Co-Nonsolvency Response of a Polymer Brush: A Molecular

Article Cite This: Macromolecules 2019, 52, 4120−4130

pubs.acs.org/Macromolecules

Co-Nonsolvency Response of a Polymer Brush: A Molecular Dynamics Study Andre Galuschko*,† and Jens-Uwe Sommer*,‡ †

Leibniz-Institut für Polymerforschung Dresden, Institute Theory of Polymers, Hohe Strasse 6, 01069 Dresden, Germany Institute for Theoretical Physics, TU Dresden, Zellescher Weg 17, 01069 Dresden, Germany

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‡

ABSTRACT: Using molecular dynamics simulations, we study the response of a polymer brush exposed to co-nonsolvent (CNS), which acts as a preferential solvent for the polymer. We investigate a broad range of attractions between CNS and monomers and of grafting densities over the full range of cosolvent volume fractions. We compare our simulation results with the recently proposed adsorption−attraction model for co-nonsolvency in polymer brushes. The brush layer collapses with increasing CNS concentration into a more compact layer and followed by a reswelling toward sufficiently high CNS concentrations. As the strength of attraction of CNS toward the brush increases, the collapse transition becomes discontinuous. Increasing the grafting density leads to higher sensitivity with respect to CNS but also to a weaker collapse behavior as predicted from the analytical model. In the narrow collapse region, two states of the brush layer coexist, as can be expected from the type-II phase-transition behavior. The coexistence states are identified by analyzing the density profiles. Here, a dense layer is formed at the substrate, which is covered by a swollen layer. Both collapse and coexistence behaviors are even more pronounced in polydisperse brushes, indicating the robust nature of the co-nonsolvency effect in polymer brushes.

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free-energy model to rationalize this concept analytically.5,7 In a recent work, one of the authors extended this idea to polymer brushes and solutions using a mean-field model for the adsorption-induced attraction between the monomers, called the adsorption−attraction model. 8,9 In this theory, a discontinuous collapse transition of the brush has been predicted, which results from a competition between effectively attractive many-monomer interactions in the presence of repulsive pair interactions similar to an early conjecture by de Gennes10 and Lifshitz et al.11 The discontinuous transition scenario is of great interest for potential application where a switchlike response of the polymer surface is required.3,12 From the theoretical point of view, the nonspecific attraction of a solvent component by the polymer leads to a new type, or type-II, phase transition, which can be mapped to a concentration-dependent Flory parameter. Interestingly, the concept of collapse and demixing of polymers induced by the presence of an attractive component of the polymer solution goes far beyond the CNS problem. Motivated by the operation of the nuclear pore complex where a brush of intrinsically disordered proteins (FG-nups) acts as a gate in the presence of transport factor proteins, Opferman, Coalson, Jasnow, and Zilman (OCJZ)13−16 developed a theory that differs from the adsorption−attraction model of ref 8 in the way the interaction between polymer and cosolvent/

INTRODUCTION Complex solvents can lead to unusual conformational transitions in polymers. One particularly interesting scenario which has attracted renewed attention recently is co-nonsolvency, where in a certain range of composition of two good and perfectly miscible solvents, a collapse of the polymer chains and a liquid−liquid phase transition toward a condensed polymer phase can be observed. A well-studied example is poly(N-isopropylacrylamide) PNiPAm in a mixture of water and alcohol.1−3 Despite a growing number of experimental studies, the origin of this phenomenon is still under debate. Using computer simulation that deals with chemically specified details,4 Mukherji, Kremer, and Marques (MKM) proposed a generic model to reproduce qualitatively the reentrant behavior of PNIPAm and PAPOMe in aqueous methanol mixtures.5 In this model, a single coarse-grained polymer chain is immersed into two explicitly simulated good solvents. While the two solvents are perfectly miscible, a bias of the solvents toward the polymer is mimicked by the attractive tail of the Lennard-Jones (LJ) potential, where one solvent is more attracted toward the polymer becoming the better cosolvent. For small volume fractions, the better solvent induces the collapse due to sticky bridging interactions between distant monomers along the chain. Given the preferential adsorption mechanism, this concept can be further simplified by considering only explicit co-nonsolvent (CNS) but implicit athermal solvent.6 Based on the simple view of CNS as attractive particles that can form temporally intramolecular bridges between the monomers, MKM proposed a © 2019 American Chemical Society

Received: March 20, 2019 Revised: May 8, 2019 Published: May 24, 2019 4120

DOI: 10.1021/acs.macromol.9b00569 Macromolecules 2019, 52, 4120−4130

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vicinity of the tagged monomer, times the probability that exactly one of the two monomers carries a CNS, 2ϕ(1 − ϕ). The energetic gain for this contact is again proportional to the adsorption energy ϵ. The prefactor γ can account for a free-energy difference between double adsorption of CNS and single (noncoupling) adsorption. Finally, we consider the Alexander−de Gennes17,18 mean-field model for the brush in terms of the (homogeneous) monomer density

cosolute is taken into account. In their model, the polymer brush takes up transport factors, which are considered as attractive nanoparticles in this context, due to the mean-field potential of the brush profile. Collapse of the brush is the result of the possibility of increase in the mean-field attraction by increasing the density of the brush. Also this model can lead to a sharp transition of the height of a polymer brush layer. The assumption of the OCJZ model leads to strong collapse only for very large coupling parameters of more than 10kBT. In this work, we study the co-nonsolvency effect in polymer brushes using coarse-grained molecular dynamics (MD) simulations. While the analytical treatment is restricted to mean-field concepts so far, the simulations can account for fluctuation effects and nonhomogeneous density distributions. Our work shows that the collapse transition of the brush can be quantitatively described by the adsorption−attraction model. On the other hand, novel aspects due to the type-II nature of the transition can be identified in our simulations. This concerns in particular the coexistence of two brush layers at the collapse transition, namely, a dense layer at the substrate that is covered by a cilia-like solvated layer. Our simulations further show that polydispersity of the polymers leads to an even sharper transition instead of softening or broadening the collapse.

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fbrush =

c=

f (c , ϕ) = fads + fattr + fbrush

(5)

(6)

For simplicity, we set γ = 1 in the following, which leaves us with three model parameters: θ, ν1, and the solvent selectivity parameter ϵ. We note that the third parameter should be proportional to the energy parameter used in the simulation model. The chemical potential of the CNS can be written as i ρ zy zz μ = lnjjjj z k1 − ρ {

(7)

where ρ is the volume fraction of CNS in the bulk phase. This corresponds to lattice gas model (or equivalently a Flory−Huggins model), i.e., an ideal mixture of both components. We note that the most interesting results are related with a low volume fraction of CNS; thus, a more realistic equation of state, such as for spherical CNS molecules, is not essential. The equilibrium state of the brush in the solvent mixture is given by the minimum of the free energy with respect to both arguments (c, ϕ). This minimization cannot be performed analytically in the general case, and we use MATLAB unconstrained multivariable function to obtain the numerical solution. Via eq 5, this provides us with the equilibrium height of the brush as a function of the chemical potential μ and the model parameters ν1, θ, and ϵ. We will usually consider the relative brush height given by h(μ)/h(μ = −∞), where h(μ = −∞) represents a brush height in the absence of CNS. The model parameters are then fixed using the simulation data. A simple analytical conclusion can be drawn for the maximally collapsed state. If we ignore the additional excluded volume due to bound CNS, ν1, this state is given by the maximum of the coupling term in eq 3, hence by the half-filled state ϕ0 = 1/2. This, on the other hand, becomes the solution for the adsorption isotherm given by eq 2 for ϕ/(1 − ϕ) = 1

(1)

where μ is the chemical potential of the CNS and ϵ is the energy gain per CNS−monomer contact. The adsorbed fraction of CNS on the polymer is denoted by ϕ. Minimizing with respect to ϕ provides the standard adsorption isotherm ϕ(ϵ, μ), which can be written in the form (2)

We note that the polymer chains become the substrate for the adsorption process. This is in contrast to the approach chosen in ref 13, where the brush has been considered as an attractive container for CNS defined by its mean density. The essential assumption of our model concerns the coupling between the adsorption of CNS and the polymer chains via bridging following the idea in ref 5. Bridging occurs because the CNS−monomer interaction is not specific, i.e., restricted to a single contact, but can involve more than one monomer per CNS. In the simplest form, one CNS becomes sandwiched between two monomers, thus providing an effective attraction between the monomers. We chose a mean-field-like approach for this coupling as

fattr = − 2ϵγϕ(1 − ϕ)c

σN h

The second term in eq 4 is the mixing free energy of the solvent components that includes the excluded volume contribution due to adsorbed CNS particles bound along the chains simply being expressed by ν1. The sum of all three terms are understood as the overall free energy per monomer

METHODS

ϕ/(1 − ϕ) = exp(μ + ϵ)

(4)

The first term represents the elastic contribution with a numerical prefactor θ relating the elastic free energy of the stretched chains in the brush to an ideal stretching of Gaussian chains with fixed ends. The averaged monomer concentration, c, is related to the brush height, h, and the grafting density, σ, in a homogeneous steplike profile by

Adsorption−Attraction Model. Here, we revisit our recently developed generic mean-field approach for the CNS problem.8 Three perfectly miscible components are considered: The volume fraction of monomers is denoted by c, and 1 − c is the common volume fraction of solvent and cosolvent. We consider the brush coexisting with the pure solvent phase. Free energies are given per monomer unit and in units of kBT if not noted otherwise. While the Flory−Huggins parameters for solvent−solvent, polymer−solvent, and solvent− cosolvent are zero, the only energetic contribution to the mixing free energy is due to the preference of CNS particles for the monomers. The resulting condensation of CNS along the chains balances the entropy loss and energy gain by adsorption. Using a simple lattice gas model for the state of CNS adsorbed on the polymers, the free energy per monomer for the adsorption of CNS is given by

fads = ϕ ln ϕ + (1 − ϕ) ln(1 − ϕ) − μϕ − ϵϕ

1 σ2 i1 y θ + jjj − 1 − ν1ϕzzz ln(1 − c − cν1ϕ) 2 c2 kc {

μ0 = −ϵ

(8)

Thus, by increase of the selectivity parameter, ϵ, we expect a shift of the collapse toward smaller values of the CNS concentration. This simple relation has to be corrected by the excluded volume, ν1, due to bound CNS. As a result, a lower value of ϕmin < ϕ0 will correspond to the state of minimal height of the brush. A first approximation of the corresponding solution leads to ϕm = 1/2 − O(ν1/ϵ), which corresponds to shift of the chemical potential at minimal brush height by μm = −ϵ − O(ν1/ϵ). More details about the analytical approach to the adsorption−attraction model can be found in ref 8.

(3)

Here, we assume that the probability of coupling (bridging) is proportional to the probability, c, that a second monomer is in the 4121


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Figure 1. Brush height and order parameters as a function of the chemical potential of the CNS in the bulk phase. Top: Brush height h normalized by the height in the absence of CNS, h0, for different interaction strengths (color coded) for brushes at moderate and high grafting densities, σ = 4σ* and 8σ*. Bottom: Order parameter ϕ and number of adsorbed CNS particles. Simulation data are shown by circles, and mean-field theory by solid lines. 2, 4, 8 with the overlap density σ* ≈ 1/πR2g, given by the radius of gyration of a free bulk chain.23 For a polydisperse brush, we use the Schulz−Zimm distribution24 for different chain lengths N

Although we implement the preferential adsorption scenario in the coarse-grained simulations directly, let us emphasize that an alternative scenario is to consider strong attraction between solvent and cosolvent.19 Both scenarios have been recently compared using lattice-based density functional formalism for the case of a polymer solution in a slit in ref 20. It should be noted that a very strong attraction between the solvent components, typically above the range of kT per solvent molecule, is necessary in this scenario since the cooperative effect of the polymer (forming loops and bridges connecting large parts of the polymer chains) is missing here. As a specific variant of the preferential adsorption scenario for lower critical solution temperature (LCST) polymers, in particular for water as common solvent, we mention the model by Tanaka,21 where a cooperative hydrogen bonding along the polymer chains is assumed. Simulation Method. Bead-spring polymer chains are represented by the Kremer−Grest model. In this model, the connectivity along the linear chains is enforced through a finite extensible nonlinear elastic potential22 ij r yzz zz U (r ) = − 0.5κR 02 lnjjj1 − j R 0 z{ k

P(N ) =

(11)

with an average chain length of ⟨N⟩ = 120, k, an exponent related to the polydispersity index PDI = 1 + 1/k, and Γ(k) the γ function. We choose a large polydispersity of PDI = 1.66. To match closer the continuous distribution, we quadrupeled the surface area. The setup procedure creates chains of minimal length N = 1 and up to a maximum length of N = 300. To simulate the co-nonsolvent effect, we follow the approach in ref 6, similar to ref 13 (canonical MD), where a variable amount of LJ spheres represent the CNS particles. We choose the size of the particles equal to the monomer size: bCNS = bp. Additionally, the interaction range between polymer and CNS is extended to rpc = 2.5bp with a variable polymer−CNS interaction energy of ϵCNS = [1.0...2.4] kBT, with kBT = 1. All energy scales are given in units of kBT. Thus, our system is controlled by a single microscopic interaction parameter, which is a measure of the selectivity of the CNS with respect to the polymer. To fix the system’s extension in z-direction, the brush height h0 is measured in the absence of CNS. For systems with explicit CNS, the upper wall is placed at Dz = 1.75 × h0 leaving enough space for the polymer brush to swell vertically at larger CNS content. The equations of motion are integrated using a velocity Verlet algorithm implemented efficiently in the graphical processing unit-accelerated software HOOMD25,26 with a time step δt = 0.005τMD and a damping coefficient Γ = 1.0τ−1 MD for the Langevin thermostat to keep the temperature constant. The MD time scale is given by

(9)

with the spring constant κ = 30ϵb2p and R0 = 1.5bp the maximum extend of the bond, where p indicates the monomer species. Nearby particles interact via the Lennard-Jones (LJ) potential | li b y12 i b y6o o o ojj zz ULJ(r ) = 4ϵLJm − jjj zzz } j z o o ok r { o r k { n ~

ij k kN k − 1 N yzz expjjj− k zz k Γ(k)⟨N ⟩ k ⟨N ⟩ {

(10)

where ϵLJ set the energy scale and b the minimal length scale. For the mutual interactions between the equal type of particles, eq 10 becomes truncated and shifted, e.g., for the monomers ϵp and bp, and set to unity with a cutoff radius rcut = 21/6bp. Such set of parameters for the beads mimic good solvent condition at any temperature T. One end monomer of each chain becomes fixed and randomly distributed on a face-centered cubic crystalline purely repulsive surface with a lateral extension A = Lx·Ly = 42b × 72.74b. Periodic boundary conditions are used in the in-plane directions. We choose the degree of polymerization to N = 120 and vary the grafting density σ/σ* = 1,

τMD = b2m/ϵCNS with mass m = 1. Initial configurations are equilibrated over 104τMD and at least observed 2.5 × 104τMD to calculate density profiles and the mean brush height, defined as a sum over all monomer positions in z-direction 4122


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Figure 2. Same as Figure 1 but for lower grafting densities, σ = σ*, 2σ*.

Figure 3. Simulation snapshots of a single brush layer (σ = 4σ*, ϵCNS = 2.4) in three different stages. Left: Swollen brush, where the chains (blue) stretch vertically from the substrate (pink), with only a small CNS fraction (yellow). Middle: A brush layer forms with an enriched phase of CNS particles near the wall and a diluted swollen brush above. Right: Entirely collapsed brush forming a thin homogeneous film. For better visual appearance, the chains are unfolded beyond the periodic boundaries, whereas the CNS particles remain in the simulation box.

⟨h⟩ =

1 N ·M

∑ zi i=1

the theoretical model and of the simulations for monodisperse brushes with different grafting densities are summarized in Figures 1 and 2 for higher grafting densities, σ = 4σ*, 8σ*, and lower grafting densities, σ = 1σ*, 2σ*, respectively. The top plots show the dependence of the equilibrium normalized brush height h(μ)/h0 on the chemical potential μ of cosolvent, see eq 7, for different attraction energies ϵCNS (different colors). The simulation data are represented by open circles. The signature of the co-nonsolvency transition in polymer brushes is the non-monotonous change of height with a pronounced minimum at low volume fractions of the CNS. When the interaction strength is increased, the point where the brush reaches its minimal vertical extension shifts toward smaller CNS concentrations. This follows the prediction of eq 8. When exceeding a critical attraction strength, ϵ*(σ), the collapse transition becomes steplike, indicating a discontinuous transition (for typical snapshots see Figure 3). The solid lines in Figures 1 and 2 represent the mean-field theory according to eq 6. In Figure 4, we display the mapping from the simulation parameter ϵCNS, which encodes the preference of the CNS for the monomers to the adsorption strength, ϵ, of mean-field model in eqs 1 and 3. The best fits of the model to the simulation results for the brush height yield a linear relation ϵ = a(ϵCNS − ϵc), which is independent of the grafting density. This linear fit leads to an offset value of ϵc. In fact, simulations for ϵ < ϵc display a monotonous increase of

(12)

where N·M is the number of monomers in the brush. The definition in terms of the first moment ⟨z⟩ of the density profile, which is often used for flexible brushes and holds for the parabolic profile of the selfconsistent field theory, is not useful for intermediate or collapsed brushes. The reference state h0 is later used to normalize the resulting brush height h(μ) in the presence of CNS particles. In the beginning of the simulation, a fixed number of CNS particles are placed above the brush and the system is run until a smooth CNS density profile is established for a fixed attraction energy ϵCNS. Once a system is equilibrated, the last configuration is taken for a measurement run. The last configuration is taken for an equilibration run for the next larger attraction energy. Such a protocol helps to reach equilibrium faster for larger attraction energies, e.g., ϵCNS ≥ 2.0 in descent run times. In particular, we calculate the bulk density only after equilibrium is fully established for comparison with the theory, i.e., calculation of the chemical potential of the bulk phase. Our simulation setup ensures that the bulk phase is well defined above the brush phase for all parameters used in this work.

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RESULTS AND DISCUSSION Monodisperse Brushes. The inclusion of CNS particles inside the brush changes the thickness, and such systems respond sensitively to the number of CNS particles and their cohesiveness (selectivity) toward the polymer. The results of 4123


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Using the adsorption−attraction model, it is implicitly assumed that the CNS is the minority component in the bulk. This is the case for the collapse transition and in a large part of the collapsed state. We note that the deviations from the model are apparently less important for high grafting densities and for smaller selectivities. The lower set of plots in Figures 1 and 2 shows the fraction of CNS particles adsorbed by the polymers via direct count in MD snapshots as a function of chemical potential. Only particles that are in direct contact with the brush rabs = 21/6b are counted uniquely. To take into account a realistic occupation around the chain, we normalize the number of absorbed particles with a maximum number of particles Nmax, which can be accommodated at all. To find the maximum value Nmax, we extrapolate the data in the reswollen state linearly and estimate Nmax at μ = 1. Therefore, the order parameter can be read as ϕ(μ) = Nabs(μ)/Nmax. At very low CNS volume fraction and before the collapse, the order parameter grows linearly with CNS concentration. This reflects the behavior of the adsorption isotherm, eq 2, for ρ ≪ 1 and ϕ ≪ 1. Here, bridging effects, eq 3, can be neglected. After the minimal layer height is reached, the number of particles populating the brush grows with smaller slope, approaching finally the saturation value in the reswollen state. For small values of the interaction parameters, ϵ < ϵ*, ϕ(μ) displays a smooth crossover and corresponds largely to that of a simple adsorption isotherm. This indicates that the coupling term in eq 3 is less important. For larger value of ϵ, where the coupling term becomes essential, the two regimes become clearly distinguishable via a kink, located at the collapse transition. When the attraction parameter exceeds the critical value, ϵ*(σ), the signature of the first-order transition is mirrored by jumplike behavior of the adsorption order parameter. This is again followed quantitatively by the theoretical model shown by the solid lines in the figures. When comparing the responses of polymer brushes with different grafting densities at a fixed energy scale, we observe a transition from discontinuous to smooth behavior with increasing grafting density (see results for ϵCNS = 2 in Figure 1). The lower grafting densities at σ/σ* = 1, 2, 4 exhibit a jumplike transition but a significant difference in minimal brush height. For our largest grafting densities, σ/σ* = 8, we observe that the transition becomes continuous. Also, the point, where the brush collapses, shifts slightly to smaller CNS concentrations with increasing grafting density. All of these observations are in agreement with our model, as seen by the fits. The origin of both effects has been discussed in detail in our previous work:8 Increasing the grafting density leads first to an increase of CNS adsorption in the brush due to increase of brush density. This leads to the earlier onset of the collapse. On the other hand, the diluted phase is increasingly destabilized.8 Responsible for this behavior is the first term in eq 4, which disfavors low concentrations. This leads eventually to a suppression of the discontinuous transition (second minimum on free energy of the brush for small concentrations disappears in the theory) and a smooth crossover is observed instead. Moreover, the weakening of the collapse transition with increasing grafting density has been predicted analytically including an estimate for the critical endpoint of the discontinuous transition according to σc ∼ (ϵ − 1)5/2ϵ1/2.8 We discuss the corresponding phase diagram below.

Figure 4. Effective adsorption strength, ϵ, of the CNS as a function of the interaction parameter, ϵCNS, of the simulations. The obtained values are the result of the best approximation of the data in Figures 1 and 2 using the adsorption−attraction model, eq 6. The solid line is a linear function as expected for the mapping between the simulation parameter and the model parameter. The finite offset indicates a critical adsorption energy of the simulation model of about ϵc ≃ 0.6.

brush height with increasing volume fraction of CNS, indicating the absence of the adsorption−attraction effect. This can be explained by the entropic loss of a cosolvent molecule when located near or in contact with a polymer chain. For an all-explicit solvent simulation, this offset would correspond to a depletion of the solvent by cosolvent at the polymer. For larger values of selectivity, ϵ, the mean-field solution predicts a discontinuous brush collapse. The origin of this discontinuous transition is the consequence of an effective Flory parameter, which increases with the concentration of the polymer phase resulting from the adsorption equilibrium of the CNS, including the nonlinear coupling/bridging contribution in eq 3. This transition has been discussed analytically in ref 8. Above the value of ϵ*(σ), the densification of the brush leads to an increase in coupling of monomers by CNS, which in turn leads to even larger densities until higher-order virial coefficients stabilize the brush. Thus, a certain window of concentration/height is unstable. One can show that this is the result of the interplay between a negative third-order viral coefficient and a still positive second-order virial coefficient.8,9 We note that this type of transition cannot be a result of a standard Flory−Huggins model. For the latter, the translational degree of freedom of the polymers is essential to obtain a discontinuous transition scenario: For constant χ-parameters, a polymer brush can undergo a smooth contraction/expansion only. Our simulations prove that a discontinuous transition is taking place also in the presence of fluctuation effects, which are not accounted for in the mean-field model. We will discuss further below that a vertical two-phase coexistence, which follows from the discontinuous transition inside the brush, also can be detected in the simulations. In the region of higher volume fractions of CNS, we observe deviations of the simulation results from the mean-field prediction. This concerns the location of the reentry transition and also an overshoot for expected layer height in the pure CNS phase. After reswelling, the CNS is the dominating volume fraction in the bulk, where the notation of adsorbed CNS and of the order parameter, ϕ, become meaningless. 4124


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Figure 5. Lateral brush density profiles for σ/σ* = 1, 2, 4 from left to right taken at their maximum collapsed states μ0 ≈ −7, −9, −12.8 for ϵCNS = 2.0. The distinct points in all profiles represent the grafted end monomers. For the smallest grafting density (left), a stable micelle/droplet structure is formed. For σ/σ* = 4 (right), a thin film with rather homogeneous density is established.

properties: N = 100, σ/σ* ≈ 7) have been obtained based on a same bead-spring model, except that the attraction was modeled via a shifted Lennard-Jones potential. In the context of ref 14, the CNS is called nanoparticles. Figure 6 shows the normalized brush height as a function of the chemical potential of nanoparticles of different sizes. In the top plot, the size of the particles was given by bnp = bp. This corresponds to our simulations. The polymer brush height progressions for different simulated energy scales ϵnp are color coded. From top to bottom, the diameter of nanoparticles is roughly doubled, i.e., dnp = 1, 2.67, 4.57bp. The solid lines represent fits of our theory. The data display a linear trend for the adsorption energy, ϵ(ϵnp), for all three cases. The threshold for adsorption, ϵc, is shifted for larger co-solutes due to excluded volume. Furthermore, the change in size of the nanoparticles (cosolute) is reflected consistently by an increase of fitted excluded volume parameter, ν1, in eq 4. To conclude this part, simulation results using a generic model of attraction between CNS and monomers are in quantitative agreement with the adsorption−attraction model. Characteristic predictions of the model, such as a jumplike collapse transition at larger values of the selectivity, a shift of the maximum collapsed state with increasing selectivity, and smoothing of the transition with increasing grafting density, are clearly observed. Moreover, size effects of the CNS can be taken into account. The formation of surface micelles as observed for small grafting densities is consistent with the idea of a quasi-poor solvent state induced by CNS. Density Profiles and Phase Coexistence within the Brush. In Figure 7, we present distribution profiles of monomers and CNS of the simulated monodisperse brushes for the grafting density σ/σ* = 4 and for ϵCNS = 2.4. For this case, a discontinuous collapse transition takes place, as predicted in the mean-field model, with increasing CNS concentration (see Figure 1 (left) and Figure 7 (inset)). The top plot shows the total density profiles (solid lines) for the monomers and the corresponding CNS distribution (dashed lines) at various concentration of the CNS in the bulk. The samples are chosen close to the transition point at about μ ≈ −15.5 (see Figure 1 top right plot). The distance from the substrate is given in LJ units. In the absence of CNS (blue solid line), a nearly parabolic profile is resembled as predicted by the strong stretching limit of the self-consistent field theory.29 For systems with large amounts of CNS (in the reentry region), the parabolic density profile is also recovered but extends further vertically due to the additional excluded volume of the cosolvent (data not shown).

The description of the simulation data by the adsorption− attraction model works better for higher grafting densities displayed in Figure 1. Particularly strong deviations from the model can be observed for σ = σ*. The reason for this behavior becomes clear by considering the lateral height profiles of the brushes in the maximum collapsed states, which are displayed in Figure 5. All monomer positions in the z-direction are projected onto a lateral grid parallel to the substrate and averaged over 5000τMD to obtain the local brush density, which is displayed by color in Figure 5. The left panel in Figure 5 displays the lateral density distribution at the transition between mushroom and brush regime. The anchored but fluctuating chains serve as condensation points for the CNS particles, which form patterns that are stable in time. Particularly, we observe a cylindrical droplet (green) that is surrounded by individual chains (light blue) reaching into the drop forming a surface miscelle. When doubling the grafting densities (middle and right plots in Figure 5), the lateral densities become smeared out, the brushes form a closed film locally fluctuating in height. Distinct bright points indicate the randomly distributed end tethered monomers of each chain. It is notable that a brush at σ = 2σ* can exhibit transitions between different surface structures depending on the volume fraction of CNS in the collapse region. For strong attractive CNS particles, the brush in the collapsed phase forms at first miscelles, but with increasing concentration of CNS, a continuous film is established. This can be seen from Figure 2 (top right panel): Here, the brown curve displays small jumps in brush height, which can be related to the morphology changes in the collapsed state. Note that the collapsed state is always a co-acervate, and thus increasing adsorption of CNS increases the volume of the dense phase, which eventually causes a transition between a micellar and a smooth surface morphology. The formation of surface micelles, so-called “octopus micelles”, is well understood for brushes under poor solvent conditions.27,28 Beyond a grafting density of about σ/ σ* ≈ 3), lateral structures are not observed and a continuous layer forms at all cosolvent concentrations. Since we consider in the mean-field model only the case of a homogeneous brush, the approximation for the average height and the corresponding fits for the order parameter become poorer the more the brush is dominated by heterogeneous structures in the collapse state. Additional data for brushes in the presence of an attractive cosolvent/cosolute have been taken from ref 14 to test the adsorption−attraction model. In this work, similar simulations have been carried out for two grafting densities, but the size of the CNS particles has been varied. The presented data (brush 4125


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Figure 7. Top: Number density of monomers in monodisperse brushes (solid lines) and CNS particles (dashed lines) at different CNS concentrations vs distance from the substrate. The color code is defined in the inset, which displays the brush height vs CNS chemical potential, extract from Figure 1. A swollen brush phase is observed without any CNS (blue solid line). With increasing CNS density (red to black), the brush concentration increases near the grafting surface until a fully collapsed brush phase is obtained (solid black line). Bottom: End monomer distribution. During the collapse, a bimodal distribution is established, while in both the entirely collapsed and the swollen phase, end monomers reside mainly at the outer edge of the film. The distance is given in LJ units.

partially collapsed film, while the other ends remain mobile in the swollen phase. In the maximum collapsed state, the singlepeaked distribution is resembled again. Vertical phase coexistence in brushes similar to our results has been predicted in ref 31 based on the assumption of an empirical concentration-dependent χ-parameter. These works have addressed the collapse behavior of LCST polymers rather than CNS. The LCST collapse is jumplike, as the CNS transition, very much in contrast to upper critical solution temperature behavior, where a discontinuous transition cannot happen for immobilized polymers. Various proposals for a specific form of concentration-dependent χ-parameters have been made to resolve the apparent contradiction with the Flory−Huggins model for LCST transitions.32 However, the molecular origin of the concentration-dependent χ-parameter for LCST-type transitions is not yet fully established. We hasten to note that our model (both theory and simulation) is not related to a specific thermal phase behavior of the polymer. In fact, the model is set up to reflect a purely athermal behavior of the polymer/solvent systems. We note, however, that LCST behavior and co-nonsolvency might be considered as classes of the same phenomenon, namely, the modification of the solubility of the polymer by

Figure 6. Normalized brush height vs chemical potential for attractive nanoparticles with different sizes dnp = 1, 2.67, 4.57bp (from top to bottom) and grafting density of σ = 7.0σ*, Data are taken from ref 14. The result for the adsorption−attraction model is given by the solid lines. The effective interaction parameter, ϵ, is indicated in the plots. The adsorption−attraction model works well for the collapse transition, but deviates in the reswelling regime.

With increasing concentration of CNS particles, we observe a coexistence of two phases within the brush: A collapsed phase near the substrate and a swollen phase with an almost triangular profile on top of the collapsed phase. In the maximum collapsed state, the density profile becomes rectangular (black line in Figure 7) with a sharp interface similar to poor solvent conditions.30 In the bottom plot in Figure 7, the terminal monomer distribution of the corresponding brush states is shown. During the emerging vertical phase separation, the localization of end monomers in the outer rim crosses over into a bimodal distribution. At approximately half way toward the collapsed phase, a significant amount of end monomers become trapped in the 4126


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Macromolecules binding of other molecules. In the case of LCST, water has a “Janus head” role: It can form strong bonds (H-bonds) with the polymer, but it remains a poor solvent in the nonbonded state. For the CNS effect, these roles are distributed between the two solvents. This explains why ad hoc models introduced for the LCST transitions lead to similar results as obtained for the adsorption−attraction model for CNS with respect to the phase behavior of polymer brushes. We note that experimental indications for the two-phase structure in a LCST polymer brush have been obtained already in ref 33. Polydispersity. Strictly monodisperse brushes are hardly available in experiments, and it is an interesting question of how polydispersity impacts the transitions described above. We restrict our study to a grafting density of σ = 4σ*, which leads to a smooth collapsed film (no micelle formation) as for the monodisperse systems and shows a discontinuous collapse for higher interaction parameters. Both brushes are compared in Figure 8.

collapse is reached: The polydisperse brush displays a stronger collapse particularly at smaller selectivites such as ϵCNS = 1.0. The observation inverts beyond ϵCNS = 1.6, where the monodisperse layer becomes slightly more compact. Moreover, for the polydisperse case, the beginning of the collapse becomes retarded toward slightly larger CNS concentrations but is even sharper compared to the monodisperse systems. These observations are also reflected by the adsorption order parameter shown in the bottom panel of Figure 8. In Figure 9, the density profiles of the polydisperse brush during the discontinuous transition for the largest simulated interaction strength ϵCNS = 2.2 is shown. The color-coded density profiles are associated with the different stages of brush height in the inset. In the absence of cosolvent, the brush

Figure 9. Top plot: Monomer number density of a polydisperse brush (solid lines) and CNS particles (dashed lines) at different CNS concentrations following the brush height curve in the inset (color coded) for ϵCNS = 2.2. The almost linear blue curve shows the monomer distribution of swollen polydisperse brush, whereas the black curve shows the collapsed film. For growing CNS concentration, the intermediate density profiles show the enrichment of monomers near the substrate, compactifying the film, whereas a decreasing fraction of chains remains in the swollen phase. Bottom plot: normalized height for given windows of averaged chain lengths ⟨N⟩ following the color code of the top inset. The shorter chains respond weakly toward the CNS concentration. The intermediate chain lengths compactify in the beginning of the collapse, but the longer chains remain partially swollen until the concentration is large enough to compactify all chains into a thin film.

Figure 8. Normalized layer height (top) and order parameter (bottom) as a function of chemical potential of CNS for polydisperse brushes (filled circles) and monodisperse brushes (empty circles), respectively, for different interaction strengths (color code follows Figure 1). The grafting density is fixed to σ = 4σ*.

The top panel of Figure 8 compares the height response of a polydisperse brush (filled circles) with the monodisperse one (empty circles) in the presence of increasing CNS concentrations for different interaction strengths (color coded). For rather weak interaction strength and small CNS concentrations, both the responses remain continuous and are close to each other. A difference can be seen when the maximum 4127


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A finite grafting density decreases the stability of the lowconcentration state. In Figure 10, we display all simulations parameters. The blue circles indicate brush−CNS systems, which display a continuous transition, while the red rectangles display systems exhibiting a discontinuous collapse transition. The dashed line is added as a guide to the eye, where the phase boundary for the collapse transition may be located. In accordance with the theoretical model, higher values of the grafting density require larger values of the selectivity parameter for a discontinuous collapse behavior.

density profile is almost triangular in shape for that PDI value in accordance with the theoretical expectation (as recently reviewed in ref 24). With increasing CNS density, the brush phase thickens near the wall building up the collapsed phase. Again coexistence of the dense layer with a swollen layer can be observed, and the swollen phase maintains the triangular shape of the density profile. We also plot the normalized height in selected windows of chain lengths, ⟨N⟩ (see the bottom plot in Figure 9 following the color code above). In the beginning of the collapse, the film height is determined by nearly all chains (red curve), with a minimal extension for intermediate chain lengths. With increasing collapse, the minimum shifts toward longer chains. The longest chains stick out of the collapsed film forming the swollen phase. In the limit of fully collapsed film (black curve), all chains build up to the collapsed phase. To conclude this part, we have detected a coexistence between a dense layer near the substrate and a swollen layer, which are formed during the collapse transition of the brush. This is a proof of the discontinuous nature of the CNS-induced phase transition. Polydispersity leads to a sharpening of the collapse transition and thus increases the switchability of the brush. Thus, the volume/height transition of brushes due to the co-nonsolvency effect shall be very robust with respect to practical limitations of preparation/synthesis of brushes. Phase Diagram. The adsorption−attraction model for the homogeneous brush can be solved analytically for limiting cases, as discussed in ref 8. Most interesting is the phase diagram in the three-dimensional parameter space, i.e., (ϵ, μ, σ). In Figure 10, we depict a reduced phase diagram as

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SUMMARY AND CONCLUSIONS We studied a polymer brush layer on a planar surface where the polymers were exposed to explicit co-nonsolvent (CNS) particles at equilibrium. We have considered the case of perfectly miscible solvents as well as athermal polymers. Only CNS particles and monomers have an attractive interaction, which reflects the selectivity of the monomers for the cosolvent. We observe collapse and reentry behavior of the brush height as a function of the CNS concentration in the bulk. The collapse behavior becomes discontinuous with increasing selectivity. Our simulation results are in quantitative agreement with the adsorption−attraction model, where the interaction between cosolvent and monomers is described in a mean-field manner. Characteristic predictions of the model are a shift of the maximum collapsed state with increasing selectivity and a smoothing of the transition with increasing grafting density. By reconsidering data from simulations of other authors, also various sizes of cosolvent particles can be taken into account. For low grafting densities, the formation of surface micelles was observed, which is consistent with the idea of a quasi-poor solvent state induced by CNS. An interesting feature of the CNS-induced collapse is that the collapsed phase is a co-acervate formed by polymers and cosolvent. Thus, with increasing concentration of cosolvent, transitions between surface micelle states and a smooth film can be induced by the increasing volume fraction of the co-acervate. In our simulations, such transitions can be detected by a jumplike change in the brush height in the collapsed phase for low grafting densities. The adsorption−attraction model predicts the volume phase transition of the brush, which is driven by an interplay of an attractive effective third virial coefficient and an effectively repulsive second virial coefficient, which has been termed as type-II transition by de Gennes. An important consequence of this type of phase transition is the possibility of coexisting states of different degrees of swelling within the brush. In our simulations, we detected the coexistence of a compact layer at the substrate (bottom layer) covered by a swollen layer (top layer) most notable by a bimodal distribution of chain ends. While this effect is known for thermoresponsive brushes, which display a lower critical solution temperature, the two-state coexistence due to the co-nonsolvency effect is an essential consequence of the concept of preferential adsorption of the cosolvent. Interestingly, polydispersity enhances the signature of the collapse transition and also that of the coexistence state. The latter is now related with a triangular shape of the monomer concentration profile (linear decay) of the top layer extending up to 3 times further into the bulk as compared to the bottom layer in our simulations. Thus, the co-nonsolvency transition is

Figure 10. Phase diagram of the brush in scaled parameters according to the prediction in ref 8. The blue circles indicate monotonous changes of the brush height (smooth crossover), and red rectangles display the parameter at which a discontinuous collapse is observed. The latter was characterized based on the behavior of the order parameters in Figures 1 and 2. The dashed line is added as a guide to the eye to visualize the phase transition in parameter space.

proposed in ref 8. Here, the effect of the grafting density is rationalized, which suppresses the discontinuous phase transition beyond a characteristic value depending on the selectivity parameter, ϵ. In the adsorption−attraction model, discontinuous collapse (effectively poor solvent conditions) can only happen for sufficiently strong selectivity, ϵ > 1 (γ = 1). 4128


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(10) de Gennes, P.-G. A second type of phase separation in polymer solutions. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers 1991, 313, 1117. (11) Lifshitz, I. M.; Grosberg, A. Y.; Khokhlov, A. R. Some problems of the statistical physics of polymer chains with volume interaction. Rev. Mod. Phys. 1978, 50, 683−713. (12) de Beer, S.; Kutnyanszky, E.; Schön, P. M.; Vancso, G. J.; Müser, M. H. Solvent-induced immiscibility of polymer brushes eliminates dissipation channels. Nat. Commun. 2014, 5, No. 3781. (13) Opferman, M. G.; Coalson, R. D.; Jasnow, D.; Zilman, A. Morphological control of grafted polymer films via attraction to small nanoparticle inclusions. Phys. Rev. E 2012, 86, No. 031806. (14) Opferman, M. G.; Coalson, R. D.; Jasnow, D.; Zilman, A. Morphology of Polymer Brushes Infiltrated by Attractive Nanoinclusions of Various Sizes. Langmuir 2013, 29, 8584−8591. (15) Coalson, R. D.; Nasrabad, A. E.; Jasnow, D.; Zilman, A. A Polymer-Brush-Based Nanovalve Controlled by Nanoparticle Additives: Design Principles. J. Phys. Chem. B 2015, 119, 11858. (16) Eskandari Nasrabad, A.; Jasnow, D.; Zilman, A.; Coalson, R. D. Precise control of polymer coated nanopores by nanoparticle additives: Insights from computational modeling. J. Chem. Phys. 2016, 145, No. 064901. (17) Alexander, S. Adsorption of chain molecules with a polar head a scaling description. J. Phys. France 1977, 38, 983−987. (18) de Gennes, P. G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069−1075. (19) Dudowicz, J.; Freed, K. F.; Douglas, J. F. Communication: Cosolvency and cononsolvency explained in terms of a Flory-Huggins type theory. J. Chem. Phys. 2015, 143, No. 131101. (20) Chen, X.; Feng, W.; Han, X.; Liu, H. Possible Way to Study Cononsolvency in Confinement: A Lattice Density Functional Theory Approach. Langmuir 2017, 33, 11446−11456. (21) Fukai, T.; Shinyashiki, N.; Yagihara, S.; Kita, R.; Tanaka, F. Phase Behavior of Co-Nonsolvent Systems: Poly(N-isopropylacrylamide) in Mixed Solvents of Water and Methanol. Langmuir 2018, 34, 3003−3009. (22) Grest, G. S.; Kremer, K. Molecular dynamics simulation for polymers in the presence of a heat bath. Phys. Rev. A 1986, 33, 3628− 3631. (23) Kreer, T.; Metzger, S.; Müller, M.; Binder, K.; Baschnagel, J. Static properties of end-tethered polymers in good solution: A comparison between different models. J. Chem. Phys. 2004, 120, 4012−4023. (24) Qi, S.; Klushin, L. I.; Skvortsov, A. M.; Schmid, F. Polydisperse Polymer Brushes: Internal Structure, Critical Behavior, and Interaction with Flow. Macromolecules 2016, 49, 9665−9683. (25) Anderson, J. A.; Lorenz, C. D.; Travesset, A. General purpose molecular dynamics simulations fully implemented on graphics processing units. J. Comput. Phys. 2008, 227, 5342−5359. (26) Glaser, J.; Nguyen, T. D.; Anderson, J. A.; Lui, P.; Spiga, F.; Millan, J. A.; Morse, D. C.; Glotzer, S. C. Strong scaling of generalpurpose molecular dynamics simulations on GPUs. Comput. Phys. Commun. 2015, 192, 97−107. (27) Williams, D. R. Grafted polymers in bad solvents: octopus surface micelles. J. Phys. II 1993, 3, 1313−1318. (28) Jentzsch, C.; Sommer, J.-U. Polymer brushes in explicit poor solvents studied using a new variant of the bond fluctuation model. J. Chem. Phys. 2014, 141, No. 104908. (29) Milner, S. T.; Witten, T. A.; Cates, M. E. Theory of the grafted polymer brush. Macromolecules 1988, 21, 2610−2619. (30) Dimitrov, D. I.; Milchev, A.; Binder, K. Polymer brushes in solvents of variable quality: Molecular dynamics simulations using explicit solvent. J. Chem. Phys. 2007, 127, No. 084905. (31) Baulin, V.; Zhulina, E.; Halperin, A. Self-consistent field theory of brushes of neutral water-soluble polymers. J. Chem. Phys. 2003, 119, 10977. (32) Afroze, F.; Nies, E.; Berghmans, H. Phase transitions in the system poly(N-isopropylacrylamide)/water and swelling behaviour of the corresponding networks. J. Mol. Struct. 2000, 554, 55−68.

robust with respect to realistic preparation conditions of brushes. We note that switchlike control of the vertical distribution of the chains will be interesting for the lubrication properties of sliding bilayer brushes in a slitlike geometry, where the overlap region defines the physical response of the system. In general, out-of-equilibrium conditions combined with the co-nonsolvency transition can give rise to novel phase behavior driven by flow conditions, for instance. To conclude, we have shown that realistic brush models including correlation and fluctuation effects as well as polydispersity display a sharp collapse transition by variation of the solvent composition. Our findings can be well understood within the framework of mean-field model, which extends the Flory−Huggins approach to co-nonsolvency.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.G.). *E-mail: [email protected] (J.-U.S.). ORCID

Andre Galuschko: 0000-0002-6384-6594 Jens-Uwe Sommer: 0000-0001-8239-3570 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SO-277/17. The authors acknowledge the ZIH of the TU Dresden for providing the computational resources and support.

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REFERENCES

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