Morphological Transformations in Polymer Brushes in Binary Mixtures

Oct 8, 2014 - Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway New Jersey 08...
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Morphological Transformations in Polymer Brushes in Binary Mixtures: DPD Study Jianli Cheng, Aleksey Vishnyakov, and Alexander V. Neimark* Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway New Jersey 08854, United States S Supporting Information *

ABSTRACT: Morphological transformations in polymer brushes in a binary mixture of good and bad solvents are studied using dissipative particle dynamics simulations drawing on a characteristic example of polyisoprene natural rubber in an acetone−benzene mixture. A coarse-grained DPD model of this system is built based on the experimental data in the literature. We focus on the transformation of dense, collapsed brush in bad solvent (acetone) to expanded brush solvated in good solvent (benzene) as the concentration of benzene increases. Compared to a sharp globule-to-coil transition observed in individual tethered chains, the collapsed-to-expanded transformation in brushes is found to be gradual without a prominent transition point. The transformation becomes more leveled as the brush density increases. At low densities, the collapsed brush is highly inhomogeneous and patterned into bunches composed of neighboring chains due to favorable polymer−polymer interaction. At high densities, the brush is expanded even in bad solvent due to steric restrictions. In addition, we considered a model system similar to the PINR−acetone−benzene system, but with the interactions between the solvent components worsened to the limit of miscibility. Enhanced contrast between good and bad solvents facilitates absorption of the good solvent by the brush, shifting the collapsed-to-expanded transformation to lower concentrations of good solvent. This effect is especially pronounced for higher brush densities.

I. INTRODUCTION Polymer brushes (PB) are formed by chain molecules grafted to solid surfaces. PBs are ubiquitous in nature (for example, they provide lubrication of joints1 and form periciliary layers that separate mucus layer from airway epithelia in lungs2) and find a broad range of practical applications, including stabilization of colloids and composites, mediating forces between solid, as cushions for supported lipid bilayers,3 modification of chromatographic supports,4,5 ion-selective electrodes,6 functionalization of surfaces for biomedical applications,7,8 and various particle and nanoparticle technologies.9 PBs respond to changes in chemical and physical environment (such as temperature, pH, solvent composition). In a thermodynamically good solvent the grafted chains are solvated and the brush swells. Depending on grafting density, the chains are stretched and form a layer of polymer solution at the substrate, which is called “expanded” state. In a thermodynamically poor solvent, the brush collapses into a dense melt-like layer on the substrate. Transition between the collapsed and expanded states is analogous to the globule−coil transition in a single tethered polymer chain. The difference is that polymer brushes are dense. That is, the distance between the attached chains is much smaller than the chain length, and therefore the structure of polymer brushes strongly depends on the grafting density. The other factors governing brush solvation are interactions of polymer with solvent and solid substrate. Conformations of chains in a brush affect the interactions of polymer-grafted surfaces with other bodies, such as solid surfaces, colloidal particles, proteins, and macromolecules. © 2014 American Chemical Society

Understanding of transitions between collapsed and expanded configurations is therefore of critical importance for practical applications, and significant efforts were dedicated to this phenomenon. Collapsed-to-expanded transformations in PBs were extensively studied both experimentally10,11 and theoretically with molecular dynamics,12,13 Brownian/Langevin dynamics,14,15 and dissipative particle dynamics (DPD)16 simulations, as well as with density functional theory (DFT)17 and self-consistent field theory (SCFT).18,19 It was found that collapsed brushes in bad solvents can be uniform (the polymer forms a homogeneous film adsorbed at the substrate) or patterned. Several regular and irregular pattern morphologies such as “octopus micelles”,14,20 “pancake micelles”,21 or stripes21,22 were observed in various systems depending on their chemistry and composition. In a singlecomponent solvent, a transition to an expanded configuration may be caused by a temperature increase, or chemically: for example, several theoretical studies of polyelectrolyte brushes16,18,23−28 including a DPD study with explicit electrostatic forces29 focus on the effect of charge density on the chains, which is controlled by the pH. Recent simulation studies of polyelectrolyte explored the effect of counterion charge15 and brush reaction to external electric fields.13 In multicomponent solvents, the solvent quality is controlled by the solvent composition: the brush transits from a collapsed Received: May 22, 2014 Revised: October 7, 2014 Published: October 8, 2014 12932

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composites,44,45 and even single-wall carbon nanotubes.46 Acetone (A) and benzene (B) are common solvents, which are mutually miscible at ambient conditions; acetone is a bad solvent47 and benzene is a good solvent48 for PINR.

to an expanded configuration as the fraction of good solvent in the solvent mixture increases. Because brushes are dense, they may selectively absorb a particular component, even when both components are miscible. This effect can substantially affect polymer configurations compared to those in single-component solvents. Although multicomponent, particularly, binary, solvents are present in most of the practical systems, the theoretical literature on PBs in binary solvents is rather sparse.11,17,30−39 PBs in binary solvents were studied by SCFT and density functional theory (DFT), but particular experimental systems were not targeted. The conditions of collapsedto-expanded transitions were found to depend on the brush grafting density, polymer molecular weight, and specifics of polymer−solvent and solvent−solvent interactions. In some systems, the conformational transitions were found to be accompanied by sharp changes of solvent composition inside the brush.35,36,38−40 Using de Gennes’ mean-field theory,41 Birshtein and Lyatskaya32 considered a brush in a bad solvent A, to which a good solvent B, miscible with A, was gradually added. The transition between collapsed and swollen states was analogous to that in a single-component solvent with changes in temperature or solvent quality. Later, Birshtein et al.35 considered solvents that had low mutual solubility. They found that the brush exhibited a first-order compositional transition characterized by spontaneous increase in height and adsorption of a good solvent. The authors also examined a conformational transition induced by an external deformation of the brush.35,37 Lyatskaya and Balasch34 considered by SCFT water sorption from oil−water mixture by dense hydrophilic brushes including a weakly acidic polymer that was allowed to dissociate. The water concentration in the mixture was shown to control the effective brush height, and the local density of polymer segments has a maximum at the interface that formed between the hydrophilic layer that consisted of water and hydrophilic polymer and the hydrophobic oil bulk. Amoskov et al.42 also theoretically modeled polymer brushes in binary solvents of limited miscibility. The equilibration between the brush and the solvent bulk was restricted using a membrane. Borowko and Staszewski17 used a classical DFT model with Lennard-Jones (LJ) interactions between polymer and solvent particles. The authors focused on selective adsorption of a better solvent by PB, whose condition was characterized by an effective brush height. Polymer length was fixed to 18 LJ monomers, while brush density, solvent composition, and depth of LJ potential between solvents, polymer, and substrate were varied. No spontaneous transitions between the collapsed and extended configurations were observed. Interestingly, the brush height showed a non-monotonic variation with the solvent composition, which was explained by the balance of solvent−polymer attraction and entropic repulsion between the polymer chains. In this paper, we investigate with dissipative particle dynamics (DPD) simulations the behavior of PBs of different grafting densities in a binary mixture of good and bad solvents. We explore the transformation of PB structure and the alteration of solvent concentration within the brush, as the solvent composition changes. Calculations are performed for a realistic system, polyisoprene natural rubber (PINR) in acetone−benzene mixture, which provides an opportunity for correlation with experiments. Polyisoprene chains are commonly used for functionalization of particles of various sizes and chemistry, including macroscopic silica particles applied in rubber tires,43 polymeric and metal particles for nano-

II. MODELS AND METHODS In a setup standard for DPD simulations,49,50 we represented the polymer as a sequence of soft beads connected by harmonic bonds, (B) (B) is the bond strength, rij is the F(B) ij = −K (rij − re)(rij/rij), where K distance between beads i and j, and re is the equilibrium interbead distance. The system dynamics and equilibration is monitored by solving the Newton equations of motion for the beads, which interact through pairwise short-range conservative repulsion, bond, drag, and random forces.

dri/dt = vi fi =

∑

midvi/dt = fi (F(C) ij

(D) (R) + F(B) ij + Fij + Fij )

(1)

j

Random and drag forces are implemented in a standard selfconsistent manner: (R) F(R) ij (rij) = σw (rij)θij(t )rij/ rij

and

(D) F(D) ij (rij, vij) = γw (rij)θij(t )(rij(vj − vi))/ rij

(2)

where θij(t) is a randomly fluctuating variable with Gaussian statistics, and the weight functions are linked though the fluctuation−dissipation theorem, w(D)(r) = [w(R)(r)]2, σ2 = 2γkT. We used standard weight functions w(D)(r) = (1 − r/Rc)2 at r < Rc, w(D)(r) = 0 at r ≥ Rc. The balance between drag and random forces serves as the Langevin thermostat that maintains constant temperature. Short-range conservative forces Fij(C) account for nonbonded interactions; they are represented conventionally as short-range harmonic repulsion F(C) ij = aIJ (1 − rij / R c)(rij/ rij) at rij ≤ R c

F(C) ij = 0 at rij > R c

(3)

where Rc is the effective bead diameter. The repulsion parameter aIJ depends on bead types I and J to which beads i and j belong. We use the most common formulation of the DPD method,50 which implies that the same effective diameter Rc for all system components and the same intracomponent repulsion parameter aII = aAA = aBB = aPP. The reduced density ρ* of DPD beads (the average number of bead centers in 1R3c ) is set to ρ* = 3, and the friction coefficient γ is set to 4.5 as recommended in ref 50 (note that dynamic properties are not considered in this paper). The system parameters were customized to mimic the properties of PINR, benzene, and acetone. Bead Diameter Rc and Intracomponent Repulsion Parameter aII. In most of the DPD simulations, the system parametrization is based on the DPD parameters for water recommended in the original paper of Groot and Warren.50 Here, we deal with a purely organic system. Keeping in mind the DPD water parameters as a reference, we derived the coarse-grained parameters of the system under consideration from the properties of organic compounds involved. We model PINR as a chain of beads P representing one monomer and the solvent with beads A and B representing one molecule of acetone and benzene, repectively. The bead diameter Rc was determined from the effective volumes of PINR monomer, benzene, and acetone molecules provided the reduced system density of 3. These volumes, calculated from the densities at ambient conditions and presented in Table S1 of Supporting Information, are relatively close and yielded the effective Rc values ranging from 0.71 to 0.76 nm. The bead size was chosen as Rc = 0.71 nm. This value corresponds to the bead diameter of water coarse-grained as four water molecules per bead.50 The intracomponent repulsion parameter was estimated from the ambient isothermal compressibility κT of the solvent by interpolation of experimental value onto the reference correlation between 1/κT and aII50 obtained in a separate series of DPD simulations (the reference 12933

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curve for Rc = 0.71 nm is given in Supporting Information to ref 51). This procedure produced values of aII = 36kT/Rc for acetone and aII = 48kT/Rc for benzene. Because the values are relatively close and compressibility has to be reproduced only approximately, we chose the mean value of aII = 42kT/Rc. Rigidity of Polymer Chain. The rigidity of molecules may substantially influence the solution properties.52 The chain rigidity determines the persistence length and therefore affects PB’s height and gyration radius. The bond strength K(B) and equilibrium distance re of the DPD model were fitted to semiquantitatively reproduce the chain conformations in the atomistic molecular dynamic (MD) simulation. Fitting DPD parameters to MD results on molecular structure is rather common (e.g., refs 53,54). To this end, we performed NPT MD simulations of PINR melt. The details are described in Supporting Information, section SII. PINR is a rubbery polymer, liquid-like at ambient conditions. We simulated a melt of PINR oligomers, each of which contained 16 monomers, using OPLS united-atom force field55 and M.DynaMix software.56 In DPD simulations, each oligomer was modeled as a linear chain of 16 beads. From MD trajectory saved to the disk, we calculated the center of mass of each monomer and obtained the distribution of distances between the neighboring monomers (1−2 distance), second and third neighbor monomers separated by one (1−3 distance) and two (1−4 distance) monomers. The obtained distributions are shown in Figure 1. It is clear that the

Table 1. Parameters of the Coarse-Grained Models for Systems I and IIa Short-Range Conservative Repulsion Parameters aIJ, kT/Rc Bead type S P A

S

P

A

42.0

42.0 42.0

42.0 49.0 42.0

B N = 100

B 42.0 43.5 42.0 43.5 49.0 42.0

(s. (s. (s. (s.

I) II) I) II)

Chain Length and Bond Parameters K(B) = 120 kT/Rc2 re = 0.49 nm

a

Denotation of beads: S - substrate, P - PINR monomer, A - bad solvent, and B - good solvent.

technique for given binary system of A and B beads to the experimental value. The correlation between aAB and IDAC is described in Supporting Information, Section S−IV, Figure S3. From this fitting we obtained the mismatch parameter ΔaAB = aAB − aII = 1.5 kT/Rc, and respectively, aAB = 43.5kT/Rc. For polymer−solvent interactions, IDAC values were not found in the literature, but the dependences of solvent activity on concentration a(c) were given in ref 48 for benzene and ref 47 for acetone. Benzene and PINR are miscible, and a(c)is accurately described by the FloryHuggins (FH) equation of state with a single FH parameter χBP = 0.42 independent of c (Figure 2). We used a standard correlation ΔaBP =

Figure 1. Model of polyisoprene natural rubber (PINR) and distributions of the distances between the near neighbor r1‑2, second neighbor r1‑3, and third neighbor r1‑4 monomers in PINR oligomer obtained in MD simulation (lines) and DPD (points).

Figure 2. Dependence of solvent activity on composition for PINR− acetone (open symbols, dashed line) and PINR−benzene (closed symbols, solid line) mixture at T = 298 K and P = 1 atm. Experimental results from refs 47,48 are fitted with the Flory − Huggins model and Monte Carlo simulations. aPA = 49kT/Rc produces the best fit to experimental data for PINR−acetone mixture.

maximum probability for the effective 1−3 (second-neighbor) and 1− 4 distances are not commensurate with nearest neighbor (1−2) distance: r1‑3 < 2r1‑2, r1‑4 < 3r1‑2. Furthermore, both distances show two maxima, indicating a complex conformational behavior. This means that straight conformation is not preferential even at the small scale, and therefore the persistence length is very short, less than 8 covalent bonds that separate the centers of second-neighbor beads. This behavior of PINR is dissimilar to alkanes and other simple chain molecules, such as perfluoroalkanes (for which these distributions are given for comparison in Supporting Information, Figure S1). Because of the short persistence length and inability of the standard models with second-neighbor bonds and covalent angles52 to reproduce the complex shape of PINR, we decided to use only the nearest neighbor 1−2 bonds with the parameters K(B) and re fitted to the MD results; see Table 1 and Figure 1. Intercomponent Repulsion Parameters. The parameter of short-range repulsion between acetone and benzene, aAB, was calculated from experimental value of the infinite dilution activity coefficient (IDAC) of benzene in acetone, γ∞ = 1.70,57 using the method suggested by us earlier.58 This method is based on fitting the theoretical IDAC determined by the Monte Carlo Widom insertion

χBP /0.29259 to obtain benzene−polymer parameter aBP = 43.5kT/Rc. It is worth noting that the correlation between χ and Δa was obtained from best correspondence of FH and DPD phase diagrams of monomeric mixtures rather than polymer solutions. For polymers, the agreement between the two models is not as good. For acetone, which is a bad solvent, the FH parametrization was not possible, because the FH coefficient χAP estimated by Booth et al.47 drastically increased with the fraction of polymer in the mixture, and our attempts to fit the experimental data with the FH equation did not produce a satisfactory fit and underestimated acetone solubility in PINR (Figure 2). For this reason, we performed the parametrization using the MC simulation with Widom insertions60 of an acetone bead into the DPD model of the solution of 60-mer PINR oligomers and acetone. We performed several series of simulations with different polymer−solvent compositions and varying aAP. We found that with aAP = 49 kT/Rc, the coarse-grained model described the experimental activities pretty well, and the solubility limit is also in good agreement 12934

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Figure 3. Dependence of normal (a) and lateral (b) gyration radii of polymer chains on the good solvent mole fraction xB in the equilibrium bulk mixture for System I. Note gradual collapsed-to-expanded transformations in polymer brushes compared to the stepwise globule−coil transition in single tethered chains. The standard deviation of the normal radius of gyration for single chain ranges from 0.1 in pure acetone to 0.8 in pure benzene.

Figure 4. Snapshots of PINR conformation in binary solvent (solvent not shown) in System I. (a) globular single tethered chain, xB = 0; (b) coiled single chain, xB = 1; (c) low density brush (ρs = 0.12 nm−2) in collapsed state, xB = 0. The polymer layer is inhomogeneous with some areas of substrate exposed to the solvent (d) semiexpanded brush at xB = 0.4 (e) fully expanded brush in good solvent, xB = 1 (f) dense brush (ρs = 0.88 nm−2) forms a uniform polymer layer in bad solvent, xB = 0 (d) dense expanded brush (ρs = 0.88 nm−2, xB = 1). Rc, which corresponds to brush densities ρs of 0.12, 0.32, 0.50, and 0.88 nm−2. That is, in the least dense brushes, the distance between the chains was compared to the radius of gyration of an individual tethered chain placed in a good solvent, while in the densest brushes this distance was comparable to the bead diameter. The simulation box was rectangular, and periodic boundary conditions were applied in all three dimensions. In z direction normal to the substrate, box size was 60Rc, that sufficiently exceeded the height of polymer brushes to ensure that a uniform layer of solvent existed between the brush and the lower face of the substrate. Other details of DPD simulations, including the scheme of the simulation setup, may be found in Supporting Information, section S−V.

with the experimental data. The details of respective MC simulations are described in the Supporting Information, section S−III. The model with the parameters fitted to experimental parameters of the PINR−acetone−benzene system will be referred to as System I. We have to note that the solvents are very miscible. In order to explore how an increase of the contrast of solvent quality affects the polymer brush solvation, we also considered a model system where the solvents are more dissimilar yet still miscible. According to ref 59, as well as our own DPD simulations, the maximum of the mismatch parameter Δa, at which a symmetric mixture shows no phase separation, is close to 7.5 kT/Rc. Therefore, we modeled the mixture of two dissimilar yet still miscible solvents employing aAB = 49kT/Rc and all other parameters remaining the same as in System I. This system is referred to as System II. All system parameters are summarized in Table 1. Simulation Setup. The solid substrate was formed by seven parallel layers of immobile beads at an effective density of 19.3 Rc−3. aS = 42 kT/Rc parameters were assigned to substrate interactions with all other bead type. Because of the high density of the substrate, it effectively repelled all other beads and had no preference for either a polymer or solvent. The polymer chains that form the brush consisted of 100 beads (which corresponds to the molecular weight of 6800 Da) and were attached to the upper face of the substrate in a square lattice order. The substrate area was about 30 × 30 Rc2 (21.3 × 21.3 nm2). The distance between the neighboring chains was varied from 4 to 1.5

III. COLLAPSED-TO-EXPANDED TRANSFORMATION IN POLYMER BRUSHES Figure 3 shows the dependence of the normal (R⊥) and lateral (R∥) radii of gyration for PINR polymers in acetone−benzene mixture (System I) at different brush densities and solvent compositions. A single PINR 100-mer-chain tethered to the substrate serves as a reference. The single chain exhibits a characteristic transition from a globule at the substrate surface to a coil, signified by the increase of average radii of gyration: 12935

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Figure 5. Density and solvent composition profiles for System I at brush density of 0.88 nm−2. Density profiles for polymer and solvents are displayed for xB = 0 (a), xB = 0.1 (b), and xB = 0.9 (c). Panel (d) displays solvent composition profiles yB(z) = ρB(z)/(ρB(z) + ρA(z)) for xB = 0.1, 0.5, 0.9. z = 0 plane contains the centers of the beads forming the top facet of the substrate.

R⊥ from 0.73Rc in pure acetone (benzene mole fraction in the solvent bulk xB = 0) to 1.91 Rc in pure benzene (xB = 1). The transition in the binary solvent is analogous to that in a onecomponent solvent with increase in temperature, for example. Normal and lateral radii of gyration R⊥ and R∥ grow monotonically with xB. The inflection point corresponds approximately to xB = 0.55. The snapshots of the collapsed and expanded configurations of single tethered chains are given Figure 4a,b. Very different picture is observed for low-density polymer brushes. At xB = 0, the average normal radii of gyration R⊥ of polymer chains at grafting densities of 0.12 and 0.32 nm−2 are about 1Rc, which suggests that the brushes are collapsed at the surface. The chains, however, are extended in lateral direction: at ρs = 0.12 nm−2, R∥ for a brush collapsed in pure benzene exceeds that for a single globular chain at xB = 1 (Figure 3b). Instead of forming individual globules, the chains are spread over the surface, forming bigger aggregates (snapshots of which are given in Figure 4c), apparently similar to the patterned configurations mentioned in the Introduction. The size of our simulation cell is not sufficient to identify the pattern. We can only say that at low brush densities the polymer does not cover the entire surface; for example, a “parting” is seen in Figure 4c. As the quality of solvent improves with the increase of the benzene fraction xB, R⊥ grows slowly, while R∥ displays a minor, but sharp decline (Figure 3b), indicating that PINR chains become less stretched along the surface. Gradually, the brush forms globules, each of which is composed of several chains and is swollen with benzene (Figure 4d), before finally transforming into a fully expanded configuration (Figure 4e). The presence of multiple chains in globules apparently delays the collapsedto-expanded transformation, which is less pronounced and happens at higher xB compared to the coil−globule transition of an individual tethered chain (Figure 3a). At the same time,

almost no chain expansion in the lateral direction is observed until xB approaches 0.9 (Figure 3b). The fully expanded configurations at low grafting densities show no pronounced interface between the polymer layer and the solvent bulk: the local density of polymer beads decreases monotonically with distance to the substrate. As the brush density increases, the collapsed-to-expanded transformation becomes even less pronounced (Figure 3a). For the denser brushes, the transition point could not be identified. Nevertheless, it appears that transformation from a collapsed to an expanded state is shifted toward lower xB than in low-density brushes. Dense brushes form melt-like uniform rather than patterned layers at low xB, with sharp interfaces between the polymer and solvent (Figures 4e). The density profiles in Figure 5a,b show practically constant polymer density within the brush layer, except for very short distances where we observed layering of polymer beads at smooth regular substrate surface. Adsorbate layering is a common phenomenon in adsorption, but cannot be adequately modeled with crude coarse-grained models employed in this work. In any case, the layering disappears at z > 1.5Rc, which is insignificant, compared to brush height. As xB increases, the brush preferentially absorbs benzene from the solvent bulk. This is demonstrated by the density and the solvent composition profiles (yB is the number of benzene beads related to the total number of solvent beads at particular z) in Figure 5: the fraction of good solvent inside the brush drastically exceeds that in the solvent bulk. The brush expands (Figure 4e,g) and polymer−solvent interface becomes less pronounced (Figure 5c). While R⊥ grows monotonically, R∥ slowly declines (Figure 3b) indicating chain stretching in normal direction. As the chains expand and become coil-like, the entropic repulsion between the neighboring chains contributes to stretching and brush height increase. The selectivity of solvent sorption in the 12936

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brush layer toward benzene can be estimated as 10.2 at xB = 0.1 and decreases as the solvent quality improves and the brush swells, because the two solvents interact favorably with each other. However, the solvent composition profile inside the brush is never truly constant (Figure 5d): in all systems, the benzene fraction of solvent monotonically declines from the substrate toward the bulk, despite a weak maximum in absolute benzene concentration being observed at the interface between the brush layer and bulk solvent (e.g., Figure 5b) which is more pronounced in System II.

increases, brushes show a stronger swelling and chain stretching, and the collapsed-to-expanded transformation shifts to lower xB, due to absorption of good solvent by the brush being strongest in densest brushes. At ρs = 0.88 nm−2, the change in solvent−solvent interaction causes the increase of the adsorption selectivity coefficient from 10.2 to 38.3 at xB = 0.1 (Figure 8a). The collapsed-to-expanded transformation in the brush occurs between xB = 0 and xB = 0.2. After that, polymer chains extend slowly as the overall solvent quality improves further. The lateral radii of gyration in System II remain practically constant in the entire range of solvent compositions. The density and solvent concentration profiles in System II are qualitatively similar to those in System I: in bad solvent the densities of components inside the polymer layer are constant and the interface with the solvent is well-defined. The fluid becomes inhomogeneous throughout the polymer layer as the solvent quality improves (Figure 8a). In System II, the density of good solvent shows a well-pronounced maximum near the interface between the polymer layer and the solvent bulk (Figure 8a), showing a certain degree of affinity of benzene to the brush−solvent interface. This effect is much stronger than in System I; it is worth noting that published theoretical studies34 showed no surface activity of good solvent, for either miscible or immiscible solvent pairs. The maximum of good solvent density, however, does not correspond to a maximum of solvent composition, which decreases monotonically with z (Figure 8b).

IV. INFLUENCE OF THE INTERACTIONS BETWEEN THE GOOD AND BAD SOLVENTS Unlike real acetone and benzene, the solvents in System II are barely miscible. As a result, the structure of the solvent bulk is locally inhomogeneous with “domain” of clustered A and B beads, although the mixture exhibits no macroscopic phase separation. While the interaction between solvents has almost no effect on the behavior of the individual chains, it strongly affects the collapsed-to-expanded transition in the brushes, especially in the dense ones: as the solvent−solvent interactions become less favorable, the absorption of good solvent by the polymer layer becomes much more pronounced. As a result, considerable brush swelling is observed even at low values of xB (Figures 6, 7). The transformation between collapsed and

V. CONCLUSION The results obtained with DPD modeling of PINR brush in benzene−acetone mixtures demonstrate the specifics of conformations of polymer brushes in binary solvents. The experimentally informed parametrization of the model provides an opportunity to test the conclusions obtained from the modeling against respective measurements once the latter become available. We found that, compared to the globule−coil transition in single chains, the collapsed-to-expanded transformation in PB occurs gradually and at the higher fractions xB of good solvent, especially at low grafting densities. This finding is in contrast with the earlier mean field modeling studies32 that predicted that the conformational transformations in brushes are rather similar to globule−coil transitions. The ability of chains to merge, forming bunches, “octopus” micelles, stripes, or other patterns, effectively stabilizes the collapsed state. The entropy gain per chain is however limited, since neighboring chains restrict available conformations of each other, effectively destabilizing the stretched state and therefore shifting the PB expansion toward higher xB. Same effect could be seen in the theoretical studies of Birshtein and Lyatskaya.29 Worsening of interactions between the solvent components shifts the collapsed-to-expanded transformation in the opposite direction toward lower xB, due to absorption of the good solvent within polymer chains. In collapsed-to-expanded transformations in PBs, just as in globule-to-coil transitions in single chains, the gain in entropy overcomes cohesive forces between the polymer segments. The difference between a single chain and a brush may be looked upon as the difference between a chain and an array of interacting chains. The interchain interference facilitated by adsorption of good solvent makes the chain expansion a collective process that occurs gradually without a prominent stepwise transition. Rounded collapsed-to-expanded trans-

Figure 6. Snapshots of PINR conformation in binary solvent (solvent not shown) at ρs = 0.12 nm2: (a) System I, xB = 0.8; (b) System II, xB = 0.35.

Figure 7. Dependence of the normal gyration radius R⊥ on bulk solvent composition and grafting density for System II.

expanded brush configurations is also observed at lower xB compared to the brushes of the same grafting density in System I. This effect is visible even for lowest brush densities. For example, at xB = 0.35, the chains of the least dense brush (ρs = 0.12 nm−2) in System II have the same Rg as they have at xB = 0.85 in System I. From comparison of the snapshots for these two systems (examples given in Figure 6), it appears that the polymer layers in System II are more uniform compared to similar configurations in System I. As the grafting density 12937

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Figure 8. Density and solvent composition profiles for System II with ρs = 0.88 nm−2. (a) Density profiles for polymer and solvents at xB = 0.1. (b) Solvent composition profiles yB(z) = ρB(z)/(ρB(z) + ρA(z)) for xB = 0.05, 0.33, 0.88. z = 0 plane contains the centers of the beads forming the top facet of the substrate.

binary solvents. S−VI: Snapshots from DPD simulations of polymer brushes and overall radii of gyration. This material is available free of charge via the Internet at http://pubs.acs.org.

formations in PB may be explained by a lower free energy barrier (per chain in the brush) associated with chain expansion, compared to globule−coil transitions. The results of this work have important implications for various systems of practical relevance. In particular, the behavior of PB in binary solvents is of a special significance for chromatographic separations on polymer grafted stationary phases. In particular, in the gradient elution chromatography,61,62 the separation is controlled by varying the solvent composition, which affects adsorption of solute components. The effect of solvent composition is expected to be more pronounced for separation of polymers, proteins, or nanoparticles, elution of which is restrained by their adhesion to grafted substrates. In this process, the collapsed-to-expanded transformation of grafted chains upon variation of solvent composition may significantly influence the solute partition between mobile and immobile phases. The problem of interactions between polymer-grafted nanoparticles on various stages of nanocomposite processing represents a challenging and practically important area for the extension of this work. While the length of grafted chains is smaller than the particle size and the particle concentration is sufficiently low, the qualitative conclusions of this work remain intact. However, the nanoparticle size and concentration bring about additional scales that may change the qualitative behavior leading to nanoparticle aggregation induced by the conformational changes of grafted chains. Another interesting set of problems that involve conformational transformations in PBs includes the interactions of grafted nanoparticles with substrates, lipid bilayers, and cells. The experimentally informed approach to the system coarse-graining and parametrization suggested in this work lays out a foundation for developing practically relevant DPD models for these much more complex systems.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by NSF through the GOALI grant “Multiscale modeling of adsorption equilibrium and dynamics in polymer chromatography” and the CBET grant “Adhesion and Translocation of Nanoparticles through Lipid Membranes”.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

S−I: Properties of pure compounds and resulting R c parameters. S−II: Details of MD simulations of skeleton rigidity. S−III: Experimental data on solvent interactions with PINR and modeling with UNIFAC Free Volume and FloryHuggins equations. S−IV: Details of obtaining acetone-PINR conservative repulsion parameters using Monte Carlo simulations. S−V: Details of DPD simulations of polymer brushes in 12938

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