Vertical Phase Segregation Induced by Dipolar Interactions in Planar

Article pubs.acs.org/Macromolecules

Vertical Phase Segregation Induced by Dipolar Interactions in Planar Polymer Brushes Jyoti P. Mahalik, Bobby G. Sumpter, and Rajeev Kumar* Computer Science and Mathematics Division and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States S Supporting Information *

ABSTRACT: We present a generalized theory for studying structural properties of a planar dipolar polymer brush immersed in a polar solvent. We show that an explicit treatment of the dipolar interactions yields a macroscopic concentration dependent effective “chi” (the Flory−Hugginslike interaction) parameter. Furthermore, it is shown that the concentration dependent chi parameter promotes phase segregation in polymer solutions and brushes so that the polymer-poor phase consists of a finite/nonzero polymer concentration. Such a destabilization of the homogeneous phase by the dipolar interactions appears as vertical phase segregation in a planar polymer brush. In a vertically phase segregated polymer brush, the polymer-rich phase near the grafting surface coexists with the polymer-poor phase at the other end. Predictions of the theory are directly compared with prior reported experimental results for dipolar polymers in polar solvents. Excellent agreements with the experimental results are found, hinting that the dipolar interactions play a significant role in vertical phase segregation of planar polymer brushes. We also compare our field theoretical approach with the two-state and other models invoking ad hoc concentration dependence of the chi parameter. Interplay between the short-ranged excluded volume interactions and long-ranged dipolar interactions is shown to play an important role in affecting the vertical phase separation. Effects of mismatch between the dipole moments of the polymer segments and the solvent molecules are investigated in detail.

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and charged polymer (polyelectrolyte) brushes38−43 and studying their response to different stimuli, polymer brushes consisting of uncharged but polar monomers in equilibrium with solvent still elude a clear understanding. In the absence of any explicit charges or dipoles, the standard approach has been to lump all pairwise short-ranged interactions into a “chi” (χ) parameter (the Flory−Huggins interaction parameter). Such an approach has been quite successful in capturing various qualitative effects of temperature and solvent quality on polymer brush structure. Effects of temperature are included by assuming an ad hoc temperature dependent functional forms for the χ parameter. Effects of solvent quality are studied by varying the χ parameter, which characterizes polymer−solvent interactions (i.e., χ = χps) so that χps < 0.5, χps = 0.5, and χps > 0.5 correspond to a good solvent, theta solvent, and poor solvent, respectively. Such an approach is suitable for nonpolar polymers in nonpolar solvents such as poly(styrene) in toluene. However, it is not clear how the presence of long-range dipolar interactions affects the structural behavior of dipolar polymer brushes in polar solvents, such as poly(ethylene oxide) or poly(N-isopropylacrylamide) in water.

INTRODUCTION Polymers have been routinely used for altering adhesive and cohesive properties of various surfaces.1−3 In particular, anchoring ends of long polymers to a surface leads to a polymer brush.4−9 It is well-established that the end-anchoring can avoid aggregation of colloids and nanoparticles in solutions and melts, termed colloidal stabilization in the literature.10−14 Necessary repulsions required to overcome short-range attractive interactions originate from the entropic penalty the chains have to endure under compression. Other significant features of a polymer brush are their responsiveness to different stimuli such as pH, temperature, and solvent quality. Brushes made up of polar polymers exhibiting lower critical solution temperature (LCST) in solution, such as poly(N-isopropylacrylamide)15−26 and poly(ethylene oxide),27−30 exhibit quite intriguing and poorly understood responses to temperature and solvent conditions. One of the unique features of the polar polymer brushes is the coexistence of two phases31−37 under certain experimental conditions. In particular, it has been shown that a polymer-rich phase near the grafting surface can coexist with a polymer-poor phase near the pure solvent at temperatures close to the LCST of the polymer solutions. Development of an understanding of this so-called vertical phase segregation has been a topic of previous reports.36,37 Despite extensive research focused on the modeling of neutral © XXXX American Chemical Society

Received: May 27, 2016 Revised: August 31, 2016

A

DOI: 10.1021/acs.macromol.6b01138 Macromolecules XXXX, XXX, XXX−XXX

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density distribution of width a, which leads to a regularized theory and captures variations of the polymer segment density in three-dimensional space. However, only the local variations in the dielectric inhomogeneities are considered in this work. Comparison of the local term in the free energy obtained in this work and our previous work79 reveals that qmax ∼ 1/a. Also, for point-like dipoles exhibiting purely local dielectric function, the leading order term capturing the nonlocal effects in the free energy is a square-gradient term and scales as qmax. In contrast, the local term in the free energy scales as qmax3 (after taking Δz ∼ 1/qmax in eq 54 in ref 79). For point-like dipoles with a → 0, qmax ∼ 1/a ≫ 1, which points out that the nonlocal effects are much smaller in comparison with the local effects. Keeping this in mind, the nonlocal effects of the local dielectric function are ignored in this work and the limit of a → 0 is studied in detail. This paper is organized as follows: in the Theory section the general formalism and details about the application of the formalism to the dipolar brushes are presented, in the Results section, numerical and semi-analytical results are presented, and we conclude with the Conclusions section.

A large body of experimental work sheds light on the structural behavior of polymer brushes containing polar neutral monomers such as poly(ethylene oxide) in water44−50 or poly(N-isopropylacrylamide) in water.51−68 Coexistence of two phases has been detected for poly(N-isopropylacrylamide) in a water−acetone mixture by specular reflectivity experiments,22 which exhibit two-step or nonmonotonic monomer volume fraction profiles, known as a vertically phase segregated brush. However, the origin of the vertical phase segregation is not completely understood. Co-nonsolvency,69 monomer−substrate, or monomer−solvent interactions may lead to the vertical phase segregation. Since classical theory6,70 for the neutral polymer brush predicts one-step or monotonic monomer volume fraction profiles, an ad hoc concentration dependent χps36,37 has been used to model the nonmonotonicity. The concentration dependence of χps can be inferred from either experimental results on colligative properties of the polymer solutions or two-state models for the polar polymer solutions.35−37,71−78 In the latter, it is assumed that the polymer can exist in different conformational states, which are stabilized by interactions with the solvent. Although useful, such an approach based on a concentration dependent χps parameter does not capture microscopic details of the solvent and temperature dependencies of the vertical phase separation. This approach assumes the concentration dependence of the χps parameters rather than obtaining it in a self-consistent manner. In this work, we have developed a theory for describing the vertical phase segregation in planar dipolar polymer brushes in equilibrium with a polar solvent. The theory is quite general and can also be applied to polymer solutions, allowing us to directly compare predictions of the theory with the experimental results on the colligative properties of the polymer solutions. A key ingredient of the theory is an explicit treatment of the dipolar interactions among permanent dipoles of the monomers and solvent molecules along with the short-ranged excluded volume interactions. It is shown that a concentration and temperature dependent χps parameter naturally arises due to the explicit dipolar interactions. The functional form for the χps parameter depends on the dipole moments of the polymer segments and solvent molecules, which hints at the dipolar interactions being responsible for the vertical phase segregation. Furthermore, it is shown that the dipolar interactions among polymer segments and solvent, in general, promote phase segregation between a polymer-rich and a polymer-poor phase. In the absence of the dipolar interactions, the theory for the polymer solutions simplifies to the classical Flory−Huggins theory, and the theory for the brushes becomes the same as the classical theory for brushes,6,70 which can only predict a monotonic polymer segment volume fraction profile for a monodisperse neutral brush. The theory developed in this paper is based on our previous work79 for understanding the effects of dielectric inhomogeneity on the phase behavior of polymer blends. We consider only the high-temperature regime of rotating dipoles (weakcoupling limit for the dipoles) without considering any frustrated states. Polymer segments and solvent molecules are assumed to have fixed permanent point dipoles. In ref 79 the point-like nature of the dipoles leads to divergences in the field theory, which was regularized for one-dimensional variations of the polymer segment densities by introducing a cutoff length (∼1/qmax). In this work, we have generalized our theoretical treatment by studying point-like dipoles with a smeared spatial

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THEORY We consider a planar polymer brush formed by n monodisperse flexible chains, each having N Kuhn segments of length b. The chains are assumed to be grafted uniformly onto an uncharged and repulsive substrate with a grafting density of σ (defined as the number of chains per square nanometer) (see Figure 1). In

Figure 1. Schematic of a planar dipolar brush containing monodisperse polymer chains immersed in a dipolar solvent medium. The dipolar chains are modeled as continuous curves (blue colored) containing electric dipoles (represented by blue arrows), and the solvent molecules are represented as discrete electric dipoles (represented by yellow arrows). The dipoles are treated as “point” dipoles with smeared density distribution so that ap and as are the widths of the distribution for the segment and the solvent dipole, respectively. b is the Kuhn segment length. The dipole moment of a segment tα along the αth chain is represented as uα(tα), and the dipole moment of the kth solvent molecule is represented as uk.

experiments, attractive monomer−substrate interactions play an important role in affecting chain conformations near the substrate.80 However, in this work, we deliberately ignore the attractive monomer−substrate interactions to more clearly highlight the effects of dipolar interactions on the vertical phase segregation. In the presence of attractive monomer−substrate interactions, nonmonotonic density profiles similar to the profiles for vertically phase segregated brushes may be induced by the presence of adsorbed monomers next to the substrate.80 For the field theoretical formulation81,82 described in this work, each chain is represented as a continuous curve of length Nb. B


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Macromolecules An arc variable tα is used to represent any segment along the backbone of αth chain so that tα ∈ [0, N]. tα = 0 and tα = N represent the grafted and the free end, respectively, of the αth chain. We use the notation Rα(tα) to represent the position vector for a particular segment, tα, along the αth chain, and Rα(tα = 0) = rα(0) is the position vector for the grafted end of the αth chain. Each segment along a chain is modeled as a “point” dipole with a “smeared” spatial density distribution of width ap. The density distribution is represented by a function ĥp(r − Rα(tα)) centered at the center of mass of the segment (Rα(tα)). Specifically, for each segment tα along the αth chain, an electric dipole of moment (in units of electronic charge, e) pα(tα) is assigned. Similarly, each of the ns solvent molecules is assigned an electric dipole moment, and we use the notation pk to represent the dipole moment of the kth solvent molecule. Similar to the Kuhn segment, each solvent molecule is assigned a “smeared” density distribution of width as. A local incompressibility constraint was imposed so that sum of volume fractions of polymer segment and solvent is unity at all locations. The global incompressibility constraint leads to Ω = nN/ρp0 + ns/ρs0, where ρp0 and ρs0 are the number densities for the pure polymer segment and solvent, respectively. Ω is the total volume of the system. We use the notation rk to represent the position vector of the kth solvent molecule. We assume that the polymer brush region is in equilibrium with the bulk solution. In this paper, we use the word “bulk” quite frequently while referring to the region in space where the electrostatic potential and densities are spatially independent. One should not confuse the region just outside the brush as the “bulk” because the fields may be inhomogeneous even in polymer segment-free regions. The theory for this system is developed in the canonical ensemble, and equating the chemical potential of solvent molecules in the polymer brush region to that in the bulk imposes equilibrium with the bulk solution. The partition function for the brush in the presence of solvent molecules can be written as Λ−3n Z= ns!

np

np

ns k=1

α=1

np

ρ̂p (r) =

∑∫ α=1

0

⎛ ∂R (t ) ⎞ dtα⎜ α α ⎟ ⎝ ∂tα ⎠

dt αhp̂ [r − R α(tα)]

0

(4)

∑ hŝ [r − rk]

(5)

(6)

where functional form of ĥj characterizes the density distribution of a molecule of type j and aj represents a measure of its spatial extent. Particular functional forms for hĵ given in eq 6 are chosen due to the fact that in the limits of aj → 0, ĥj(r) → δ(r), representing point-like solvent molecules and infinitely thin continuous polymer chains. Such choices for ĥj(r) allow direct comparisons with the earlier field theoretic treatments of polymer solutions by simply taking the limit of aj → 0 in the current work. In eq 1, Hdd is the electrostatic contribution to the Hamiltonian, which takes into account the dipole−dipole interactions. In particular, for point dipoles, Hdd can be written as83 Hdd{R α, uα(tα), rk , uk} =

α = 1 tα = 0

lBo 2

∫ dr ∫ dr′

̂ (r)][∇r ′ ·Pave ̂ (r′)] [∇r ·Pave |r − r′|

(7)

where lBo = e2/4πϵ0kBT is the Bjerrum length in vacuum, e is the charge of an electron, ϵ0 is the permittivity of vacuum, kB is the Boltzmann constant, and T is the temperature. Furthermore, P̂ ave(r) = ∫ du P̂ (r,u)u is the angularly averaged local polarization density (which is a vector) so that P̂ (r,u) = ∑j = p,s pj ρ̅j(r,u) is the magnitude of the local polarization density at r in the direction specified by unit vector u on the surface of a sphere. Formally, ρ̅p(r,u) and ρ̅s(r,u) are microscopic number densities of the polymer segment and solvent dipoles, respectively, defined by

(1)

where n = npN + ns is the total number of particles and Λ is the de Broglie wavelength. H0{Rα} is the well-known Wiener measure for a flexible polymer chain,82 given by N

N

⎡ ⎤3/2 ⎡ ⎤ π |r|2 1 hĵ = p , s(r) = ⎢ 2 ⎥ exp⎢ − 2 ⎥ ⎢⎣ 2aj ⎥⎦ ⎢⎣ 2aj ⎥⎦

k=1

3 H0{R α} = 2 2b

(3)

k=1

∫ ∏ duk exp[−H0{R α} − Hw{R α, rk}

np

∫ dr wjj′ρĵ (r)ρĵ′ (r)

ns

ρ̂s (r) =

N

⎡ ⎤ ρĵ (r) − Hdd{R α, uα(tα), rk , uk}]∏ δ ⎢ ∑ − 1⎥ ⎢⎣ j = p , s ρj0 ⎥⎦ r

∑∫ α=1

ns

×

∑ ∑ j = p , s j ′= p , s

where wjj′ is the excluded volume parameter describing the strength of interactions between particles of type j and j′. It should be noted that eq 3 is written with the assumption that the form of interaction potentials between different pairs can be approximated by delta functions so that range of interaction is infinitesimal. Furthermore, ρ̂p(r) and ρ̂s(r) represent the microscopic number density of the polymer segments and solvent, respectively, at a certain location r. They are defined as

∫r (0) ∏ D[R α] ∫ ∏ drk ∫ ∏ ∏ duα(tα) α

1 2

Hw{R α, rk} =

np

ρp̅ (r, u) =

2

∑∫ α=1

0

N

dtα hp̂ [r − R α(tα)]δ[u − uα(tα)] (8)

(2) ns

Hw{Rα, rk} takes into account the energetic contributions originating from the short-range repulsive hard-core interactions and the attractive dispersive interactions excluding the permanent dipole−dipole interactions. Hw can be expressed using the Edwards’s formulation82 for local short-ranged interactions as

ρs ̅ (r, u) =

∑ hŝ [r − rk]δ[u − uk] k=1

(9)

where uα(tα) and uk represent the dipole moment of a segment tα along the αth chain and the kth solvent molecule, respectively. C


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approximation while evaluating functional integrals over ψ requires working in the weak coupling limit (WCL) for dipoles, defined by pj|∇rψ(r)| → 0 for j = p, s. Physically, such a treatment should be representative of weakly correlated dipoles realizable in solutions of polar polymers such as poly(ethylene oxide) (PEO) in water near room temperature or at elevated temperatures. In the following, we present our approach for evaluating the functional integral over ψ in the WCL for dipoles. Regularized Field Theory in the Weak Coupling Limit (WCL) for Dipoles. In the WCL of dipoles, pj|∇rψ(r)| → 0 for j = p, s and H̅ elec can be simplified using ln(sin x/x) → −x2/6 for x → 0. These simplications lead to

Using field theoretical transformations81,82 described in Appendix A of the Supporting Information, we can write the partition function for the brushes as Z=

Z0 ζψ

⎡

H̅ ⎤ ⎥ ⎣ kBT ⎦

∏ {∫ D[ϕj] ∫ D[wj]} ∫ D[ψ ] ∫ D[η] exp⎢− j=p,s

(10)

where ζψ is a normalization factor and Z0 = exp[−F0/kBT] is the partition function in the absence of interparticle interactions. Explicit expressions for ζψ and F0 are presented in Appendix A of the Supporting Information. ψ, wp, and ws are the collective fields introduced to decouple electrostatic interactions, short-range dispersion interactions among polymer segments, and short-range dispersion interactions among solvent molecules, respectively. η is the pressure field introduced to impose the local incompressibility condition. ϕp and ϕs are the collective density variables for the polymer segments and solvent, respectively. H̅ is the field-theoretic Hamiltonian of the system, and it can be written as H̅ = H̅ neu + H̅ elec, where those terms that explicitly depend on ψ are clubbed together in H̅ elec so that H̅ neu = αps ̅ kBT +i

∫

∫ dr

ρp (r) ρ (r) s ρp0

ρs0

−i

H̅ elec,WCL H̅ elec 1 ≃ = kBT kBT 2

(14)

where E(r, r′) = −

∑ ln Q̅ p, α{wp}

∫ dr″ϕj(r″) (15)

(16)

where

j=p,s

Zelec,WCL{ϕj} =

1 ζψ

⎤

⎡

∫ D[ψ ] exp⎣⎢− 12 ∫ dr ∫ dr′ψ (r)E(r, r′)ψ (r′)⎦⎥ (17)

(12)

Evaluating the functional integral over ψ (see Appendix B of the Supporting Information), eq 17 can be written as

In these equations, α̅ps (having dimension of length−3) is defined by

⎡ 1 Zelec,WCL{ϕj} = exp⎢ − ⎣ 8π

wppρp0 2 + wssρs0 2 2

3

⎡ H̅ neu ⎤ ⎥Zelec,WCL{ϕj} ⎣ kBT ⎦

j=p,s

∫ dr ψ (r)∇r ψ (r) − ∑ ∫ dr′ϕj(r′)

αps̅ = wpsρp0 ρs0 −

j=p,s

pj 2

∏ {∫ D[ϕj] ∫ D[wj]} ∫ D[η] exp⎢−

Z ≃ Z0

α=1

2

⎡ sin{p | ∫ dr ψ (r)∇ h ̂ (r − r′)|} ⎤ r j j ⎥ × ln⎢ ⎢ p | ∫ dr ψ (r)∇ h ̂ (r − r′)| ⎥ r j ⎦ ⎣ j

∑

We evaluate the functional integral over ψ after writing eq 10 as

np

(11)

H̅ elec 1 =− kBT 8πlBo

1 ∇r 2 δ(r − r′) + 4πlBo

× {∇r hĵ (r − r″) ·∇r ′hĵ (r′ − r″)}

∑ ∫ dr wj(r)ϕj(r) j=p,s

⎡ ⎤ ρj (r) − 1⎥ − ns ln Q̅ s{ws} − dr η(r)⎢ ∑ ⎢⎣ j = p , s ρj0 ⎥⎦

∫ dr ∫ dr′ψ (r)E(r, r′)ψ (r′)

(13)

⎤

∫ dr f (r) ln ϵ(r)⎥⎦

(18)

where

Furthermore, we have defined ρj(r) = ∫ dr′ hĵ (r − r′)ϕj(r′), and Q̅ p,α{wp} is the normalized partition function for a polymer chain grafted at point Rα(0) = rα. Similarly, Q̅ s is the partition function for a solvent molecule. Explicit expressions for Q̅ p,α and Q̅ s are presented in Appendix A of the Supporting Information. Evaluation of functional integrals over the field variables ϕj, wj, η, and ψ requires novel sampling schemes for the Hamiltonian, which is numerically challenging. A standard approach has been to approximate the functional integrals by the values of the integrand at the saddle point (saddle point approximation), which amounts to vanishing functional derivatives of the Hamiltonian with respect to the collective fields and densities.81 However, our numerical results based on the standard saddle-point approximation do not capture the vertical phase separation in dipolar brushes. Hence, we took an intermediate approach for evaluating the functional integrals using a partial saddle-point approximation. In this approach, the functional integral over ψ is evaluated by going beyond the saddle point, and the other functional integrals are subsequently approximated by the saddle-point approximation. The mathematical treatment for going beyond the saddle-point

ϵ(r) = 1 +

4πlBo 3

∑

pj 2 ρj (r)

(19)

j=p,s

is the local dielectric function and f(r) is given by f (r) =

∑j = p , s pj 2 ∫ dr′ϕj(r′)gj(r − r′) ∑j = p , s pj 2 ∫ dr′ϕj(r′)hĵ (r − r′)

(20)

so that gj(r) = −

⎡ ⎤ ⎧ ⎪ ⎪ 1 π ̂ ⎢ 2 ̂ π |r| ⎫ ⎥ ⎬ ⎨ − h ( r ) 4 a h ( r ) erf j ⎪ ⎪ ⎢⎣ j j ⎥⎦ | | 2 a r aj 2 ⎩ j⎭

(21)

In deriving eq 18, we have ignored nonlocal (or derivative) terms due to spatial variations in ϵ(r). Advantages of introducing smeared density distributions (i.e., hĵ ) in the theoretical treatment become clearer at this stage: (i) the Hamiltonian stays free of divergences such as those encountered in the field theory of point dipoles with density distributions of zero width,79 (ii) allows treatment of multicomponent systems without any ambiguity in choosing D


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Macromolecules the cutoff required to regularize the divergent theory, and (iii) allows three-dimensional simulations alleviating the need to invoke one-dimensional variations. In the following, we use eq 16 and apply the theory in the limit of aj ≡ a → 0 to study a laterally homogeneous planar polymer brush in equilibrium with a solution bath. It turns out that in the limit of aj ≡ a → 0 the function f(r) becomes independent of ϕj, and only the functional dependence of ϵ(r) on ρj(r) (cf. eq 19) affects the thermodynamics of polymer brushes and solutions. A Planar Dipolar Polymer Brush in Equilibrium with a Solution Bath. Laterally Homogeneous Brush: One-Dimensional Model. After evaluating the functional integral over ψ, we can study an inhomogeneous polymer brush in equilibrium with a solution bath using the saddle-point approximation81 for the field variables wj, ϕj, and η. The approximation evaluates the functional integrals over the field variables by the value of the integrand at the saddle point and leads to coupled nonlinear equations representing the saddle point. For a laterally homogeneous brush, these equations can be further simplified by integrating out the in-plane degrees of freedom. In particular, taking the x-axis along normal to the grafting plane of area A, and optimization of the integrand in eq 16 with respect to wp, we obtain

ϵ*(x) = 1 +

∑ α= 1

where q(x, ̅ N − t) satisfies ⎤ ∂q ̅ (x , t ′) ⎡ b2 ∂ 2 =⎢ − w*p (x)⎥q ̅ (x , t ′) 2 ∂t ′ ⎣ 6 ∂x ⎦

ns exp[−ws*(x)] A ∫ dx exp[−ws*(x)]

ρp0

+

ρs0

∫

ws*(x) =

+ (22)

⎪

(29)

∫

⎡ α ρ *(x′) ̅ p ps η*(x′) + dx′ hs(x − x′)⎢ ⎢⎣ ρp0 ρs0 ρs0

lBops2 f * (x′) ⎤ l p2 ⎥ + Bo s 6 ϵ*(x′) ⎥⎦ 6

∫ dx′ [gs̅ (x − x′) − f * (x′)

⎡ ln ϵ*(x′) ⎤ × hs(x − x′)]⎢ ⎥ ⎣ ϵ*(x′) − 1 ⎦

(30)

For a dipolar brush in equilibrium with a solution bath, we have to equate the chemical potentials of all the components that can be exchanged between the bulk solution and the brush region. Also, mechanical equilibrium requires that the osmotic pressure must be the same everywhere. However, due to the incompressibility constraint, equating the chemical potentials of the solvent molecules is sufficient to define the equilibrium state of the system. Using the thermodynamic relation between the free energy and chemical potential in the canonical ensemble (i.e., μj = (∂F/∂nj)Ω,T = −kBT(∂ ln Z{wj*, ϕj*, η*}/ ∂nj)Ω,T), we equate the chemical potential of solvent molecules in the brush and the solvent bath via

(23)

(24)

kBT

=1 =

(25)

wssρ0

=

2

wssρ0 2

⎡ ⎤ ns ⎥ − ln 4π + ln⎢ ⎢⎣ ∫ dr exp[−iws*(r)] ⎥⎦

− ln 4π + ln ρso + ws*(∞)

(31)

where the last expression is obtained by considering the case of spatially independent field far from the brush, representing the solvent bath. Numerical Methods. We have solved the nonlinear set of equations presented above numerically after writing them in dimensionless form. All of the quantities having dimensions of length are made dimensionless by dividing them by Rg = (Nb2/ 6)1/2. The numerical results presented in this work have been obtained by using a box length of 25Rg with 256 grid points, which is large enough to ensure that the fields far from the brush region approach zero. The modified diffusion equation represented by eq 23 has been solved by using implicit-explicit scheme known as the extrapolated gear method.84,85 In order to use the extrapolated gear method, we rewrite eq 23 in the form

⎡ α ρ*(x′) η*(x′) ̅ s ps dx′ hp(x − x′)⎢ + ⎢ ρ ρ ρp0 ⎣ p0 s 0

lBopp 2 f * (x′) ⎤ lBopp 2 ⎥ + + 6 ϵ*(x′) ⎥⎦ 6

⎪

Similarly, optimization with respect to ϕs yields

where ρj*(x) = ∫ dx′ hj(x − x′)ϕj*(x′) for j = p, s so that hj(x) = ∫ dr∥ĥj(r) = exp[−πx2/2aj2]/√2aj and r = {x, r∥}. Optimizing the integrand in eq 16 with respect to ρ*p , we obtain wp*(x) =

(28)

⎡ ⎧ π |x − x′| ⎫⎤ π2 ⎢ π 2 ⎨ ⎬⎥ − − 1 erf 4 ⎪ ⎪ ⎢ ⎥ aj ⎭⎦ 4aj ⎣ 2aj 4 ⎩ 2

⎡ ⎤ π (x − x′)2 ⎥ × exp⎢ − ⎢⎣ ⎥⎦ aj 2

μs

ρs*(x)

(27)

j=p,s

∑j = p , s pj 2 ∫ dx′ hj(x − x′)ϕj*(x′)

gj̅ (x − x′) =

and ρp*(x)

pj 2 ρj*(x)

so that gj̅ (x − x′) = ∫ dr∥′ gj(r − r′). Explicitly

with the condition q(x,0) = 1 for t′ = N − t = 0 and we have ̅ used the notation w*p = iwp. Similarly, qxα(x,t) satisfies the same equation but with the initial condition qxα(x,0) = δ(x − xα(0)). The superscript ∗ highlights the fact that the quantities are representative of the saddle point. Optimizing the integrand in eq 16 with respect to ws (w*s = iws) and η (η* = iη), respectively, we obtain ϕs*(x) =

∑

∑j = p , s pj 2 ∫ dx′ gj̅ (x − x′)ϕj*(x′)

f * (x ) =

∫0 dt qxα(x , t )q ̅ (x , N − t ) A ∫ d x q x (x , t )q ̅ (x , N − t ) α

4πlBo 3

and

N

np

ϕp*(x) =

where

∫ dx′

⎡ ln ϵ*(x′) ⎤ × [gp̅ (x − x′) − f *(x′)hp(x − x′)]⎢ ⎥ ⎣ ϵ*(x′) − 1 ⎦ (26) E


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Macromolecules ∂q ̅ (x ̅ ) λ ∂ 2q ̅ (x ̅ ) ⎡ λ ⎤ ∂ 2q ̅ (x ̅ ) − Nw*p (x ̅ )q ̅ (x ̅ ) + ⎢1 − max ⎥ = max 2 ⎣ 2 2 ⎦ ∂x ̅ 2 ∂t ̅ ∂x ̅ (32)

phase in polymer solutions are then applied to a more complex polymer brush system for predicting vertical phase segregation. Polymer Solution. In a homogeneous polymer solution, different components are uniformly distributed in space so that ϕ*j (r) = ϕ*j = ρ*j (r) = ρ*j , ϕ*p /ϕp0 + ϕ*s /ϕs0 = 1, and ϵ*(r) = ϵh(ϕ*p ) = ϵp ϕ*p + ϵsϕ*s , where ϵj=p,s = 1 + 4πlBopj2ρj0/3 so that ϵp and ϵs are the dielectric constants for the pure polymer and the pure solvent, respectively. For a homogeneous polymer solution, eq 18 can be simplified to

where t ̅ = t/N, x̅ = x/Rg, and we have chosen λmax = 1/2. In the extrapolated gear method, the first term on the right-hand side of eq 32 is treated implicitly and the rest is treated explicitly. The Crank−Nicholson scheme is used to initialize the gear method for the modified diffusion equation. Dirichlet boundary conditions are used for qxα and q̅ at the substrate, i.e., qxα(x = 0, t) = q(x ̅ = 0, t) = 0 for all values of t. The grafted ends are displaced to the first grid point while solving the equations to avoid conflict between the initial and boundary conditions. Also, Δt ̅ = 10−3 was used for the time stepping while solving the modified diffusion equations. We started from random numbers as an initial guess for the fields (wp*, ws*) and compute the values of the fields by solving the modified diffusion equation and the set of equations described in the previous section. η* was updated using the computed values of w*p and w*s . The guessed and the computed values for wp* and ws* are mixed using the simple mixing scheme81 to obtain new values for the next iteration. The iterative procedure was continued until the absolute difference between the free energy of the brush (with respect to the solution bath) between consecutive iterations reaches a tolerance value of 10−8. The height of the brush was calculated by the relation

∫ υp*(x)x 2 dx H = ∫ υp*(x) dx

H̅ elec,WCL kBT

= −ln[Zelec,WCL{ϕj*}] = Ω

⎡ πl × ⎢ Bo ⎢ 12 ⎣

ln ϵh(ϕp*) [ϵh(ϕp*) − 1]

pj 2 ϕj* ⎤ ⎥ aj 3 ⎥⎦

∑ j=p,s

(34)

where Ω is the total volume of the system. The total free energy of the homogeneous polymer solution within the saddle-point approximation is given by F = H̅ neu + H̅ elec ≃ H̅ neu + H̅ elec,WCL (cf. eqs 11 and 34) so that the free energy per unit volume (f = F/Ω) can be expressed as f (υp*, T ) ρ0 kBT

=

υp* N

ln υp* + (1 − υp*) ln(1 − υp*)

+ χeff (υp*, T )υp*(1 − υp*)

(35)

Here, υ*p (= ϕ*p /ρp0) represents the volume fraction of the polymer segments. Also, we have considered solutions with symmetric spread (ap = as = a) and symmetric molar volume (ρp0 = ρs0 = ρ0 = 1/b3) for simplicity. Also, we have defined

2

(33)

where υ*p (= ϕp*/ρp0) is the volume fraction of the polymer segment.

χeff (υp*, T ) =

■

RESULTS The theoretical framework presented above is general and can be applied to any polar macromolecular system. In this paper, we have used this framework to study a dipolar polymer brush immersed in a polar solvent. The main goal of this paper is to identify the parameter space, where the polymer brush is likely to show vertical phase segregation, i.e., coexistence of a concentrated phase and a supernatant phase so that the supernatant phase consists of finite measurable polymer concentration. Moreover, the goal is also to understand the effects of different parameters characterizing dipolar interactions on the structure of a polymer brush. Varying these parameters to study vertical phase segregation in the polymer brushes requires exhaustive numerical work. Rather than varying the parameters arbitrarily to map out a phase diagram for the brushes, we have taken an approach based on the hypothesis that if a homogeneous polymer solution becomes destabilized and undergoes phase segregation into a concentrated and a supernatant phase for a given set of parameters, a brush made up of the same polymers immersed in the same solvent may also undergo phase segregation. In the case of the brush, the phase segregation will occur along the normal to the grafting plane. Below we report on the phase segregation in homogeneous polymer solutions and construct coexistence curves. This study of polymer solutions allows us to compare the predictions of the theory directly with the prior experimental results reported in the literature. The parameters exhibiting coexistence between a concentrated and a dilute

αps ̅ ρ0

+

ln[1 + Δϵυp*] 1 3 16(a /b) υp*(1 − υp*)

(36)

so that Δϵ = (ϵp − ϵs)/ϵs characterizes mismatch between the dipole moments of the segments and the solvent molecules. Free energy is not directly measurable in experiments. Instead, osmotic pressure measurements provide indirect access to the underlying free energy. Computation of the osmotic pressure using the free energy given by eq 35 leads to another parameter χ that is related to χeff by the relation36,37 χ (υp*, T ) = χeff (υp*, T ) − (1 − υp*)

∂χeff (υp*, T ) ∂υp*

(37)

which generates χ (υp*, T ) =

αps ̅ ρ0

+

⎡ ln(1 + Δϵυ*) ⎤ Δϵ 1 p ⎢ ⎥ − 3 2 * * * υp (1 + Δϵυp ) ⎥⎦ 16(a /b) ⎢⎣ (υp )

(38)

The concentration dependence of χ in eq 38 allows us to model the experimental values of χ reported in the literature for different dipolar polymers−polar solvent pairs as shown in Figure 2. It is clear that the concentration dependence of the χ parameter measured experimentally is accurately captured by eq 38. The comparisons were done by varying three parameters: a/b, α̅ ps/ρ0, and Δϵ. The parameters corresponding to the theoretical curves shown in Figure 2 are presented in Table 1. (i) Physically, a is the spread of the Kuhn segment density distribution about its center of mass and b is the length of a Kuhn segment so that the ratio a/b is expected to be less than F


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where a22 = e4(pp2 − ps2)2ρ02/(9ϵ02kB2) and b22 = e2ps2ρ0/ (3ϵ0kB). The second term on the right-hand side in eq 39, which results from the dipolar interactions, is positive. From the functional form of a22, it can be readily inferred that the dipolar mismatch always leads to an increase in χ0 in comparison with the case of zero dipolar mismatch. Inverse square dependence on T appearing in eq 39 reveals that a decrease in temperature should lead to an increase in the contributions from the dipolar interactions. Furthermore, one can describe the lower critical solution temperature (LCST) or the upper critical solution temperature (UCST) behavior using eq 38 or 39 by using temperature dependence of α̅ ps/ρ0. For example, one can invoke different approximations pertaining to the changes in conformational states of polar polymers and binding of solvent molecules such as those done in a model introduced by Karlstrom71 and its extensions74 to describe the temperature dependence of α̅ ps/ρ0 in the form α̅ ps/ρ0 = a1 + b1/ T, where b1 < 0 for the LCST and b1 > 0 for the UCST. Equation 38 provides a microscopic description for the higher order temperature dependence of χ(υp*,T) and reveals that the dipolar interactions affect the transition temperature. In order to determine the conditions at which the polymer solution will most likely show coexistence of a concentrated and a dilute phase with nonzero polymer concentration, a/b obtained for PEO−water and PMMA−acetone solutions (∼a/b for poly(N-vinylcaprolactam)−water) were used as fixed parameters in eq 35 generating spinodal as well as binodal curves (cf. Figure. 3). For the PEO−water a/b ∼ 1, whereas for PMMA−acetone a/b ≪ 1. We considered these two systems to investigate the coexistence curves for realistic polymer solutions at the extreme limits of a/b. A value of N = 150 was selected for all the calculations. As shown in Figure 3a,b, the binodal and spinodal curves show asymmetric features as expected for an asymmetric system like polymer and solvent. However, the minima in the coexistence curves shift downward and toward higher υp* as Δϵ is increased in magnitude. In other words, an increase in the dipolar mismatch enhances phase segregation in the polymer solutions. Comparisons of the phase diagrams in Figures 3a and 3b reveal that the dipolar interactions tend to have stronger effect for a/b ≪ 1. Physically, b/a is proportional to the number of monomers per Kuhn segment. We observed that a high mismatch in the dipole moments (i.e., high values of Δϵ) along with small values of a/b, and α̅ ps favors segregation of polymer solution into a concentrated and a dilute phase with nonzero concentration. Planar Polymer Brush. Strong Stretching Limit: Semianalytical Treatment. For a polymer brush and solvent molecules with point-like density distribution (i.e., aj → 0), hj(x−x′) = δ(x−x′) and f*(x) = π/2a3 for aj = a. Assuming, ρs0 = ρp0 = ρ0 and imposing local incompressibility constraint, ρs*(x)/ρ0 + ρp*(x)/ρ0 = υs*(x) + υp*(x) = 1, eqs 26 and 30 can be written as

Figure 2. Experimental results for χ based on the osmotic pressure measurements reported in the literature are used to extract α̅ ps/ρ0, a/b, and Δϵ using eq 38 for three different dipolar polymer and polar solvent pairs. The three systems are (a) PEO in water at 25 °C,36 (b) poly(N-vinylcaprolactam) in water at 25 °C,88 and (c) PMMA in acetone at 50 °C.89 The extracted parameters are presented in Table 1.

Table 1. Parameters Obtained after Fitting Eq 38 for Comparisons with the Concentration Dependence of χ Reported in the Literature system PEO−water at 25 °C36 poly(N-vinylcaprolactam)−water at 25 °C88 PMMA−acetone at 50 °C89

α̅ps/ρ0 0.4334 0.1979

0.886 0.184

−0.81390 −0.292

−0.0034

0.170

−0.27691,92

a/b

Δϵ

unity for the most polymers. The parameter a/b for the three systems is within reasonable values (0 < a/b < 1). (ii) α̅ ps/ρ0 characterizes the strength of dispersive interactions between the Kuhn segments and the solvent molecules. The values for α̅ps/ ρ0 are reasonable compared to the reported literature values of the Flory−Huggins interaction parameter for most of the polymer−solvent pairs. (iii) Δϵ = (ϵp − ϵs)/ϵs characterizes the mismatch between the dipole moments of the Kuhn segments and the solvent molecules. For most polymeric solutions prepared with water as the solvent, −1 < Δϵ < 0, highlighting that the polymer segments are less polar than the water molecules. Furthermore, values of Δϵ are negative as expected and have expected magnitudes. These comparisons highlight the importance of dipolar interactions in polymer solutions containing polar polymer segments and solvent molecules. The dependence of χ on υp* in eq 38 vanishes as Δϵ approaches zero. We should point out that in this limit the concentration dependence of χ can appear if the contributions from the density fluctuations in the free energy are taken into account.86,87 These contributions are ignored in this work to focus on the effects of dipoles. The concentration dependence appearing in eq 38 results from the fluctuations of electrostatic potential about the saddle point for the case of dipolar molecules. The dipolar interactions lead to new temperature dependent terms in χ(υ*p ,T) (cf. eq 38) in addition to α̅ps. The temperature dependence in these additional terms appear via Δϵ = (ϵp − ϵs)/ϵs and the fact that lBo ∼ 1/T. In particular, if we consider the limit of Δϵυp* → 0, eq 38 leads to χ (υp* → 0, T ) ≡ χ 0 =

αps ̅ ρ0

+

w*p (x) − ws*(x) =

ρ0

[1 − 2υp*(x)] +

ϵs

Δϵ * ϵ 16(a /b) (x) 3

(40) 36,37

In the strong stretching limit, semi-analytical calculations can be done to obtain polymer density profiles besides the use of numerical techniques required for solving the SCFT equations. Using the relation ws*(x) = −ln[1 − υp*(x)] in eq 40 (cf. eqs 24 and 25), we obtain

a2 2 (b2 2 + T )2

αps ̅

(39) G


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Vertical Phase Segregation in a Dipolar Brush. Equipped with approximate ranges of the three parameters obtained from the study of polymer solutions (i.e., a/b, α̅ ps/ρ0, and Δϵ), we explored conditions for vertical phase separation in polymer brushes using the semi-analytical treatment and numerical solution of the SCFT equations. Keeping N fixed at 150, we investigated the effect of σb2 on the phase behavior of polymer brushes. Typical results for one such case are presented in Figure. 4a. At low value of σb2 ( −0.59, and poor-solvent-like conditions are seen for Δϵ ≤ −0.68. For the intermediate values of Δϵ vertical phase segregation is observed.

(41)

Subtracting the field for the reference frame, taken to be the pure solvent, we can express wp* as w*p (x) − w*p (∞) = −ln[1 − υp*(x)] − −

2αps ̅ ρ0

υp*(x)

Δϵ2υp*(x) 16(a/b)3 [1 + Δϵυp*(x)]

(42)

In the strong stretching limit, w*p (x) − w*p (∞) = B(H − x2) so that B = 3π2/(8N2b2) and H is the height of the brush. Using the boundary conditions υp*(H) = 0 and υp*(0) = υp0 * , we can compute the density profile for the brush using 35

x(υp*) =

2

density profiles exhibiting a depletion region next to the substrate and monotonically decaying density profiles afterward. The semianalytical calculations do not capture the depletion region, but beyond this region, monotonically decaying density profiles almost identical to the numerical calcualtions are predicted. However, for σb2 ≥ 0.10, two-step profiles are obtained beyond the depletion region. The two-step profile is a signature of vertical phase separation (a uniform concentrated polymer phase near the substrate coexisting with a nonuniform dilute phase near the solvent side). An increase in the grafting density leads to an increase in the brush height as

w*p {υp*0} − w*p (υp*) B

(43)

along with the mass balance condition, given by Nσρ0 =

υp*(H )

∫υ*

p0

∂x dυp* υp* ∂υp*

(44)

Here, wp*(υp*) is given by eq 42. H


Article

Macromolecules well as widening of the uniform concentrated phase and shrinkage of the nonuniform dilute phase. Qualitatively similar results have been reported in ref 37 using an ad hoc concentration-dependent χ parameter and the strong stretching limit. It has been demonstrated35−37 that the polymer concentration dependence of χeff has to be at least quadratic for vertical phase segregation to be feasible in a brush. Our work provides a microscopic description of the concentration dependence of the χ parameter given by eq 36 and captures the concentration dependence even beyond the quadratic order. Equation 36 reveals that the dipolar mismatch parameter Δϵ can be used to tune the χ parameter and in turn modulate the coexistence curves as shown in the previous section. This, in turn, means that Δϵ can also affect the vertical phase segregation. We have investigated the effect of Δϵ on the polymer segment density profiles by keeping σb2 fixed (Figure 4b). The brush shows a monotonically decaying density profile for Δϵ > −0.59 and vertical phase segregation for −0.68 < Δϵ < −0.59 and exhibits uniform density profile for Δϵ ≤ −0.68. This shows that the brush transitions from good-solvent-like conditions to poor-solvent-like conditions via a coexistence regime as Δϵ is gradually increased in magnitude. In other words, an increase in magnitude of Δϵ makes the quality of the solvent poorer. Analysis of the changes in free energy corresponding to the structural transitions shown in Figure 4 reveals that the transition from one phase to the coexistence regime is smooth, exhibiting no discontinuities in the first or the second derivatives of the free energy with respect to the height of the brush defined by eq 33. Such smooth changes in the free energy are similar to those observed for collapse of neutral nonpolar planar polymer brushes.93,94 However, a detailed analysis of the free energy is required to show generality of this result, which is beyond the scope of this work. The semi-analytical calculations corresponding to the strongstretching limit capture the qualitative trends but differ quantitatively from the numerical results. The origin of this discrepancy lies in the assumptions made in the semi-analytical calculations. For example, in the numerical analysis, use of the Dirichlet boundary conditions leads to a depletion region next to the substrate, but in semi-analytical calculations the polymer volume fraction is maximum next to the substrate. Because of the mass balance, such a treatment of polymer segment− substrate interactions leads to deviations from the numerical results. From our study of polymer solutions and brushes, it is clear that an interplay between the short-ranged excluded-volume interactions (i.e., α̅ ps/ρ0) and the permanent dipole−dipole interactions (i.e., Δϵ) affects the coexistence of different phases in the solutions and the brushes. Furthermore, Figure 4b reveals that Δϵ affects the brush height, and in particular, an increase in magnitude of Δϵ leads to shrinakge of the vertically phase-segregated brush. The shrinkage with an increase in magnitude of Δϵ is predicted even for a dipolar brush, which is not vertically phase segregated as shown in the next section. Effects of the Dipolar Mismatch (Δϵ) on the Brush Height. In this section, we consider a dipolar brush, which is not vertically phase segregated, and show the effects of dipolar interactions on the brush height. Δϵ is the parameter, which characterizes mismatch in the dipole moments of the Kuhn segments and the solvent molecules. In the limit of small dipolar mismatch parameter, i.e., Δϵ → 0, wp*(x) − ws*(x) in eq 40 can be further simplified to

w*p (x) − ws*(x) ≃

αps̅ ρ0

− 2χ 0 υp*(x)

(45)

where χ0 =

αps ̅ ρ0

+

Δϵ2 32(a /b)3

(46)

is the renormalized α̅ps/ρ0 parameter. It is to be noted that eq 46 is the zero concentration (i.e., υ*p → 0) limit of eq 38. Equations 45 and 46 reveal that the solvent quality becomes poorer with increase in the dielectric mismatch (Δϵ) irrespective of its sign and the extent of shrinkage of a polymer brush is purely determined by the magnitude of Δϵ. The numerical results presented in Figure 5 qualitatively confirm these predictions. In particular, Figure 5 shows that the

Figure 5. (a) Effect of Δϵ on the segment density profiles demonstrating that the brushes shrink with an increase in magnitude of Δϵ, independent of the sign of Δϵ. The semi-analytical results based on the strong stretching approximation agree reasonably well with the numerical results based on the SCFT. (b) The height of the brush as a function of the mismatch parameter showing quadratic dependence for low Δϵ and asymmetric dependence for higher Δϵ. The parameters used in the calculation are σb2 = 0.25, N = 150, α̅ ps/ρ0 = 0.4, a/b = 0.2, and ϵs = 80.

brush shrinks in the presence of nonzero mismatch (relative to the case of zero mismatch), and the height of the brush decreases with an increase in the magnitude of the mismatch. For low mismatch parameter (Δϵ ≤ 0.1), the brush height depends only on the magnitude of the mismatch as predicted by eq 46. However, with an increase in the mismatch, the height of the brush not only depends on the magnitude of the mismatch but also on the sign of the mismatch. The brush shows more shrinkage for negative mismatch compared to the positive mismatch for any given magnitude of Δϵ. As a result, an asymmetric dependence of brush height on Δϵ is observed as shown in Figure 5b. The origin of this asymmetry lies in the I


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Macromolecules asymmetric dependence of w*p (x) on Δϵ for higher values of Δϵ (eq 45). Qualitatively similar results were observed by Kumar et al.79 for dipolar polymer blends. In ref 79 it was demonstrated that the phase segregation can be enhanced by increasing the mismatch parameter. A simpler approach based on the freely rotating dipoles was used to derive a renormalized χ parameter similar to eq 46. However, the theory presented in ref 79 had divergences and was regularized by introducing a cutoff length. The asymmetry in phase segregation was also observed for polymer blends of asymmetric size. Furthermore, the polymer segment density profiles (Figure 5a) obtained using semi-analytical treatment in the strong stretching limit agree reasonably well with the numerical results despite the absence of the depletion region near the substrate in the former.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC05-00OR22725. This research was conducted at the Center for Nanophase Materials Sciences, which is a Department of Energy Office of Science User facility. J.P.M. acknowledges support from the Laboratory Directed Research and Development program at ORNL.

■

■

CONCLUSIONS Field theory was developed for a planar dipolar polymer brush in a polar solvent to understand its macroscopic structural behavior, starting from a microscopic description of the system. It is shown that the dipolar interactions among the polymer segments and solvent molecules can stabilize a polymer-rich phase coexisting with a polymer-poor phase consisting of nonzero polymer concentration in the polymer solution. Similarly, the dipolar interactions can stabilize a vertically phase-segregated planar polymer brush with the polymer-rich phase next to the grafting plane coexisting with the polymerpoor phase near the pure solvent. It is shown that a mismatch in the dipole moments between the polymer segment and solvent (i.e., Δϵ) always leads to shrinkage of the brush. However, reduction in the brush height is dependent on the sign of the mismatch due to the nonlinear asymmetric dependence of the effective interaction parameter on Δϵ at higher values of Δϵ. Lastly, the chemical structure of the polymer segment defined as the spread of the Kuhn segment (a) over its length (b) is shown to play an important role. Interplay between the short-ranged excluded volume interactions and long-ranged attractive dipolar interactions is found to play a significant role in affecting the structure of a polymer brush. Overall, we have formulated a new field theory approach that enables study of dipolar brushes in polar solvents in threedimensional space. The theory is free from any divergences negating the need for any regularization procedure and can be generalized to multicomponent systems in a straightforward manner.

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(1) Hammond, P. T. Form and Function in Multilayer Assembly: New Applications at the Nanoscale. Adv. Mater. 2004, 16, 1271−1293. (2) Hussain, F.; Hojjati, M.; Okamoto, M.; Gorga, R. E. Review Article: Polymer-matrix Nanocomposites, Processing, Manufacturing, and Application: An Overview. J. Compos. Mater. 2006, 40, 1511− 1575. (3) Currie, E. P. K.; Norde, W.; Cohen Stuart, M. A. Tethered Polymer Chains: Surface Chemistry and Their Impact on Colloidal and Surface Properties. Adv. Colloid Interface Sci. 2003, 100, 205−265. (4) Alexander, S. Adsorption of Chain Molecules with a Polar Head a Scaling Description. J. Phys. (Paris) 1977, 38, 983−987. (5) Degennes, P. G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069−1075. (6) Milner, S. T. Polymer Brushes. Science 1991, 251, 905−914. (7) Rühe, J.; et al. Polyelectrolytes with Defined Molecular Architecture I; Springer-Verlag: Berlin, 2004; Vol. 165, pp 79−150. (8) Ballauff, M.; Borisov, O. Polyelectrolyte Brushes. Curr. Opin. Colloid Interface Sci. 2006, 11, 316−323. (9) Ballauff, M. Spherical Polyelectrolyte Brushes. Prog. Polym. Sci. 2007, 32, 1135−1151. (10) Chen, K.; Ma, Y. Q. Interactions Between Colloidal Particles Induced by Polymer Brushes Grafted onto the Substrate. J. Phys. Chem. B 2005, 109, 17617−17622. (11) Claesson, P. M.; Poptoshev, E.; Blomberg, E.; Dedinaite, A. Polyelectrolyte-Mediated Surface Interactions. Adv. Colloid Interface Sci. 2005, 114, 173−187. (12) Fritz, G.; Schadler, V.; Willenbacher, N.; Wagner, N. J. Electrosteric Stabilization of Colloidal Dispersions. Langmuir 2002, 18, 6381−6390. (13) Marla, K. T.; Meredith, J. C. Simulation of Interaction Forces Between Nanoparticles: End-grafted Polymer Modifiers. J. Chem. Theory Comput. 2006, 2, 1624−1631. (14) Striolo, A.; Egorov, S. A. Steric Stabilization of Spherical Colloidal Particles: Implicit and Explicit Solvent. J. Chem. Phys. 2007, 126, 014902. (15) Yamauchi, H.; Maeda, Y. LCST and UCST Behavior of Poly(Nisopropylacrylamide) in DMSO Water Mixed Solvents Studied by IR and Micro-Raman Spectroscopy. J. Phys. Chem. B 2007, 111, 12964− 12968. (16) Xu, J.; Ye, J.; Liu, S. Synthesis of Well-defined Cyclic Poly(Nisopropylacrylamide) via Click Chemistry and Its Unique Thermal Phase Transition Behavior. Macromolecules 2007, 40, 9103−9110. (17) Plummer, R.; Hill, D. J. T.; Whittaker, A. K. Solution Properties of Star and Linear Poly(N-isopropylacrylamide). Macromolecules 2006, 39, 8379−8388. (18) Lutz, J.-F.; Akdemir, O.; Hoth, A. Point by Point Comparison of Two Thermosensitive Polymers Exhibiting a Similar LCST: Is the Age of Poly(Nipam) Over? J. Am. Chem. Soc. 2006, 128, 13046−13047. (19) Hore, M. J. A.; Hammouda, B.; Li, Y.; Cheng, H. Cononsolvency of Poly(N-isopropylacrylamide) in Deuterated Water/ ethanol Mixtures. Macromolecules 2013, 46, 7894−7901.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01138. Details of the theoretical derivations; Appendix A: the details of the field theoretical formulation; Appendix B: weak electrostatic coupling approximation (for the dipoles) is applied to obtain a numerically tractable form for the electrostatic part of the Hamiltonian (ZIP)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (R.K.). J


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Vertical Phase Segregation Induced by Dipolar Interactions in Planar

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