Spinodal Decomposition of a Polymer and Ionic Liquid Mixture: Effects

Article pubs.acs.org/Macromolecules

Spinodal Decomposition of a Polymer and Ionic Liquid Mixture: Effects of Electrostatic Interactions and Hydrogen Bonds on Phase Instability Issei Nakamura* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China S Supporting Information *

ABSTRACT: We studied the spinodal decomposition of a homopolymer and ionic liquid mixture. Our theory accounts for the dielectric contrast and hydrogen bond between the polymer and the ionic liquid and the effect of fluctuations in the local density and electrostatic potential. We attempted to rationalize the observed shift in the critical point and the asymmetry of the observed spinodal curve by applying the selfconsistent field theory and Langmuir adsorption model. The dielectric contrast between the polymer and the ionic liquid produces a shift in the critical point toward polymer-rich regions. The fluctuation effect yields drastic changes in the trend of the phase boundary. We show that both effects are marked by the appearance of inflection points in the spinodal curve. Although hydrogen bonding also yields similar effects, the spinodal curve rather exhibits a double-well structure or relatively flat structure when combined with the solvation energy of ions. Hydrogen bonding, ion solvation, and the fluctuation have equal significance on the magnitude and trend of the spinodal curve. Our theory provides strategies to dissolve low-dielectric polymers in ionic liquids by altering the dielectric constant of ionic liquids and employing hydrogen bonding. shifts toward polymer-rich regions, ϕc ∼ 0.8, when the hydrogen bond between the cation and PEO is significantly suppressed by replacing the relevant H atom with a methyl group. Thus, such an asymmetric spinodal curve against the curve predicted by the Flory−Huggins theory appears to address new challenges in understanding polymer-ion interactions. However, experiments have indicated that hydrogen bonding is not likely to be a single key factor that causes the peculiar phase boundary. Despite being a relatively new issue, numerous studies have already emerged on the subject of the relationship between a polymer and an ionic liquid in a mixture. A recent lattice-based theory accounting for the natures of the compressibility and nonrandom mixing has suggested that the key effects driving the unusual shifted critical composition are the stronger cohesive energy of the ionic liquid and the reduced free volume compared to the polymer.2 Although this information is critical to rationalize the unusual spinodal curve, further microscopic details are required. To date, few theoretical studies have considered the effect of the correlation between hydrogen bonding and electrostatic interactions for a polymer and ionic liquid mixture. Given that the fluctuation around the

1. INTRODUCTION Dissolving polymers in ionic liquids has attracted considerable attention over the past several decades because of the many important applications for biorenewable materials and energy storage. For example, the development of green processing of cellulose is of critical importance1 for sustainability and environmental protection; the discovery of the dissolution of cellulose in ionic liquids has resulted in a new paradigm for prospective sustainable strategies in the fields of electrochemical science and technology. The hydrogen bond between a polymer and an ionic liquid is likely to be a critical factor to account for their solubility. However, despite in-depth theoretical analysis,2 the complete mechanism of the solubility between a polymer and an ionic liquid, particularly in terms of the correlation between hydrogen bonding and electrostatic interactions, remains elusive.3,4 The effect of hydrogen bonding was shown to significantly affect the temperature−composition phase diagrams of poly(ethylene oxide) (PEO) dissolved in imidazolium-based tetrafluoroborate ionic liquids.5 The spinodal curve obtained in that experiment exhibited an approximately symmetric structure around the volume fraction of PEO ϕPEO ∼ 0.5. However, this feature substantially deviates from the critical point ϕc ∼ 1/√N with the chain length N given by the Flory− Huggins theory for polymer solutions, which is expected to lie at a polymer-poor region. Moreover, the critical point even © 2016 American Chemical Society

Received: October 5, 2015 Revised: December 29, 2015 Published: January 13, 2016 690

DOI: 10.1021/acs.macromol.5b02189 Macromolecules 2016, 49, 690−699

Article

Macromolecules homogeneous phase via the strong ion−ion correlation6−8 and the dielectric contrast between a polymer and an ionic liquid9 are critically important, we should further develop theoretical frameworks for the dissolution of polymers in ionic liquids. The effect of fluctuations in the local density and electrostatic field is of particular concern in polymer physics.10−16 Within the lowest-order contribution of mean-field approximations for a homogeneous phase, the free energy density does not reflect the short wavevector component of the electrostatic interaction.15,17−22 Accordingly, any physical quantity related to the electrostatic screening length is missed; at high salt concentrations, this is clearly a gross approximation. Indeed, a recent theoretical study based on a hybrid of the self-consistent field theory combined with the integral-equation theory revealed that the effects of a strong correlation between ions and between an ion and a charged polymer changed the geometrical trend of the phase diagrams considerably. Thus, higher-order contributions arising from the local fluctuation should be considered for a polymer and ionic liquid mixture. It should also be noted that the integral-equation theory resorts to a closure relation, such as the Percus−Yevick approximation23−25 and the hypernetted-chain equation,25 to solve the Ornstein−Zernike equation. However, the choice of the approximation is relatively heuristic25 and hence uncertain in terms of its applicability to a polymer and ionic liquid mixture with different dielectric constants. Therefore, the qualitative features of the phase behaviors from different theoretical frameworks should be examined. This paper qualitatively rationalizes the origin of the shift in the critical point5 and provides insight into the effects of the hydrogen bonds and electrostatic interactions on the phase instability of a homopolymer and ionic liquid mixture. Specifically, we highlight the difference between the dielectric constants of the polymer and the ionic liquid. These key interactions can account for the notable features of the observed phase diagrams.5 We pose the question of whether the characteristic phase behaviors in the case of a polymer with hydrogen bonding can also be commonly observed in a polymer and ionic liquid mixture. 26−28 For example, sulfobetaine polymers dissolved in water containing sodium ions have both intra- and intermolecular hydrogen bonding.26 The phase boundary of sulfobetaine polymers dissolved in water exhibits highly nonmonotonic behavior that can be characterized by notable concave and convex structures. However, whether ionic liquids also exhibit similar phase behaviors intrinsically remains unclear. The remainder of this paper is organized as follows. We begin with a simple mean-field theory for a homopolymer and ionic liquid mixture by developing a Langmuir adsorption model (section 2.1) in light of previous theories for the hydrogen bonds between PEO and water.29 Cations can be hydrogen-bonded with the polymer. Importantly, the dielectric constants of the polymer and ionic liquid are typically different. Because we must account for the preferential solvation of ions due to the dielectric inhomogeneity,30−33 we invoke the Born solvation energy as a simple mean-field form. In section 2.2, we illustrate that the effect of the hydrogen bond produces a double-well or relatively flat shape in the spinodal curve. Furthermore, the dielectric inhomogeneity causes the shift in the critical point from a polymer-poor to a polymer-rich region. Note that the spinodal curve is plotted as χ vs composition. Thus, it does not indicate the lower critical solution temperature (LCST). We then consider the fluctuations in

the local density and electrostatic potential as a correction to the lowest-order mean-field approximation. Accordingly, we write the partition function of the polymer and the ionic liquid and invoke the saddle-point approximation via the selfconsistent field theory (section 3.1). By integrating the quadratic-order contributions to the free energy, we derive the spinodal curve from the zero of the second derivative of the effective free energy. We then demonstrate that the characteristic spinodal curve exhibiting an inflection point appears as the dielectric constants of polymer and/or ionic liquid are decreased (sections 3.2 and 3.3). This phase behavior is analogous to that predicted from a hybrid of the self-consistent field theory combined with the integral-equation theory for charged polymers.6−8 Thus, we anticipate that the inflection of the spinodal curve is a signal of the strong correlation between ions through electrostatic interactions. In Sec. 3.3, we also demonstrate that the solubility of polymers can be enhanced when the dielectric constant of polymers is higher than that of ionic liquids. In practice, however, the dielectric constant of polymers is often considerably lower than that of ionic liquids. Nevertheless, our theoretical results suggest that the alteration in the dielectric constant of ionic liquids should serve as a good strategy to increase the solubility of polymers.

2. HYDROGEN BONDING AND DIELECTRIC INHOMOGENEITY 2.1. Simple Mean-Field Theory for the Complexation between a Polymer and an Ionic Liquid (IL). We consider an incompressible, homogeneous mixture of np homopolymers with a total degree of polymerization N and ionic liquid consisting of n0 cations and n0 anions. The volumes of a monomer and the total system are vp and V, respectively. For theoretical conciseness, the volumes of the cation and anion are set to be the same value, v0. We denote the dielectric constant of the homopolymer as ϵPEO, taking PEO as a model system. We describe the dielectric constant of the cation and anion as a single variable, ϵIL. Although the difference in the dielectric response of the cation and anion under external electrostatic fields were shown to be essential,9 it does not alter the conclusive point of our theory in this study. The cations and anions are set to be monovalent ions. Because the numbers of cations and anions are the same, the system is charge-neutral overall. The significant effect of the hydrogen bonds between the cation (1-ethyl-3-methylimidazolium) (EMIM) and PEO on the phase instability of a disordered phase was reported in ref 5. To model such a system, in our theory, we allow nb cations to associate with the oxide groups of PEO through hydrogen bonding. A recent mean-field theory has suggested that in ionic liquids, tightly bound ion pairs are short-lived and outnumbered by the free ions.34 Consequently, the effect of ion pairing on the dielectric response was shown to be insignificant, in accordance with experimental measurements.35−37 Similarly, we do not consider tightly bound ion pairs in our current study. We first start with a system involving the hydrogen bonds between a cation and the polymer (PEO) but without fluctuations in the local density and electrostatic potential. The anions are not capable of binding to the polymer and are hence free. Thus, we consider the nb bound cations and n0 − nb unbound (free) cations. Thus, the number of ways to choose nb cations is W = NnpCnb. The free energy for the association is then written as 691


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Macromolecules Fhb = − ln(W ) + E hbnb kBT

vpftot kBT

= − Nnp ln(Nnp) + (Nnp − nb) × ln(Nnp − nb) + nb ln nb + E hbnb

+ (1)

ϕp N vp v0

ln ϕp −

ϕp N

+

vp v0

ϕ0 ln(ϕ0 − ϕb) −

ϕ0 ln ϕ0 − vp v0

vp v0

ϕ0

(ϕ0 − ϕb)

⎛ vpϕb ⎞ ⎟ + χϕ (1 − ϕ ) + ϕp ln⎜⎜1 − p p ⎟ v ϕ 0 p⎠ ⎝

The energetic gain caused by the formation of the hydrogen bond is given by Ehb in the units of kBT. The translational entropy for the homopolymers (PEO), unbound cations, and anions is given by F0/(kBT) = np ln[np/(ξpV)] − np + (n0 − nb) ln[(n0 − nb)/(ξ0V)] − (n0 − nb) + n0 ln [n0/(ξ0V)] − n0. ξp is the partition function due to the kinetic energy and the internal degrees of freedom of the homopolymer (PEO). For theoretical conciseness and to avoid asymmetry between ions, the counterparts of both cations and anions are set to be the same value, ξ0. For the solvation energy of both cation and anion, we employ the Born solvation energy, VBorn = 2l0n0/ (2aϵr) because the theoretical treatment is succinct. l0 = e2/ (4πϵ0kBT) is the vacuum Bjerrum length. Previous studies based on a local lattice Monte Carlo simulation38 and field-theoretical approaches39 suggested that the reorientation of solvent dipoles near closely neighboring ions can be synergistically correlated. Moreover, the solvation energy of ions dissolved in polymers may depend notably on the chain length of polymers.40 However, the simple mean-field form for the solvation energy ignores these complexities and can be a significant simplification of the true solvation energy. Thus, we simply adopt the commonly accepted formulation for the solvation energy to gain a qualitative understanding of the phase behavior. The fraction of bound cations can be obtained by minimizing (F0 + Fhb)/(kBT) with respect to nb. This leads to the cationpolymer complexation equilibrium via hydrogen bonding, nb exp( −E hb) = (Nnp − nb)(n0 − nb) ξ0V

=

+

l0vp(1 − ϕp) 2aϵr v0

, (3)

where ϕp = Nvpnp/V and ϕ0 = v0n0/V are the volume fractions of the homopolymer and anion, respectively. We dropped the thermodynamically inconsequential terms in ϕp and ϕ0 using eq 2. Here, we have the incompressibility condition, ϕp + 2 ϕ0 = 1. The Flory−Huggins χ parameter indicates molecular interactions unrelated to hydrogen bonding and the electrostatic interactions, such as the van der Waals’ interaction. Thus, this χ parameter is expected to be written in the form A/T + B. We casted χ+ ϕpϕ0 + χ−ϕpϕ0 for the Flory−Huggins interactions of the cation and anion into χϕp (1 − ϕp). The vanishing of the second derivative of the free energy defines the spinodal curve. Thus, we can plot the spinodal curves by calculating ∂2[vp f tot/(kBT)]/∂ϕp2 = 0. In this study, we assume a simple volume-fraction-weighted average, ϵr = ϵPEOϕp + 2ϵILϕ0. N is set to be 100. T is set to be 200 °C for the vacuum Bjerrum length l0, which falls on the temperature range in the experiment for PEO and EMIM.5 For theoretical conciseness, we also assume that the energy of the hydrogen bond Ehb is independent of temperature. We consider the Born radius a = 0.38 nm and set vp = v0 for simplicity. Our theoretical framework for hydrogen bonding is an analogue of that in ref 29, in which the conventional Flory− Huggins χ parameter A/T + B is used. The counterpart of eq 2 in ref 29 depends on temperature. Accordingly, the number of the hydrogen bonds between the solvent and PEO significantly decreases as the temperature increases. The LCST transition is caused by the fact that the dissociation of the hydrogen bonds yields increases in the translational entropy of the solvents. Therefore, the aggregation of PEO in the solvents is induced as the temperature increases. Along similar lines, our current theory can also provide the LCST transition. In this case, the calculation of the spinodal curves requires the values A and B. However, we are not aware of the experimental values for the mixture of PEO and the ionic liquid. Thus, we do not consider the nature of the LCST in this study but attempt to suggest a mechanism for the asymmetry of the observed spinodal curve in terms of χ. Thus, we consider the energy of the hydrogen bond Ehb in the units of kBT to be independent of temperature; therefore, eq 2 does not depend on temperature. The global charge neutrality condition of the system is known to result in no contribution to the free energy of the homogeneous mixture.15 Thus, in eq 3, we have not added the electrostatic part of the free energy that involves the electrostatic screening length. However, in section 3.1, we demonstrate that the fluctuation of the local electrostatic potential yields the effect of the short wavevector component of the electrostatic interaction on the free energy. 2.2. Effects of the Dielectric Contrast and Hydrogen Bonding between a Polymer and an Ionic Liquid on the Critical Point. Figure 1 shows that the dielectric contrast

(2)

This formula indicates the adsorption equilibrium for [EO] + [C+] ⇄ [EO−C+], [EO−C+]/([EO][C+]) = K, where K is the association constant given by exp(−Ehb)/ξ0. [EO], [C+], and [EO−C+] denote the density of free ethylene oxide (EO) groups, cations, and cation-complexed EO groups, respectively. We can obtain nb/V and hence the volume fraction of bound cations ϕb = v0nb/V by solving eq 2 with respect to K. Here, we are not aware of an experimental value for K to account for the PEO-IL complexation via hydrogen bonding. In general, however, K may range over several orders of magnitude.41 Thus, we examine experimentally suggested values for K in the range from 0.1 to 3 M−1, which are reported in the case of the complexation between the pyridine and phenol units in ionic liquids.42 We now combine F0, Fhb, and VBorn with the Flory−Huggins interaction to write the free energy density of the liquid mixture, f tot, as follows: 692


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Figure 1. Critical point as a function of the dielectric contrast between a polymer and ionic liquid, Δϵ = ϵIL − ϵPEO. The colors of the lines correspond to ϵIL = 15 (black), ϵIL = 10 (red), and ϵIL = 7 (blue). No hydrogen bonding is considered, thus the shift in the critical point is purely caused by the dielectric inhomogeneity. The insets illustrate the spinodal curve corresponding to (ϵIL, Δϵ) = (15, 10) and (7, −2).

Figure 2. Effects of the dielectric inhomogeneity and hydrogen bonds between a cation and PEO on spinodal curves. Concentration fluctuations are not considered. The colors of the lines correspond to K/v0 = 0 (black), K/v0 = 3 (blue and green), K/v0 = 10 (red), and K/v0 = 20 (purple). (a) Hydrogen bonding only; the lines are the spinodal curves for different values of the equilibrium constant K. (b) Both dielectric inhomogeneity and hydrogen bonding; the black, blue, red, and purple lines correspond to ϵIL = 13.9 and ϵPEO = 7.5. The green line corresponds to ϵIL = 10 and ϵPEO = 7.5.

between a polymer and an ionic liquid is critically important for the phase instability of the mixture. In this case, the system includes no hydrogen bonding (i.e., K = 0) but involves the dielectric inhomogeneity. The critical point increases as the dielectric constant of the polymer is decreased. Because the experimental value of ϵPEO ranges from 5 to 1243−46 (or possibly ϵPEO ∼ 22 for short PEOs47), the critical point reads ϕc ≥ 0.5. Thus, our results suggest that the effect of the dielectric inhomogeneity may account for the observed shift in the critical point of PEO−IL mixtures toward polymer-rich regions.5 A commonly accepted dielectric constant of PEO is ϵPEO ∼ 8. If the polymer (PEO) is then replaced by a lower-dielectric species, such as cellulose (ϵcell = 2−848−50), then Δϵ becomes larger. Thus, our theory predicts further shifts in the critical point toward polymer-rich regions. Further experimental results or computational simulation are required to validate this theoretical prediction. Next, we illustrate that the effects of the dielectric inhomogeneity and hydrogen bonding on the spinodal curves are of the same order of magnitude and hence are equally important. Here, the value ϵIL = 13.9 corresponds to the dielectric constant of 1-butyl-3-methylimidazolium tetrafluoroborate.37 In Figure 2a, we show the result for the system involving hydrogen bonding with no dielectric inhomogeneity. The critical point in the case of K = 0 (the black line is for no hydrogen bonding) is 0.09, corresponding to the polymer-poor (or IL-rich) region. When we increase the strength of hydrogen bonding with K, the spinodal curve forms a nonmonotonic shape having both convex and concave curvatures. We are not aware of this unconventional curve for the phase diagram of a polymer and ionic liquid mixture in previous studies. However, this peculiar shape has been reported in an experimental study of sulfobetaine polymers in an aqueous solution;26 the authors suggested that the “W” (or double-well) shape is caused primarily by hydrogen bonding and can be flattened by reducing the strength of hydrogen bonding. Thus, our theory is consistent with the conclusions derived from such a study, suggesting a similar phase behavior in the case of a polymer−IL mixture. Figure 2a implies that the effect of hydrogen bonding alone is unlikely to rationalize the observed significant shift in the critical point toward polymer-rich regions. If hydrogen bonding

is the only factor causing the shift in the critical point, the “W” (or double-well) shape is also expected to appear. However, the experimental spinodal curve in ref 5 does not show such a characteristic feature. Thus, the effects of the dielectric contrast between the ionic liquid and the polymer may reasonably be one of the primary reasons for the observed shift in the critical point. Figure 2b illustrates that the effects of both hydrogen bonding and ion solvation equally deform the original spinodal curve having the critical point at the polymer-poor concentration (see the black line in Figure 2a). Thus, these two effects can be of equal importance to account for the phase instability of the polymer−IL mixtures. Specifically, a certain case for a small dielectric contrast (green line for ϵIL = 10 and ϵPEO = 7.5) produces a shift in the critical point toward polymer-rich regions and a relatively symmetric, flat spinodal curve. This structure is an analogue of the spinodal curves observed in the case of a EMIM and PEO mixture.5 The current results show that the immiscibility between PEO and ionic liquid can be weakened as the dielectric contrast decreases (see the blue and green lines); however, this observation is not a universal property. We will demonstrate the opposite case with and without fluctuation effects in section 3.3.

3. FLUCTUATION AROUND THE HOMOGENEOUS PHASE 3.1. Partition Function of a Polymer and Ionic Liquid Mixture. Of further interest is the effect of fluctuations in the local density and electrostatic potential around the homogeneous phase on the spinodal curves. Indeed, a field theoretical study of the phase instability of ions dissolved in solvents demonstrated that at the Gaussian level, fluctuations in the local electrostatic potential cause a substantial shift in the spinodal curve.51 The effect of fluctuations is significant even in the case of charge-neutral polymer mixtures.11,14,52 Furthermore, a recent hybrid self-consistent field theory combined with the integral-equation theory suggested that the effect of the strong correlation between the concentrations of the ion and the 693


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Macromolecules polymer can be of critical importance6−8,33,53,54 for the phase behavior of ion-containing polymers. This effect modifies the phase diagram of the electrolytes considerably and may even deform the spinodal curve in an unconventional manner. Thus, to account for the fluctuations in the local density and electrostatic potential, we draw upon the grand canonical partition function of a polymer and ionic liquid mixture as follows: λpnp λ+n+ λ−n− np! n+! n−!

∑

Z=

np , n+ , n−

×

d r j⃗(+)

d rk⃗(−)

np

n0

Q p[ωp] 1 = V

Hp =

∑∫

0

i=1

N

propagator qp:

∫ ∏ ∏ ∏ +R⃗ i(t )

1 V

Q p[ωp] =

i=1 j=1 k=1

δ[vpcp̂ ( r ⃗) + v0c+̂ ( r ⃗)

2 ⎛ ⎞ 3 ⎟⎡ dR⃗ i(t ) ⎤ ⎜ ⎥ d t ⎜ 2 ⎟⎢ ⎝ 2bp ⎠⎣ dt ⎦

2 ⎤ ⎫ ⎧ ⎪ ⎪ 3 ⎡ d R ⃗ (t ) ⎤ ⎢ ⎥ + iωp⎬⎥ dt ⎨ 2 ⎪ ⎪⎥ ⎩ 2bp ⎣ dt ⎦ ⎭⎦

Qp can be written in terms of the one-end-integrated

∫ d r ⃗ qp( r ⃗ , N )

(9)

where qp is obtained by solving the modified diffusion equation (4)

∂qp( r ⃗ , t )

VFH = ∫ dr ⃗ χĉp(r)⃗ [ 1 − vpĉp(r)⃗ ] is the Flory−Huggins part of the interaction energy. The δ function enforces the incompressibility of the liquid mixture. λp, λ+, and λ− are the fugacities of the homopolymer, cation, and anion, respectively. The position of the tth monomer of the ith polymer is denoted by R⃗ i(t), and the positions of the ith cation and anion are n0 (s) and r(−) denoted by r(+) i⃗ i⃗ , respectively. ĉs(r)⃗ = ∑i=1 δ(r ⃗ − ri⃗ ) is np N the number density for ion s. ĉp(r)⃗ = ∑i=1 ∫ 0 dt δ[r ⃗ − R⃗ i(t) ]is the number density for the monomer. Thus, the volume fraction of species s is given by ϕ̂ s(r)⃗ = vs ĉs(r)⃗ . Here, we express the elastic energy of the homopolymer in terms of a Gaussian chain model with the Kuhn length bp. The total charge density ρ̂(r)⃗ is given by ρ̂(r)⃗ = q+ ĉ+ (r)⃗ + q−ĉ−(r)⃗ . The Hamiltonians are then given by np

∫0

N

(8)

n0

+ v0c −̂ ( r ⃗) − 1] exp( −Hp − Hc − VFH)

∫

⎡ ⃗ +R(t ) exp⎢ − ⎢ ⎣

=

∂t

bp2 6

∇2 qp( r ⃗ , t ) − iωp( r ⃗ , t )qp( r ⃗ , t )

(10)

with the initial condition qp(r,⃗ 0) = 1. Q± is the partition function for the cation and the anion as follows: Q ±[ω±] =

1 V

∫ d r ⃗ exp[−iω±( r ⃗) − iq±ψ ( r ⃗)]

(11)

Extremizing the free-energy functional F with respect to the field variables yields the mean-field equations. For a homogeneous phase, the mean-field free energy is known to simply become the free energy of the Flory−Huggins theory vpFFH kBTV

(5)

=

ϕp N

ln ϕp −

ϕp N

+

⎡ vpϕs

∑⎢ s =±

⎣ v0

ln

vpϕs v0

−

vpϕs ⎤ ⎥ v0 ⎦

+ χϕp(1 − ϕp)

and 1 2

Hc =

This lowest-order free energy does not include the electrostatic

∫

d r ⃗ d r ′⃗ ρ ̂( r ⃗)v( r ⃗ − r ′⃗ )ρ ̂( r ′⃗ )

screening length scale. Thus, we must account for the

(6)

correction to FFH that arises from fluctuations in the field

where v(r ⃗ − r′⃗ ) = l0/(ϵr |r ⃗ − r′⃗ |) is the Coulomb potential. The dielectric value is again given by the simple volume-fractionweighted average, ϵr(r)⃗ = ϵPEO vp ĉp(r)⃗ + ϵILv0 ĉ+(r)⃗ + ϵILv0 ĉ−(r)⃗ . Using standard field-theoretical techniques in polymer physics,55 we derive the free-energy functional from the partition function in terms of the field variables: 1 F = 2 kBT −i

variables around the homogeneous phase. We now write the field variables in terms of the deviation from their values in the homogeneous state as follows: ρ( r ⃗) = ρ ̅ + δρ( r ⃗) cp( r ⃗) = cp + δcp( r ⃗)

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

∑

c±( r ⃗) = c0 + δc±( r ⃗) ωs( r ⃗) = ωs̅ + δωs( r ⃗)

∫ d r ⃗ cs( r ⃗)ωs( r ⃗) − i ∫ d r ⃗ ρ( r ⃗)ψ ( r ⃗) − λpVQ p

ψ ( r ⃗) = ψ̅ + δψ ( r ⃗)

s=p,±

−

∑ λsVQ s + ∫ dr ⃗ χcp( r ⃗)ϕIL( r ⃗)

(12)

Without loss of generality, we may set the electrostatic (7)

potential as ψ = 0. For charge neutrality, the spatial average of

Interested readers are referred to the calculation details in Supporting Information. In this equation, cs and ωs are the density field and its conjugate field for species s, respectively, and ψ is the electrostatic potential. Qp is the configuration partition function for a single-chain homopolymer:

the charge density ρ is zero. The incompressibility condition is

s =±

given by vp cp(r)⃗ + v0c+ (r)⃗ + v0 c−(r)⃗ = 1, or vpδcp(r)⃗ + v0δc+ (r)⃗ + v0δc−(r)⃗ = 0. We then expand Qp and Q± to quadratic order in the deviatory field variables in Fourier space as16 694


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Macromolecules ⎡ N2 Q p ∼ Q̅ p⎢1 − 2V ⎣ ⎤ × δωp( −k ⃗)⎥ ⎦ q±

−

−

⃗

∫ (2dkπ )3 δω±(k ⃗)δψ (−k ⃗) 2V

Q± ∼ 1 − q±2

current study, we complete the Gaussian integral over all field variables by considering the normalization factors to obtain the effective free energy as a function of the bulk polymer density. A similar analysis for the mixture of charge-neutral polymers can be found, for example, in refs 11, 14, and 52. The effective free energy can thus be expressed as

⃗

∫ (2dπk)3 gp(R g 2k2)δωp(k ⃗)

Feff = FFH +

2V 1 2V

∫ (2dkπ )3 δω±(k ⃗)δω±(−k ⃗)

⃗

(13) −x

where gp(x) = 2(x − 1 + e )/x is the Debye function for a homopolymer. Here, R g = bp N/6 is the mean square radius of gyration for the homopolymer. Inserting Qp and Q± into eq 7 and expanding F to the quadratic order, we obtain ΔF (2) = kBT

∫

2

Gel(k) = 1 +

∑ λsqsδωs(k ⃗)δψ (−k ⃗) ∑ s =±

+

∑

λs δωs(k ⃗)δωs( −k ⃗) 2 λsqs2 2

s =±

−i

∑

δψ (k ⃗)δψ ( −k ⃗)

δcs(k ⃗)δωs( −k ⃗)

s=p,±

+

λpN 2Q̅ p 2

gp(R g 2k 2)δωp(k ⃗)δωp( −k ⃗)

− χδϕp(k ⃗)δϕp( − k ⃗)⎤⎦

(14)

To obtain the effective free energy, we perform the Gaussian functional integral in ⎡ F ⎤ Z ≈ exp⎢ − FH ⎥ ⎣ kBT ⎦

8πl0q0 2c0 ϵr k 2

where we set q+ = −q−= q0 for charge neutrality. Gchain(k) and Gel(k) arise from fluctuations in the local concentration of the polymer and the electrostatic potential, respectively. Importantly, the solvation energy of ionic liquids may be strongly correlated with the strong ion−ion correlation through ϵr in Gel(k) . We must now regularize the divergence arising from the k-⃗ integral in eq 15. We introduce the ultraviolet cutoff Λ for the ⃗ k-integral, casting ∫ dk⃗ into ∫ 2π/Λ0dk 4πk2. The cutoff should establish a minimum size scale in this system.11 Here, we consider the case with v0 = vp. Because the sizes of the monomer, cation, and anion are the same, it is reasonable that we choose the cutoff for the integral of ln Gel(k) that is of the same order of the magnitude as the diameter of those species. Specifically, the integral of ln Gel(k) should yield the conventional Born solvation energy (q+2c0 + q−2 c0)l0/(2aϵr) when the system is a dilute solution consisting of ions dissolved in a nonpolymeric solvent. With c0 ∼ 0 taken in ln Gel(k), we find that Λel = 4a fulfills this requirement precisely. In addition, we set the cutoff for the integral of ln G c h a i n (k) to Λchain = R g = b N/6 .56,57 Thus, we can obtain the spinodal curve by numerically solving ∂2Feff/∂ϕp2 = 0. For the model parameters, we set the Kuhn length to bp = 0.56 nm for PEO. The volumes of the species are given by 4π v0 = vp = 3 a3, with a = 0.38 nm. 3.2. Shift in the Spinodal Curve with No Dielectric Contrast between the Polymer and the Ionic Liquid. We first consider no dielectric contrast by setting ϵIL = ϵPEO. Thus, the phase diagram is altered purely by the effects of the fluctuations in the local density and electrostatic potential. Figure 3 illustrates the significant shift in the phase boundary caused by the fluctuation effects. The spinodal curve near the critical point tends to be relatively flat as the dielectric constant decreases. When the ionic liquid is enriched, the effect of the dielectric constant on the spinodal curve is insignificant. Specifically, the spinodal curve exhibits an inflection point (i.e., both convex and concave structures) and intersects the xaxis when the dielectric constant becomes on the order of unity. This characteristic structure was also reported in the case of polyelectrolytes by developing a hybrid of the self-consistent field theory combined with the integral-equation theory.6−8 We

dk ⃗ ⎡ 1 ⃗ ⎢ v(k )δρ(k ⃗)δρ( −k ⃗) (2π )3 ⎣ 2

s =±

+

∫ dk ⃗ [ln Gchain(k) + ln Gel(k)]

(15)

− iδρ(k ⃗)δψ ( −k ⃗) +

2(2π )3

⎡ N 2c v 2g (R 2k 2) p p p g Gchain(k) = ⎢ + 2c0v0 ⎢ v0 ⎣ ⎤ − 4χvpcpv0c0N 2gp(R g 2k 2)⎥ ⎥ ⎦

⃗

∫ (2dkπ )3 δψ (k ⃗)δψ (−k ⃗)

kBTV

∫ +Ωδ[vpδcp( r ⃗) + v0δc+( r ⃗) + v0δc−( r ⃗)]

⎡ ΔF (2) ⎤ ⎥ exp⎢ − ⎣ kBT ⎦

over field variables,11,14,51,55 where Ω symbolically denotes all the field variables (Supporting Information). The absence of the first-order contribution reflects the saddle-point nature of the free energy at the homogeneous state. In ref 55, a similar integration for ion-containing block copolymers was performed with field variables, except for the polymer density field, to obtain the structure function. In that case, the Gaussian integral can be completed simply by extremizing the integrand with respect to the field variables because the relevant normalization factor does not involve field-variable quantities. However, in the 695


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Figure 3. Effects of fluctuations in the local density and the electrostatic potential on spinodal curves. Dielectric inhomogeneity or hydrogen bonding is not considered. The curves correspond to ϵIL = ϵPEO = 20 (black), ϵIL = ϵPEO = 10 (blue), and ϵIL = ϵPEO = 5 (red). The black dashed line indicates the spinodal curve for no fluctuation.

Figure 4. Effects of the dielectric inhomogeneity and the fluctuations in the local density and electrostatic potential on spinodal curves. Hydrogen bonding is not considered. The lower insets correspond to the counterparts of the spinodal curves for no fluctuation effect. (a) (ϵIL, ϵPEO) = (6.5, 2) (blue), (ϵIL, ϵPEO) = (6.5, 6.5) (black), and (ϵIL, ϵPEO) = (6.5, 10) (red). (b) (ϵIL, ϵPEO) = (20, 2) (black) and (ϵIL, ϵPEO) = (6.5, 2) (blue). The right-hand phase boundary for the fluctuation effect is shown in the upper inset. In the lower inset, the right-hand black and blue lines correspond to ϕ ∼ 0.9975 and ϕ ∼ 0.99, respectively.

thus suggest that this type of phase behavior is commonly observed in ion-containing polymers and indicates the strong effect of the fluctuations. From our theoretical perspective, such a curvature is expected to exist in the experimental spinodal curve of a PEO and ionic liquid mixture,5 in which the hydrogen bond between the polymer and the ionic liquid is substantially suppressed. However, the experimental curvature is uncertain and cannot be clearly evidenced. Therefore, the present predictions require quantitative validation based on molecular dynamics simulations or further experimental studies. Furthermore, our present method is self-contained within the conventional self-consistent field theory and thus does not involve the hard-core nature of ions. Along similar lines, the tightly bound ion pairs are not considered in our current theory. However, this assumption is also controversial because a recent experimental analysis suggests that the ion pairs play an important role in the electrostatic screening length.58 In fieldtheoretical jargon, our theory accounts for the fluctuation effects at the one-loop level. 3.3. Effects of the Dielectric Inhomogeneity on the Spinodal Decomposition. We now consider the dielectric contrast between a homopolymer (PEO) and an ionic liquid. Figure 4 illustrates the substantial change in the spinodal curves caused by the dielectric inhomogeneity. Here, we use ϵIL = 6.5 for the nearly lowest value of the dielectric constant of ionic liquids.59 The polymers of interest for a biorenewable material or energy storage, such as PEO (ϵPEO = 5−1243−46), polystyrene (ϵPS = 2.6−460), and cellulose (ϵcell = 2−848−50), typically have a dielectric constant between 2 and 12. Thus, the current choice of ϵIL allows us to clearly contrast the two cases with ϵIL < ϵPEO and ϵIL > ϵPEO. Our theory for the fluctuation effect predicts a higher solubility of the polymer (PEO) and the ionic liquid as the dielectric constant of PEO is increased (Figure 4a) when PEO is enriched. Alternatively, when an ionic liquid is relatively enriched, the change in the spinodal curve caused by the dielectric contrast is insignificant. The inset of Figure 4a indicates that without the fluctuation effect, the increase in the dielectric constant of PEO leads to the enhancement of the miscibility of the mixture. We conceive that this case with no fluctuation may arise when the fluctuation effect is suppressed by the strong hydrogen bonds between the cation and the anion, as in the case of ref 5. However, the resultant geometrical

trends of the curves significantly contrast that for the fluctuation effect. The current case for the fluctuation effect with dielectric inhomogeneity is even an analogue of that without dielectric inhomogeneity presented in Figure 3. This correspondence thus indicates that the strong ion−ion correlation is more dominant than the solvation energy in the formation of the phase boundary. Note the left-hand phase boundaries in Figure 4b and its lower inset. Both theories with and without the fluctuation effect predict that decreases in the dielectric contrast cause increases in the immiscibility between PEO and the ionic liquid. However, in Figure 2b, for no fluctuations, we demonstrated the opposite case; the miscibility of the mixture can be enhanced by decreasing the dielectric contrast when PEO is a lower-dielectric component. Thus, the preferential solvation of ions cannot be simply predicted by the degree of the dielectric contrast between the polymer and the ionic liquid. In the case of no fluctuation effect (see the lower inset in Figure 4b), the opposite response of the phase behavior can be explained by the fact that the second derivative of the Born solvation energy, d 2[vpVBorn /(kBT )]/dϕp2 =

l 0Δϵ(ϵIL − Δϵ) 2a(ϵIL − Δϵϕp)3

, exhibits nonmonotonic

behavior with respect to the volume fraction of PEO and the dielectric contrast. In the case of a fluctuation effect, the change in the left-hand phase boundary is caused predominantly by the strong ion−ion correlation. The tendencies of changes in the left-hand and right-hand phase boundaries with respect to the dielectric contrast are qualitatively different, as shown by Figure 4b and its upper inset: Here, the solvation energy of ions is primarily responsible for the trend of the right-hand phase boundary. The ion−ion correlation then yields quantitative changes in the phase boundary. Incidentally, this tendency caused by the dielectric contrast is similar to that obtained from the meanfield theory of lithium-containing polymer mixtures in low salt concentrations.55,61 Our notable finding in this subsection is that the spinodal curve obtained from a simple hybrid of the Flory−Huggins 696


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summarize these insights as follows. (1) The difference between the dielectric constants of PEO and the ionic liquid can cause the significant shift in the critical point toward the polymer-rich region (Figure 1). A large dielectric contrast may produce an inflection point in the spinodal curve. (2) The hydrogen bonds between the polymer and the ionic liquid increase the miscibility of the mixture (Figure 2a). The magnitude of this increase is equivalent to the change in the spinodal curve caused by the dielectric inhomogeneity (Figure 2b). (3) The highly nonmonotonic behavior of the spinodal curve may be caused by the strong hydrogen bonds between the polymer and the ionic liquid (Figure 2). (4) The fluctuation of the local density and the electrostatic potential causes drastic changes in the spinodal curve, driving the trend of the phase boundary to be highly nonmonotonic (Figure 3). When the dielectric constant of the mixture is on the order of unity, the spinodal curve exhibits an inflection point and intersects the xaxis for χ = 0. This characteristic behavior is analogous to that for polyelectrolytes recently suggested by a hybrid of the selfconsistent field theory (SCFT) and integral-equation theory.6−8 However, our theoretical framework is at the one-loop level and within the conventional SCFT. (5) When the dielectric constant of polymers is higher than that of an ionic liquid, the miscibility of the mixture may be enhanced (Figure 4a). Although this enhancement is highly desired in the case of cellulose (ϵcell = 2−848−50), the dielectric constant of ionic liquids is typically higher than ϵcell. Nevertheless, when the volume fraction of ionic liquids is small (e.g., 0.005%), the selection of a lower-dielectric IL would lead to a better situation due to the reduction in the dielectric contrast (see the insets in Figure 4b). However, when the volume fraction of ionic liquids is relatively large (e.g., 20%), lowering the dielectric contrast with a low-dielectric IL leads to the enhancement of the ion− ion correlation, which increases the immiscibility of the mixture (Figure 4b). Our theory indicates that the left-hand phase boundary is predominantly affected by the ion−ion correlation, whereas the solvation energy of ions is substantially responsible for the trend of the right-hand phase boundary. We plotted the spinodal curve as χ vs the polymer composition ϕ in Figure 2. Although our current results do not present the LCST, our theoretical framework for hydrogen bonding may yield the LCST using χ = A/T + B.29 However, few studies have considered the temperature dependence of χ for the mixture of polymer and ionic liquid. Thus, further analysis yet requires the detailed information about the χ parameter based on experiments and molecular dynamics simulations. Our results obtained from the mean-field theory for no fluctuations are qualitatively consistent with the observed shift in the spinodal curve of the PEO and the ionic liquid with hydrogen bonds5 (Figure 2). This correspondence suggests that further in-depth studies of the correlation between the ion−ion correlation and hydrogen bonding are required. For example, the hydrogen bonds between a cation and anion in ref 5 may produce charge-neutral ion pairs and reduce the fluctuation effect. Moreover, the inflection point of the experimental spinodal curve appears to be marginal and thus cannot be clearly observed. We conceive that further experiments and computer simulations are required to validate the current theoretical prediction as well as the highly nonmonotonic behavior due to hydrogen bonding. Finally, the fluctuation effect can be notably weakened by the parallel alignment of the dipoles of the cation and anion

theory and the Born solvation energy tends to be significantly altered both qualitatively and quantitatively when fluctuations in the local density and electrostatic potential are considered. Although both fluctuation and dielectric inhomogeneity are of equal significance for the phase behavior, the fluctuation effect is relatively dominant in the formation of the left-hand phase boundary. Importantly, the strong ion−ion correlation is significantly correlated with the solvation energy of ionic liquids. Furthermore, the dielectric inhomogeneity does not substantially alter the geometrical trends of the left-hand spinodal curves for no dielectric contrast, as noted from the comparison between Figures 3 and 4. If we must dissolve a large amount of low-dielectric polymers, such as cellulose, in an ionic liquid, then our present results concerning the right-hand phase boundary suggest minimization of the dielectric contrast between the polymer and ionic liquid by virtue of the solvation energy. However, when an ionic liquid is also fairly enriched, the left-hand phase boundary must be considered; thus, the miscibility of the mixture should increase by increasing the dielectric constant of ionic liquids. In addition, we likely require the hydrogen bonds between the polymer and the ionic liquid to overcome their large immiscibility caused by the strong ion−ion correlation and the solvation energy of ions. Certainly, the complexation between the polymer and the ionic liquid is of significance to enhance their miscibility. However, we also refer to the mechanism to enhance the miscibility of charged polymers, as demonstrated in previous theoretical studies:21,22,32,62,63 ioncomplexed PEO via hydrogen bonding should behave as charged polymers. In this case, the phase separation between the polymer-complexed ion and its counterion causes an entropic penalty due to electroneutrality and thus tends to be inhibited. Accordingly, the miscibility of charged polymers tends to be enhanced as the degree of charges is increased. We thus suggest that PEO complexed by ionic liquids via hydrogen bonding is also an analogue of polyelectrolytes.

4. CONCLUSIONS In conclusion, we studied the effect of the dielectric contrast and hydrogen bonds between a polymer and an ionic liquid as well as the effect of the fluctuations in the local density and the electrostatic potential on the spinodal decomposition. We first constructed a mean-field theory to account for the correlation between the dielectric inhomogeneity and hydrogen bonding by combining the Born solvation energy and the Langmuir adsorption model. Our theory can show that the number of the hydrogen bonds between the cation and PEO decreases as the temperature increases. This effect yields increases in the translational entropy of the ions. However, we have not considered such an entropic effect in this study (see section 2.1). For the fluctuation effect, we calculated the effective free energy to the quadratic order of the field variables using the self-consistent field theory. The spinodal curve is derived from the zero of the second derivative of the effective free energy. Taking the PEO and ionic liquid mixture as a model system,5 we demonstrated that all of the effects are of significance in altering the spinodal curves. Our theory consists of a minimal set of parameters; the dielectric constants and molecular volumes of PEO and the ionic liquid, the Flory−Huggins χ parameter, the Kuhn length, the degree of polymerization, and the equilibrium constant for PEO-cation complexation. Nevertheless, our simple theory qualitatively accounts for some of observed features and provides profound insights. We 697


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Macromolecules through ion pairing.64,65 This collective dipole reorientation accounts for remarkably large dielectric constants of some ionic liquids (ϵr > 25). In our current calculations, these large dielectric constants do not cause an inflection point in the spinodal curve due to the fluctuation (Figure 3). However, the effect of ion solvation should become pronounced because the dielectric contrast between the polymer and the ionic liquid or between the ion pair and ion is increased. These counteracting features would likely lead to a more complex phase behavior. Importantly, modeling the dielectric response through the bulk dielectric constant ϵr could miss the true nature of the correlations in the molecular interactions. Nevertheless, an atomistic simulation for ion solvation remains a challenge in improving the computational performance, especially for polymerized species. In this context, further development of the theory for strongly correlating dipoles and ions is necessary for in-depth studies of polymer and ionic liquid mixtures.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02189. Calculation details of the self-consistent field theory for a polymer and ionic liquid mixture (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(I.N.) E-mail: [email protected]. Funding

This work was supported by the National Natural Science Foundation of China (21474112). Notes

The authors declare no competing financial interest.

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REFERENCES

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Spinodal Decomposition of a Polymer and Ionic Liquid Mixture: Effects

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