Determining the Regimes of Dielectric Mismatch and Ionic Correlation

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Determining the Regimes of Dielectric Mismatch and Ionic Correlation Effects in Ionomer Blends Ha-Kyung Kwon,† Boran Ma,† and Monica Olvera de la Cruz*,†,‡,§,⊥ Department of Materials Science and Engineering, ‡Department of Chemistry, §Department of Chemical and Biological Engineering, and ⊥Department of Physics, Northwestern University, Evanston, Illinois 60208, United States

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†

ABSTRACT: We examine the effects of ionic correlations and dielectric mismatch on the miscibility of ionomer blends with a hybrid ionomer blend thermodynamic liquid state theory approach. The ionomer (A) has a dielectric constant ϵA and low charge fraction ( 0; that is, the charged polymer can always solvate ions more easily than the neutral polymer. The charge-containing polymer has a tunable charge fraction fq,A along its backbone. This charge fraction is varied between 0 and 0.1 to maintain a low concentration of charge, which is later shown to play an important role in eliciting specific phenomena in strongly correlated systems. With these contributions, the total free energy of mixing can be written as

where z+ and z− are valency of charged ions, e

is the elemental charge, ϵr is the dielectric constant of the medium, ϵ0 is the vacuum permittivity, and a is the radius of an ion. In these systems, the ionic correlation strength Γ can be as high as 30 kT, necessitating a careful consideration of local ordering of ions and their effect on the overall phase behavior of polymers. However, these previous works neglected the dielectric mismatch even though in many ternary blends containing ionic liquids, plasticizers, or salt ions, the role of dielectric mismatch cannot be ignored, as the dielectric constant of the constituents can vary widely. To separate the effect of dielectric mismatch from that of ionic correlations, we first perform molecular dynamics (MD) simulations of ionomer−neutral polymer blends with χ = 0 and Δϵ = 0, which are similar to blends of poly(styrenesulfonate) (PSS) and polystyrene (PS). With the MD simulations we demonstrate the “chimney effect” predicted by ionic correlations included via the Debye−Hückel extended mean spherical approximation (DHEMSA) closure 17 in the thermodynamics of charged−neutral polymer systems.15,16,35 Because it is complicated to introduce dielectric mismatch in coarse-grained molecular dynamics simulations,36−42 we investigate the effects of dielectric mismatch on the interplay between solvation effects and ionic correlations in chargecontaining polymer blends using the DHEMSA approach that we verified with the simulations in the absence of dielectric mismatch. Specifically, by including the dielectric mismatch in the DHEMSA and polymer blend thermodynamics, we identify regimes where ion−ion correlations dominate, solvation effects dominate, and both effects must be properly accounted for. To do so, we investigate electrostatic effects in a system when (1) one of the polymers contains charge, (2) there is a dielectric mismatch between the two polymers, ϵA ≠ ϵB, and (3) a tertiary component is introduced to enlarge the dielectric contrast in blends of charged and neutral polymers as well as in salt-doped neutral blends. This results in a compositiondependent dielectric constant ϵr(ϕ) and a compositiondependent ionic correlation strength Γ(ϕ), and a nontrivial behavior arises that is dependent on both the absolute value of the dielectric constants ϵr and the dielectric mismatch Δϵ. A benefit of analyzing charged−neutral and salt-doped ternary blends is the ability to decouple the effects of surface polarization from mixing thermodynamics. Surface polarization in coexisting phases with dielectric contrast can strongly modify interactions42 and can also lead to modifications to the Poisson−Boltzmann adsorption of ions at liquid−liquid interfaces.41 These effects may be significant in charged− neutral block copolymers when the blocks are microphasesegregated32 but do not modify the thermodynamics of

ftot (ϕk , Nk , χkl , fq , k , am , Γmn) = fFH (ϕk , Nk , χkl ) + fPM (ϕk , fq , k , am , Γmn) + fBorn (ϕk , Δϵ)

(1)

where k and l are indices for molecular components (polymers A or B) and m and n are indices for ionic components (positive and negative ions); ϕk is the volume fraction of component k, Nk is the degree of polymerization of k, χkl is the Flory− Huggins parameter between k and l components, fq,k is the fraction of charge along component k, am is the radius of ion m, and Γmn is the ionic correlation strength between m and n ions. For simplicitly, we assume a symmetric blend with monovalent, symmetric ions. In addition, we assume that only the counterions (of same size as the backbone charge) are available for solvation. Subsequently, many of the indices can be dropped, and the Flory−Huggins contribution f FH, the contribution to f PM from translational entropy of the ions, and solvation term f Born can be written in simple analytical forms: fFH =

ϕA ln ϕA NA

+

ϕB ln ϕB NB

+

ϕC ln ϕC NC

+ χAB ϕA ϕB

+ χAC ϕA ϕC + χBC ϕBϕC

(2)

fPM = (fq ,A ϕA + fq ,C ϕC) ln(fq ,A ϕA + fq ,C ϕC) + fexc (fq ,A ϕA + fq ,C ϕC , Γ(ϕA , ϕC)) fBorn =

(3)

z 2e 2(fq ,A ϕA + fq ,C ϕC) 8π ϵ0ϵr(ϕA , ϕB , ϕC)akBT

(4)

For binary blends, all terms with ϕC, χAC, χBC, and fq,C are dropped. For ternary blends, we assume for simplicity that the counterions to backbone charge and counterions to the salt are nondistinguishable. Pair correlation functions of the charges are calculated numerically using the liquid state theory with Debye−Hückel extended mean spherical approximation (DHEMSA) closure17 on the Ornstein−Zernike equation.43,44 Here, the volume fraction of charged polymer ϕA is coupled to the volume B

DOI: 10.1021/acs.macromol.8b02376 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. A breakdown of total free energy into f FH, f PM‑id, and f PM‑exc. f FH increases with χ, and f PM‑id decreases with fq. f PM‑exc increases with Γ (solid lines), but the location of the maximum shifts to a higher ϕ with an increase in Γ. f PM‑exc further increases with fq (dashed lines), and the location of the maximum shifts back to a lower ϕ. Depending on the depth of the minima, multiple phase coexistence may be obtained. Here, ϕ represents ϕA. Reproduced with permission from ref 15.

Figure 2. (a) Chemical potential contribution from ionic correlations as a function of charge concentration. Chemical potential is normalized by Γ (b) for ease of viewing. Note that the dip in chemical potential well is found at fq,AϕA ≈ 0.002, which indicates that the effects of ionic correlations are especially dominant at low charge concentrations.

fraction of charges fq,AϕA. The Ornstein−Zernike equation is solved: h(r ) = c(r ) + fq,A ϕA c∗h

The pair correlation function is related to the total correlation function: g=h+1

(5)

(where ∗ is a convolution) with the so-called Debye−Hückel extended MSA closure (DHEMSA) c = h ̃ − ln(h ̃ + 1) − u/kT

(7)

Excess free energy from ionic correlations, fexc, can be calculated by thermodynamically integrating the pair correlation function:

(6)

where u is the pair potential between the charges, c is the direct correlation function, and h̃ is an estimation for the total correlation function h made by nonlinear Debye−Hückel theory. The estimation includes hard-core corrections to the nonlinear Debye−Hückel theory, thereby capturing shortrange interactions as well as long-range interactions. Thus, we can obtain results that are very consistent with results obtained using the hypernetted chain (HNC) method but are convergent and numerically efficient, in contrast to the HNC.15,17,45

fexc =

(fq ,A ϕA )2 2

∫0

1

dλ

∫ ∫ dr dr′ gλ(r , r′)u(r , r′)

(8)

When the dielectric mismatch between the two polymers is small, the medium can be effectively considered as homogeneous with a uniform distribution of dielectric constant. This assumption is valid for many charge-containing polymers such as polystyrene-block-poly(styrenesulfonate), where a small degree of sulfonation is not expected to appreciably change the dielectric constant. In these cases, it is C


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Figure 3. (a) Schematic phase diagram of charged−neutral polymer blends (NA = NB = 40, fq,A = 0.1) at lB = 12σ. (b) Simulation snapshots of polymer blends with ϕA = 0.2 (in black rectangle) and ϕA = 0.8 (in green rectangle). Green beads represent charged monomers along the backbone of polymer A, pink beads represent the neutral monomers of polymer A, yellow beads represent the counterions, and blue beads represent the neutral polymer B. (c) Pair correlation functions of charged polymer. Inset: scattering functions of charged polymer. (d) Local A monomer density distribution functions.

mildly dependent on the ionic correlation strength and independent of the ion radius but that the depth increases as the ionic correlation strength increases. If the charge fraction is increased, the effect due to ionic correlations is significantly reduced, as an increased charge concentration moves the system out of the chemical potential dip. Therefore, at higher values of charge fraction, the chimney-type phase separation can no longer be accessed. It is important to make the distinction that even when the system moves out of the chemical potential dip, the effects of ionic correlations do not disappear (note that μex scales with Γ in Figure 2a). However, this may explain why the chimney-type phase segregation is only accessible at low charge concentrations in systems with no dielectric mismatch. However, when the polymeric medium has a large dielectric constant, solvation effects need to be considered to accurately describe the mixing thermodynamics. Furthermore, the addition of salt ions can significantly change the dielectric constant of the medium. The dielectric constant of saltcontaining media is a complex function that depends on ionspecific effects, solution viscosity, and binding of ions to the solvent particles. Over the years, theoretical and experimental investigations have shown that the dielectric constant of saltcontaining media can decrease with salt concentration due to dielectric decrement49−51 or increase with concentration at high concentrations and high valency of salts.52 A recent study has employed a field theoretic approach to capture the nonlinear and linear dielectric decrement that arises from iondipole correlations and fluctuations.53 For simplicity, we follow linear mixing rule to investigate the effect of introducing a third component into the blend. Further investigations are necessary to delineate the effect on the overall dielectric constant with

important to note that effects due to ionic correlations are particularly significant at dilute concentrations of charge. Previous studies have found a chimney-type phase segregation in systems with Γ ≊ 16 at fq,A ≊ 0.15 where the composition difference between coexisting phases is very small (Δϕ < 0.1).15,16,35 By conducting a full free energy analysis, it was found that this chimney-type phase segregation was due to the behavior of the fexc term. This composition-dependent term results in a total free energy of mixing with multiple minima, as shown in Figure 1. This term was analyzed in detail in a previous publication.15 It is important to note that the chimney-type phase segregation is characteristic of weakly charged systems due to a steep dip in the chemical potential, μex =

∂fexc ∂ϕA

, found at

small concentrations of charge. The chemical potential contribution from ionic correlations is plotted in Figure 2a, with the normalized chemical potential plotted as a function of charge concentration in Figure 2b. The dip is found at volume fraction of charges fq,AϕA = 0.002. We note that the deep minima in the excess chemical potential (over the ideal) as a function of charged monomer volume fraction fq,AϕA is responsible for the phase transition of electrolyte solutions in the primitive model46 as well as that of charged polymer and counterion solutions47 in low dielectric media. The initial decrease is due to screening, which is captured by Debye screening, and as the concentration of charges increases, the contributions from hard core of the charges become prevalent, especially at high ionic correlation strengths (Γ). The correlations induce clustering of charges, which reduces the screening as the concentration of charges increases.48 Our calculations show that the position of the minimum is only D


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Figure 4. (a) Schematic phase diagram of charged-neutral polymer blends (NA = NB = 40, ϕA = 0.5, fq,A = 0.1). (b) Simulation snapshots of the blends at lB = 12σ (in black rectangle with solid lines) and lB = 10σ (in rectangle with dashed lines). Green beads represent charged monomers along the backbone of polymer A, pink beads represent the neutral monomers of polymer A, yellow beads represent the counterions, and blue beads represent the neutral polymer B. (c) Pair correlation functions of charged polymer. (d) Local A monomer density distribution functions.

−0.5kR02 ln(1 − r2/R02)) with k = 30.0 and R0 = 1.5, both in LJ units and the purely repulsive Weeks−Chandler− Andersen potential55 (with ϵ = 1.0 and σ = 1.0) between all the beads and Coulombic interactions among charged

the addition of salt and solvent, taking into account complex factors such as local dielectric saturation and valency. For our system: ϵr(ϕ) = ϵA ϕA + ϵBϕB + ϵCϕC

(9)

monomers and counterions (UC(r ) =

where ϕC = 0 in binary blends. Because the dielectric constant is a function of composition, both ionic correlations and solvation effects must be written as a function of composition ϕA: Γ(ϕ) =

z+z −e 2 8π ϵ0kTaϵr(ϕ)

z+z −e 2 ), 4πϵ0ϵrr 56

computed with

particle-particle particle-mesh methods; the 1000 polymer chains with counterions randomly placed in cubic boxes with bead number density 0.85σ−3 are equilibrated for 2 × 107 time steps of 0.005τ, where τ is the dimensionless LJ time unit. Figure 3 shows a sketch of a phase diagram of a symmetric charged−neutral polymer blend (NA = NB = 40) as predicted by the liquid state theory described in the previous section with low charge fraction fq,A = 0.1 at lB = 12σ (see Figure 3a) along with snapshots of simulated systems with various ϕA values (ϕA = 0.2 and 0.8 in black rectangle and green rectangle in Figure 3b, respectively). Phase separation is observed in blends with ϕA = 0.2, while those with ϕA = 0.8 display a homogeneous phase. The pair correlation function g(r) between monomers of polymer A is calculated from the trajectory of the simulations; since the simulation box size is 36σ, the range of r is from 0 to 18σ. For blends with ϕA = 0.8, g(r) reaches 1 at a relatively short distance, consistent with the miscibility observed in the snapshots. However, for blends with ϕA = 0.2, g(r) fluctuates even when approaching the maximum value of r, which indicates the structure is heterogeneous over large distances and the system undergoes phase separation, as shown in the snapshots. A larger scale simulation of a system with the size 4 times that of the original simulations was performed; the monotonically decaying tail remains in the pair correlation function g(r) calculated from the larger system, which confirms that the phase separation in systems with 1000 chains is not affected by finite box size effects. The scattering

(10)

This means that unlike the cases where a fixed concentration of salt is added into a polymer blend, charge concentration is proportional to the concentration of polymer A. As a result, forming a phase rich in polymer A may be advantageous in increasing the dielectric constant within the phase, and thereby lowering solvation energy, but entropically unfavorable as it increases the amount of ions confined in the phase.

■

RESULTS AND DISCUSSION Chimney-Type Phase Diagram of Charged−Neutral Polymer Blends. Neglecting the dielectric mismatch in the theory described in the previous section, we obtain an ionomer blend thermodynamic model that includes ion correlations via liquid state theory studied earlier.12,15 MD simulations of charged−neutral polymer blends are performed here to analyze and validate the predictions of the theory in regard to ionic correlations alone (that is, without dielectric mismatch). NVT simulations are performed using the coarse-grained bead− spring model54 with the finitely extensible nonlinear elastic (FENE) potential between bonded beads (UFENE(r) = E


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Figure 5. (a) Evolution of the critical point as dielectric mismatch, Δϵ increases. Multiple critical points are observed for ϵB = 3, 5 at low dielectric mismatch, but only one critical point is observed for ϵB = 7,10. Critical points for Δϵ < 5 correspond to correlation-induced multiple criticality seen in ref 15. (b) As charge fraction fq,A increases, critical points due to ionic correlations disappear as the charge fraction increases, indicating the disappearance of the “chimney” type region. Inset shows a zoomed-in version marked by the rectangle.

Figure 6. Phase diagram of binary blends where ϵB of the neutral polymer varies from 3 (a), 5 (b), and 7 (c), where fq,A = 0.1. ϵA of the chargecontaining polymer is increased to increase dielectric mismatch. For all of the phase diagrams, we see that the degree of phase separation increases with small increases in dielectric constant but decreases when the mismatch is increased further. Lines have been drawn to guide the eye.

function S(q) was calculated from g(r) and plotted as the inset of Figure 3c; the periodic peaks with relatively strong intensities of ionomer blends with ϕA = 0.2 compared to those of ionomer blends with ϕA = 0.8 also indicate that phase separation takes place in ionomer blends with ϕA = 0.2 at lB = 12σ. To further characterize the phases of the systems, the local A monomer density distribution was obtained by dividing up the simulation box into cubic cells and calculating ρA in each cell. The kernel density estimation method was applied to smooth the density distribution plot. As shown in Figure 3d, the green line and error bars show the frequency of finding a specific density of A monomer in the blend with ϕA = 0.8. Here, the single peak at ρA ≈ 0.8 represents the miscible nature of the system. However, for the blend with ϕA = 0.2 (black line

and error bars in Figure 3d), the two peaks correspond to the two phases that the blend separates into; the ratio of intensities of the two peaks is in accordance with the lever rule and can be used to trace back to where the binodal intersects with χN = 0. The size of the cubic cells was chosen to balance the number of particles per cell required for good statistics of local density calculation (therefore big enough cubic cells) and the number of cubic cells required for good statistics of the frequency calculation. Local density distribution functions calculated from cubic cells with size within a range (∼3σ to ∼4.5σ) show consistent results. The size of the cubic cells used to plot Figure 3d is ∼4σ. The effect of the ionic correlation strength Γ = lB/σ on the phase behavior of charged−neutral polymer blends is revealed F


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Macromolecules by comparing MD simulations of two systems with the same composition (ϕA = 0.5) but at different Bjerrum lengths (shown in Figure 4). By comparing the simulation snapshots of a blend with ϕA = 0.5 at lB = 12σ (solid line rectangle in Figure 4b) with those of a blend with the same volume fraction ratio at lB = 10σ (dashed line rectangle in Figure 4b), it is evident that the blend miscibility decreases as the strength of the ionic correlations increase, since an increases in lB leads to a more strongly segregated system. The pair correlation function g(r) of charged polymers (Figure 4c) shows that the system is strongly correlated at a higher Bjerrum length; the fluctuations are stronger at half of the box size corroborating the stronger phase separation observed in the snapshots as lB increases. The local monomer A density distribution functions are plotted to further characterize the effect of the ionic correlation strength on the blend. As shown in Figure 4d, the single peak on the dashed line with error bars (lB = 10σ) and the two peaks on the solid line with error bars (lB = 12σ) show that the blend with ϕA = 0.5 separates into two phases at lB = 12σ, while it is miscible at lB = 10σ. Effect of Dielectric Mismatch in Binary Ionomer Blends. Here, we investigate the criticality of a chargecontaining binary blend where Δϵ is slowly increased. Results are summarized in Figure 5 for values of ϵB = 3, 5, 7, and 10 and Δϵ = 0−20. Here, ϵA is always larger than ϵB, and the charge fraction fq,A is kept at 0.1. As shown in Figure 5, the location of the critical point(s) is determined by both the absolute value of the dielectric constants ϵA and ϵB and the dielectric mismatch Δϵ. When ϵA and ϵB are both low, the behavior of the system is dominated by ionic correlations, whose strength Γ is inversely proportional to the dielectric constant of the system ϵ(ϕ). Multiple critical points are found, and their locations correspond to the critical points previously found in highly correlated systems exhibiting a chimney.15 When both ϵA and ϵB are high (and Δϵ is small), ion entropy dominates the behavior of the system, and only one critical point is found. This critical point is located at ϕA ≊ 0.6, which is typical of weakly correlated systems.12,15 When Δϵ is large (ϵA ≫ ϵB), the critical point moves to a lower value of ϕA. In these systems, Born solvation drives the phase behavior. It is important to note that at high charge fractions seen in Figure 5b the critical points that emerge from the chimney-type phenomena due to ionic correlations disappear. This result is consistent with our observations in Figure 2, where an increase in charge fraction moves the system away from the chemical potential dip. It is also consistent with a previous study,15 where the emergence of chimney-type phase separation was seen at moderate values of Γ at low charge fraction fq,A = 0.2. As the charge concentration is increased, the system moves away from the dip. At high values of charge concentration, the blend is strongly segregated, and coexistence is found between two almost-pure phases. This type of phase separation is driven by ionic correlations and other related electrostatic effects but is observed to be different from a chimney-type phase separation, where the two coexisting phases are both dilute in charge.15,35 Having established regimes where ionic correlations or solvation effects dominate, we calculate and plot the phase boundaries in Figure 6, where fq.A = 0.07. Gray lines denote the phase boundary for Δϵ = 0, for the selected values of ϵA and ϵB. As seen in Figure 6a−c, when Δϵ = 0, the system is driven by ion−ion correlations alone when Δϵ = 0. The increase in ϵA and ϵB results in increased miscibility, as the ionic correlation

strength Γ is inversely proportional to ϵr: Γ =

z+z −e 2 . 8πϵ0kTa ϵr

In

weakly correlated charged systems, where ϵA and ϵB are high, ion entropy causes the system to become more miscible.15 When Δϵ is increased, the blend exhibits re-entrant miscibility, where the miscibility is initially seen to decrease before increasing. This behavior is found across all values of ϵB. The details of the trend are more obvious in Figure 6c, where we plot the effect of Δϵ on the phase behavior of a blend with ϵB = 7. The switch in trend occurs at Δϵ ≈ 5. That is, for Δϵ < 5, an increase in dielectric mismatch decreases miscibility, and critical χN moves to a lower value. At Δϵ = 5, phase separation can be found at χN = 0. For Δϵ > 5, increasing ϵA enhances miscibility, and the chimney becomes significantly narrower and shifts to a lower ϕA, especially in regions of high ϕA. The contributions to the free energy of mixing from ionic correlations (fexc) and Born solvation ( f Born) are compared in Figure 7. These curves show that re-entrant behavior arises

Figure 7. Free energy of mixing contributions from fexc (a), f Born (b), and fexc + f Born (c) with increasing dielectric mismatch when ϵB = 7. Arrows indicate increasing Δϵ. Free energy from ionic correlations (a) decreases with increased dielectric mismatch, but Born solvation (b) shows a nonmonotonic trend. This results in an increased phase separation followed by a decrease in phase separation.

from the composition-dependent nature of solvation energy. fexc (Figure 7a) decreases as ϵA is increased, as Γ is inversely proportional to ϵA. For ionic correlations, the absolute value of ϵA is important, not the magnitude of Δϵ. For solvation effects, the increase in Δϵ is significant in increasing the energy of mixing. As the mismatch increases, the ions have a stronger preference to be solvated in the higher dielectric medium. This energetic gain provides a driving force toward phase segregation, so that the ions can be concentrated within a phase with a higher ϵr. On the other hand, obtaining a uniform distribution of dielectric constant and of ions across the bulk is entropically favorable. In addition, the energy G


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with Figure 6. However, the re-entrant miscibility for Δϵ > 5 is shown to be dependent on the mixing rule and does not appear when an inverse-additive mixing rule is used. This is because this mixing rule accentuates contributions from the low-ϵr polymer, which is consistent with our analysis for re-entrant miscibility. We expect that experimental polymer phase behavior can be captured between the two limits; however, further investigations into dielectric properties of the mixtures are necessary for direct comparisons with experiments. In particular, future investigations of ionically heterogeneous systems can benefit from consideration of dielectric effects at an atomistic level, such as ion-dipole correlations and backbone dipolar moments, which give rise to more complex concentration-dependent dielectric phenomena.61,62 Effects of Dielectric Mismatch in Ternary Ionomer Blends. In previous work, we have shown that the phase behavior of ternary ionomer blends containing charged polymer A, neutral polymer B, and solvent C is driven by the competition between ionic correlations and ion entropy in the absence of dielectric mismatch Δϵ.16 In such a case, ionic correlations drives the blend toward increased immiscibility, especially at dilute concentrations of polymer A. Here, we show that in the presence of dielectric mismatch Δϵ the inclusion of a solvent always leads to increased miscibility. To investigate systems with a large dielectric mismatch (i.e., those containing a high dielectric solvent), we extend our analysis to three-component systems, where one of the components is a solvent with a high dielectric constant. The system is incompressible, and ϕA + ϕB + ϕC = 1. For this analysis, we limit ourselves to spinodals to study qualitative trends. Figure 9 shows the effect of increasing the dielectric constant ϵC of the solvent in a ternary system with ϵA = 7.2, ϵB = 3.2 (a) and ϵA = 7.2, ϵB = 7 (b), which correspond to high and low dielectric mismatch cases, respectively. Polymer A is charged at fq,A = 0.1 while polymer B is neutral. In both cases, the increase in solvent dielectric constant generally increases miscibility. For a polymer blend with a higher Δϵ between the two polymers (Δϵ ≈ 4), the changes in miscibility upon solvent addition are far more dramatic. It is important to note that the increase in solvent dielectric constant causes the disappearance of the chimney (marked with a star) in Figure 9a. Even with the disappearance of the chimney, the spinodals remain wide, indicating that phase separation occurs between two almost pure phases. This is in contrast to a low dielectric mismatch case (Figure 9b), where the spinodals occur at less pure concentrations of each polymer. When ϵA ≊ ϵB, the dielectric constant of the solvent has a reduced effect on the overall phase behavior, as seen in the closeness of the spinodal lines. Overall, the increase in ϵC reduces blend immiscibility, as seen in the movement of the critical point to lower ϕC, indicated by an arrow. However, at dilute solvent concentrations (ϕC < 0.05), increasing ϵC decreases the miscibility of the polymer blend. When there is little solvent, it is favorable to phase separate by incorporating the solvent into the charge dense phase (A) and increase the solvation of the ions. However, at higher concentrations of solvent ϕC > 0.05, it becomes entropically unfavorable to concentrate the solvent in one phase. Past theoretical studies have focused on the role of salt on the phase behavior of polymer blends with a dielectric mismatch,13,18 finding that small amounts of salt can shift the phase boundaries vertically by changing the effective χN.

required to solvate the ions begins to increase as the phase increases in ionomer volume fraction ϕA. At low values of dielectric mismatch, Δϵ ≤ 5, we observe enhanced phase segregation with increasing Δϵ. Increasing Δϵ drives the blend toward phase segregation even though the contribution from ionic correlations decreases because it is energetically favorable to concentrate the ions within a high ϵr phase. However, when Δϵ > 5, the energetic gain from solvating the ions within the high ϵr phase cannot overcome the entropic penalty of confining these ions, and we get increased miscibility, especially in regions of high ϕA. This re-entrant miscibility is a characteristic feature of a blend where the ionomer carries charge on its backbone, in contrast to blends that are doped with a fixed concentration of salt. Because the charge concentration is proportional to the concentration of ionomer A, the energetic gain from solvating in a higher ϵr drops off at high ϕA, and the solvation energy can no longer be decreased by phase segregating to obtain a high ϵr phase. This effect is seen in f Born for Δϵ > 5, where the curvature of free energy changes at high Δϵ, leading to a nonmonotonic change in the total free energy of mixing. Coupled with entropic effects, this leads to re-entrant miscibility seen in Figure 6c. We have thus far assumed for simplicity that the dielectric constant of the medium is linearly dependent: ϵr(ϕ) = ϵAϕA + ϵBϕB. Several mixing rules have been employed for theoretical and experimental investigations, such as Lichtenecker’s mixture formula,33,57,58 the Clausius−Mossotti equation,18,59 or a power law-type dependence.22 Experimental systems can deviate from the linear approximation used in this study; however, a study based on low-molecular-weight liquids has shown that deviations from the linear approximation can be moderate.60 In Figure 8, we show the phase diagram of binary blends using the inverse-additive mixing rule for ϵB = 7 and ϵA varied.

Figure 8. Phase diagram of binary blends where ϵB = 7 and ϵA is varied. Here, an inverse-additive mixing rule is used: 1/ϵr(ϕ) = ϕA/ϵA + ϕB/ϵB. Increasing Δϵ decreases the miscibility across all ϕA. Noticeably, re-entrant miscibility found in Figure 6 is not observed with this mixing rule.

When the mixing rule is changed to an inverse-additive, 1 ϵr(ϕ)

=

ϕA ϵA

+

ϕB ϵB

(corresponding to Lichtenecker’s mixture

formula with ν = −1) we find that the behavior for Δϵ < 5 is consistent across two different mixing rules. That is, an increase in Δϵ leads to increased phase segregation and a shift of the critical point to low ϕA that is in qualitative agreement H


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Figure 9. Calculated spinodals of a ternary system of charge-containing polymer A, neutral polymer B, and high dielectric solvent C, where ϵA = 7.2 and ϵB = 3.2 (a) and ϵB = 7 (b). The solvent dielectric constant has a greater effect when the Δϵ is greater. Unstable region within the spinodal lines, marked “unstable”, decomposes spontaneously via spinodal decomposition. Arrow shows the movement of critical point to a lower ϕC.

Figure 10. Calculated spinodals of a ternary system consisting of charge-containing polymer A (ϵA = 7.2), neutral polymer B (ϵB = 3.2 in (a) and ϵB = 7 in (b)) and salt C. Spinodals in high Δϵ case (a) show that one of the coexisting phases consists of B + C with no A. The yellow highlighted region shows region of the phase diagram corresponding to dilute concentrations of ϕA, where the change in miscibility is more prevalent.

arrow in Figure 10a). The movement of the spinodal is dramatic in the region of the diagram dominated by the low dielectric polymer (low ϕA, marked by yellow) as this dielectric constant drives the thermodynamics of mixing. When the dielectric constants of the two polymers are similar (Figure 10b), increasing the dielectric constant of the salt has a minimal effect, and all spinodals collapse onto a single spinodal line that is symmetric. This indicates that the salt might be forming its own phase and that the phase behavior of the blend is unaffected by the solvation of the salt, except at very low concentrations of salt. At low concentrations (ϕC < 0.05), an increase in the dielectric constant leads to decreased miscibility, which is similar to that found in the solvent-containing case (Figure 9). Effects of Dielectric Mismatch in Salt-Doped Neutral Ternary Blends. In the last section of this work, we extend calculations to a salt-containing blend of neutral polymers, where the dielectric constants of the two polymers are 7.5 and 4, respectively. This allows for a comparison of our work with experimental studies of salt-doped polymer blends and block copolymers,27,63,64 where the solvation energy is shown to increase effective χ and substantially decrease the miscibility of the system. Theoretical calculations have qualitatively

Here, we extend the study to include the effect of salt concentration on the dielectric constant of the media. Figure 10 shows the role of salt on the spinodal lines of a ternary blend with a high dielectric mismatch (Figure 10a) and a low dielectric mismatch (Figure 10b). Here, we treat salt as a tertiary component that can change the dielectric constant of the media using the following mixing rule: ϵ(ϕ) = ϵAϕA + ϵBϕB + ϵCϕC. When the dielectric mismatch in the polymer is high (Δϵ = ϵA − ϵB = 4), the increase in ϵC leads to a re-entrant behavior; the unstable region bounded by spinodal lines is observed to decrease at dilute concentrations of ϕA (highlighted in yellow) and increase at higher concentrations of ϕA. The change highlighted in yellow is more interesting, as it indicates that the unstable region can decompose into two phases where one of the phases is pure in B (ϕA = 0), right along the B−C axis. We hypothesize that here salt can selectively swell into pure B phase to achieve a uniform distribution of dielectric constant across two phases. The resultant uniform distribution of ions also makes it entropically favorable to form a phase rich in A and a phase containing B and C. As ϵC increases, it becomes more favorable for the salt to form its own high ϵr phase instead of absorbing into the B phase (as indicated by the I


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Macromolecules

correlations. Very low concentrations of charge allow access to a chimney-type phase separation, where the composition difference between the two coexisting phases is small. When ϵA and ϵB are both high, the criticality can be described by ion entropy. When the dielectric mismatch is substantial (ϵA > ϵB), the competition between solvation energy and ionic correlations leads to a shift of the critical point to a lower ϕA. Furthermore, the inclusion of the Born solvation energy leads to a re-entrant miscibility, where the increase in Δϵ initially induces immiscibility, followed by increased miscibility. This is shown to be an effect of both the mixing rule and the fact that charge concentration is proportional to the charge-containing polymer A. For charge-containing blends in neutral high dielectric media, where the dielectric mismatch can be as large as 70, inclusion of media with a higher dielectric constant always leads to decreased miscibility. The solvent effect is greater for polymer blends with a greater dielectric mismatch and subdued for polymers that have negligible difference in the dielectric mismatch. On the other hand, increasing the dielectric mismatch in a polymer blend with a high dielectric mismatch by introducing salt can induce re-entrant behavior similar to one found in binary blends. In such a case, salt induces a strong segregation in blends with a symmetric composition but increases miscibility in blends with a low concentration of charged polymer. In systems with a low dielectric mismatch, the introduction of salt decreases miscibility but only at very low concentrations. For salt-doped neutral polymer blends, the dielectric mismatch is believed to lead to a qualitatively different B−C segregation, where salt partitioning occurs in blends containing dilute concentrations of the high dielectric polymer, which is consistent with salt partitioning seen in experimental systems.27,64 Our results provide a framework for understanding and identifying different regimes governed by ionic correlations (at low Δϵ) and by solvation effects (at high Δϵ). In this study, we employ an integral method for calculating ionic correlations and couple this with the Born solvation term, which assumes a continuum media whose dielectric constant is given by a linear mixing rule. However, when the mixing rule is changed to inverse-additive (corresponding to Lichtenecker’s mixture formula with ν = −1), we find that the behavior for low dielectric mismatch (Δϵ < 5) is consistent across the two different mixing rules, though the re-entrant miscibility at large dielectric mismatch does not appear when an inverse-additive mixing rule is used. One can improve the models by considering dielectric effects at an atomistic level, such as ion-dipole correlations and backbone dipolar moments, which give rise to more complex concentration-dependent dielectric phenomena.67−70 Regardless of the limitations, charged− neutral polymer blends with or without added salt are ideal ionomers to study the effect of dielectric mismatch. This is because they segregate macroscopically avoiding complex polarization effects, such as those that arise in dieletrically heterogeneous microphase-segregated diblock copolymers, where attractions among domains,42 especially in the presence of macroions,71 can arise.

predicted these results by calculating the phase behavior of a pseudobinary system, where the amount of salt added is negligible and does not take up appreciable volume within the system.18,65 Here, we treat salt as a tertiary component, the addition of which can change the dielectric constant of the system. Figure 11 shows the calculated spinodals of a salt-doped neutral polymer blend, where ϵr of the polymers A and B is 7.5

Figure 11. Calculated spinodals of a ternary system consisting of two neutral polymers (A, ϵA = 7.5), neutral polymer B (ϵB = 4) and salt with varying dielectric constant. Inset shows a broadening of the unstable region with increasing ϵC to lower salt concentrations, marked with an arrow.

and 4, respectively. The dielectric contribution from the salt is varied between ϵC = 7.5 and ϵC = 13.5. We find that the inclusion of salt into a neutral blend with a dielectric mismatch leads to a narrowing of the unstable region enveloped by spinodal curves. At very low concentrations of A, the envelope expands (shown in inset) to lower concentrations of salt C (indicated with an arrow). Furthermore, the spinodal region now fully touches the B−C axis in addition to the A−B axis, indicating that the spinodal decomposition in this region occurs along A−B and B−C axes. If it decomposes along the B−C axis, it indicates that the decomposed phases will contain substantially different concentrations of component C (salt). This would be consistent with salt partitioning that has been theoretically predicted66 and experimentally observed in many salt-doped block copolymer and blend systems.27,63,64

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CONCLUSION By simulating charged−neutral polymers blends with no dielectric mismatch, we show that ionic correlations alone leads to the chimney effect predicted in blends with low concentrations of charged groups in the charged polymer (ionomer). That is, phase separation is observed in a narrow range of small concentrations of the ionomer in the blend, even when χ = 0, as long as the ionic correlations are strong enough. The concentration range of immiscibility decreases when the strength of the ionic correlations decrease. We further show that the dielectric mismatch between charge-containing and neutral polymers has far-reaching effects on the phase behavior. In binary charge-containing blends, dielectric mismatch necessitates a composition-dependent dielectric constant ϵ(ϕ) resulting in a composition-dependent ionic correlation strength Γ(ϕ). When ϵA and ϵB are both low, multiple criticality of the blend is dominated by ionic

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Ha-Kyung Kwon: 0000-0002-9351-4806 Boran Ma: 0000-0002-5172-1196 Monica Olvera de la Cruz: 0000-0002-9802-3627 Present Address

H.-K.K.: Toyota Research Institute, 4440 W El Camino Real, Los Altos, CA 94022. Author Contributions

H.-K.K. and B.M. contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was performed under the financial assistance award 70NANB14H012 from the U.S. Department of Commerce, National Institute of Standards and Technology as part of the Center for Hierarchical Materials Design (CHiMaD) and NSF Grant DMR-1611076. The authors thank V. A. Pryamitsyn for helpful suggestions and feedback.

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Determining the Regimes of Dielectric Mismatch and Ionic Correlation

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