases. For stars, a reduction of the number of weakly stretched stars is equivalent to increasing the relative weight of the upper population stars and, correspondingly, its contribution to ⟨zse⟩. C. Structure of Mixed Brush in the Long Chain Regime. In the long chain regime (in Figure 3 results for f = 3 and M = 5 = 3n are shown) the ends of linear chains are located above those of the stars, and we encounter the situation that linear chains dominate over “short” stars. The analysis of the contribution of the density profiles of stars and linear chains to the overall profile (Figure 6) proves that the space close to the grafting plane is now predominantly occupied by stars. The profile of the linear chains

chains the distributions of the ends are unimodal; with increasing q it extends, and the position of the maximum shifts from the grafting surface toward the periphery. For the branched polymers, a “dead zone” in the distributions of ends and branching points appears near the grafting surface; the size of the “dead zone” grows with increasing fraction of linear chains. In other words, the short linear chains cut off the part of the profile adjacent to the grafting plane. Interestingly, the bimodal character of both the end point and branch point distributions of the stars corresponding to the two-population structure of the star brush at σ = 0.3 persists as q increases up to rather large values. The growth of the “dead zone” is implemented at the expense of the part of the profile that corresponds to the population of weakly stretched stars. The position of the peak on the branching points distribution corresponding to the almost fully stretched stem remains unchanged with increasing q. 2268

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Figure 8. Fraction of stars in the stretched population, α, in a mixed brush composed of stars with f = 3 arms and linear chains as a function of fraction of linear macromolecules q calculated using eq 3 (solid symbols) and eq 4 (open symbols). Linear chain length is (a) M = 2n = 200 and (b) M = 3n = 300; grafting density σ is indicated at each curve.

zmax corresponds to the full extension of the stem (zmax = n). The sum in the nominator is the area below the upper population peak, and the sum in the denominator is the total area below φbp(z) profile. Similarly, α can be estimated from the φse(z) distribution

features a near-surface well, and a maximum in the distribution of the free chains appears at the periphery of the brush. The well depth can also be compared to the minimum value φlinear,min = σq (that is, φmin = 0.06 and 0.24 at q = 0.2 and 0.8, respectively) corresponding to the maximum possible extension of the “transit parts” of the linear chains. We see from Figure 6 that the well depth is very close to φlinear,min at q = 0.2, whereas at large q the well is somewhat less deeper. At small and intermediate fractions of linear chains in the brush the overall profile is clearly divided into two parts: the inner part contains stars and stretched “transit” parts of linear chains while the outer part contains only “tails” of linear chains. As a result, the overall profile has a two-step staircase shape not with rectangular but rather very rounded steps. Branching points and end point distributions profiles evolve with increasing q differently than in the case of short linear chains, too (see Figure 7). For the profiles of end points of linear chains a “dead zone” near the grafting surface appears at q < 1, its width decreases with increasing q (this leads to the decreasing ⟨zle⟩(q) dependence; see Figure 3). The distribution for the branching point of the stars and the profiles of the free ends are squeezed from above as q increases. From the two-population structure point of view, the position of the maxima on the profiles (Figure 7b,c), related to extended stars, shifts toward the grafting surface; its absolute and relative contribution (i.e., the area below this peak and its ratio to the area below the whole profile) both decrease. Hence, we can conclude that linear chains substitute the stars that belong to the extended population. This explains the decrease of the average height of the free ends of the stars as q increases (Figure 3). D. Two-Population Structure. Let us now discuss in more detail how the substitution of stars by linear chains in the brush changes the two-population distribution of stars mentioned in the Introduction and discussed in refs 13−19. The fraction of stars in the extended, “upper”, population can be found from either the profiles of the branching points or the free ends. If the stars are divided into two populations, these distributions are bimodal, and the fraction of extended stars corresponds to the area below the “upper” (peripheral) peak. More specifically, the fraction of extended stars α can be quantified by α=

z max

z max

∑

φbp(z)/ ∑ φbp(z)

z = z min

z=0

z max

α=

z max

∑

φes(z)/ ∑ φes(z)

z = z min

z=0

(4)

where the lower limit of the summation z = zmin corresponds to the rightmost minimum of the φse(z) curve and the upper limit z = zmax corresponds to the limit of the full extension of the longest path in the star (zmax = 2n). Both ways of calculating α should give, in principle, comparable results (the differences are due to the discreteness of the lattice model and the diffusiveness of the boundary between two populations in the brush). Figure 8 shows the dependence of α on the brush composition q for brushes made of 3-arm stars mixed with short (M = 2n, panel a) and long (M = 3n, panel b) linear chains. One can see that the α(q) dependences exhibit opposite behaviors in these two cases in full accordance with the results presented above in Figures 5 and 7: the fraction of extended stars grows when the stars are replaced by short chains while substitution of stars by long chains leads to a decrease in α. Short chains, suppressed by the stars, occupy a position close to the grafting surface, thus replacing the stars of the lower, nonstretched, population; hence, α increases. When long chains are present in the brush, they pass through the layer of stars and place their ends at the brush periphery, thereby pushing the stars toward the grafting surface, out of the extended population. Therefore, we see that in the mixed brush the ratio of molecular masses and longest paths of stars and chains plays a crucial role in determining the character of organization of stars and chains in the brush. In the two cases considered up to this point we saw both a domination of stars over linear chains when the chains are short and a domination of linear chains over stars when the chains are long. It is to be noted that in the former case both linear chains and stars have the same longest path but the molecular mass of the star is larger, whereas in the latter case both the star and the chain have the same molecular mass but the chain is longer that the longest path in the star. That is, in each case either the star or the chain had an “advantage” over its “companion” in the brush in one characteristic (molecular weigh or the longest path length) and the “parity” in the other. In other words, the first case belongs to M ≤ 5 ≤ N situation while

(3)

where the lower limit in the summation z = zmin corresponds to the rightmost minimum on φbp(z) curve and the upper limit z = 2269

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Figure 9. Average height of free ends of stars and linear macromolecules (solid lines) with their variations (as error bars) and the average height of all free ends (dashed line) as functions of the fraction of linear chains q in a mixed brush composed of stars with f = 6 arms and linear chains with M = 3n = 300 monomers with grafting density σ = 0.1 (a), 0.15 (b), and 0.25 (c).

Figure 10. Overall (solid lines), star (dashed lines), and linear chain (dotted lines) density profiles in a mixed brush composed of stars with f = 6 arms and linear chains with M = 3n = 300 monomers at a grafting density σ = 0.2 for various fraction of linear macromolecules, q, as indicated. Long dashed lines in (a) and (c) show overall polymer density profiles for a brush of stars and linear chains, respectively.

the latter case to 5 ≤ N ≤ M. Moreover in both cases we considered “boundary situations” (equality instead of strict inequality), but it is obvious that the chain shorter than the longest path of the star (M < 5 ) will be suppressed by stars and

long and heavy chains such that M > N will expose free ends at the brush periphery, i.e., above the stars. Therefore, it is interesting to consider an intermediate case 5 < M < N where the advantage of the linear chain over the star in the longest path 2270

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Figure 11. (a−c) Number distributions of free ends of linear chains in a mixed brush composed of stars with f = 6 arms and linear chains with M = 3n = 300 monomers at grafting densities σ = 0.1 (a), 0.2 (b), and 0.25 (c), for various fractions of linear macromolecules, q, as indicated. (d) Fraction of transit chains as a function of the mixed brush composition q for various values of the grafting density as indicated.

of larger fraction of stars is necessary in order to get enough free space for placing the linear chains near the grafting surface. This picture is well supported by the density profiles of stars and linear chains shown in Figure 10. At a low content of linear chains (q = 0.2, Figure 10a) the profiles are similar to those presented in Figure 6: the segment density is depleted near the grafting surface. The chains are pushed to the periphery where their density profile has a maximum. As q increases (q = 0.4, 0.8), the contribution of the linear chains to the density profile becomes bimodal: it has both surface and peripheral maxima. This corresponds to two populations of linear chains: weakly and strongly stretched. Stars contribute to the density contribution at intermediate positions. Therefore, from this point of view the brush has a three-layer structure: there are two layers, surface and peripheral, enriched by linear chains, and the layer in between is enriched by stars. With increasing fraction of linear chains in the brush, the width of the first (interfacial) layer grows, the size of the second (middle) decreases, and the size of the third (peripheral) one changes weakly. The reason for the switching of positions in the brush of the linear chains is related to the fact that at small q overlapping of the stars is still high, and it is advantageous to allow the linear chain to pass through the layer of stars (it is quite easy to implement since the chain is linear and not branched), thus relaxing the crowding of the stars. However, when the fraction of stars is further decreased and the fraction of linear chains is increased, there is enough space close to the grafting surface to place light (twice lighter, in our case) linear chains there, in order to optimally fill the brush space. The emergence of two populations of linear chains in a mixed brush is well illustrated by the distribution of their ends (Figure 11a−c). At low grafting density (Figure 11a), linear chains “sit” near the grafting surface, and their end distribution is close to that one for short chains presented in Figure 5a. As the number of

will be counterbalanced by the advantage of the star over the linear chain with repect of the overall molecular weight. E. Brushes Composed of Multiarm (f = 6) Stars and Long (M = 1.55 ) Linear Chains: An Intermediate Chain Regime. Consider a mixed brush made of 6-arm stars and linear chains of M = 3n monomer units. In this case the star is twice as heavy as the chain while the chain has a 1.5 times longer “longest path” than the star. As before, we start with dependences of the average position of ends and branching points; these are presented in Figure 9 for various grafting densities. At sparse grafting (σ = 0.05) longer but lighter chains behave as effectively short (cf. Figure 2): the chains prefer to “hide” under the stars (the chains are “suppressed” by stars) at any composition of the mixed brush, the average height of the star and the chain ends above the surface increase with increasing q, and the average height of all end groups decreases with increasing q (although this decrease is weak because the difference in the brush height in two limits at q = 0 and 1 is rather small). Increasing σ changes this picture dramatically: at small amount of chains, it turns out that it is more preferable for the chains to pass through the layer of stars and place their ends above this “star layer”, at the brush periphery. However, surprising it may seem, in spite of small amount of linear chains, the greater longest path factor dominates here. As the fraction of linear chains increases, the situation changes cardinally: now the linear chains prefer to “sit” near the grafting surface and push the stars to the brush periphery. At the same time, in the range of the brush composition corresponding to switching between two regimes, the fluctuations of the position of the ends of the linear chains enhances. The nature of this effect will become clear later when we consider density profiles and chain ends distributions. As the grafting density σ increases, both the range of q where the chains dominate over the stars and the value of q at which the switching between two regimes occurs increase. This is because at dense grafting the brush density is very high, and substitution 2271

DOI: 10.1021/acs.macromol.5b00357 Macromolecules 2015, 48, 2263−2276

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Figure 12. Number distributions of free ends of stars for a mixed brush composed of stars with f = 6 arms and linear chains with M = 3n = 300 monomers at grafting density σ = 0.05 (a) and 0.2 (b), for various fraction of linear macromolecules, q, as indicated.

Figure 13. Number distributions of star branching points for a mixed brush composed of stars with f = 6 arms and linear chains with M = 3n = 300 monomers at grafting density σ = 0.05 (a) and 0.2 (b), for various fraction of linear macromolecules, q, as indicated.

other hand, if a small amount of stars is admixed to the brush, it perturbs the distribution of the chain ends making it (formally) bimodal (in Figure 11b,c we see that even 1% of stars is enough to produce such effect): there are two maxima, but these are not well separated (as compared, for example, to the distribution at q = 0.8). This results in the nonzero limit of β at q → 1 and the nonmonotonicity of the β(q) dependence (when the maxima were fully separated, then β would monotonously tend to zero). The distribution of free ends of the stars (Figure 12) exhibits the same tendency as the density profile of the stars (Figure 10): it is “squeezed” between two populations of surface and peripheral linear chains. With increasing q it shrinks from above and from below while the branching point distribution (Figure 13) shrinks only from below. However, the shrinkage does not violate the partition of stars between two populations (if the pure star brush initially has the two-population structure). The distribution of stars between two populations behaves nonmonotonously (Figure 14): the fraction of stars in the upper population, α, decreases at small q but increases at large q. At a low grafting density α increases monotonously. F. Free Energy and Stability of the Mixed Brush. Thus far we have assumed that the mixed brush is laterally homogeneous; i.e., grafting points of stars and linear chains are uniformly mixed on the grafting surface. For the experimental realization of such mixed brush it is important to know how realistic this assumption is. Nonideal mixing is likely to result in large regions rich in stars intermixed with regions rich in linear chains, and the vertical segregation does not make any sense. We realize that in a segregated, or demixed state, the grafting density of stars (ρs, the number of stars per unit area in the star-rich region) and that of linear chains (ρl, similarly) will likely differ

linear chains increases, the distribution broadens, and its maximum shifts to the brush periphery. With an increase in the grafting density, a second population of extended “transit” chains appears and the distribution becomes bimodal (Figure 11b,c). The range of the brush composition where the chain end distribution is bimodal corresponds to “switching” between these two regimes for the linear chains. As q increases, the height and the statistical weight of the “surface” maximum both increase while the weight of the peripheral maximum decreases. That is, a redistribution of the linear chains between two populations occurs. The distribution bimodality also explains the anomalously large fluctuations in the transition range of the brush composition q seen in Figure 9b,c. Upon integrating the distribution of the ends, one can calculate the fraction of of transit chains β. Figure 11d shows the result of this integration β as a function of the mixed brush composition. These dependences are in full accordance with the above discussion: there are no transit chains at any q in sparely grafted brushes; there are only transit chains at small q in densely grafted brushes, the range of q where only transit chains are found in the brush is broader for larger σ; a decrease of the transit chains fraction as q increases further. It is interesting that at moderate grafting density (σ = 0.15) “transit” chains are always found in coexistence with a population of weakly stretched “surface” chains, even at very small fraction of linear chains. Another point to notice is that β does not tend to zero as q → 1, as one might expect. Indeed, at q = 1 we have a brush made of linear chains which is characterized by a broad continuous distribution of chain ends (see the corresponding curves in Figure 11a−c), and it is impossible to set boundary between “surface” and “transit” chains. In other words, at q = 1 the value of β is indefinite! On the 2272

DOI: 10.1021/acs.macromol.5b00357 Macromolecules 2015, 48, 2263−2276

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1. Stars with f = 3 arms (N ≡ f n = 3n = 300) and linear chains with M = 2n = 200 units (5 = M < N). 2. Stars with f = 3 arms (N = 3n = 300) and linear chains with M = 3n = 300 units (5 < M = N). 3. Stars with f = 6 arms (N = 6n = 600) and linear chains with M = 3n = 300 units (5 < M < N). We see that the free energy of the mixed brush is a convex function of the brush composition in all presented cases, and the curve F(q) lies below the curve Fdemixed(q) (except the extreme points at q = 0 and q = 1 corresponding to pure star or linear brush). This means that the mixing of stars and chains is thermodynamically favorable, and therefore, the lateral segregation of stars and linear macromolecules in a mixed brush will not occur. Qualitatively very similar free energy dependences are observed in polymer brushes made of short and long chains.22 Figure 15 shows that this result is general, and the main conclusion is not affected by a variation of the number of arms of a star, the chain lengths, or the grafting density.

Figure 14. Fraction of stars in the stretched population in a mixed brush composed of stars with f = 6 arms and linear chains as a function of fraction of linear macromolecules q. Linear chain length is M = 3n = 300; grafting density σ is indicated at each curve. Values of α were calculated using eq 3 (solid symbols) and eq 4 (open symbols).

from σ, even when the overall grafting surface area remains invariant upon partitioning. Of course the ρs and ρl are related to the average grafting density in the brush as a whole as 1/σ = (1 − q)/ρs + q/ρl

IV. DISCUSSION AND CONCLUSIONS We have presented a detailed SCF study of the structure and thermodynamics of mixed brushes made of arm-grafted starlike and linear macromolecules. We have demonstrated that the mixing of star-branched and linear components in a single brush is advantageous at all proportions with respect to their partitioning in lateral segregated states. A detailed study of the structure of mixed brushes allowed us to establish that two important characteristics of linear and branched starlike macromolecules; namely, the overall molecular weight and the longest path length play a key role in determining the conformations of the stars and the linear chains in a mixed brush. For the stars, the overall molecular weight is the product of the number of arms and arm length, N = fn, while the longest path comprises two arm length, 5 = 2n. For the linear chain the longest path is equivalent to the degree of polymerization 4 = M. Depending on the ratios of these parameters, different scenarios are possible. We have considered the following cases: 1. N > M, 5 = 4 (short chains regime). Linear chains occupy the space adjacent to the grafting surface, thus pushing the stars toward the brush periphery and replacing, in fact, the stars of the lower (unstretched) population in the twopopulation structure.15 Close to the grafting surface there are “dead zones” for star ends and branching points. With increasing fraction of linear chains, the fraction of stars in the upper

(5)

The free energy per chain of demixed state is expressed via the free energies of “pure” star and linear brushes grafted at the densities ρs and ρl as follows Fdemixed = (1 − q)Fstar(ρs ) + qFlin(ρl )

(6)

The free energy, eq 6, should be minimized with respect to ρs and ρl taking into account the condition 5. By solving this constrained optimization problem with the aid of the Lagrange multiplier, we arrive at the following equation F ′star (ρs )ρs2 = F ′lin (ρl )ρl 2

(7)

where Fstar ′ and Flin ′ denote the derivative of the star and linear brush free energies with respect to the grafting density, respectively. From the system of eqs 7 and 5 we get equilibrium values of ρs and ρl, which give the equilibrium free energy of demixed brush, eq 6. Consider the dependences of the free energy in units kBT per one grafted macromolecule (star or chain) F on the brush composition q, in mixed and segregated states. Examples are shown in Figure 15. In accordance with the above discussion, three cases are considered:

Figure 15. Free energy per one macromolecule in units of kBT in a mixed brush (solid lines) and in the demixed state (dashed lines) as a function of the fraction of linear macromolecules q calculated at various grafting densities σ as indicated for mixed brushes with (a) f = 3, M = 2n = 200, (b) f = 3, M = 3n = 300, and (c) f = 6, M = 3n = 300. Dashed straight line in (a) indicates the linear dependence F = (1 − q)Fstar + qFlinear at σ = 0.4 for the sake of comparison. 2273

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Figure 16. Average height of the linear chain end as a function of the fraction of chains q in a mixed brush composed of stars with f = 3 arms and linear macromolecules with M monomers as indicated, for brushes with grafting density σ = 0.1 (a), 0.2 (b), 0.3 (c), and 0.4 (d).

moderate and high grafting density linear chains pass through the (dense) layer of stars and expose their free ends at the brush periphery (as in the case 2), if the fraction of linear chains is small. On the contrary, if linear chains are the major component of the brush, they occupy the space near the grafting plane pushing the stars to the brush periphery (as in the case 1). Transitions between two regimes occur at intermediate brush composition and are characterized by a bimodal distribution of linear chains and large fluctuation in the position of their free ends. Interestingly, at such intermediate composition of the mixed brush both stars and the linear chains are divided into two populations: one weakly and one strongly stretched. Hence, such mixed brush can may be referred to as a “double two-population” brush. Note that for the sake of convenience we have chosen above for specific lengths of the linear chain to be a multiple of the length of an arm of the star. The third and most interesting scenario with 5 ≤ M < N was obtained from the second one (5 < N ≤ M) by increasing the number of arms in the brushforming stars. Alternatively, we could implement the transition 5