Ion Solvation in Polymer Blends and Block Copolymer Melts: Effects of

Article pubs.acs.org/JPCB

Ion Solvation in Polymer Blends and Block Copolymer Melts: Effects of Chain Length and Connectivity on the Reorganization of Dipoles Issei Nakamura* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China ABSTRACT: We studied the thermodynamic properties of ion solvation in polymer blends and block copolymer melts and developed a dipolar self-consistent field theory for polymer mixtures. Our theory accounts for the chain connectivity of polymerized monomers, the compressibility of the liquid mixtures under electrostriction, the permanent and induced dipole moments of monomers, and the resultant dielectric contrast among species. In our coarse-grained model, dipoles are attached to the monomers and allowed to rotate freely in response to electrostatic fields. We demonstrate that a strong electrostatic field near an ion reorganizes dipolar monomers, resulting in nonmonotonic changes in the volume fraction profile and the dielectric function of the polymers with respect to those of simple liquid mixtures. For the parameter sets used, the spatial variations near an ion can be in the range of 1 nm or larger, producing significant differences in the solvation energy among simple liquid mixtures, polymer blends, and block copolymers. The solvation energy of an ion depends substantially on the chain length in block copolymers; thus, our theory predicts the preferential solvation of ions arising from differences in chain length.

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INTRODUCTION Ion-containing polymers have attracted considerable attention in recent decades because of their applications in electrochemical devices. Typically, ion−ion, ion−monomer, and monomer−monomer interactions play a key role in the thermodynamic properties of the system. The effects of these interactions depend on the electrochemical system, but in many cases, their energetic balance can significantly affect the system, causing, for example, microphase and macrophase separations of polymer mixtures. In general, the local packing structures of molecules near an ion and in the bulk phase take very different forms.1−3 Although a vast amount of literature has emerged on this subject, interest in the effects of electrostatic interactions on ion-containing polymers has developed relatively recently. For example, an atomistic simulation has predicted a variety of unanticipated morphologies in lithium-neutralized ionomers that are primarily driven by electrostatic interactions.4 Furthermore, an increase in temperature has, counterintuitively, been found to result in the formation of ion pairs in ionomers as a result of the fact that the dielectric constant of an ionomer rapidly decreases as the temperature increases.5 To study the fundamental nature of ionic liquids composed of large cations and anions with weak interactions, the effects of the orientational polarization and polarizability of polymerized ionic liquids on their dielectric properties have been studied.6 Among others, both theoretical7−10 and experimental11,12 studies have focused on several features of the thermodynamic properties of lithium-containing block copolymers. The results of these studies have suggested that the solvation energy of an © 2014 American Chemical Society

ion, which is generally associated with the dielectric properties of the material, plays a key role in ion-dissolving polymers. In experiments concerning the salt-free, low-molecularweight poly(ethylene glycol)s, a decrease in the bulk dielectric constant has been observed to accompany an increase in the molecular weight of the polymer.13 For ionic liquids with alkyl groups, the bulk dielectric constant at room temperature has been found to decrease as the chain length of the alkyl residue of cations increases. On the theoretical side, Onsager14 and Kirkwood15 theories are widely employed to evaluate bulk dielectric constants. Their theories account for the electrostatic responses of the dipole moments of molecules to external electrostatic fields, but they do not account for the polymerized degrees of freedom, and the chain connectivity and architecture of polymers. However, we intuitively expect that these features may be correlated with the orientational organization of the dipoles, as suggested by early theories regarding polar polymers.16,17 Specifically, if the system consists of a binary component, the spatial inhomogeneity of the compositional fraction should be incorporated into the correlation. In these perspectives, it is necessary to study the dielectric contrast among simple liquid mixtures, homopolymer blends, and block copolymer melts. For simplicity, the solvation energy of an ion is often modeled phenomenologically at the linear dielectric level by a crude Born expression, ΔGBorn = [(ze)2/8πaε0][1/ε(r)⃗ − 1], where e is the elementary charge, a is the radius of the ion, and Received: March 26, 2014 Revised: May 6, 2014 Published: May 7, 2014 5787

dx.doi.org/10.1021/jp502987a | J. Phys. Chem. B 2014, 118, 5787−5796

The Journal of Physical Chemistry B

Article

z is the valency.18−22 For a liquid mixture, the local dielectric constant is given by a simple compositional weighted average, ε(r)⃗ = εAϕA(r)⃗ + εBϕB(r)⃗ .7,9,23 However, this Born solvation energy is in poor agreement with the experimental values for liquid mixtures.2,24 From a theoretical perspective, ΔGBorn and the local dielectric constant ε do not account for various characteristic features of polymers, such as the polymerized degrees of freedom, and the chain connectivity and architecture of polymers. Furthermore, solvent molecules near an ion should experience a strong electrostatic field that compresses their volumes. This effect is referred to as electrostriction and is known to be correlated with the electric permittivity.25 Thus, we require a theory that accounts for the effects of electrostriction on the solvation energy of an ion. It is important to note that a strong electrostatic field in close proximity to an ion significantly reorganizes the orientation of the dipoles, leading to a saturated dipole orientation.26 Thus, the dielectric function near an ion in a simple liquid changes substantially with the position from the center of the ion in a nonlinear manner. This saturation effect causes the observed dielectric constant to decrease as the ion concentration is increased.27,28 Moreover, the dipolar self-consistent field theory for ion-containing incompressible liquids indicates that the dielectric function in the dielectric-saturation regime is the primary factor that determines the solvation energy of an ion in both a simple liquid and a simple liquid mixture.24 In this paper, we study the dielectric properties of ion solvation in homopolymer blends and block copolymer melts. In the Theory section, we develop the dipolar self-consistent field theories for an ion immersed in a block copolymer melt and in a homopolymer blend, taking into account the reorganization of dipoles caused by the electrostatic field. As the theoretical frameworks for the block copolymer melt and the homopolymer blend are essentially the same, we provide details of the calculation only for the block copolymer melt. In our theory, a dipole with permanent and induced dipole moments is attached to each monomer. In general, dipolar groups in chain molecules are classified into three types;17 a dipole can be (A) parallel to the chain direction (type A), (B) rigidly attached to the chain backbone but perpendicular to the chain direction (type B), or (C) located on a flexible side chain (type C). In the current study, we consider type C dipoles that can freely rotate about polymer backbones. This case occurs when polar groups are attached to side groups capable of internal rotation. Examples are poly(methyl methacrylate) and poly(vinyl acetate), although these polymers also have type B components. However, it should also be noted that our theory does not include the steric effects of the side groups. Our treatment of type C dipoles is an extension of the dipolar selfconsistent field theory presented in ref 24 for simple liquid mixtures and hence enables us to highlight important effects of the chain connectivity on the dielectric contrast among simple liquid mixtures, polymer blends, and block copolymer melts. Our results are presented in the Results section . In Figures 2, 4, and 8, the data points for the solvation energy of an ion are connected by lines to guide the eye.

cavity of radius a with a point charge ze in the center. The position of the tth monomer of the ith block P is denoted by R⃗ pit, and its dipole moment is denoted by p⃗pit. We can then write the grand canonical partition function as follows Z BC =

λcnc nc!

∑ nc

nc

1 2

A

B

+R⃗ it dpit⃗ p δ[R⃗ iNA − R⃗ i0 ]

i = 1 p = A,B

⎡ nc × exp⎢ −∑ ⎢⎣ i = 1 −

p

∫∏ ∏

Np

∑ ∑ p = A,B t = 1

p p 3 (R⃗ it − R⃗ it − 1)2 2bp2

⎤

∫ d3r d3r′ ρ ̂( r ⃗)ν( r ⃗ − r ′⃗ )ρ ̂( r ′⃗ ) − VH − Vhp⎥⎥ ⎦

(1)

where ν(r ⃗ − r′⃗ ) = e2/(4πε0|r ⃗ − r|⃗ ) is the Coulomb potential in vacuum and λc is the fugacity of the dipolar block copolymer. We express energies in units of kBT. The δ function in eq 1 enforces the chain connectivity between the two blocks, where c n̂p(r)⃗ = ∑ni=1 ∑Nt=1p δ[r ⃗ − R⃗ pit] is the number density for the monomer P. The total charge density ρ̂(r)⃗ is given by29

∑

ρ ̂( r ⃗) = zδ( r ⃗) +

ρp̂ ( r ⃗)

p = A,B nc

Np

p

ρp̂ ( r ⃗) = −∑ ∑ pit⃗ p ·∇δ( r ⃗ − R⃗ it ) i=1 t=1

(2)

where ρ̂p(r)⃗ is the charge density resulting from the dipoles. The dipoles on the monomers consist of an intrinsic (permanent) contribution pp̅ and a contribution that is induced by the electric field. Using the harmonic approximation, we describe the deformation energy of the latter as follows30,31 nc

VH =

Np

∑ ∑ ∑ i = 1 p = A,B t = 1

1 (|p ⃗ p | − pp̅ )2 2αp it

(3)

where αp is the molecular polarizability of a monomer P. Here, we have expressed the elastic energy of the block copolymer in terms of a discrete Gaussian chain model. This formulation enables us to more naturally assign the dipole moment for each monomer, which is modeled as a discrete site. However, we use a continuous description of the Gaussian chain model to calculate the chain propagators. We also write the nonbonded interactions in terms of a harmonic penalty32,33 Vhp =

∫ d r ⃗ 21κ [ ∑

p = A,B

νpnp̂ ( r ⃗) − 1]2 (4)

where κ denotes the compressibility of the A−B block copolymer. This energy accounts for the deviation in the density from that in the incompressible state with ∑p=A,B νpn̂p(r)⃗ = 1, primarily because of electrostriction near an ion in the current system. Here, we note that, in general, we must employ the equation of state to fully account for the nature of the compressibility in liquid mixtures. For example, we could have borrowed the DFT excess free-energy density functionals from other studies34−36 in our self-consistent field formalism. However, the description based on the harmonic penalty described by eq 4 is simple to use and is widely accepted for use in theoretical formulations37 and computer simulations.38,39 Thus, we use this potential in this paper.

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THEORY We consider an ion fixed at the origin immersed in a system of nc A−B diblock copolymers with a total degree of polymerization Nc. The degrees of polymerization for blocks A and B are NA and NB, respectively. The monomer volumes for blocks A and B are vA and vB, respectively. The ion is modeled as a 5788



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In this study, we consider fully miscible liquid components with the dipolar interactions. Thus, we ignore the other differential energetic interactions between the two liquid species in the mixture. However, in ref 24, it is demonstrated that the dipolar interactions in simple liquid mixtures lead to the preferential solvation of specific components near an ion. Below, we prove that for a mixture of polymers, this effect becomes more marked at large length scales (1−2 nm), significantly altering the electrostatics of the system. Using standard field-theoretical techniques in polymer physics40 and following the procedure described in ref 24, we derive the free-energy functional from the partition function in eq 1 in terms of field variables (see the Appendix) F=

1 8πl0

∫

× qp+( r ⃗ , t )

np( r ⃗) = λc ωp( r ⃗) =

(5)

In this equation, l0 = e2/(4πε0) is the vacuum Bjerrum length, np and ωp are the density field and its conjugate field for polymer P, respectively, and ψ is the electrostatic potential scaled by the elementary charge. Qc is the configuration partition function for a single-chain block copolymer Qc =

1 V

∫ +R⃗(t ) δ[R⃗ A(NA) − R⃗ B(0)]

⎧ ⎪ × exp⎨− ∑ ⎪ p = A,B ⎩

∫0

Np

2 ⎫ p ⎤⎪ ⎡ 3 ⎛ d R ⃗ (t ) ⎞ ⎟ + ωpeff ⎥⎬ dt ⎢ 2 ⎜ ⎥⎪ ⎢ 2bp ⎝ dt ⎠ ⎦⎭ ⎣

ΔG =

(6)

=

⃗p ⃗p where ωeff p = ωp[R (t)] − ln Ip[R (t)]. The function Ip(r)⃗ is given by 2 p

∫ dpp⃗ e−(1/2α )(|p⃗ |−p̅ )

Ip( r ⃗) =

p

p

(7)

Qc can be written in terms of the one-end-integrated propagators qp and q+p as follows41 Qc =

where

q+p

1 V

νp κ

∫0 [

Np

t) =

6

∇2 qp+( r ⃗ , t ) − ωpeff ( r ⃗ , t )qp+( r ⃗ , t )

q+A(r,⃗ 0)

Z HB =

νpnp( r ⃗) − 1] (12)

1 2

∫ d r ⃗ [D⃗( r ⃗)·E⃗( r ⃗) − D0⃗ ( r ⃗)·E0⃗ ( r ⃗)]

z 2l0 2

⎡

⎤

∫ dr ⎢⎣ ε(1r) − 1⎥⎦ r12

∑ ∏

λpnp

(13)

np!

np

∫ ∏ +R⃗ i p(t ) dpit⃗ p i=1

⎡ Nc × exp⎢ − ∑ ∑ ⎢ ⎣ p = A,B i

(9)

with the initial conditions = qB(r,⃗ NB) and = qA(r,⃗ NA). A similar equation holds for qp(r,⃗ t), with the initial condition qp(r,⃗ 0) = 1. Extremizing the free-energy functional with respect to the field variable ψ(r)⃗ yields ∇·[ε( r ⃗)∇ψ ( r ⃗)] = −4πl0zδ( r ⃗)

p

dt qp( r ⃗ , Np − t )qp+( r ⃗ , t )

∑

nA , nB p = A,B

bp2

p

p = A,B

(8)

is obtained by solving the modified diffusion equation

∂qp+( r ⃗ , ∂t

∫ d r ⃗ qp+( r ⃗ , Np)

2 p

∫ dpp⃗ e−(1/2α )(|p⃗ |−p̅ )pp2 F(pp |∇ψ ( r ⃗)|)

where the subscript 0 denotes the corresponding variables in the vacuum. Note that D⃗ (r)⃗ remains unchanged from D⃗ 0(r)⃗ because the charge remains constant. For a spatially uniform dielectric constant, eq 13 yields the standard Born expression ΔGBorn. Here, however, ε(r) is position-dependent and also depends nonlinearly on the electrostatic field, the compressibility, and the chain connectivity through eqs 9−12. We now write the partition function of the AB homopolymer blend

sinh[pp |∇ψ ( r ⃗)|] pp |∇ψ ( r ⃗)|

dt qp( r ⃗ , Np − t )

Equations 10−12 constitute a modified Poisson−Boltzmann (PB) equation for polarizable block copolymers in which the compressibility of the fluid is incorporated via the conjugate field ωp(r)⃗ that is associated with the volume of the monomers. From eq 10, we can identify the electric field and the electric displacement as E⃗ (r)⃗ ≡ −∇ψ(r)⃗ and D⃗ (r)⃗ ≡ ε(r)⃗ E⃗ (r)⃗ , respectively. Because of the spherical symmetry, integrating eq 10 once with respect to the radial distance r reduces the equation to dψ(r)/dr = −zl0/[ε(r)r2]. This equation must be numerically solved in combination with eqs 9, 11, and 12. The solvation energy for transferring an ion from vacuum to a dielectric medium is given by

νpnp( r ⃗) − 1]2

p = A,B

Np

Fp is related to the Langevin function, 3 (x) = 1/tanh x − 1/x, as follows: F = 3 (x) sinh x/x2. Similarly, the extremization of eq 5 with respect to other field variables leads to the following set of equations:

d r ⃗ np( r ⃗)ωp( r ⃗)

∫ d r ⃗ 21κ [ ∑

∫0

(11)

p = A,B

+

Ip( r ⃗)−1

p = A,B

∫ d r ⃗ ψ ( r ⃗)∇2 ψ ( r ⃗) + zψ (0) − λcQ cV

∑

−

∑

ε( r ⃗) = 1 + 4πl0λc

q+B(r,⃗ 0)

−

1 2

∫

∫0

Np

2 p 3 ⎛ d R ⃗ i (t ) ⎞ ⎟⎟ dt 2 ⎜⎜ 2bp ⎝ dt ⎠

⎤ d3r d3r′ ρ ̂( r ⃗)ν( r ⃗ − r ′⃗ )ρ ̂( r ′⃗ ) − VH − Vhp⎥ ⎥ ⎦ (14)

where we have used the same notations as those in eq 1, which is the analogue of the equation given above. The δ function for the constraint of the chain connectivity between the two blocks in eq 1 does not appear in eq 14; therefore, we require the configuration partition functions for single-chain homopol-

(10)

Note that eq 10 is the Poisson equation for the solvated ion, where ε(r)⃗ represents the local dielectric function, which is given by 5789



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ymers of both polymer A and polymer B. These partition functions are given by Qp =

1 V

⎡

∫ +R⃗ p(t ) exp⎢⎢−∫0 ⎣

Np

2 ⎤ p ⎧ ⎫ ⎪ 3 ⎡ dR⃗ (t ) ⎤ ⎪ ⎥ + ωpeff ⎬⎥ dt ⎨ 2 ⎢ ⎪ ⎪⎥ ⎩ 2bp ⎣ dt ⎦ ⎭⎦ (15)

The dielectric function of the polymer blend then becomes

∑

ε( r ⃗) = 1 + 4πl0

∫0

λpIp( r ⃗)−1

p = A,B

× qp( r ⃗ , t )

Np

dt qp( r ⃗ , Np − t ) 2 p

∫ dpp⃗ e−(1/2α )(|p⃗ |−p̅ )pp2 F(pp |∇ψ ( r ⃗)|) p

p

(16)

The calculations used to obtain the mean-field equations and the dielectric function by extremizing the free-energy functional are quite similar to those used to obtain eqs 1−12 and the equations in the Appendix and are therefore not repeated here. Note that the dielectric functions for the block copolymer and the polymer blend, given by eqs 11 and 16, are not the same when these functions are spatially dependent. The dielectric functions include integrals over the one-end-integrated propagators qp and qp+; therefore, both the degree of polymerization and the chain connectivity affect the reorganization of dipoles near an ion.

Figure 1. Volume fractions of a simple liquid mixture and polymer blends. r is the distance from the ion. For all mixtures, κ = 2.77 × 10−13 Pa−1, and ϕ̅ A = ϕ̅ B = 0.5. The solid and dashed lines represent ϕA(r) and ϕB(r), respectively. The colors of the lines corresponds to the different systems considered, that is, the simple liquid mixture (black) and polymer blends with NA = 20 and NB = 1 (green), NA = NB = 20 (red), and NA = NB = 50 (blue). The inset shows the dielectric functions for the simple liquid mixture (black line) and for the polymer blend with NA = NB = 50 (blue line).

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the volume fractions of the polymers near an ion becomes more marked. The volume fractions of the polymers change drastically over a distance of 1 nm or longer, whereas the simple liquid mixture exhibits a variation in the volume fraction over only a few angstroms. Because the degrees of reorganization of the dipoles depend on the local composition of each molecule, the spatial change in the dielectric value ε(r) of the polymer mixture exhibits nonmonotonic, “overshooting” behavior over 1 nm length scales. This length scale for the polymer blend is also notably larger than that for the simple liquid mixture. Figure 2 shows that the strong dependence of the volume fraction of a polymer on the chain length affects the solvation energy of an ion. In these calculations, we set NA = NB. Note that there is a substantial difference between the ion solvation energies for a simple liquid mixture and those for a polymer blend (Figure 2a). This difference may be ≃1 kcal/mol (1.69 kBT). Within the formulation based on a harmonic penalty for compressible fluids, the solvation energy is rather sensitive to the compressibility (Figure 2a). Note that for liquids, the compressibility κ typically ranges from 10−6 to 10−10 Pa−1. For example, the values of κ for water and polyethylene are 4.6 × 10−1043 and 1.43 × 10−9 Pa−1,44 respectively. Using these values for κ in our theory produces a variation in the solvation energy between 7 and 20%.45 Thus, in more quantitative studies, we are likely to need a precise form of the equation of state that accounts for the effects of electrostriction and the dispersion force near an ion. However, the dipolar self-consistent mean-field theory for an incompressible system presented in ref 24 provides qualitatively accurate results for the solvation energies of both simple liquids and simple liquid mixtures. Thus, our corse-grained, incompressibility-based approach should provide a reasonable description of ion solvation in liquid mixtures. Along the same lines, we perform a linear extrapolation using the compressibility κ to determine the solvation energy ΔG for the incompressible liquid mixtures as a function of the chain length of polymer A

RESULTS Polymer Blends. We first present our results for homopolymer blends to analyze the effects of the chain length on ion solvation. The volume fraction of polymer P (P = A, B) at a distance r from the ion is given by ϕP(r) = νpnp(r). We set bA = bB = 5 [Å] and express the volume fraction of the polymer P in the bulk phase as ϕ̅ P. To facilitate comparison with an earlier study of simple liquids in ref 24, we use the same model parameters as those used for the solvation of a Ag+ ion (a = 1.15 [Å]) in a mixture of acetonitrile (MeCN) and methanol (MeOH). However, because we are concerned with polymerized monomers here, the terms MeOH polymer and MeCN polymer should not be taken literally but rather should be considered to refer to a MeOH-like polymer (polymer A) and an MeCN-like polymer (polymer B). Our rationale in employing these hypothetical molecules is to highlight the significant difference in electrostatic response between a polymer and a simple liquid mixture. The molecular parameters for MeOH, MeCN, and the other molecules considered in this study are listed in Table 1. The first significant difference between a polymer blend and a simple liquid mixture that is observed is the behavior of the corresponding volume fractions at a distance r from the ion (Figure 1). As the chain becomes longer, the spatial change in Table 1. Values of the Molecular Parameters42 Used in This Worka components

p̅p [D]

αp [10−24 cm3]

νp[Å3]

methanol acetonitrile chloroform ethyl acetate

1.70 3.84 1.04 1.78

3.29 4.4 9.5 8.62

67.3 87.7 133.7 163.0

ε̅ 33 36.64 4.81 6.08

a ε̅ is the experimental value of the bulk dielectric constant. T = 293.15 K for methanol, and T = 298.15 K for all other substances. This table was reproduced from ref 24.

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spatial changes in the volume fractions of the block copolymers and the polymer blends near an ion occur on similar length scales. Nevertheless, for a given set of model parameters, we find an appreciable difference in the solvation energy ΔG between the two systems (Figure 4). Figure 4a shows that the

Figure 2. Solvation energies of a Ag+ ion in a simple liquid mixture and in polymer blends of various chain lengths, where ϕ̅ A = ϕ̅ B = 0.5 for all mixtures. (a) Solvation energy versus compressibility κ for a simple liquid mixture, that is, NA = NB = 1 (purple squares), NA = NB = 20 (black squares), and NA = NB = 50 (red diamonds). (b) Solvation energies for incompressible liquids obtained via linear extrapolation to κ = 0. The blue diamond corresponds to the case in which NA = 50, NB = 1.

Figure 4. Solvation energies of a Ag+ ion in a simple liquid mixture and in block copolymers of various chain lengths. For all mixtures, ϕ̅ A = 0.8, and ϕ̅ B = 0.2. (a) Solvation energy versus compressibility κ for a simple liquid mixture (purple squares), Nc = 25 (yellow diamonds), 30 (black squares), 36 (blue upside-down triangles), 42 (red triangles), and 100 (green diamonds). (b) Solvation energies for incompressible block copolymers of chain length Nc obtained via linear extrapolation to κ = 0. The solvation energies for polymer blends with NA = NB, ϕ̅ A = 0.8, and ϕ̅ B = 0.2 are plotted at Nc = NA + NB (red triangles) for comparison, and the value for the simple liquid mixture is plotted at Nc = 2 (purple square).

(or equivalently, polymer B) (Figure 2b). Our results for incompressible polymer blends indicate that the solvation energy ΔG changes notably with the chain length NA because the spatial profile of the volume fraction of a polymer near an ion can be considerably altered. In the following discussion, we show that this change qualitatively differs from that of block copolymers. Block Copolymer Melts. Next, we consider ion solvation in block copolymer melts using the same model parameters as those used for the polymer blends. The block copolymers consist of a MeOH-like block (block A) and an MeCN-like block (block B). We express the volume fraction of the block P in the bulk phase as ϕ̅ P. Once more, we find a considerable contrast between the volume fraction of a simple liquid mixture and that of a block copolymer melt (Figure 3). Note that the

solvation energy ΔG changes notably with the compressibility κ45 and may therefore be significantly altered by the local coordination of monomers near an ion. Figure. 4b shows a substantial difference between the solvation energies of an ion in a simple liquid mixture and an ion in a block copolymer melt. The solvation energy decreases by 1−2 kcal/mol (or 1.69−3.38 kBT) when the chain length increases. Our results suggest that the solvation energy for a block copolymer changes nonmonotonically with respect to that for a simple liquid when the chain length is relatively short (Nc < 30). However, a discrete chain model should be used instead of a continuous chain model for such short chains or oligomers to achieve a more quantitative analysis. Further experiments and computer simulations are needed to validate these predictions for ion solvation. Note that the difference between the solvation energies of a polymer blend and a block copolymer can be on the order of 1 kcal/mol (or 1.69 kBT) or larger (Figure 4b). Here, we find that the solvation energy of the block copolymer is larger than that of the polymer blend. Although this behavior is associated with the local composition near an ion, we can intuitively explain this result by analogy to the classical Born theory, where the solvation energy can be expressed as ΔGBorn(ε)̅ = [l0/2a](1/ε̅ − 1): the lower-dielectric block (block A) and the higherdielectric block (block B), which are chemically connected to each other, must exist near an ion in similar proportions. This fact causes the solvation energy for a block copolymer to tend to be less negative than that for a polymer blend. This difference can also be increased by increasing the volume

Figure 3. Volume fractions of a simple liquid mixture (black) and a block copolymer (red) with a chain length of Nc = 100. r is the distance from the ion. For all mixtures, κ = 2.77 × 10−13 Pa−1, ϕ̅ A = 0.8, and ϕ̅ B = 0.2. The solid and dashed lines represent ϕA(r) and ϕB(r), respectively. 5791



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fraction of the lower-dielectric block (block A). As an example, we consider a nearly incompressible mixture with κ = 2.77 × 10−13 Pa−1, ϕ̅ A = 0.95, and ϕ̅ B = 0.05. In this case, the solvation energies of the block copolymer (Nc = 100) and the polymer blend (NA = NB = 50) are −96.75 and −105.58 kcal/mol, respectively. The difference in the ion solvation energy is thus ≃10 kcal/mol (or 16.9 kBT). Figure 5a presents a comparison of the dielectric functions ε(r) of a simple liquid mixture, a polymer blend, and a block

Figure 6. (a) The solvation energy of a Ag+ ion in a block copolymer melt versus the volume fraction of the lower-dielectric block A for κ = 2.77 × 10−13 Pa−1 and Nc = 100. The black, red, and blue symbols denote sets of (ϕ̅ A, ϕ̅ B) values of (0.9244, 0.0756), (0.9488, 0.0512), and (0.9532, 0.0468), respectively. (b) A plot of the dielectric function ε(r), where the colors of the lines correspond to the colors of the symbols in (a). r is the distance from the ion. The inset shows an enlarged section of the plot.

Figure 5. Local dielectric functions with respect to the distance from a Ag+ ion. For all mixtures, κ = 2.77 × 10−13 Pa−1, ϕ̅ A = 0.8, and ϕ̅ B = 0.2. (a) Local dielectric functions of a simple liquid mixture (black solid line), a polymer blend with NA = 50 and NB = 50 (red dotted− dashed line), and a block copolymer melt with Nc = 100 (blue dashed line). (b) Local dielectric functions of block copolymer melts with Nc = 100 (blue dashed line), 42 (red dotted−dashed line), and 25 (purple dotted line) and of a simple liquid mixture (black solid line).

copolymer melt. Here, the same parameters and volume fractions are used as those used for Figure 3. In this case, the dielectric functions of all three types of materials behave differently. Note that the dielectric functions of the block copolymer and the polymer blend become highly nonlinear and significantly larger than that of the simple liquid mixture because of the dielectric saturation and exhibit the nonmonotonic behavior. Figure 5b demonstrates that the nonmonotonic, overshooting behavior of the dielectric function of the block copolymer is relatively sensitive to the chain length. This change in the dielectric function near an ion causes the solvation energy to vary with the chain length in Figure 4b. Thus, we anticipate many theoretical and experimental challenges in future studies of ion-containing polymers. For example, the screening functions of two charges in different chain architectures are of immediate importance with respect to recent interest in ion-containing nanomaterials. The observed bulk dielectric constant at high ionic concentrations is strongly affected by the reorganization of solvent dipoles (i.e., the dielectric saturation);28,46 therefore, the present results suggest that the bulk dielectric constant may also be significantly affected by the chain architecture. Our theory produces a peculiar feature in the solvation energies of block copolymers. Figure 6a illustrates that the solvation energy ΔG changes drastically when the volume fraction of the lower-dielectric block A is changed. This behavior occurs because the dielectric function in the integral of eq 13 near an ion changes nonmonotonically with respect to the volume fraction of the lower-dielectric block A (Figure 6b).

Figure 7. Volume fraction of a simple liquid mixture and block copolymers consisting of highly polarizable blocks. r is the distance from the ion. For all mixtures, κ = 2.77 × 10−13 Pa−1, and ϕ̅ A = ϕ̅ B = 0.5. The solid and dashed lines correspond to ϕA(r) for the block with the higher polarizability and ϕB(r) for the block with the lower polarizability, respectively. The black lines represent the simple liquid mixture, the blue lines represent the block copolymer with Nc = 91, and the red lines represent the block copolymer with Nc = 45.5.

Figures 7 and 8 show more marked difference among a simple liquid mixture, polymer blends, and block copolymer melts. We now consider a Na+ ion with a = 0.98 Å for comparison with ref 24. We consider a simple liquid mixture of chloroform and ethyl acetate. The block copolymers consist of a chloroform-like block (block A) and an ethyl acetate-like block (block B), whereas the polymer blends consist of mixtures of the homopolymer analogues to blocks A and B. In ref 24, an ion immersed in such a simple liquid mixture was preferentially solvated by the lower-dielectric molecule (chloroform). This unconventional ion solvation behavior results from the effects of highly polarizable dipoles in chloroform and cannot be explained by the classical Born theory. 5792



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Article

CONCLUSION

In summary, we developed a dipolar self-consistent field theory for ion solvation in homopolymer blends and block copolymer melts. Our theory accounts for the effects of chain connectivity on ion solvation. Compressibility is also taken into account using a harmonic penalty in the Hamiltonian. The key parameters in our theory are the permanent dipole moment, the polarizability, the molecular volume, the chain length, and the Kuhn length, which are typically determined from experiments or quantum chemistry simulations. In this study, the dipoles attached to each monomer are allowed to freely rotate and are reorganized in response to an external electrostatic field. Our results indicate that an ion is preferentially solvated by polymers with a higher permanent dipole moment and higher polarizability, as in the case of simple liquid mixtures.24 However, the reorganization of dipoles in polymerized molecules can be significantly affected by the chain connectivity, the chain length, and electrostriction. Thus, the spatial variation in the volume fractions of polymers near an ion with different dielectric properties differs from that in simple liquid mixtures (see Figures 1, 3, and 7). Therefore, there is a considerable and nonlinear difference among the dielectric functions for a simple liquid mixture, a polymer blend, and a block copolymer melt (see Figures 1 and 5). This behavior is analogous to the remarkably large nanometer-scale variation in the screening function of two like charges in an aqueous solution; 47 however, the effect of the chain connectivity in our theory is caused by single-ion solvation. Thus, the screening function of two charges immersed in polymer solvents should be studied further. Moreover, our theory predicts a substantial contrast among the ion solvation energies in a simple liquid mixture, a polymer blend, and a block copolymer melt because the reorganization of dipoles on the polymer backbone is restricted by the chain connectivity. In particular, we demonstrated that the solvation energy of an ion may be drastically changed by increasing the volume fraction of the lower-dielectric block in a block copolymer melt (Figure 6), which is in accordance with the nonmonotonic change in the dielectric function near the ion. We also demonstrated that the solvation energy can depend on the chain length and electrostriction. The solvation energy is more noticeably dependent on the chain length when an ion is solvated by a block copolymer (Figures 4 and 8). This result suggests that differences in the polymer lengths cause the preferential solvation of ions. Our theory demonstrates that the compressibility substantially alters the solvation energy of an ion within the scope of the harmonic-penalty potential (Figures 2, 4, and 8).45 Ion solvation in liquid mixtures has been a long-standing subject of research for many decades, and it is particularly important for recently developed types of energy storage.48 For example, the importance of the effects of the degree of polymerization on the mean-square dipole moment was established many decades ago by Mark and Flory using a rotational isomeric state model.49 Nevertheless, there have been few in-depth theoretical studies and simulations of the dipolar nature of salt-doped polymers. Our current coarse-grained theory remains at the mean-field level, and therefore, the effects of the higher-order correlations from ion−dipole and dipole− dipole interactions50 are not accounted for. Specifically, the selfconsistent field theory for polymers is known to describe dense polymeric systems when the length scale of the systems is the

Figure 8. Solvation energies of a Na+ ion in a simple liquid mixture and in block copolymers consisting of highly polarizable blocks. For all mixtures, ϕ̅ A = ϕ̅ B = 0.5. (a) Solvation energy versus compressibility κ for a simple liquid mixture (purple squares) and for block copolymers with Nc = 18.2 (yellow diamonds), 36.4 (green diamonds), 45.5 (black squares), 54.6 (blue upside-down triangles), 91 (red triangles), and 109.2 (light blue triangles). (b) Solvation energies for incompressible mixtures obtained via linear extrapolation to κ = 0. The solvation energies for polymer blends with NA = NB and ϕ̅ A = ϕ̅ B = 0.5 are plotted at Nc = NA+NB (red squares) for comparison, and the values for the simple liquid mixture are plotted at Nc = 2 (purple square).

For a given concentration and compositional fraction, the block copolymer volume fraction varies nonmonotonically over a distance of 1−2 nm, whereas that of a simple liquid mixture varies on a scale of angstroms (Figure 7). Note that the ion solvation energy ΔG in the block copolymer differs substantially from that in the simple liquid mixture (Figure 8a and b). The solvation energy for the block copolymer varies significantly with the chain length Nc, whereas the variation in the solvation energy with the chain length for the polymer blend is relatively small. Note that the solvation energies of the block copolymer and the polymer blend are increasing and decreasing functions of 1/Nc, respectively, (Figure 8b) when the chain length Nc is relatively large. These distinct behaviors suggest that ion-containing polymer blends and block copolymer melts should exhibit a variety of opposing thermodynamic properties and phase behaviors for varying chain lengths (or molecular weights),8 even when blends and melts are of chemically similar compositions. It should be noted that a simple mean-field theory for polymer blends based on the crude Born solvation energy ΔGBorn at the linear dielectric level does not provide the dependence of the ion solvation energy on the chain length,18 even when a perturbation expansion is performed around the homogeneous phase of a homopolymer mixture. The results of the current study indicate that for homopolymer blends with weakly polarizable components, the dependence of the ion solvation energy on the chain length is indeed very small, consistent with the simple Born theory (Figure 2b). However, for block copolymer melts and polymer blends with highly polarizable components, the chain connectivity causes a strong correlation between the reorganization of the dipoles and the chain conformation, leading to a substantial dependence of the ion solvation energy on the chain length. Thus, it is unlikely that this effect can be captured by the simple Born theory at the linear dielectric level. 5793



Article

polymer size. However, the relevant length scales for the local dielectric function and the spatial variation in the volume fractions of polymers near an ion are of the atomic-scale order. In this context, it would be interesting to perform a molecular dynamics simulation of ion-containing polymer mixtures with polarizable force fields.

■

∫ +np δ[np̂ ( r ⃗) − np( r ⃗)] = ∫ +np +ωp × exp{i ∫ d r ⃗ ωp( r ⃗)[np̂ ( r ⃗) − np( r ⃗)]}

1=

(17)

where the right-hand side of the equation results from the Fourier representation of the δ function with ωp(r)⃗ being the Fourier conjugate field to np(r)⃗ . A similar procedure is performed for the total charge density ρ̂(r)⃗ with the charge density field ρ(r)⃗ , which introduces the conjugate field ψ(r)⃗ . The partition function ZBC in eq 1 can then be cast into a functional integral as follows

APPENDIX: DERIVATION OF THE FREE-ENERGY FUNCTIONAL FOR BLOCK COPOLYMERS

We first introduce the coarse-grained number density field np(r)⃗ for monomer P using the identity

Z BC =

⎧ ⎪

∫ +ρ+nA +nB+ψ +ωA +ωB exp⎨− 12 ∫ d r ⃗ d r ′⃗ ρ( r ⃗)ν( r ⃗ − r ′⃗ )ρ( r ′⃗ ) + i ∫ d r ⃗ ρ( r ⃗)ψ ( r ⃗) − izψ (0) ⎪

⎩

∑ ∫

+i

p = A,B

⎫ ⎪ λ nc dr ⃗ np( r ⃗)ωp( r ⃗)⎬∑ c ⎪ ⎭ nc nc!

⎧ nc ⎪ × exp⎨−∑ ⎪ i=1 ⎩ =

∫

∑ ∫ 0

p = A,B

Np

nc

A

2 p nc 3 ⎡ d R ⃗ i (t ) ⎤ ⎢ ⎥ − i∑ dt 2 2bp ⎢⎣ dt ⎥⎦ i=1

Np

∑ ∫ 0

p = A,B

⎫ ⎪ p p p ⃗ ⃗ dt {ωp[R i (t )]+pi⃗ (t ) ·∇ψ [R i (t )]} − VH − Vhp⎬ ⎪ ⎭

∫ d r ⃗ d r ′⃗ ρ( r ⃗)ν( r ⃗ − r ′⃗ )ρ( r ′⃗ )

∫ d r ⃗ ρ( r ⃗)ψ ( r ⃗) − izψ (0) + i ∑ ∫ d r ⃗ np( r ⃗)ωp( r ⃗) − 21κ [ ∑ p = A,B

+ λc

∫

B

+R⃗ i (t )+pi⃗ p (t ) δ[R⃗ i (NA ) − R⃗ i (0)]

i = 1 p = A,B

⎧ ⎪ 1 +ρ +nA +nB +ψ +ωA +ωB exp⎨− ⎪ 2 ⎩

+i

p

∫∏ ∏

νpnp( r ⃗) − 1]2

p = A,B

⎧ ⎪ A B ⃗ ⃗ ⃗ +R(t )+p ⃗ (t ) δ[R (NA ) − R (0)] × exp⎨ ∑ ⎪ p = A,B ⎩

∫0

Np

2 p ⎡ p p 3 ⎛ d R ⃗ (t ) ⎞ ⎢ ⎟ ⎜ dt − 2 − iωp[R⃗ (t )] − ipp⃗ (t ) ·∇ψ [R⃗ (t )] ⎢ 2bp ⎝ dt ⎠ ⎣

⎫ ⎤⎫ ⎪⎪ 1 2⎥ − (|p ⃗ (t )| − pp̅ ) ⎬⎬ ⎥⎪⎪ 2αp p ⎦⎭⎭

(18)

⎡

∫ +ρ exp⎢⎣− 12 ∫ d r ⃗ d r ′⃗ ρ( r ⃗)ν( r ⃗ − r ′⃗ )ρ( r ′⃗ )

Using the identity operator in eq 17, we have replaced the instantaneous particle density n̂p(r)⃗ by the coarse-grained

+i

(average) density np(r)⃗ . Moreover, we have performed the

⎤

∫ d r ⃗ ρ( r ⃗)ψ ( r ⃗)⎥⎦

⎡1 = 5 −ν 1 exp⎢ ⎣2

following calculation

∫ d rn⃗ p̂ ( r ⃗)ωp( r ⃗) = ∫

nc

Np

(20)

p

d r ⃗ ∑ ∑ δ( r ⃗ − R⃗ it )ωp( r ⃗)

where ν−1 (r ⃗ − r′⃗ ) is the inverse of the Coulomb operator ν−1

i=1 t=1 nc

=

Np

(r ⃗ − r′⃗ ) = −(4πl0)−1∇2δ(r ⃗ − r′⃗ ), and 5ν is the thermodynami-

p

∑ ∑ ωp(R⃗ it ) i=1 t=1

⎤

∫ d r ⃗ d r ′⃗ iψ ( r ⃗)ν−1( r ⃗ − r ′⃗ )iψ ( r ′⃗ )⎥⎦

(19)

cally inconsequential normalization term resulting from the Gaussian functional integral. Thus, eq 18 can be written in the

and the summation over the particle number using the identity

form

N x ∑∞ N=0 (x /N!) = e . Performing the Gaussian integral over the charge density

Z BC =

∫ +nA +nB+ψ +ωA +ωB exp(−F)

field ρ(r)⃗ (the Hubbard−Stratonovich transformation) trans-

(21)

where the free-energy functional of the system F is

forms the Coulomb interaction term to 5794


The Journal of Physical Chemistry B F=

1 8πl0 −

Article

(18) Wang, Z.-G. Effects of Ion Solvation on the Miscibility of Binary Polymer Blends. J. Phys. Chem. B 2008, 112, 16205−16213. (19) Onuki, A. Ginzburg−Landau Theory of Solvation in Polar Fluids: Ion Distribution around an Interface. Phys. Rev. E 2006, 73, 021506. (20) Onuki, A. Surface Tension of Electrolytes: Hydrophilic and Hydrophobic Ions near an Interface. J. Chem. Phys. 2008, 128, 224704. (21) Kung, W.; Solis, F. J.; de la Cruz, M. O. Thermodynamics of Ternary Electrolytes: Enhanced Adsorption of Macroions as Minority Component to Liquid Interfaces. J. Chem. Phys. 2009, 130, 044502. (22) Wang, R.; Wang, Z.-G. Effects of Ion Solvation on Phase Equilibrium and Interfacial Tension of Liquid Mixtures. J. Chem. Phys. 2011, 135, 014707. (23) Ben-Yaakov, D.; Andelman, D.; Harries, D.; Podgornik, R. Ions in Mixed Dielectric Solvents: Density Profiles and Osmotic Pressure between Charged Interfaces. J. Phys. Chem. B 2009, 113, 6001−6011. (24) Nakamura, I.; Shi, A. C.; Wang, Z.-G. Ion Solvation in Liquid Mixtures: Effects of Solvent Reorganization. Phys. Rev. Lett. 2012, 109, 257802. (25) Marcus, Y. Electrostriction, Ion Solvation, and Solvent Release on Ion Pairing. J. Phys. Chem. B 2005, 109, 18541−18549. (26) Debye, P. Polar Molecules; Dover Publications, Inc.: Mineola, NY, 1929. (27) Wei, Y. Z.; Chiang, P.; Sridhar, S. Ion Size Effects on the Dynamic and Static Dielectric-Properties of Aqueous Alkali Solutions. J. Chem. Phys. 1992, 96, 4569−4573. (28) Levy, A.; Andelman, D.; Orland, H. Dielectric Constant of Ionic Solutions: A Field-Theory Approach. Phys. Rev. Lett. 2012, 108, 227801. (29) Abrashkin, A.; Andelman, D.; Orland, H. Dipolar Poisson− Boltzmann Equation: Ions and Dipoles Close to Charge Interfaces. Phys. Rev. Lett. 2007, 99, 077801. (30) Coalson, R. D.; Duncan, A. Statistical Mechanics of a Multipolar Gas: A Lattice Field Theory Approach. J. Phys. Chem. 1996, 100, 2612−2620. (31) Jackson, J. D. Classical Electrodynamics, 3rd ed.; John Wiley and Sons, Inc: New York, 1998. (32) Helfand, E. Block Copolymer Theory. 3. Statistical-Mechanics of Microdomain Structure. Macromolecules 1975, 8, 552−556. (33) Wu, D. T.; Fredrickson, G. H.; Carton, J. P.; Ajdari, A.; Leibler, L. Distribution of Chain-Ends at the Surface of a Polymer Melt  Compensation Effects and Surface-Tension. J. Polym. Sci., Part B: Polym. Phys. 1995, 33, 2373−2389. (34) Schmid, F. A Self-Consistent-Field Approach to Surfaces of Compressible Polymer Blends. J. Chem. Phys. 1996, 104, 9191−9201. (35) Muller, M.; MacDowell, L. G. Interface and Surface Properties of Short Polymers in Solution: Monte Carlo Simulations and SelfConsistent Field Theory. Macromolecules 2000, 33, 3902−3923. (36) Muller, M.; MacDowell, L. G.; Yethiraj, A. Short Chains at Surfaces and Interfaces: A Quantitative Comparison between DensityFunctional Theories and Monte Carlo Simulations. J. Chem. Phys. 2003, 118, 2929−2940. (37) Daoulas, K. C.; Theodorou, D. N.; Harmandaris, V. A.; Karayiannis, N. C.; Mavrantzas, V. G. Self-Consistent-Field Study of Compressible Semiflexible Melts Adsorbed on a Solid Substrate and Comparison with Atomistic Aimulations. Macromolecules 2005, 38, 7134−7149. (38) Wang, Q. Studying Soft Matter with “Soft” Potentials: Fast Lattice Monte Carlo Simulations and Corresponding Lattice SelfConsistent Field Calculations. Soft Matter 2009, 5, 4564−4567. (39) Milano, G.; Kawakatsu, T. Hybrid Particle-Field Molecular Dynamics Dimulations for Dense Polymer Systems. J. Chem. Phys. 2009, 130, 214106. (40) Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers; Oxford University Press: New York, 2006. (41) Hong, K. M.; Noolandi, J. Theory of Inhomogeneous Multicomponent Polymer Systems. Macromolecules 1981, 14, 727− 736.

∫ d r ⃗ ψ ( r ⃗)∇2 ψ ( r ⃗) + zψ (0) − λcQ cV

∑ ∫ d r ⃗ npωp( r ⃗) p = A,B

+

∫ d r ⃗ 21κ [ ∑

νpnp( r ⃗) − 1]2

p = A,B

(22)

Here, we have replaced iψ(r)⃗ and iωp(r)⃗ by ψ(r)⃗ and ωp(r)⃗ because the original fields ψ(r)⃗ and ωp(r)⃗ are purely imaginary at the saddle point. Qc is given by eq 6 in the main text.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS We are grateful to the Computing Center of Jilin Province for essential support. REFERENCES

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(42) Haynes, W. M. CRC Handbook of Chemistry and Physics, 92nd ed.; CRC Press: Boca Raton, FL, 2011. (43) Fine, R. A.; Millero, F. J. Compressibility of Water as a Function of Temperature and Pressure. J. Chem. Phys. 1973, 59, 5529−5536. (44) Mavrantzas, V. G.; Boone, T. D.; Zervopoulou, E.; Theodorou, D. N. End-Bridging Monte Carlo: A Fast Algorithm for Atomistic Simulation of Condensed Phases of Long Polymer Chains. Macromolecules 1999, 32, 5072−5096. (45) For example, we obtain ΔG = − 126.74 kcal/mol for κ = 10−9 Pa−1 for a Ag+ ion in a simple-liquid mixture with ϕ̅ A = ϕ̅ B = 0.5. For a block copolymer with Nc = 42, ϕ̅ A = 0.8, and ϕ̅ B = 0.2, we obtain ΔG = − 111.85 kcal/mol for κ = 1.62 × 10−10 Pa−1. Here, the solvation energies of the incompressible analogues of the simple-liquid mixture and the block copolymer are ΔG = −106.11 and −104.27 kcal/mol, respectively. Thus, the solvation energy can change by 7−20%, depending on the compressibility. (46) Maribo-Mogensen, B.; Kontogeorgis, G. M.; Thomsen, K. Modeling of Dielectric Properties of Aqueous Salt Solutions with an Equation of State. J. Phys. Chem. B 2013, 117, 10523−10533. (47) Gong, H. P.; Freed, K. F. Langevin−Debye Model for Nonlinear Electrostatic Screening of Solvated Ions. Phys. Rev. Lett. 2009, 102, 057603. (48) Tarascon, J. M.; Armand, M. Issues and Challenges Facing Rechargeable Lithium Batteries. Nature 2001, 414, 359−367. (49) Mark, J. E.; Flory, P. J. Dipole Moments of Chain Molecules. I. Oligomers and Polymers of Oxyethylene. J. Am. Chem. Soc. 1966, 88, 3702−3707. (50) Levy, A.; Andelman, D.; Orland, H. Dipolar Poisson− Boltzmann Approach to Ionic Solutions: A Mean Field and Loop Expansion Analysis. J. Chem. Phys. 2013, 139, 164909.

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Ion Solvation in Polymer Blends and Block Copolymer Melts: Effects of

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