Anomalous Phase Behavior of Ionic Polymer Blends and Ionic

Article pubs.acs.org/Macromolecules

Anomalous Phase Behavior of Ionic Polymer Blends and Ionic Copolymers Victor A. Pryamitsyn,*,† Ha-Kyung Kwon,† Jos W. Zwanikken,‡ and Monica Olvera de la Cruz*,† †

Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Cook 2036, Evanston, Illinois 60208, United States ‡ Department of Physics, University of Massachusetts Lowell, Lowell, Massachusetts 01854, United States ABSTRACT: Nonionic diblock copolymers, with a highly asymmetric relative composition (f), microphase segregate into structures in which the minority component always forms cylindrical or spherical domains that are embedded in the majority component continuous matrix phase. Recently, a hybrid liquid state theory (called the DHEMSA approximation) and self-consistent field approach for ionic diblock copolymers have demonstrated the possible existence of “inverted” phases in which the minority ionic component forms the continuous matrix phase and the majority nonionic component forms cylindrical domains. We find that such anomalous behavior is closely related to the thermodynamics of phase segregation found in a blend of an ionic polymer and a nonionic polymer at the electrostatic coupling values typical of polymers in the molten state, at which nonionic and ionic polymers segregate into two partially miscible phases. This partial miscibility holds even across infinite molecular weights of the polymers. Such partial miscibility causes swelling of the minority component and a “switch” between minority and majority phases in ionic block copolymer melts. By combining the DHEMSA approximation with the Helfand−Tagami theory, we calculate the interfacial tension γ between coexisting phases of ionomers. The full phase diagram for ionomer blends and γ allows us to construct the phase diagram of block copolymers. In addition to the conventional microphases found in nonionic diblock copolymers, we find microphases with “inverted” cylindrical and spherical domains. We also predict an “inverted” phase at high values of f where the nonionic minority component becomes swollen by the ionic component and forms the matrix phase. Threedimensional self-consistent field theory modeling confirms the existence of the “inverted” bicontinuous phases between the lamella and the inverted cylinder microphase regions of the phase diagram.

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INTRODUCTION Developing ion conducting membranes for batteries, fuel cells, and water desalination/purification is challenging because of two normally conflicting requirements: high mechanical strength and high ionic conductivity (or permeability and selectivity).5,6 Block copolymers composed of ion-conductive and nonionic blocks6−8 are prospective materials capable of satisfying both requirements. The development of novel, robust theoretical methods to analyze ion-containing block copolymer morphology, combined with experimental observations, will accelerate development of these materials. Recent numerical developments have demonstrated a highly unusual phase behavior of ion containing block copolymers,2−4 such as a strong immiscibility when the ionic component is highly dilute that results in locally segregated structures in which the minority ionic component forms a continuous phase. Experimental observation9,10 has confirmed a large degree of incompatibility in ion containing copolymers. Another theoretical result is the partial miscibility of nonionic and ionic polymers at high concentration of the ionic component.1,2,4,11 Interestingly, recent experimental observations8 have shown an apparent negative χ-parameter for an ionomer diblock copolymer, in © XXXX American Chemical Society

agreement with this theoretical observation. In this paper, we aim not only to elucidate the mechanisms responsible for such high degree of immiscibility or partial miscibility but also to make predictions of microphase morphologies in ion containing copolymers and of the physical properties of equivalent (nonionic)−ionic polymer blends via theoretical models. We analyze the results and discuss the contrast between our observations and results for equivalent nonionic copolymers12,13 and blends.14,15 In the simple case of nonionic monodisperse AB diblock copolymers, symmetric compositions of copolymers form lamellar phases, while gyroid, hexagonally packed cylinders, and BCC ordered sphere phases form in compositionally asymmetric diblock copolymers.13,16,17 Polydispersity in the copolymer molecular mass and composition may change phase boundaries, causing the formation of more complex bicontinuous phases or macrophase segregation between different ordered phases.18−23 Nevertheless, these phases all share a common feature in nonionic Received: April 3, 2017 Revised: June 1, 2017

A

DOI: 10.1021/acs.macromol.7b00523 Macromolecules XXXX, XXX, XXX−XXX

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and the volume fractions of the ions in the pure A phase denoted by fq. For the corresponding ionic diblock copolymer, it is the set {χ; Γ; fq; f; N}. We demonstrate that if the parameter set {χ; Γ; fq; ϕA} for the IB system corresponds to a miscible blend, the diblock copolymer with {χ; Γ; fq; f = ϕA} is disordered at any N value. If the IB system segregates at {χ; Γ; fq; ϕA}, the corresponding diblock copolymer at {χ; Γ; fq; f = ϕA} will form a microphase segregated morphology only if the molecular mass N is larger than a transitional Ncrit(χ, Γ, fq, f) (i.e., N > Ncrit). Liquid Theory Approach and Model Parameters. The theoretical approach based on the liquid state theory and named Debye−Hückel extended mean spherical approximation (DHEMSA)2−4,11,24,26 accounts for the effects of ionic correlations caused by the interplay of strong, short-range Coulombic attraction between opposite charges and hard core repulsion between ions on the thermodynamics and phase behavior of ion containing polymer blends and copolymer melts. This approach allows calculation of the excess chemical

diblock systems: the lamellar phases are always nearly symmetric (i.e., A and B layers have almost the same thicknesses). In nonionic copolymer microphases with “curved” domains, the minority component is always the “filler” phase and the majority component is always the “matrix” phase. Different block copolymer structures (e.g., lamella with high disparity in the thickness of the A and B layers or cylindrical/spherical domains with unexpected volumes fractions or microstructures) have technological importance and represent an interesting scientific challenge.23 Control of the morphology is important for development of novel ion-conducting polymeric materials for applications including batteries with combined high conductivity and high mechanical strength. The recent theory of ion containing block copolymers predicts a phase inversion in which the majority component system forms inverted, hexagonally packed cylinders in a matrix composed of the minority component. This is in contrast to the typical case in nonionic diblock copolymers where the minority component forms cylindrical domains and not the matrix phase.3,4 Another striking prediction of the theory of ion containing copolymers and blends4,11,24 is a “chimney” regime at high values of the electrostatic coupling between ions expected to occur in polymers in the molten state due to the low dielectric constant of the medium. The chimney in the phase diagram of a polymer blend of an ionomer and a nonionic polymer is closely related to the “phase inversion” found in ionic diblock copolymers.3,4 The “chimney” effect is characteristic of a phase coexistence between two partially mixed phases far from the critical point. These effects are quite unconventional compared with the conventional “Flory−Huggins” (FH) type systems. Let us consider a symmetric incompressible blend of nonionic A and B homopolymers, with equal degrees of polymerization NA = NB = N and a Flory interaction parameter χ. If N is large, one can neglect the polymer composition fluctuations and use a mean-field approach.25 In this symmetric blend, the phase segregation between the two nonionic polymers is controlled only by the product χN with the critical point located at χN = 2 and equal volume fractions of A and of B, ϕA = ϕB = 1/2. At any positive χ value, if N → ∞, the blend segregates into two pure phases, in which the volume fractions of A and B tend to one and zero, respectively, in one phase and to zero and one in the other phase. The separation of components into domains with compositions of pure A and B components also occurs in nonionic A−B symmetric copolymer melts with N = NA + NB and f = NA/N in the limit of N → ∞. We show here that such behavior is not expected for the ionic A and nonionic B homopolymers blends and for the corresponding ionic A and nonionic B diblock copolymer melt. The ionic component A contains Ni negative charges neutralized by Ni positively charged monovalent counterions. The counterion translational entropy and the ionic correlations at the electrostatic coupling values (denoted by Γ) characteristic of ionomers cause the appearance of coexisting phases at finite χ where the volume fractions of A and B both remain larger than zero at the phase boundaries (“chimney” phases), even when N → ∞. This means that χN itself is not the appropriate universal parameter for ionic polymer blends and ionic diblock copolymer melts. We show that the set of parameters which control phase segregation in ionic polymer blends at N → ∞ (IB system) are χ, Γ, the overall fractional compositions ϕA and ϕB = 1 − ϕA,

potential

μex (Γ, ϕq) kBT

of the ions in a symmetric system of positive

and negative ions with a hard core potential, as a function of the dimensionless strength of the electrostatic interaction, Γ=

|z1z 2| e 2 , 4πε0εrdkBT

where e is the electron charge; zn is the valency

of the ions n = 1, 2; d is the contact distances between the ions as a function of the dimensionless volume fraction of the ions, π ϕq = 6 d3(ρ+ + ρ− ); and ρ+ = ρ− are the number densities of the positive and negative ions. The results of the DHEMSA were merged with the self-consistent field theory (SCFT) and a traditional Flory−Huggins mean-field theory of the polymer blends to analyze the phase coexistence in the ionic polymer blends and in ionic block copolymer melts.3,4,11,24 We use the notation typically used in Flory−Huggins models for incompressible polymer melts.15 In this model, the length of 1 the polymer segment (Kuhn segment) is defined as b = 6ρ R 2 , c

g

where ρc is the polymer chain number density, and the radius of b2 N

gyration of a polymer chain of N segments is R g 2 = 6 , where each segment has the mean-square end-to-end distance b2 and volume b3. Here, we assume the Kuhn segment b are the same for both the A and B segments. The number of segments in every A chain is NA, in every B chain is NB, and for a copolymer AB chain is N = NA + NB. The enthalpic segment−segment interactions are characterized by the Flory interaction parameter between A and B segments χ > 0. The composition N of the AB copolymer is f = NA . The volume fractions of the A and B chains are ϕA = b3ρA and ϕB = b3ρB, respectively, where ρA and ρB are the number densities of polymer segments. The incompressibility ansatz can be presented as ϕA + ϕB = 1. Finally, we use kBT units for the units of energy and kB2T for the b

units for the interfacial tension γ, and we omit kBT in all dimensionless energy expressions. The volume fraction of the ions in the pure A phase is fq =

Niπd3 λNb3

, where the packing

coefficient λ = 0.634 is used to keep consistency with earlier publications.3,4 We assume here that all phases have the same dielectric permittivity εr. Others have considered the dielectric mismatch between phases;27 it was shown that these effects could be included in the model by rescaling the Flory χ-parameter.3

B


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be obtained25 from the diffusion equations

SIMPLE SELF-CONSISTENT FIELD FORMALISM FOR COMPLEX POLYMERIC SYSTEMS A rigorous self-consistent field theory (SCFT) formalism for some polymeric systems can be found in ref 25. A simpler recipe for the mean-field theories can be traced to ideas of Flory14 and DeGennes.15 In this approach, the polymer SCFT is formulated as a density functional theory. The free energy of a system, composed as a functional fluid/ polymer density, is split into the entropy of an “ideal” noninteracting polymer chains with a given density profile and the excess free energy of short-range interactions calculated in the approximation of local homogeneity of the system.28,29 Such an approach exactly reproduces the rigorous SCFT models but is also suitable for the systems where field theoretical methods25 are impractical. One can calculate the entropy of the system of Np noninteracting polymer chains in an external field, which form an ordered structure. The entropy of a system in an external field w(r) that is a function of a generalized coordinate r is given by Fideal[w] = −Np log Q [w]

(1)

δFideal[w] δw(r)

(2)

ρ(r) =

∂q(r, s) = R G 2∇2 q + w(r, κ(s))q(r, s) ∂s ∂q†(r, s) = R G 2∇2 q† + w(r, κ(1 − s))q†(r, s) ∂s q(r, 0) = q†(r, 0) = 1;

s ∈ [0, 1]

(6)

RG2

where is the radius of gyration of an unperturbed Gaussian chain and q(r,s) is the partition function for a piece of the chain of length s with the sth segment located at r, and q† is the partition function for the chain with the ends switched. The single chain partition function Qp[w] is Q p[w] =

∫V q(r, 1) dr = ∫V q†(r, 1) dr

(7)

where V is the volume of the system. The single chain density is given by ϱi(r) = −

δ log Q p δwi(r)

1

=

∫0 δi , κ(s)q(r, s)q†(r, 1 − s) ds Qp

∫ ϱi(r) dr = ∫ δi ,κ(s) ds; ∑ ∫ ϱi(r) dr = 1 i

(8)

TSideal[w] =

∫ ρ(r)w(r) dr − Fideal[w]

where δi,κ is the Kronecker delta. The segmental number density distribution ρi(r) for the ith monomer type is given by

(3)

where Q[w] is the partition function of an “ideal” polymer chain in the external field, F[w] is the free energy of the “ideal” system, and ρ-[w(q)] is the generalized density of the system. The free energy of the system in the self-consistent field approach is FSCF[ρ] = −TSideal[w[ρ]] + Fint[ρ]

ρi (r) = NNp ϱi(r) and the volume fraction of the ith component by ϕi(r) = V ϱi(r)

The conformational entropy (canonical ensemble) as a functional of the density distribution of the polymer segments is

(4)

where Fint[ρ-(r)] is the density functional of the excess free energy of nonbonded interactions; w[ρ-(r)] is defined by inverting eq 2. Minimization of eq 4 with respect to ρ-(r) gives δF [ρ] w(r) = int δρ(r)

TScomf [ρ(r)] = NpTSideal[ρ(r)] = Np(log Q p[w] +

∑ ∫ wi(r)ϱi[w(r)] dr)

(9)

i

= Np log Q p[w[ρ(r)]] + N −1 ∑

(5)

∫ wi[ρ(r)]ρi (r) dr

i

Note that both ρ and w represent the generalized multicomponent densities and fields for all components of a polymer system. The right-hand side of eq 5 is the local excess chemical potential of the system. Equations 2 and 5 form the complete set of equations to compute w, ρ, and FSCF. These are usually referred to as the polymer self-consistent field equations. Statistics of an “Ideal” Polymer Chain. There are many available models for analysis of the conformation of an “ideal” polymer chain in an external field.15,25,28 Here, we use a model of Gaussian polymer threads and follow the Fredrickson’s notation.25 Consider a piecewise Gaussian copolymer chain consisting of m types it = { 1...m} of segments; for m = 2, we use A segments and B segments, instead of {0, 1}. Let us consider a set of m potentials, w(r,it) each acting on a segment of type it and a type select function κ-(s), s ∈ [0, 1], which gives the type of the sth segment of the chain. For such a system, the chain partition function and segmental density distribution can

where wi[ρ-(r)] is formally computed by an inversion of eq 8. Numerical methods for solving eq 6 and for computing the polymer density given by eq 8 and the conformational entropy from eq 9 are well developed and exhaustively described by Fredrickson.25 A Flory−Huggins Type Model for Ionomer Systems. The Flory−Huggins (FH) free energy per one cell of volume b3 of a homogeneous AB blend is given by log ϕB log ϕA b3FFH = + + χϕA ϕB V NA NB

(10)

The generalized FH theory applied to the case of ion containing polymers2−4,11,24,26 uses the excess chemical potential μex(ϕq,Γ) of the ions computed from DHEMSA closure and can be expressed as log ϕB log ϕA b3FΓ = + + fint (ϕA , ϕB) V NA NB C

(11)


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∫0

In Zwanikken et al.1 it was also emphasized that the DHEMSA closure does not show close agreement with simulations inside the “forbidden region”, where the system of free ions phase separates and consists of two macroscopic phases of different densities. This is an inherent aspect of theoretical methods that rely on an assumption of global homogeneity but should not limit the determination of the spinodals nor the calculation of the thermodynamic properties of each of the phases, once their densities have been determined. The nonmonotonic behavior of the chemical potential (see Figure 1), as predicted by the DHEMSA, is indeed directly related to the observed effects but does not offer a simple physical explanation. Such a simple explanation is difficult to formulate because the phase behavior depends on the second

(12)

ϕ

μex (ϕq , Γ) dϕq

(13)

where fex(ϕq,Γ) is the contribution of the excess free energy of ion−ion interaction, and fq log ϕA is the translational entropy of counterions. Equations 11−13 allow us to write the density functional of the excess free energy eq 4 of an ionomer system with the approximation of local homogeneity of the ionomer system: Fint[ρ] =

∫ fint (ρ(r)) dr

(14)

The validity of such approximation for an ionomer system (which was referred to as LS-SCFT previously3,4) was carefully analyzed in the aforementioned works. The terms fex( fqϕA,Γ) is the central points of LS-SCFT approximation for an ionomer system developed previously.3,4 This term can be regarded as an extension of the Debye− Hückel contribution in the Voorn−Overbeek model30 to the Flory−Huggins formalism for the ionomers blends. The liquid state theory1 based on the DHEMSA closure accurately incorporates ionic correlations that are the convoluted result of cohesive Coulombic interactions and repulsive short-range forces. A conventional method of linearized Debye−Hückel theory (DH) only incorporates Coulombic interactions at a mean field level and ignores any finite-size effects. DH theory predicts that the excess chemical potential is strongly convex and decaying according to μex ∝ −√φ. DH deviates strongly from the results of the DHEMSA, in particular at higher densities, because it ignores the correlations that arise from hard-core repulsions (see Figure 1). Therefore, the

derivative of the free energy f int. The derivative

δ 2fint (ϕ) δϕ2

can be

measured in disordered (nonsegregated) polymer systems from scattering experiments. With the random phase approximation (RPA)15,31 one can obtain the structure factor S(q) for disordered systems of polymer with average density ⟨ϕ⟩ by using δ 2fint (ϕ) 1 1 = + S(q) S0(q) δϕ2

(15)

ϕ =⟨ϕ⟩

where the “ideal” structure factor S0(q) can be easily computed for a given polymeric system (see refs 15 and 31 and references therein) and

δ 2fint (ϕ) δϕ2

is given by eq 17. Note that for the classical

FH model, where f int = χϕAϕB,

δ 2fint (ϕ) δϕ2

= −2χAB. Experimen-

talists routinely interpret the interaction parameter

δ 2fint (ϕ) δϕ2

obtained from fitting scattering data to eq 15 as an effective density dependent χ-parameter

δ 2fint (ϕ) δϕ2

= −2χeff (ϕ). Because

S0(q) > 0 for any q, the spinodals can be computed from the condition: min (S0−1(q)) = 2χeff(ϕ). If χeff(ϕ) is negative, the system is always stable. In our LS-SCFT approximation χeff (ϕ) = χAB −

fq 2ϕA

− fq 2

∂μex (ϕ , Γ) ∂ϕ

ϕ = fq ϕA

(16)

One can see that there are two density dependent factors which can change χeff: the couterions translational entropy term

Figure 1. Excess chemical potential μex/Γ as a function of the ions volume fraction of ions species ϕq computed within DHEMSA approximation. Numbers in the legend indicates different values of the Γ. Dashed line shows Debye−Hückel type dependence μex ∝ −√φ.

−fq 2ϕA

,

which always reduces χeff and stabilize homogeneity and −fq 2


, which is positive at low fqϕA and negative ϕ = fq ϕA

at high fqϕA. This means that at low ionic densities electrostatic interactions increase χeff along with increase of Γ may initiate a phase segregation at lower χAB. At high fqϕA the term

DHEMSA predicts significantly smaller values of |μexc| than DH, in that regime. However, at intermediate densities, the DHEMSA predicts larger values of |μexc| because DH theory strongly underestimates the contribution of the Coulombic attractions and completely ignores cluster formation. In the paper by Zwanikken et al.1 the DHEMSA closure was used to calculate thermodynamic potentials and correlation functions of the restricted primitive model, which were then compared to the results of MD and MC simulations. The closure was found to be highly accurate for low to moderate values of Γ ≲ 8kBT over the entire density regime and accurate for higher values of Γ, outside the “forbidden region” where the system is supposed to phase separate.

−fq 2


is highly negative (see Figure 1) especially at ϕ = fq ϕA

high Γ, which may turn χeff to negative value and suppress the polymer phase segregation at higher values of χAB. Recent measurements of an effective χ-parameter from SAXS/WAXS for a PEO−P-[(STASIS)Li] system8 demonstrate the apparently negative χ-parameters for certain copolymer compositions. Such measurements indirectly confirm the qualitative validity of our LS-SCFT model, which also predicts an apparently negative χ-parameter in a certain ranges of the D


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Figure 2. An example of the Maxwell construction for an IB ionomer system with Γ = 24, χ = 0.4, and fq = 0.1 (a) Chemical potential μA for this {1} < 1. (c) The “shortcutted” convex parts of the system. (b) The “shortcutted” convex parts of the free energy f {0} int (ϕ) (see eq 20) for the binodal ϕ {1} = 1. free energy f {1} int (ϕ) for the binodal ϕ

composition of the ionomer system. In our model, such a range of an apparently negative χ-parameter corresponds to intervals where μA(ϕ) in Figure 2a is an increasing function of ϕ. For the actual numerical dependencies of μex(ϕq,Γ), we have used numerically precomputed values from the DHEMSA closure, which are shown in Figure 1 and the cubic spline interpolation. Phase Segregation in a Blend of the Infinitely Long Polymer Chains. The thermodynamic stability condition of a system is defined by its free energy density. For an ionomer blend at N → ∞ (IB system), the translational entropy of polymer chains is negligible compared to the effective excess free energy, f int(ϕ) = f int(ϕA = ϕ, ϕB = 1 − ϕ). This means that f int(ϕ) is the convex function of ϕ. If in some interval {ϕ{0}, ϕ{1}} the function f int(ϕ) is a concave function, it must be “shortcutted” by a common tangent line. A simple recipe for how to compute a common tangent line for f int(ϕ) is to use the chemical potential. μA (ϕ) =

6

(21)

where a flexible chains ansatz ρ0b = 1 is used. Let us present a generalization of the Helfand−Tagami (HT) approximation of the computed interfacial tension at a polymer−polymer interface for an arbitrary functional form of the density of the excess free energy of polymer−polymer interaction f int(ϕ) in a local homogeneity approximation. At the phase segregation point {ϕ{0}, ϕ{1}} of an incompressible two component polymer blend at N → ∞ (where translational entropy of polymer chains can be neglected), one can write the SCF free energy of the polymer−polymer interface for the HT approximation15,32 also known as the ground state dominance approximation:15 3

−3

γ=b

⎫ ⎧ ⎛ dϕ(x) ⎞2 b2 {x} ⎨ [ϕ(x)]⎬ dx ⎜ ⎟ + f int −∞ ⎩ 24ϕ(1 − ϕ) ⎝ dx ⎠ ⎭

∫

∞

⎪

⎪

⎪

⎪

(22)

dfint (ϕ)

where ϕ = ϕA = 1 − ϕB, ϕ-(x) → ϕ at x → − ∞, and ϕ-(x) → ϕ{1} at x → ∞. The variational minimization of eq 22 gives {0}

dϕ

(17)

The extrema of μA(ϕ) are the spinodals. The binodals {ϕ{0}, ϕ{1}} can be defined from the condition analogous to the Maxwell construction: μA (ϕ{0}) = μA (ϕ{1})

χAB

γHT = b−2

and

∫ϕ

{x} 24ϕ(1 − ϕ)f int [ϕ(x)] ⎛ dϕ ⎞ 2 ⎜ ⎟ = 2 ⎝ dx ⎠ b

Substitution of eq 23 into eq 22 gives

ϕ{1}

{0}

(23)

μA (ϕ) dϕ = 0

(18)

−3

γ = 2b

if ϕ{1} < 1 or if ϕ{1} A = 1 μA (ϕ{0}) = μA (1)

∫

{x} f int [ϕ(x)]

dx =

1 b2 6

∫ϕ

ϕ2

1

{x} f int [ϕ]

ϕ(1 − ϕ)

dϕ (24)

(19)

Obviously, the substitution of the Flory−Huggins model f {x} int [ϕ] → χϕ-(1 − ϕ) into eq 24 reproduce the HT result

Figure 2a shows an example of such a construct. The “shortcutted” convex parts of the free energy shown in Figures 2b,c and defined as

given by eq 21, γHT = b−2

χAB 6

.

Phase Diagrams of a Blend of Infinitely Long Chains. Here we discuss the phase diagrams for the IB system. Our numerical analysis shows that the IB system can be miscible at a given set of {χ; Γ; fq}, or it can have one {ϕ{0}, ϕ{1}} or two binodals {{ϕ{0,1}, ϕ{1,1}}, {ϕ{0,2}, ϕ{1,2}}}. If the average composition of the IB system f = ⟨ϕA⟩ is inside a binodal ϕ{0,x} ≤ f ≤ ϕ{1,x}, the IB system segregates into two phases ϕA = ϕ{0,x} and ϕA = ϕ{1,x}. If f is outside any pair of binodals, the IB system is miscible. The phase diagrams for the IB systems are shown in Figure 3. One can see that at the low electrostatic coupling value Γ = 0.24 the IB systems have one critical point at {χ > 0; ϕA = 1, ϕB = 0} for all fq values. This is in clear contrast with a nonionic

{x} f int (ϕ) = fint (ϕ) − fint (ϕ{0}) − (ϕ − ϕ{0})μA (ϕ{0})

(20)

are used to compute the interfacial tension γ between coexisting phases as shown below. Interfacial Tension of an A|B Interface. For the Flory− Huggins model of an incompressible polymer melt ϕB = 1 − ϕA, the density of the excess free energy in polymer−polymer interaction can be written as f FH = χϕAϕB = χϕA(1 − ϕA). For such a model at the limit N → ∞, Helfand and Tagami32 computed the interfacial tension of a AB interface γ from Fint , A int

where Aint is the interfacial area, they obtained E


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Figure 3. Phase diagram of ionic polymer blends in the N → ∞ limit for fq = 0.1 (a), fq = 0.05 (b), fq = 0.02 (c), and fq = 0.01 (d). Points below the coexistence lines correspond to the nonsegregated phase, while points above the coexistence lines correspond to the macrosegregated phase. Red, horizontally dash-dotted arrows demonstrate examples of coexisting of macrophase segregated systems. The legends are dashed lines for Γ = 0.24, solid lines for Γ = 9.23, and dash-dotted lines for Γ = 23.67. Yellow lines indicate “chimney” type segregation.4,11

increase miscibility (reduce χeff) of ionic and nonionic polymer which results in the increase of χcr with the increase of Γ at the right critical point. Note that the critical concentration ϕAcr at right critical point is N-dependent: ϕAcr → 1 if N → ∞. Another important feature of the IB system is the role of the Γ parameter in the mixing and demixing of components. At low ϕA, high Γ reduces the miscibility of components and can lead to phase segregation of the components even at χ = 0 (see Figure 3). At high ϕA and fq, high Γ has an opposite effect: it increases the miscibility of the ionic and nonionic components and causes an increase of the value of the χ parameter required for phase segregation. Figure 4 shows the computed values of the interfacial tension between coexisting phases corresponding to the phase diagram in Figure 3. The interfacial tension is normalized by the

system where the critical point is at {χN = 2-(χ → 0); ϕA = ϕB = (1/2)}. Such behavior of the critical point (a shift from

{ϕ

A

= ϕB =

1 2

} to {ϕ

A

→ 1, ϕB → 0}) and partial miscibility

of the IB system at small positive χ is caused mainly by the translational entropy of the counterions. At high values of the electrostatic coupling Γ > 9, a “chimney” regime at lower ϕA appears. Such a “chimney” has its origin in a deep minimum in μex(ϕq) at high Γ and low ϕq (see Figure 1). The interplay between “chimney” regimes as well as the presence of a critical point at {χcr > 0; ϕA = 1, ϕB = 0} for all fq cases (except fq = 0.01 where “chimney” occupies all the width of the phase diagram) causes formation of the eutectic type phase diagram with two critical points and one three-phase coexistence point. Such phase diagrams may resemble a cartoon of a house with a chimney drawn upside down. Here after we refer to as the “roof” the right part of the phase diagram near the right critical point. The physical origin of the left critical point is the same as for Coulombic phase separation of flexible polyelectrolytes33 at low charge densities and the position of this critical point is independent from the ionomer molecular mass N. At high values of Γ and low ϕA (“chimney” regime), the electrostatic interactions decrease miscibility (increase χeff) of ionic and nonionic polymer which results in the decrease of χcr with the increase of Γ, which is consistent with the experimental data on PS−PSS ionomers blends.9,10 The position of the right critical point (“roof” regime) at lower values of Γ is controlled by the interplay of the translational entropy of the counterions and the Flory χ-parameter. At high values of Γ and high ϕA, the electrostatic interactions

HT interfacial tension γHT = b−2

χAB 6

. One can see that the

interfacial tension between phases in the “chimney” and for the eutectic phases is much lower than γHT for nonionic phases at the same χ. The main reason for this is the low value of δϕ = ϕ2 − ϕ1 (see eq 24). Self-Consistent Field Free Energy of a Copolymer. The free energy of an incompressible system of AB ionomer copolymer chains in eq 425 with a local homogeneity approximation is a functional of the polymer density profile, ϕA(r) = 1 − ϕB(r), and for our system can be written as Nb3FΓ[ϕA (r), ϕB(r)] V dr F

= −TSideal[ϕA (r), ϕB(r)] +

∫ fint (ϕA(r), ϕB(r)) (25)


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Figure 4. Normalized interfacial tension between coexisting phases

γ(χ ) γHT

=

6 b2γ(χ ) χAB

for fq = 0.1 (a), fq = 0.05 (b), fq = 0.02 (c), and fq = 0.01 (d).

Dashed lines are for Γ = 0.24, solid lines for Γ = 9.23, and dash-dotted lines for Γ = 23.67. L lines with marker γ1 indicate regular binodals, and yellow lines with the marker γ2 indicate “chimney” type binodals.

ϕA (r) + ϕB(r) = 1;

V −1

Strong Stretching Theory for a Microsegregated Copolymer System. To explain the aforementioned anomalous behavior of ionic diblock copolymers, we have chosen a version of the strong stretching theory (SST) of microphase segregation of diblock block copolymers, based on a theory developed by Semenov.12 The technical details of the application of Semenov’s method to the variety of complex copolymers system can be found elsewhere.23,34 Such theory is proven to be asymptotically equivalent to SCFT at large N;13 however, it is computationally less demanding because it only requires solution of some algebraic equations (and sometimes even analytical solutions are available), while the SCFT requires a numerical solution of the multidimensional nonlinear partial differential equations.25 For our system, we must compute a phase diagram of an ionic diblock copolymer in a parametric space of at least five dimensions: Γ, χ, fq, f, and N. In this case, the computational efficiency is a necessity. In essence, the SST is a coarse-grained version of the SCFT based on certain assumptions: (1) The thickness of the interface between microsegregated phases is much smaller than the thicknesses (or size) of each microsegregated layer (or micella). (2) The molecular composition of the coexisting microsegregated phases are exactly the same as the molecular composition of macrosegregated phases in the blend of the of homopolymers which form the copolymer at N → ∞. (3) The interfacial tension γAB between microsegregated phases is exactly the same as the interfacial tension for macrosegregated phases of homopolymers. (4) The translational entropy of the copolymer chains is negligible in comparison with other terms in the free energy of the system. All these assumptions are proven quantitatively satisfied for nonionic diblock copolymers when χN → ∞.16 We have tested the validity of our SST version by computing the full SCFT morphologies for few points in the parameter space of Γ, χ, fq, f (see section

∫ ϕA(r) dr1 = f

where f is the composition of the system and Sideal (see eq 3) is the conformational entropy of an ideal chain in an ensemble of noninteracting chains in auxiliary fields {wA, wB}. This entropic term depends on the specificity of the system. For an arbitrary polymeric system (blend, copolymers, etc.), it can be computed by standard methods.25 The last term in eq 25 is the effective local contribution to the total free energy, computed in the LS-SCFT approximation,4 eq 14, and is given by fint (ϕA , ϕB) = N (χϕA ϕB + fq ϕA log ϕA + fex [fq ϕA , Γ]) (26)

The self-consistent field equations (eq 5) become wA(ϕA , ϕB) = N (χϕB + fq (log ϕA + μex [fq ϕA , Γ])) wB(ϕA , ϕB) = χNϕA

(27)

The numerical solution of eqs 25−27 for the ionomer systems have been performed in previous publications2−4 for onedimensional (1D) and two-dimensional (2D) systems and for a limited set of points in the five-dimensional parameter space of {χ; Γ; fq; f; N}. The 1D and 2D models are only capable only of reproducing the lamella and hexagonal cylinders morphologies of the diblock copolymers. Below, we give a few examples of the full numerical solutions of the SCFT eqs 25−27 for the three-dimensional (3D) case. Unfortunately, the numerical solution of the SCFT equations in 3D is extremely computationally demanding. The high dimensionality of the parametric space makes the analysis of the full phase diagram of the ionic diblock copolymers unrealistic. For this is necessary to develop a simpler coarse-grained version of the SCFT for ionic diblock copolymers, outlined in the next section. G


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Figure 5. Schematic of the different morphologies considered for our analysis and the notations used in the main text: (a) cylindrical or spherical morphologies with an A core; (b) lamellar morphologies; (c) cylindrical or spherical morphologies with the core of pure A component, and the corona compose of a mixture of B + AB components. I and II denote the different regions, which are considered individually in the main text; h1 and h2 denote the respective thicknesses of the regions I−II, and s = σ−1 is the interfacial area per chain (inverse phases can be obtained by switching A and B). There are five phases to analyze: direct and inverse spheres and cylinders and lamellae. Both the core and corona parts can be formed by pure A or pure B components or a mixture of A + AB or of B + AB components.

“Full SCFT Modeling Results”), and the results of SST and SCFT are in qualitative agreement. The main disadvantage of our SST version is that it is restricted only to microphases with lamellar, cylindrical, and spherical domains, omitting possible bicontinuous morphologies. We evaluate the possibility of the appearance of bicontinuous morphologies at the phase border between the 1D lamellar and 2D cylindrical microphases. Free Energy for the SST Approximation. Let us start from the analysis of a disordered state in a diblock copolymer. If f for the diblock is outside the binodal of the corresponding IB system, the diblock stays in disordered homogeneous phase. If f is inside the binodal 0 ≤ ϕ0 < f < ϕ1 ≤ 1 one can define an excess free energy of the disordered nonsegregated phase, Fdis, given by b3Fdis {x} = f int (f ) V

mpc =

f (1 − f )(ϕ1 − ϕ0) (f − ϕ0)(ϕ1 − f )

(29)

where 1 − mpc is the number of nonpassing chains in the “elemental cell” of SST (see Figure 5c). The free energy per one passage chain (“elemental cell” of SST) can be written as12,23,34 γ Fcell = FI + FII + (30) σ where s = σ−1 is the interfacial area per passage chain, σ is the grafting density of the passage chains, γ is the interfacial tension between microphases, and FI and FII are the elastic free energies for flat, convex, and concave brushes of the A and B parts of the passage chains (grafted onto a surface of radius R) as calculated by Semenov:12

(28)

where f {x} int (f) is defined in eq 20. If the free energy of the disordered nonsegregated state is larger than the free energy of the ordered segregated state, the system must form a microphasesegregated morphology with the minimal free energy from all possible morphologies. Next, we must highlight the main difference between the system under consideration and Semenov’s model.12 In Semenov’s model, the microphases are perfectly segregated {ϕA → 0; ϕB → 1} and {ϕA → 1; ϕB → 0}, and all diblock chains in the microsegregated system are the passage chains shown in Figures 5a,b. In our ionic system, phase segregation occurs for the phases {ϕA → ϕ0; ϕB → 1−ϕ0} and {ϕA → ϕ1; ϕB → 1 − ϕ1}, where 0 ≤ ϕ0 < f < ϕ1 ≤ 1. This is only possible if some chains are the passage chains between microphases, and some chains are dissolved in the microphases. The cartoon corresponding to such a morphology is shown in Figure 5c, where ϕ0 > 0 and ϕ1 = 1. To compute the SST free energy for a partially miscible copolymer, one must introduce the inverse fraction of the passage chains in a ionic diblock system:

FI,II =

⎧ π2 ⎪ Nbb2σ 2 ⎪8 ⎪ ⎛ ⎞ ⎪ 3 Rbσ log⎜ 2bNbσ+ + 1⎟ + ⎪4 ⎝ R ⎠ ⎪ ⎛ ⎪3 ⎪ Rbσ+⎜⎜1 ⎪2 ⎝ ⎨ ⎪ ⎛ 3Nbbσ+ ⎞−1/3⎞ ⎟ ⎪ −⎜ + 1⎟ ⎟ ⎝ R ⎠ ⎪ ⎠ ⎪ 2 ⎪π 2 2 ⎪ Nbb σ ⎪4 ⎪ 27 2 2 2 ⎪ π Nbb σ ⎩ 80

flat brush cylindrical convex brush

spherical convex brush

cylindrical concave brush spherical concave brush

(31)

Nb is the length of the piece of copolymer chain which belongs to the brush (Nb = f N or Nb = (1 − f)N). One must minimize the free energy Fcell with respect to σ with the constraints of conservation of the volumes of parts I and II of the cell, and taking into account that each passage chain also carries N(mpc − 1) extra segments of dissolved chains. This leads to the expressions for the free energies per cell of volume b3 H


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Figure 6. Δϕ = ϕ1 − ϕ0: (a) left “chimney” and (b) right “chimney”; S and Sinv are the spheres and inverse spheres phases; C and Cinv are the cylinders and inverse cylinders; L is the lamella phase.

⎛ γ ⎞2/3 = ⎜ ⎟ fx̂ (f , ϕ0 , ϕ1) b ⎝N⎠ V

where f x̂ ( f,ϕ 0 ,ϕ 1 ) stand for one of the five allowed morphologies of spheres, cylinders, lamella, inverse cylinders, and inverse spheres, and

3 Ford

(32)

5/3

3 ̂ (f , ϕ , ϕ ) = fspr 0 1

⎛ ⎛ (ϕ0 − f )⎜⎜f 2 ⎜ 1 + ⎝ ⎝

(

3π 2 40

)−(

̂ (f , ϕ , ϕ ) = flam 0 1

)

(

(

2/3

2(f (1 − f ))

ϕ0 − f ϕ0 − ϕ1

1/3

)

⎞1/3 ⎞ − 1⎟(2f − ϕ1)⎟⎟ ⎠ ⎠

f − ϕ0

)⎞⎠

1/3

⎟

(ϕ0 − ϕ1)

25/3(f (1 − f ))2/3 (ϕ0 − ϕ1)

(33)

One can see that the right-hand sides of eq 33 are dependent only on f, ϕ0, and ϕ1. We also know from the phase diagrams for the IB system that the phase boundaries are asymmetrically tapered in from the FH boundaries of {0, 1} to 0 < ϕ0 < ϕ1 ≤ 1 for the “chimney” or 0 ≤ ϕ0 < ϕ1 = 1 for the right part of the phase diagram in Figure 3. In the “chimney” case, ϕ0 is always very small and can be safely set to zero. Let us introduce the asymmetric partial miscibility parameter Δϕ = ϕ1 − ϕ0, where Δϕ ≈ ϕ1 if ϕ1 ≫ ϕ0, or Δϕ ≈ 1 − ϕ0 if ϕ1 ≈ 1. This parameter is Δϕ ≈ 1 for a regular FH system but can be Δϕ < 1 for ionic systems (see Figure 3). Figure 6 shows the distortion of the Semenov’s standard f sequence for spheres, cylinders, lamella, inverse cylinders, and inverse spheres as f grows from 0 to 1 for the regular diblock copolymers at Δϕ = 1. If the partial miscibility parameter Δϕ decreases, the inverse cylinders become the most preferable phase, even at values of f that correspond to the direct cylinders or even direct spheres in a nonionic diblock copolymer. This explains the fascinating prediction made earlier.3 The period of the microphase-segregated structure is computed by minimizing the free energy Fcell:

̂ (f , ϕ , ϕ ), fspr 0 1

̂ (1 − f , 1 − ϕ , 1 − ϕ ), fcyl 1 0 ̂ (1 − f , 1 − ϕ , 1 − ϕ )} fspr 1 0

ϕ1 − ϕ0

3π 2/3((f 2 (ϕ0 − f )3 + (f − 1)2 (f − ϕ1)3 ))1/3

̂ (f , ϕ , ϕ , ), feq̂ (f , ϕ0 , ϕ1) = min{flam 0 1

(34)

and compare the result with the free energy of the disordered state in eq 28 to calculate Ncrit

⎛ ⎟ + ϕ1⎜ ⎠ ⎝

⎛ π 2f 2 3(ϕ0 − f )⎜ 2 + 3(f − ϕ1)2 log ⎝

The inverse cylinder and sphere morphologies can be obtained by substitution f → 1 − f, ϕ1 → 1 − ϕ0, and ϕ0 → 1 − ϕ1. Now one can choose a morphology with the minimal free energy:

⎛ f ̂ (f , ϕ , ϕ ) ⎞3/2 0 1 ⎟ eq = γ⎜ {x} ⎜ f int (f ) ⎟⎠ ⎝

ϕ0 − ϕ1

1/3 ⎞

2(f (1 − f ))2/3 (ϕ0 − ϕ1)

̂ (f , ϕ , ϕ ) = fcyl 0 1

̂ (f , ϕ , ϕ , ), fcyl 0 1

ϕ0 − f

(35)

which is the maximum value of N where the disordered phase is stable. If N < Ncrit, the copolymer system is disordered. If N > Ncrit, the ionic copolymer microphase segregates to the phase with the minimal fx̂ ( f,ϕ0,ϕ1). I


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Figure 7. Phase diagrams in the {f, χ} plane for fq = 0.1. Γ values are shown on the plots; (a−c) are the morphology maps, where S and Sinv are the spheres and inverse spheres phases. C and Cinv are the cylinders and inverse cylinders. L lamella phase (d−f) are the maps for the log Ncrit. L lam ∝ Rg Lcyl Rg Lspr Rg

∝

∝

compressed lamella phase, it could be an indication of a possible bicontinuous morphology at the phase border between lamellar and cylindrical microphases. Phase Diagram of the Diblock Copolymers. We can now use eqs 32, 34, and 35 to analyze phase diagrams of ionic diblock copolymers. These diagrams are shown in Figures 7−9. White space in all diagrams corresponds to the disordered state. The order−disorder line follows the one in the IB blend phase diagram. One can see from Figures 7−9b,c that the inverse cylinder phases Cinv dominates within the “chimney” regions, even for f values which would correspond to the cylindrical C or spherical S morphologies in nonionic copolymers. In the right side of the phase diagram, inside the “roof” region, the dominant phase is the C phase, which forms at values of f corresponding to the Cinv and Sinv phases in nonionic copolymers. At high values of χ, the ionic system reverts to the behavior typical of nonionic polymers. The Ncrit maps are significantly asymmetric with respect to f values. Copolymers with a high fraction of the ionic block f segregate at lower N compared to those with a low f. Ncrit is significantly lower for the middle part of the chimney, corresponding to the Cinv phase. Full SCFT Modeling Results. To numerically solve the diffusion equation (eq 6), we use a pseudospectral operator splitting method.25 We use a 3D spatial box of size 12 × 12 × 12 Rg with a discretization δx = δy = δz = 3/16 and δs = 0.01. The SCFT eqs 26 and 27 are solved iteratively by using a simple mixing method.25 Because of the low interfacial tension γ between the microphases, and the low free energy differences between

f (1 − f )(ϕ1 − ϕ0)γ 1/3N1/6 3

(1 − f )f 3 (f − ϕ0)3 − (1 − f )3 f (ϕ1 − f )3

((1 − f )f )2/3 (ϕ1 − ϕ0)(f − ϕ0) γ 1/3N1/6

(

(ϕ1 − f ) 3 π 2f 2 + 6(f − ϕ0)2 log

ϕ1 − ϕ0 ϕ1 − f

)

((1 − f )f )2/3 3 ϕ1 − ϕ0 (f − ϕ0)2/3 γ 1/3N1/6 (ϕ1 − f ) 3 (40 + 3π 2)f 2 − 80fϕ0 −

40(f − ϕ0)2 3 ϕ1 − f 3 ϕ −ϕ 1 0

+ 40ϕ0 2

(36)

Equation 36 demonstrates that the period of the structures is proportional to γ1/3N1/6Rg, which is the common behavior for diblock copolymers. Bicontinuous Phases and Unstable Lamella. In ionic systems, the interfacial tension γ can be extremely small in the “chimney regime” (see Figure 4). This means that the relative size of the copolymer chains at the order−disorder transition point predicted from the SST,

Leq Rg

∝ γNcrit1/6 , can be very small.

If one of the values h1 or h2 becomes less than the Gaussian size of the subchain A or B segment (depending on which subchain belongs to the brush), then A and/or B chains are not stretched relative to their Gaussian size but become compressed, which makes the assumptions of the strong stretching theory inapplicable. Brushes or lamella phases of chains compressed relative to their Gaussian size are known to be unstable.34−36 The strong stretching theory is not applicable for such systems, but it can be used as an indicator of the lamellar phase instability and of the presence of unusual phases.34,36 If the SST predicts a J


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Figure 8. Phase diagrams in the {f, χ} plane for fq = 0.02. Γ values are shown on the plots; (a−c) are the morphology maps where S and Sinv are the spheres and inverse spheres phases. C and Cinv are the cylinders and inverse cylinders. L is the lamella phase. (d−f) are the maps for the log Ncrit.

N → ∞ (IB system) are the bare Flory−Huggins segment− segment interaction parameter χ, the dimensionless strength of the electrostatic attraction between oppositely ionic ions

morphologies ∝ γ2/3, the convergence of the simple mixing algorithm is poor. The iterations tend to converge to the various metastable local minima of the free energy. The existence of the multiple local minima of the SCFT free energy which are close to a global minimum, usually indicates that the thermal fluctuations would prevent this system from settling to the morphology corresponding to global SCFT minimum, and keep it in some less ordered state with defects. In the current work we do not try to find exact global equilibrium of the SCFT free energy, but instead use the SCFT to qualitatively study a variety of possible metastable morphologies of ionic copolymers. Figures 10−12 show snapshots of the late stages of the SCFT equations evolution. Very low values of the excess free energy f {x} int inside the “chimney” range of interaction parameters and very low interfacial tension between microphases result in pure convergence of the iteration scheme. The observed differences between free energy of the metastable states obtained from different initial morphologies are less than the computation error. Qualitatively, results of the morphologies obtained from the SCFT simulations are consistent with the predictions of the SST model: the composition of the microphases are close to the compositions of the coexisting macrophases for the corresponding IB blend and the phase of lager volume forms the matrix phase.

Γ=

|z1z 2| e 2 , 4πε0εr dkBT

and the volume fraction of the ions in the ionic

polymer fq. We have numerically computed the thermodynamic equation of state for the IB system for a wide set of parameters {χ; Γ; fq} and found that such behavior is qualitatively different from the behavior of a nonionic system. At high fraction of ionic polymer ϕA and low values of the electrostatic coupling, the ionic IB system has one critical point at {χcr > 0; ϕA = 1, ϕB = 0} at all fq values. This is in clear contrast with a nonionic system IB system where the critical point is at χcr N = 2;

N → ∞ : χcr = ϕA = ϕB =

2 →0 N

1 2

{

1

}

Such behavior of the critical point (a shift from ϕA = ϕB = 2 to {ϕA → 1, ϕB → 0}) and partial miscibility of the IB system at small positive χ is known to be due to the translational entropy of counterions preventing phase segregation of polyelectrolytes. At low fraction of ionic polymer ϕA and high values of the electrostatic coupling Γ > 9, one can see the appearance of the “chimney” regime at lower volume fraction ϕA of the ionic polymer A and the volume fraction of ions in the ionic polymer fq (see Figure 3). Such “chimney” regime have their origin in a deep minimum in the excess chemical potential of the ions μex

■

CONCLUSIONS We have shown that the set parameters which control phase segregation in ionic polymer blends at large degrees of polymerization K


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Figure 9. Phase diagrams in the {f, χ} plane for fq = 0.01. Γ values are shown on the plots. (a−c) are the morphology maps where S and Sinv are the spheres and inverse spheres phases. C and Cinv are the cylinders and inverse cylinders. L is the lamella phase. (d−f) are the maps for the log Ncrit.

Figure 11. 3D morphologies corresponding to the phase map in Figure 8 with Γ = 23.67 and fq = 0.02. For (a) f = 0.15, N = 8000, and χ = 0, a very defective inverse cylinder-like phase is obtained at zero χ parameter. This f corresponds to a nonionic polymer with red spheres inside the nonionic polymer. For (b) f = 0.15, N = 2000, and χ = 0.05, a lamellar phase is obtained. Both snapshots correspond to the lower part of the “chimney”; γ values in this area are extremely low.

Figure 10. 3D morphologies corresponding to the phase map in Figure 7 for Γ = 23.67, fq = 0.1, and N = 500. For (a) f = 0.5 cylinderlike phases for the symmetric ionic copolymer are obtained, instead of the lamellar microphase obtained in nonionic copolymers with f = 0.5. For (b) f = 0.8 inverse cylinders are obtained in the place where the inverse cylinders are observed in nonionic copolymers.

at high Γ and low ions volume fraction, predicted from the DHEMSA liquid state theory. The interplay between the “chimney” regime and the presence of a critical point at high fractions of ionic polymer causes formation of the eutectic type phase diagram with two critical points and one three-phase coexistence point. The physical origin of the critical point at low fraction of ionic polymer ϕA is the same as for Coulombic phase separation of flexible polyelectrolytes33 at low charge densities, and the position of this critical point is independent from the ionomer molecular mass N. At high values of Γ and low ϕA (“chimney” regime) the electrostatic interactions decrease miscibility: high negative value of fq 2


the increase of χeff (see eq 16) of ionic and nonionic polymers, which results in the decrease of χcr with the increase of Γ, which is consistent with the experimental data on PS−PSS ionomers blends.9,10 The critical point at high fraction of ionic polymer ϕA is controlled by the interplay of the translational entropy of the counterions, ionic correlations, and the Flory χ-parameter. At high values of Γ and high ϕA the electrostatic interactions increase miscibility of ionic and nonionic polymers because high value of fq 2

at low ϕA and high Γ causes ϕ = fq ϕA


at higher ϕA and high Γcauses a ϕ = fq ϕA

decrease down of χeff to the negative values (see eq 16), which L


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fractions of the ionic component and the inversion of the nonionic phase at high fractions of the ionic component. We have yet to find any direct experimental confirmations of our predictions, but the recent experimental observation of an apparent negative χ-parameter for the PEO−P-[(STASIS)Li] system8 from SAXS/WAXS of the disordered phase at certain PEO−P-[(STASIS)Li] compositions raises our confidence in the validity of the LS-SCFT model for the partial miscibility of ionomers and nonionic polymers at intermediate ionomer compositions, near the eutectic point. There are some indications of the existence of the inverted phases in the ionic triblock copolymers,37 but the exact identification of such phases is not yet clear.

Figure 12. 3D morphologies corresponding to the phase map in Figure 9 for Γ = 23.67. For (a) f = 0.15, fq = 0.02, N = 8000, and χ = 0. For (b) f = 0.15, fq = 0.02, N = 2000, and χ = 0.05. Both these snapshots correspond to the lower part of the “chimney”.

■

AUTHOR INFORMATION

Corresponding Authors

results in the increase of χcr with the increase of Γ at the critical point. The critical concentration is N-dependent, and ϕAcr → 1 if N → ∞. The interfacial tension between coexisting phases in the “chimney” and for the eutectic phases is much lower than the interfacial tension of nonionic phases at the same χ values. The main reason for this is the small value of δϕ = ϕ2 − ϕ1. In contrast, the interfacial tension between coexisting phases for ionic blends at high χ value where δϕ ≈ 1 is much higher than the interfacial tension of nonionic phases at the same χ. Another important feature of the ionic IB system is the role of Γ in the mixing and demixing of the components. At low fraction of the ionic polymer ϕA, the increase of Γ reduces the miscibility of the components and can lead to phase segregation of the components even at χ = 0 (Figure 3). At high fractions of ionic polymer ϕA and volume fraction of ions in the ionic polymer fq, the increase of Γ has an opposite effect: it increase the miscibility of the ionic and nonionic components and causes an increase in the χ parameter required for the phase segregation. As it was explained above, such behavior of χeff is preconditioned by the highly nonmonotonous behavior of μex as a function of density of ions (see eq 16 and Figure 1). The phase diagram for microphase segregation of diblock copolymers can be inferred from the phase diagram for the IB system. If an IB blend does not segregate for the set {χ, Γ, fq, ϕA}, the diblock copolymer with composition f = ϕA is disordered for the set {χ, Γ, fq, f = ϕA} at any value of N. If an IB blend segregates for the set {χ, Γ, fq, ϕA}, the diblock copolymer with composition f = ϕA is disordered for the set {χ, Γ, fq, f = ϕA} if the molecular mass N of the ionic diblock is less than the transitional molecular mass N < Ncrit(χ, Γ, fq, f). If N > Ncrit, the disordered phase segregates to an ordered phase with the morphology defined by the set {χ, Γ, fq, f}. The outcome of the strong stretching theory approximation is that for N ≫ Ncrit the morphology of the microsegregated diblock is independent of N and only depends on two parameters: the composition of the block copolymer f and the width of the binodal between partially miscible microphases Δϕ-[χ,Γ,fq,f ] (see Figure 6). If Δϕ decreases, the inverse cylinders become the most preferable phase even at the values of f corresponding to cylinders or even spheres in a nonionic diblock copolymer melt. This is the explanation of the most fascinating prediction made for these systems.3 The asymmetric shift of the phase boundaries and the partial miscibility effect can happen at both sides of the phase diagrams (see Figures 3 and 7−9). This means that one can expect two types of phase inversion in ionic diblock copolymers: the inversion of the ionic phase at low

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Monica Olvera de la Cruz: 0000-0002-9802-3627 Notes

The authors declare no competing financial interest.

■

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). H.K. is supported by the Center of Computation and Theory of Soft Materials at Northwestern University.

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REFERENCES

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Anomalous Phase Behavior of Ionic Polymer Blends and Ionic

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