Density Functional Theory Approach - American Chemical Society

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Lock/Unlock Mechanism of Solvent-Responsive Binary Polymer Brushes: Density Functional Theory Approach Yuli Xu,† Xueqian Chen,*,‡ Xia Han,† Shouhong Xu,† Honglai Liu,*,† and Ying Hu† †

State Key Laboratory of Chemical and Engineering and Department of Chemistry and ‡Department of Physics, East China University of Science and Technology, Shanghai 200237, China ABSTRACT: A density functional theory (DFT) approach based on a weighted density approximation has been employed to study the perpendicular microphase separation of symmetric binary polymer brushes with weak incompatibility in explicit solvents with different selectivities. Characterized by the relation between the grand potential and vertical structures (including nonlayered and layered structures), a dry binary brush can be categorized as W-type or U-type according to whether the characteristic relation contains a structure that undergoes spontaneous symmetry breaking. A W-type brush can memorize the selectivity of the induced solvent in one of its two layered structures after the removal of solvent, which can be seen as a kind of lock state with the nonselective solvent used as its key to unlock. A U-type brush is lockless but can adapt to the environment without the nonselective solvent’s triggering. Also, the boundary described in chain-length-incompatibility space is investigated by the DFT approach, which also verifies that the spontaneous symmetry breaking of the W-type brush originates from the molecular contributions to asymmetry, such as the enthalpic contribution of incompatibility and the entropic contribution of chain connectivity. Sidorenko19 first synthesized a mixed surface by grafting both polystyrene (PS) and poly-2-vinylpyridine (PVP) to a substrate; they found that the substrate was hydrophobic after exposure to toluene and hydrophilic after exposure to HCl by measuring its contact angle. These processes are reversible. Minko et al.20,27 synthesized several binary polymer brushes, such as poly(styrene-co-2,3,4,5- pentafluorostyrene) (PSF)/ PVP, PSF/poly(methyl methacrylate) (PMMA), and PS/ PMMA. They reversibly tuned the surface properties by exposing the surface to solvents that are selective to one of the components of the brush, and they observed that in a nonselective solvent different species segregate into a “ripple” structure. After exposure to a selective solvent, the ripple structure transformed to a “dimple” structure in which the unfavored component formed clusters with simultaneous perpendicular segregation enhancement. Recently, Yang et al.31 and Zhao et al.5 synthesized PS/poly[2-(N,N-dimethyl amino)ethyl methacrylate] (PDMAEMA) and Y-shaped PS/ PMMA brushes, respectively, both of which could undergo spontaneous chain reorganization in response to solvent changes. In recent years, many theoretical works32−36 and simulation37−45 studies were carried out to investigate the influence of grafting density, solvent quality, chain length, and compatibility on the lateral and perpendicular segregation in

1. INTRODUCTION Polymer brushes play an important role in a diverse range of applications1 because of their responsive behavior in a specific environment. In recent years, more and more attention has been paid to incompatible binary brushes grafted to a material surface, of which the two components can respond to an external stimulus (such as solvent quality,2−5 pH,6 humidity,7 or temperature8) differently to form a layered structure with one component being enriched near the surface and the other being on top. Therefore, the surface properties of the material such as the adsorption, adhesion, friction, and wettability can be determined by the outer component. The switching between the two components in the outer layer provides a way to create smart surfaces adopting switching properties. Now, binary polymer brushes have various applications in nanotechnology and “smart” materials, such as chemistry in microfluidic channels,9 environmentally responsive lithography,10 smart coatings,11,12 responsive colloids,13−16 protein adsorption,17 supports for the directed assembly of nanoparticles,18 and wettable surfaces.19,20 When a solvent-responsive binary polymer brush is immersed in a selective solvent, perpendicular phase separation will occur near the solid substrate. The solvophilic chains swell to the top whereas the solvophobic chains twist at the bottom, which results in the surface being solvophilic even when the solvent is evaporated. If the binary polymer brush is composed of hydrophilic and hydrophobic polymers, then the surface will show hydrophilicity/hydrophobicity when treated with a hydrophilic/hydrophobic solvent.21−30 In the experiment, © 2013 American Chemical Society

Received: December 17, 2012 Revised: February 28, 2013 Published: March 25, 2013 4988

dx.doi.org/10.1021/la304984s | Langmuir 2013, 29, 4988−4997

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binary polymer brushes. Soga et al.39 simulated the equilibrium structures of mixed brushes with two species sufficiently immiscible under various nonselective solvent conditions. Lateral microphase separation is observed over a wide range of solvent conditions, which can be delayed as the solvent quality improves. Müller40 further studied the structure and phase behavior of binary polymer brushes in nonselective solvents by self-consistent field theory and found that a rippled phase forms at small incompatibilities and two “dimple” phases form at large incompatibilities or asymmetric compositions. Wu and co-workers41,42 adopted a Langevin dynamics method to investigate the switching of binary mixed brushes in various solvents and found that perpendicular microphase separation occurs when the brushes are exposed to a selective solvent, with the switching relaxation times display a scaling of N2. Recently, Xue43 used DPD simulations to study the layered structure of compatible binary brushes with a high graft density, and they found that a slight change in the solvent selectivity favoring the short chains will greatly change the film morphology. Designing or synthesizing new mixed polymer brushes with pronounced switching properties in a specific solvent environment still attracts interest in experimental research. Ionov et al.44 recently reported mixed polymer brushes consisting of hydrophobic polystyrene and highly hydrophilic poly(acrylic acid) with the property of locking switching. The wetting properties of the mixed brush can be switched between the wetting properties of individual polymer species by the appropriate selection of solvent. It is interesting that the mixed polymer brushes’ wetting behavior can be locked into the hydrophobic state, which can be unlocked via treatment by a proper “unlocking” solvent. Inspired by Ionov et al.’s experiment, we used density functional theory (DFT) to investigate the solvent responsive behavior of mixed polymer brushes. Derived from statistical mechanics, DFT provides an efficient tool to simulate the responsive behavior of polymeric system including grafted polymers. In addition to computer simulation, it has the ability to analyze the relationship between possible responsive states and their free energy, which is useful in understanding the switching between two responsive states. Previous studies focused on homopolymer brushes45,46 or copolymer brushes47,48 rather than mixed polymer brushes. Gong et al.49 studied the behavior of mixed polymer brushes in an implicit solvent by DFT. However, they have not investigated the layered structure after the removal of solvent, which determines the surface property to be preserved.19 In this work, to investigate the brush structure with evaporated solvent, we treat the solvent as an independent component. Wet or dry brushes can be discriminated by whether the solvent is present. Considering the pronounced surface switching properties of the layered structure, we focus on the vertical structure and neglect the possible lateral structure. A DFT approach based on the weighted density approximation (WDA) developed by Ye et al.50 is extended to binary polymer brushes. The excess Helmholtz free energy functional is divided into two parts: the repulsive term due to athermal chains and the attractive term due to segment−segment attractions. The equation of state for a hard-sphere-chain fluid developed by Liu et al.51 is employed to calculate the repulsive term, and the equation of state for square-well chain fluids with variable range developed by Li et al.52 is employed to calculate the attractive term. Using this theoretical approach, we investigated the perpendicular microphase separation and solvent-responsive behavior of binary polymer brushes in nonselective and

selective solvents. The structure of binary polymer brushes with evaporated solvent is also studied by minimizing the grand potential with a zero solvent density. This study provides a theoretical basis for the lock/unlock process found in Ionov et al.’s experiment44 and divides dry binary brushes into lockable ones and lockless ones. To the best of our knowledge, no similar theoretical observation for the distinction between the two types of polymer brushes has been reported before. This Article is organized as follows: section 2 presents the DFT model, section 3 shows our results and corresponding discussion, and section 4 contains concluding remarks.

Figure 1. Schematic representation of solvent-responsive behavior for binary polymer brushes.

2. DENSITY FUNCTIONAL THEORY 2.1. Model. In this work, we consider a binary polymer brush (mixture of A and B) in a solvent S. The solventresponsive polymers are modeled as flexible chains of spherical segments immersed in spherical solvent molecules. The diameters of polymer segments and solvent molecules are the same (σA = σB = σS = σ), and the interaction among them is given by the square-well potential ⎧0 rij > 1.5σ ⎪ ⎪ uij(rij) = ⎨−εij σ < rij < 1.5σ ⎪ ⎪∞ rij < σ ⎩

(1)

where rij and εij stand for the distance and interaction strength, respectively, between segments i and j. In our work, εAA = εBB = εSS = ε and reduced temperature T* = kT/ε = 3.0. The interaction between the hard wall and the chain segment is given by ⎧ 0 ≤ zi − (σ /2) ⎪ 0 u wall(zi) = ⎨ ⎪ ⎩∞ zi − (σ /2) < 0

(2)

where zi denotes the distance between the center of the segment and the wall. To graft polymers onto a surface, the bonding potential between the first segment of polymers and the wall is set as ⎧−∞ z1 = σ /2 ugra(z1) = ⎨ ⎩ 0 z1 ≠ σ /2 ⎪ ⎪

4989

(3)


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This expression implies the translational invariance of the segment density on any plane parallel to the grafting surface. 2.2. Density Functional Theory. We considered the system to be a mixture of binary grafted chains with chain length MA = MB = M and solvent molecules that can also be treated as special chains with chain length MS = 1. According to the density functional theory, the grand potential Ω can be expressed as a functional of the density profile of the fluid

ex (1) F ex[ρM(1) , ρM(2) , ρM(3)] = Fhc [ρM , ρM(2) , ρM(3)] ex [ρM(1) , ρM(2) , ρM(3)] + Fattr

where Fex hc is the contribution due to hard-sphere repulsions and hard-sphere-chain formation and Fex attr is the contribution due to the square-well attractions and its effect on chain formation. The DFT approaches for the attraction term reported in the literature can generally be divided into two categories: the perturbative53,54 and weighted density55−58 approaches. These approaches have been successfully applied to simple fluid systems such as square-well fluids,59 Lennard-Jones fluids,60 Yukawa fluids,61,62 and Sutherland fluids.63 Considering the correlation between chain connectivity and attraction, the extension of the above DFT approaches to a polymer brush system64 still needs to be studied, and usually the mean-field approximation is used for simplicity. With respect to the correlation effect, following our previous work on inhomogenous polymer mixtures,50 we combined the equation of state52 (which accounts for the correlation effect in the bulk) with WDA to treat the Helmholtz free energy for attraction. The expression of the excess-free-energy functional is obtained by adopting WDA as

Ω[ρM(1) , ρM(2) , ρM(3)] = Fint[ρM(1) , ρM(2) , ρM(3)] ⎤

3

+

(i) (R i) ρM(i)(R i) dR i − μi ∫ ρM(i)(R i) dR i⎥ ∑ ⎡⎣⎢∫ V ext

⎥⎦

i=1

(11)

(4)

where subscripts 1 and 2 stand for grafted polymer chains A and B, respectively, and subscript 3 stands for the solvent, μi and V(i) ext are the chemical potential and external field of component i, respectively, Ri = (r1,...,rMi) denotes the configuration of an Mi-mer with rj being the position of the jth segment, R3 = r for solvent S, and Fint is the intrinsic Helmholtz free energy functional that can be separated into two contributions: the ideal part and the excess part,

3

Fint[ρM(1) , ρM(2) , ρM(3)] = F id[ρM(1) , ρM(2) , ρM(3)]

ex (1) Fhc [ρM , ρM(2) , ρM( 3)] =

+ F ex[ρM(1) , ρM(2) , ρM(3)]

∑ ∫ ρi (r)f hc(i) [ρhc̅ (i) (r)] dr

(12)

i=1

(5)

3 ex Fattr [ρM(1) , ρM(2) , ρM(3)] =

3

F

id

[ρM(1) ,

ρM(2) ,

ρM(3)]

=

∑∫

ρM(i)(R i)[ln

ρM(i)(R i)

−1

f (i) hc (ρ̅)

where is the excess Helmholtz free energy density of a hard-sphere chain for a given density ρ̅ of component i in the (i) bulk phase, fattr (ρ̅) is the difference between the excess Helmholtz free energy for square-well chains and that for hard-sphere chains, and

(6)

Here, β = 1/kT, k is the Boltzmann constant, T is the temperature, and Vintra is the intramolecular interaction that vanishes for solvent S. At equilibrium, the grand potential Ω reaches its extremum with respect to the density profile,

δΩ δρM(i)(R)

⎛ ∂F ex ⎞ (i) = ⎜⎜ hc ⎟⎟ , f hc ⎝ ∂ρi ⎠T , p , ρ

=0

k

(7)

⎧ ⎛ ⎪ (i) (i) ρM(i)(R) = exp⎨β ⎜⎜μi − V ext (R i) − V intra (R i) ⎪ ⎩ ⎝ −

δF

[ρM(1) ,

ρM(2) , δρM(i)

⎪ ρM(3)] ⎞⎫ ⎟

⎟⎬ ⎪ ⎠⎭

(8)

(9)

δρM(i)(R i)

(15a)

(i) ρattr ̅ (r) =

∫ ρi (r′) wattr(|r − r′|) dr′

(15b)

3 (i) (i) = f hc [ρhc ̅ ]+

δρi (r) mi

=

∑∫

δF ex[ρM(1) , ρM(2) , ρM(3)] δρi (r′)

l=1

(r′ − rl) dr′

(i) δFattr

δ

δρM(j)(R j)

(10)

The excess Helmholtz free energy F [ρM] can be separated into two parts

(k) δρhc ̅ (r′)

(16a) 3

=

(i) (i) f attr [ρattr ̅ ]

+

3

∑ ∑ ∫ ρi (r′) j=1 k=1

δρi (r) 4990

(j) (j) δf hc [ρhc ̅ (r)]

dr′

(k) δρattr ̅ (r′)

ex

3

∑ ∑ ∫ ρi (r′) j=1 k=1

(k) δρhc ̅ (r′)

Using eq 9, we have δF ex[ρM(1) , ρM(2) , ρM(3)]

(14)

k≠i

∫ ρi (r′) whc(|r − r′|) dr′

δρM(j)(R j)

for i = 1, 2, 3

l=1

k

(i) ρhc ̅ (r ) =

(i) δFhc

mi

∫ ∑ δ(r − rl)ρM(i)(R i) dR i

k≠i

where whc(r) = 3Θ(σ − r)/(4πσ3), wattr(r) = 3Θ(1.5σ − r)/ (4π(1.5σ)3), and Θ is the Heaviside function. Combining eqs 9, 12, 13, and 15, we have

where the chain density is related to the segment density as ρi (r) =

⎛ ∂F ex ⎞ (i) = ⎜⎜ attr ⎟⎟ f attr ⎝ ∂ρi ⎠T , p , ρ

The weighted densities are given by

The equilibrium density distribution is then given by

ex

(13)

i=1

i=1 i + βV intra (R i)] dR i

(i) (i) [ρattr ∑ ∫ ρi (r) f attr ̅ (r)] dr

(j) (j) δf attr [ρattr ̅ (r)] (k) δρattr ̅ (r′)

dr′ (16b)


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The segment density distribution for component i can be expressed as ⎡ ⎛ (i) (r ) − Vintra(R ) ∑ δ(r − rl)exp⎢⎢β ⎜⎜μi − V ext l=1 ⎣ ⎝ mi

ρi (r ) =

∫

mi

−

⎛ δF (i) hc

∑ ⎜⎜ j=1

+

(j)

⎝ δρM (R)

V(i) ext(r)

u(i) g (r)

(i) ⎞⎞⎤ δFattr ⎟⎟⎥ d R i ⎟ δρM(j)(R) ⎠⎟⎠⎥⎦

(17)

u(i) wall(r).

with = + Approximating the intramolecular potential with the bonding potential between nearest-neighbor segments, we obtain a numerical solution of mi

ρi (z) = exp(βμi ) ∑ exp( −β Ψl (z))GLl(z) GRl (z) l=1

for i = 1, 2

Figure 2. Segmental density profiles of 40-mer A or B (red lines) in nonselective solvent S (black lines). εAB = 0.8ε, ρgra‑Aσ2 = ρgra‑Bσ2 = 0.1, and ρbulk‑Sσ3 = 0.3. Solid lines: in a good solvent (εSA = εSB = 1.0ε). Dashed lines: in a poor solvent (εSA = εSB = 0.0).

(18)

where GLl(z) =

1 2σl − 1, l

min(Hσ , z + σ )

∫max(0,z−σ)

3.2. Symmetrical Binary Polymer Brushes in a Selective Solvent. We now consider the effect of selective solvents on binary brushes. Figure 3 shows the density profiles

dz′exp( −β Ψl − 1(z′))GLl− 1(z′) (19a)

GRq (z) =

1 2σq , q + 1

min(Hσ , z + σ )

∫max(0,z−σ)

dz′exp( −β Ψq + 1(z′))

GRq+ 1(z′)

for l = 2, ..., m and q = 1, 2, ..., m − 1, Ψl (z) =

(19b)

G1L(z)

ex ex δFhc δFatt (l) + + uext (z ) δρl (z) δρl (z)

= 1,

GmR (z)

= 1, and (20)

For component i = 3, ρ3 (z) = exp(βμ3 ) exp(−β Ψ(z))

(21)

3. RESULTS AND DISCUSSION 3.1. Symmetrical Binary Polymer Brushes in Nonselective Solvents. We first investigate the effect of nonselective solvents on binary brushes. Below, unless marked otherwise, the interaction between 40-mer A and B is εAB = 0.8ε (weak incompatibility), the total grafting density is ρgraσ2 = 0.2 with mole fractions of xA = xB = 0.5, and the bulk density of the solvent is ρbulk‑Sσ3 = 0.3. Figure 2 shows the density distributions of nonselective solvent S and a grafted polymer (A or B) under the conditions of a poor solvent (εSA = εSB = 0.0) and a good solvent (εSA = εSB = 1.0ε), respectively. It can be found that the brush and the solvent separate into two regions as a result of the incompatibility between polymer and solvent. The brush twists within 0.5 < z/σ < 15 because of the squeezing of solvent molecules, and the solvent molecules decreases rapidly at z/σ = 15 when encountering the brush layer and nearly disappear inside the brushes (z/σ < 12). In the nonselective good solvent, the boundary between the regions of the brush and solvent vanishes and a certain amount of solvent penetrates the grafted film, which causes the grafted polymers to stretch. Whether the film is in a good or poor nonselective solvent, perpendicular phase separation is not found in the binary brush with the above parameters. It is necessary to mention that our calculation cannot judge whether lateral microphase separation occurs even for strong incompatibility.

Figure 3. Segmental density profiles of A (red line) and B (black line) in a selective solvent S (blue line, εSA = 0.0, εSB = 1.0ε). The other parameters are the same as for Figure 2.

of brushes A and B with the same parameters as for Figure 2 in a selective solvent S (εSA = 0.0, εSB = 1.0ε). It can obviously be seen that, when exposed to a selective solvent, the grafted polymer film separates perpendicularly into two layers, of which the inner layer (0.5 < z/σ < 10) contains almost all of A with a small amount of B and the outer layer (10 < z/σ < 20) is filled with B. The solvent molecules penetrate the outer layer with the help of solvophilic B but hardly approach the hard wall because of the protection of neutral A. The result implies that the solvent-induced perpendicular segregation of the binary brush can form a solvophobic surface and an inner layer to protect the wall from solvent whether the selective solvent is hydrophilic or hydrophobic. Such a layered structure also has been reported in experiments27,28 and simulations.42,43 When Figure 3 is compared to Figure 2, it can be concluded that the selectivity of solvent is a necessary factor in causing perpendicular segregation in binary brushes, but how strong 4991


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structure shown by Figure 4) as the initial state of the calculation in DFT with ρ*S(z) set to zero anywhere and then minimize the grand potential of the binary brush via Picard iteration to obtain an equilibrium structure characterized by each polymer brush species’ density profile, which is called a dry-brush structure pretreated with a solvent. Figure 5 shows

the selectivity should be to induce the segregation still needs investigation. 3.3. Effect of the Selectivity of Solvents on Symmetrical Binary Polymer Brushes. To investigate the effect of solvent selectivity on binary brushes, parameter εSA is set from 0.0 to 1.0ε gradually with the other parameters fixed, of which a higher value means a weaker selectivity. Figure 4 shows

Figure 5. Segment density profiles of A (red line) and B (black line) with the solvent removed. Solid lines: pretreated with selective solvent (εSA = 0.0ε, εSB = 1.0ε). Dashed line: pretreated with nonselective solvent (εSA = εSB = 1.0ε). The other parameters are the same as for Figure 2.

Figure 4. Variation in the density difference profile of the two species along the z direction with different selectivity parameter, εAS, values. The arrow indicates the direction of greater εSA. Inset: curve of S versus εSA. The other parameters are the same as for Figure 3.

the segment density profiles of A or B pretreated with selective or nonselective solvent. For the situation with nonselective solvent, it is no surprise that the dry brush retains its nonlayered structure in solvent because there is no external stimulus to trigger the phase separation. For the situation with selective solvent, it is interesting that the dry brush can memorize its layered structure (with a solvophobic outer layer) in solvent. This phenomena has been reported in some experiments,7 but few theoretical studies and simulations have been carried out to explain it. By comparing the results for pretreatment with nonselective and selective solvents, it can be found that one dry binary brush can adopt both nonlayered and layered structures. To investigate the effect of solvent selectivity on the drybrush structure (layered structure), we also calculate the thicknesses of the binary brushes pretreated with different solvents (denoted by different strengths of selectivity), and the thickness of each polymer species with the solvent removed is defined as65

the variation in the density difference profile of the two species along the z direction with different solvent selectivities. Here the density difference profile of the two species δρ*(z) is defined as43 δρ*(z) = ρB*(z) − ρA*(z)

(22)

As shown in Figure 4, when the selectivity is strong (εAS = 0.0), the value of δρ*(z) is negative at 0.5 < z/σ < 8 and positive at 8 < z/σ < 22, indicating that the inner layer and the outer layer are dominated by A and B, respectively. The location z0 specified by δρ*(z0) = 0 can be defined as a boundary between the two layers, which is approximately at z/σ = 8 for εAS = 0.0. With increasing εSA, the absolute value of δρ*(z) decreases in both the inner layer and the outer layer, which illustrates that weakening the selectivity will result in a decrease in the degree of phase separation. Meanwhile, the outward shift of z0, which implies a more stretched conformation of A, is due to the enhancement of the compatibility between A and S. Until εAS reaches 1.0ε, δρ * (z) vanishes anywhere with layered segregation eliminated. It can be found that when an incompatible (εAB = 0.8ε) binary brush is exposed to a selective solvent, layered segregation will occur regardless of the strength of selectivity. The difference between the two curves at εSA = 0.9ε and 1.0ε is obvious, showing that the vertical structure is sensitive to whether the solvent is selective; however, it is less sensitive to the strength of selectivity, which also can be seen clearly from the relation between the vertical structure parameter (S = ∫ |δρ*(z)| dz) and εSA. 3.4. Symmetrical Binary Polymer Brushes with the Solvent Removed. To investigate the solvent-induced structure (layered or nonlayered structure) of a binary brush with the solvent removed (dry binary brush), we choose the structure of the binary brush immersed in a solvent (such as the

Hi* =

2 ∫ zρi (z) dz σ ∫ ρi (z) dz

(23)

Figure 6 shows the dependence of the brush thickness on the selectivity of the pretreatment solvent. It can be found that the thickness of each polymer species was not affected by the selectivity of the pretreatment solvent (H*A = 13.2, H*B = 19.5), indicating that the layered structure of dry brushes pretreated with different solvents is unique as long as the solvents prefer the outer-layer species. 3.5. Grand Potential Ω and Lock/Unlock Mechanism. It is known from Figure 5 that a weak incompatible (εAB = 0.8ε) symmetrical binary brush has two vertical structures with the solvent removed: layered and nonlayered structures. Here 4992


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Figure 8. Schematic diagram of the relation between the grand potential and the vertical structure of a binary brush. Figure 6. Dependence of brush thickness on the selectivity of the pretreatment solvent for A (red line) or B (black line). The other parameters are the same as for Figure 3.

equilibrium structure. As illustrated by the cartoons in Figure 8, the maximum point corresponds to the nonlayered structure, and each of the two minimum points corresponds to a layered structure. Because of the exchange symmetry between A and B, two states are degenerated at their maximum grand potentials with the unique nonlayered structure and exhibit two different structures named the A-outer layer and B-outer layer at their minimal grand potentials. The nonlayered structure is unstable because of its higher grand potential. Stimulated by the environment (such as a selective solvent), it would easily transfer to a structure with a lower grand potential. In other words, with a small perturbation, the surface compositions of the brush would experience spontaneous symmetry breaking from a higher grand potential to a lower grand potential. In fact, a similar phenomenon can be seen in the phase diagram of Gibbs free energy versus polymer composition for the corresponding homogeneous binary polymer mixture,52 where the average composition of the mixture corresponds to the surface composition of the end-tethered polymer mixture here. In addition, it can be seen from Figure 8 that the energy barrier hinders the binary brush switch from one layered structure to the other layered structure; that is to say, the binary brush with a specific layered structure (A-outer layer or B-outer layer) can be locked by the energy barrier. However, if a layered structure is induced by a selective solvent, then the energy barrier will guarantee the brush’s memory of the layered structure after solvent removal. Because the solvent-induced perpendicular segregation is irreversible for the brush characterized by Figure 8, an approach to unlock the layered structure except for removing the induced solvent is needed. Two cases are studied in Figure 9. The results show that even in a solvent selective for A (compatible with A but neutral to B), the binary brush retains the structure of the B-outer layer rather than switching to the structure of the A-outer layer, whereas nonselective solvent (being compatible with both A and B) makes the same binary brush form a nonlayered structure. It can be concluded that a binary brush with a specific layered structure (such as a B-outer layer) is not necessarily unlocked when treated with a solvent selective for an inner component (such as A) but can be unlocked by a nonselective solvent. According to the previous analysis, a nonlayered structure with solvent removed can easily transform into an A-outer or B-outer layered structure because of its instability, which implies a relock process. After the layered structure is unlocked by a nonselective solvent, a selective solvent can be used to stimulate the perpendicular phase separation and relock the brush in a new required layered

the reduced grand potential Ω* of the two structures is quantitatively compared and is defined as Ωσ 2 Ω* = kTA

(24)

where Ω is the grand potential of dry brushes and A is the surface area. Figure 7 is the grand potential Ω*dry of two structures versus compatibility parameter εAB. It can be seen that Ω*dry of the

Figure 7. Reduced grand potential Ω* vs incompatibility parameter εAB. Blue lines: nonlayered structure. Red lines: layered structure. Solid lines: ρ*gra‑total = 0.16. Dashed lines: ρ*gra‑total = 0.25. The other parameters are the same as for Figure 6.

nonlayered structure is larger than that of the layered structure, indicating that the layered structure is more stable, especially for strong incompatible binary brushes. The analysis of the grand potential of the dry brush reveals the spontaneity in the transition from a nonlayered structure to layered structure as a result of the incompatibility in different polymer brush species and makes the results in Figures 4 and 6 understandable in which the layered structure is sensitive to the selectivity of the solvent. To explain, we outline a schematic in Figure 8. Figure 8 is the schematic diagram of the grand potential versus the vertical structure of the binary brush. The horizontal axis represents an order parameter denoting the vertical structure, and each extreme point corresponds to an 4993


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Figure 11. Schematic diagram of the relationship between the grand potential and the vertical structure of a binary brush.

For the binary brush characterized by Figure 11, there is only one minimum point on the schematic curve of the grand potential that corresponds to the nonlayered structure. For convenience, we call the binary brushes with the same character as those in Figure 11 U-type brushes and those with the same character as in Figure 8 W-type brushes. The W-type brush has a memory of solvent selectivity, and its layered structure can be locked after the removal of solvent or the reversion of solvent selectivity. To clear the memory or unlock and reverse the layered structure, one needs a nonselective solvent as a key. The U-type brush can adapt to the reversion of solvent selectivity and spontaneously recover to a state with nonlayered structure after the solvent is removed, which implies that the Utype brush has no memory of solvent selectivity. We also investigate the conditions of chain length and εAB for the transition between the W-type brush and U-type brush with fixed ρ*gra in Figure 12, which shows the boundary line

Figure 9. Variation in the density difference profile of the two species along the z direction with a different solvents. Here, ρ*gra = 0.25, εAB = 0.5ε, and M = 40. Red line: εSA = 0.7, εSB = 0.0ε. Black line: dry brushes. Blue line: εSA = 1.0ε, εSB = 1.0ε.

structure. The above theoretical calculation properly interprets the molecular mechanism of the lock/unlock process in view of Ionov et al.’s experimental results.44 Also as presented in their paper, when the locked layered structure does not adapt to the environment (such as a solvent selective for an inner component), the nonselective solvent can be used as a trigger. It is no doubt that the presence of the energy barrier in Figure 9 requires a certain condition and not all of the symmetrical binary brushes can be characterized by Figure 9. Figure 10 is the segment density profiles of a tethered 15-mer

Figure 10. Segment density profiles of A (red line) and B (black line). Solid lines: in a selective solvent (εSA = 0.0ε, εSB = 1.0ε). Dashed line: with solvent removed. Here ρ*gra = 0.2, εAB = 0.8ε, M = 15.

Figure 12. Effect of incompatibility on the transition (between W-type and U-type) chain length for ρ*gra = 0.2.

in a selective solvent (A, red line; B, black line; εSA = 0.0ε; εSB = 1.0ε) and without solvent (A or B, dashed line) (both with ρ*gra = 0.2, εAB = 0.8ε). The results reveal that this kind of binary brush cannot maintain a layered structure after the removal of solvent, which reveals the lack of the lock state in Figure 8. Actually, the brushes with a layered structure will spontaneously return to the state with a nonlayered structure having a lower grand potential, which means that solvent-driven segregation is a reversible process for the brushes. For this kind of binary brush, we can outline another schematic in Figure 11.

between the two types. As seen in Figure 12, increasing chain length or enhancing incompatibility favors the W-type brush and shortening chain length or weakening incompatibility favors the U-type brush. It is already concluded from calculations that the W-type brush is lockable and the U-type brush is lockless. From another perspective, a short chain length or weak incompatibility indicates the smart brushes’ adaptability to the environment whereas la ong chain length or strong incompatibility indicates the smart brushes’ memory of 4994


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brushes. Essentially, the information on spontaneous symmetry breaking is already contained in the equation of state for mixed polymer solutions. However, a more reliable approximation such as the weighted density approximation defined by fundamental measure theory64,66,67 is still needed to improve the numerical results when the model is used to provide a prediction for a real system. This will be investigated in our future work.

the environment. This also verifies that the spontaneous symmetry breaking of the W-type brush originates from the molecular contributions to symmetry, such as the enthalpic contribution to incompatibility and the entropic contribution to chain connectivity.

4. CONCLUSIONS We employed a DFT approach to study the vertical structures of symmetric binary polymer brushes with weak species incompatibility immersed in solvents with different selectivities. We consider the explicit solvent and neglect lateral microphase separation. It is found that the solvent selectivity is a necessary factor to trigger perpendicular segregation in binary brushes no matter how weak the selectivity is. The sensitivity of the vertical structure to the solvent selectivity stems from the spontaneity of symmetry breaking in the transition from a nonlayered structure (with a higher grand potential and better symmetry) to a layered structure (with a lower grand potential and worse symmetry) of the dry binary brush. From the perspective with respect to symmetry, the difference between the grand potentials of nonlayered and layered structures originates from the enthalpic contribution of incompatibility and the entropic contribution of chain connectivity. For the convenience of the description, a dry binary brush is characterized by the relationship between the grand potential and vertical structure and is categorized as W-type or U-type according to whether the relation contains a structure of spontaneous symmetry breaking. The W-type brush can adopt both nonlayered and layered structures and memorize the information on the selectivity of the induced solvent in one of its two layered structures with a solvophobic outer layer after the removal of the induced solvent. Nonselective solvent can be used to clear the memory and prepare it for a new layered structure, which means that the W-type brush is lockable with nonselective solvent used as its key. The key also play the role of trigger in the switching between the A-outer layer and the Bouter layer. This lock/unlock mechanism is also found in the experiment. Whereas the U-type brush is lockless because of the lack of a layered structure in its dry brush state (i.e., the lack of memory effect), the U-type brush is adaptable to its environment: it can automatically recover its nonlayered structure after the removal of induced solvent without the nonselective solvent’s triggering and generate a new layered structure with a solvophobic outer layer when immersed in a new induced solvent. To discriminate W-type and U-type brushes through their molecular parameters, we performed DFT calculation to paint the boundary line between the two types in the chain-length-incompatibility space. The results coincide with the above analysis from the point of view of symmetry providing more physical insight. This work can provide some guidance as to the design of solvent-responsive mixed polymer brushes with the properties of locking switching or adaptive switching but still has room for extension. We neglect the possible lateral structure to investigate the pronounced surface switching properties of mixed polymer brushes with a layered structure. A detailed discussion of the lateral segregation (such as the rimple or the dimple state) requires a 3D version of our DFT. The calculation of surface tension is useful in determining the wetting properties of a specific layered structure. The observation of spontaneous symmetry breaking in the vertical structure of symmetrical mixed polymer brushes is the key point in determining the locking/unlocking behavior of the

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Basic Research Program of China (2013CB733501), the National Natural Science Foundation of China (project nos. 21176065 and 21076071), the 111 Project of the Ministry of Education of China (no. B08021), and Fundamental Research Funds for the Central Universities.

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