Model Amphiphilic Polymer Conetworks in Water: Prediction of Their

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Model Amphiphilic Polymer Conetworks in Water: Prediction of Their Ability for Oil Solubilization Constantina K. Varnava and Costas S. Patrickios* Department of Chemistry, University of Cyprus, P.O. Box 20537, 1 University Avenue 2109 Aglantzia, 1678 Nicosia, Cyprus

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S Supporting Information *

ABSTRACT: In this work, we computationally explored the ability of water-swollen, model ionizable ABA triblock copolymer-based amphiphilic polymer conetworks (APCNs) to solubilize a water-immiscible organic solvent (oil), via Gibbs free energy minimization. This was done as a function of the conetwork hydrophobe (A-blocks) mol fraction and the degree of ionization of the hydrophilic B-blocks. Expectedly, highest oil solubilization capacities were calculated for the most hydrophobic and least ionized APCNs, which could absorb up to 6.4 times more oil than water and exhibited a lamellar morphology. Our results also included a phase diagram, which indicated transitions from spheres to cylinders, lamellae, and unimers in oil, as the hydrophobe content increased and the degree of ionization decreased. All of these transitions were accompanied by discontinuous changes in the degrees of swelling in the aqueous and oil nanophases, discontinuous changes in the asymmetry ratios (for the anisotropic morphologies), and discontinuous changes in the oil solubilization capacities. This is the first time that a dual discontinuous volume phase transition is reported within a polymer gel.

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soft contact lenses,20 drug delivery,21 phase transfer catalysis,22 and solid or gel polymer electrolytes.23,24 Despite the great interest in the experimental study of the properties and applications of APCNs, much less effort has been undertaken by the scientific community to investigate theoretically the behavior of these intriguing polymer systems, especially well-defined (“regular”) APCNs. The ability to predict via theory or simulations the properties of these materials may facilitate the design of new-generation APCNs with better attributes and superior performance in their applications. In a relatively recent important self-consistent field approach, Schmid25 observed that a regular APCN based on end-linked diblock copolymers (in the melt) undergoes with temperature a discontinuous transition from the disordered state to lamellae. This is in contrast to the noncross-linked analogue in which the transition is continuous. Our team used phenomenological Gibbs free energy descriptions and investigated end-linked ABA triblock copolymer-based APCNs in the melt over a broad polymer composition range and found that, in addition to lamellae, these systems can also organize into spheres and cylinders, depending on their comonomer composition and cross-linking functionality.26−28 Earlier theoretical work by our group29 showed that APCNs based on end-linked ABA triblock copolymers with a hydrophilic, ionizable mid-block and hydrophobic nonionic

INTRODUCTION This year marks the 50th anniversary of Dušek’s1,2 theoretical prediction of the discontinuous volume phase transition in polymer gels and the 40th anniversary of its experimental confirmation by Tanaka.3,4 Although this discontinuous transition may also take place within uncharged polymer gels,5 it always takes place when the gel is even slightly charged and when the solvent is not too good for the polymer. Solvent quality, temperature, pH, salt concentration, light irradiation, electric field, biochemical stimuli, and stress were subsequently shown to be able to trigger this volume phase transition in gels; furthermore, multiple discontinuous transitions were observed in certain polyampholytic gel systems.6 In all of the above experimental systems,3−6 the gel structure was poorly controlled, with the length of the chains between cross-linking points being broadly distributed and, in the case of copolymer gels, the two or more types of monomer repeating units being randomly distributed. In contrast, the polymeric hydrogel experimental system we developed 20 years ago possesses a much better structure.7−10 In particular, our polymer gels comprised polymers of well-defined length and composition, end-linked at cross-linking cores with a distribution of functionality.11 In enriching the behavior of our hydrogel experimental system, we also introduced hydrophobic segments, which, together with the hydrophilic ones, rendered the material amphiphilic.12−16 Amphiphilic polymer conetworks (APCNs)17−19 represent a rapidly emerging class of polymeric materials with important properties, the most important of which are swelling in both water and oil, microphase separation, and a variety of applications, including © 2019 American Chemical Society

Received: December 28, 2018 Accepted: February 13, 2019 Published: March 4, 2019 4721

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Figure 1. Unit cells for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers microphase-separated to spheres, infinite cylinders, and infinite lamellae. In each unit cell, the conformation of three copolymer chains, each toward one of the three Cartesian coordinates, is indicated.

one gives tables revealing the main Gibbs free energy components responsible for some of the volume phase transitions; the fourth one provides several extra figures to better illustrate the connection among the degrees of swelling, the asymmetry ratios, and the oil solubilization capacity; and the last one derives approximate analytical expressions to help explain some of the swelling trends presented in the Results and Discussion section.

end-blocks in water are in a disordered state under certain conditions but could microphase-separate into spheres upon changing solvent conditions or/and copolymer composition. This disorder-to-sphere transition was accompanied by discontinuous volume phase transitions. Subsequent work30 revealed a richer phase diagram for this APCN system. In particular, in addition to the disordered state and spheres, cylinders and lamellae also appeared at very low degrees of ionization and intermediate to high hydrophobe contents. Again, transitions between different structures were accompanied by large changes in the aqueous degrees of swelling. Large changes in the aqueous degrees of swelling of APCNs upon small changes of an external condition are known from experiments,31−33 whereas APCN organization in water into morphologies with long-range order has recently been recorded using scattering and microscopy techniques.34−38 In the present investigation, we further extend our previous modeling work on the phase behavior of APCNs based on endlinked ionizable amphiphilic ABA triblock copolymers by including, in addition to water, a water-immiscible organic solvent (“oil”). Thus, each of the nanophases of the presently explored system can be swollen with the respective solvent. The model calculates exactly the extent of this swelling in each nanophase, from which the oil solubilization capacity (SC) of the system is determined, a property highly relevant to applications such as water-remediation (removal of oil from contaminated water39−41) and the delivery of hydrophobic drugs.21 The model also provides a phase diagram and the dependence of the swelling degrees in each nanophase, both of which present discontinuous changes when a morphological transition takes place. To our knowledge, this is the first report on a dual discontinuous volume phase transition within the same polymer gel material. The rest of the article is organized as follows. First, the Geometry and Forces section describes the geometry of the considered morphologies, the peculiarity in the configuration of some of the polymer chains within these morphologies, and the components used to calculate the total Gibbs free energy. Next, the Theory section provides all mathematical expressions used in the article that led to the main findings, which are presented in the Results and Discussion section. The Conclusions section is followed by five appendices in the Supporting Information, the first two of which further analyze the anisotropic morphologies, lamellae and cylinders; the third

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GEOMETRY AND FORCES We consider ionizable amphiphilic polymer conetworks (APCNs) with an ideal structure, comprising ionizable amphiphilic ABA triblock copolymers of well-defined size and composition, end-linked at cores of exact functionality. This APCN system can exist in an ordered morphology or in a disordered state. Three ordered morphologies are considered, the spherical, the cylindrical, and the lamellar. The disordered state concerns the unimers either in oil or in water. Figure 1 illustrates the unit cells for spheres, cylinders, and lamellae, which are a cube, a regular hexagon parallelepiped, and a square parallelepiped, respectively. There is one cross-link at the center of each unit cell. Three copolymer chains (although a number of 20 is used in the calculations that follow) appear to emanate from each cross-link. Each of these three polymer chains is part of an ABA triblock copolymer, half of which belongs to a given unit cell (i.e., an AB diblock copolymer), whereas the other half belongs to a neighboring unit cell. Each of the three diblock copolymer chains illustrated points toward one of the three Cartesian coordinates. For the purposes of this investigation, the overall length (total degree of polymerization) of the constituting AB diblock copolymer chains was kept constant, but the relative lengths of the A- (hydrophobic, in red) and B- (hydrophilic ionizable, in blue) blocks were systematically varied so as to thoroughly explore the effect of copolymer composition on the properties of the system. Considering now more than one unit cells, we can recognize a peculiarity in the anisotropic morphologies, i.e., cylinders and lamellae, illustrated in Figure 2. Some chains act as bridges, interconnecting hydrophobic cores in different lines in cylinders (two out of three chains) and in different planes in lamellae (one out of three chains), whereas the rest of the chains are in the form of loops, connecting hydrophobic cores in the same line in cylinders (one out of three chains) and in the same plane in lamellae (two out of three chains). This 4722

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minimization. In particular, for the more symmetrical spherical morphology, exactly two independent variables were necessary, and these were chosen to be the polymer volume fractions in the aqueous hydrophilic and organic solvent hydrophobic nanophases, φshell and φcore A B , respectively. On the other hand, for the anisotropic morphologies, cylinders and lamellae, two further independent variables were required, and these were the two asymmetry ratios, one in each of the nanophases. Asymmetry Ratios for the Anisotropic Morphologies. In this approach, the total Gibbs free energy for each of the three ordered morphologies consists of six Gibbs free energy components: the Gibbs free energies of mixing in the core and in the shell, the electrostatic Gibbs free energy in the shell, the elastic Gibbs free energies in the core and in the shell, and the interfacial Gibbs free energy. These Gibbs free energy components are dependent on the polymer volume fractions in the two nanophases and, for the two anisotropic ordered morphologies, are also dependent on the asymmetry ratios of the nanophases (lamellar stretching and cylindrical compression), with these asymmetry ratios appearing only in the last three above-mentioned Gibbs free energy components. Their appearance in the expressions of only some of the Gibbs free energy components allowed for the derivation of closed-form expressions for these asymmetry ratios by appropriate Gibbs free energy minimization, as detailed in Appendices A and B for lamellae and cylinders, respectively. These expressions correlated the asymmetry ratios to the two polymer volume fractions, thereby keeping the number of independent variables for numerical minimization down to 2. The asymmetry ratio for the hydrophobic nanophase in lamellae is the ratio of the stretched to unstretched heights of the lamellar core, h/h0. The expression for h/h0 is given by ÄÅ ÉÑ1/3 1/3 ÅÅ Ñ χAB 1/2 Ñ 4ζ 2/3 ÅÅ 1 + ζ 2/3 ijj φAshell yzz + ÑÑ ÅÅ ÑÑ core z core 1/3 j η φB 6 (βφ0φB ) ÅÅ ÑÑ h k { = ÅÅÅ ÑÑÑ 8/3 4/3 i φ core y ÅÅ ÑÑ h0 η B ÅÅ ÑÑ j z jj shell zz 1+ ζ ÅÅ ÑÑ φ ÅÅÇ ÑÑÖ k A {

Figure 2. Bridges and loops in the anisotropic morphologies (cylindrical and lamellar). Bridges interconnect hydrophobic cores in different lines in cylinders and in different planes in lamellae, whereas loops connect hydrophobic cores in the same line in cylinders and in the same plane in lamellae.

distribution of loops and bridges is imposed by the initially cross-linked state of the conetwork assumed to be formed in a nonselective solvent and, consequently, in a disordered state. Although also encountered in infinite cylinders and infinite lamellae formed by linear block copolymers, the occurrence of loops is less frequent in the systems comprising linear block copolymers.42 Thus, the following calculations implicitly consider the occurrence of a large percentage of polymer chains in a loop configuration, 33% in cylinders and 67% in lamellae. The prevailing morphology for each conetwork composition and degree of ionization of the hydrophilic units is the one with the lowest value in minimized total Gibbs free energy. The total Gibbs free energy comprises the following components: the Gibbs free energies of mixing in the core and in the shell, the electrostatic Gibbs free energy in the shell, the elastic Gibbs free energies in the core and in the shell, and the interfacial Gibbs free energy.

()

( )

()

(1)

whereas the derivation is given in Appendix A. The asymmetry ratio for the hydrophilic nanophase in lamellae is the ratio of the stretched to unstretched heights of the lamellar shell, l/l0. The expression for l/l0 is given by

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THEORY The mathematical expressions for the components of the Gibbs free energies for the considered structures are listed in Table 1. These expressions were derived for one unit cell containing one cross-link at its center, as illustrated in Figure 1, whereas all symbols are defined in the footnote to Table 1. For the disordered state (unimers), absorbing only one type of solvent, either oil or water, only one independent variable was necessary for minimization, and this was chosen to be the (overall) polymer volume fraction, φ.29,30 In contrast, for the three ordered morphologies, absorbing both immiscible solvents, at least two independent variables were required for

iηy l = jjjj zzzz l0 kζ {

2/3

ÅÄÅ ÅÅ Å +

Å1 φBcore ÅÅÅÅ ÅÅ φAshell ÅÅÅÅ Å ÅÅ ÅÅÇ

2/3 i φshell y1/3

() ζ η

jj A zz j φ core z k B {

1+

4ζ 2/3 (βφ0φBcore)1/3

4/3 i φ core y8/3

() η ζ

+

jj B zz j φshell z k A {

χAB 1/2

(6)

ÑÉÑ1/3 ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÖ

(2)

and the derivation is also given in Appendix A. The asymmetry ratio for the hydrophobic nanophase in cylinders is the ratio of the compressed to uncompressed heights of the cylindrical core, H/h0. The expression for H/h0 is given by

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Table 1. Equations for the Gibbs Free Energy Components for the Various Structures Formed by the Ionizable Amphiphilic Polymer Conetworks in Equilibrium with Water and Oila equations for Gibbs free energy components for the various structures Gibbs free energy of mixing

ÄÅ ÉÑ ÅÅ ÑÑ ζ+η ΔG = β(1 − φ)ÅÅÅÅζχB − oil + ηχA − oil + ln(1 − φ)ÑÑÑÑ ÅÅÇ ÑÑÖ (T1.1) φ kT Core of Spheres, Cylinders, and Lamellae ÅÄÅ ÑÉÑ ÅÅ ÑÑ ΔG 1 = βζ(1 − φBcore)ÅÅÅÅχB − oil + core ln(1 − φBcore)ÑÑÑÑ ÅÅÇ ÑÑÖ (T1.3) φB kT Unimers in Oil

ΔG kT

ΔG kT

electrostatic Gibbs free

i φηι yz ΔG zz = βηι lnjjjj z kT kη + ζ {

Unimers in Water ÉÑ ÄÅ ÑÑ ÅÅ ζ+η ln(1 − φ)ÑÑÑÑ = β(1 − φ)ÅÅÅÅζχB − water + ηχA − water + ÑÑÖ (T1.2) ÅÅÇ φ Shell of Spheres, Cylinders, and Lamellae ÅÄÅ ÑÉÑ ÅÅ ÑÑ 1 = βη(1 − φAshell)ÅÅÅÅχA − water + shell ln(1 − φAshell)ÑÑÑÑ ÅÅ ÑÑ (T1.4) φA ÅÇ ÑÖ energy

Unimers

Shell of Spheres, Cylinders, and Lamellae

ΔG = βηι ln(φAshellι) kT

(T1.5)

ÑÉÑ ÅÄÅ 2/3 3 ÅÅi φ y 1 i φ yÑÑ ΔG = βÅÅÅÅjjjj 0 zzzz − 1 − lnjjjj 0 zzzzÑÑÑÑ kT 4 ÅÅÅk φ { 3 k φ {ÑÑÑ (T1.7) ÅÇ ÖÑ Core of Spheres ÄÅ ÉÑ 2/3 ÅÅ ÑÑ ΔG 3 ÅÅÅijjj φ0 yzzz 1 ijjj φ0 yzzzÑÑÑ = βÅÅÅjj core zz − 1 − lnjj core zzÑÑÑ kT 4 ÅÅÅjk φB z{ 3 jk φB z{ÑÑÑ ÅÇ ÑÖ (T1.8)

Core of Cylinders ÉÑ ÄÅ 2/3 | l Ñ Å oij φ yz ÅÅi y2 ij φ yzo o h0 ÑÑÑÑ βo ΔG ojj 0 zz ÅÅjj H zz jj 0 zzo j z 2 3 ln = m + − − } Ñ Å z j z j j z Ñ Å core z o ojj φ core zz ÅÅj h z j Ñ o z j kT 4o H φ ÑÑ o 0 o o k B {o ÑÖ ~ nk B { ÅÅÇk { Core of Lamellae ÉÑ ÄÅ 2/3 | l Ñ Å oij φ yz ÅÅi y2 ij φ yzo o h0 ÑÑÑÑ βo ΔG ojj 0 zz ÅÅjj h zz jj 0 zzo j z 2 3 ln = m + − − } Ñ Å z j z j j z Ñ ÅÅj h z core z core z o j j Ñ o o z j z j kT 4o h φ φ ÑÑ Å 0 o o o k B {o ÑÖ ~ nk B { ÅÅÇk {

(T1.6)

elastic Gibbs free energy Unimers

Shell of Spheres ÅÄÅ ÑÉÑ ÅÅi φ y2/3 Ñ ΔG 3 ÅÅÅjjj 0 zzz 1 jijj φ0 zyzzÑÑÑÑ = βÅÅjj shell zz − 1 − lnjj shell zzÑÑ kT 4 ÅÅÅj φA z 3 j φA zÑÑÑ {ÑÑÖ (T1.9) k { ÅÅÇk Shell of Cylinders ÉÑ 2/3Ä | li Å 2 Ñ Å o o jij φ0 zyzo jj φ0 zyz ÅÅÅijj H yzz o o l0 ÑÑÑÑ βo ΔG o Å zzo jj z j j z 2 3 ln = m + − − } Ñ Å z j j z Ñ Å z j z j oj φshell z ÅÅj l z shell zo Ñ j o kT 4o H ÑÑ o o ok A { ÅÅÇk 0 { k φA {o ÑÖ ~ n

(T1.10)

Shell of Lamellae ÉÑ 2/3Ä | li Å 2 ÑÑ Å o o Ñ ji φ zyo jj φ0 zyz ÅÅÅijj l yzz o o l βo ΔG o 0 Å zz Åjj zz + 2 ÑÑÑ − 3 − lnjjj 0 zzzo jj = m jj shell zz} jj shell zz ÅÅj z o o o kT 4o l ÑÑÑÑ φ o o ok φA { ÅÅÅÇk l0 { A {o k ÑÖ ~ n

(T1.12)

(T1.11)

(T1.13)

interfacial Gibbs free energy Unimers

ΔG = 0

Spheres

jj 3βζ yzz jj core zz jj φ zz k B { Lamellae 1/2 i

ΔG iχ y = (4π )1/3 jjj AB zzz kT k 6 {

(T1.14) Cylinders

(βζ )2/3 ji πχ yzz ΔG zz = 2 1/6 jjjj AB j 6φBcore zz kT φ0 { k

1/2

ij H yz jj zz jj zz k h0 {

ΔG iχ y = 2φ01/3(βζ )2/3 jjj AB zzz kT k 6 {

1/2

1/2

(T1.16)

2/3

(T1.15)

h0 hφBcore

(T1.17)

Symbols. Constants: φ0, polymer volume fraction in the unperturbed state, taken as that upon synthesis, set equal to 0.2; β, number of arms per cross-link, taken equal to 20 throughout the manuscript; χA−water, Flory−Huggins interaction parameter between water and the hydrophilic units, set equal to 0.45; χB−oil, Flory−Huggins interaction parameter between oil and the hydrophobic units, also set equal to 0.45; χA‑oil, Flory−Huggins interaction parameter between oil and the hydrophilic units in unimers, set equal to 2.0; χB−water, Flory−Huggins interaction parameter between water and the hydrophobic units in unimers, also set equal to 2.0; χAB, Flory−Huggins interaction parameter between the water-swollen hydrophilic nanophase and the oil-swollen hydrophobic nanophase, set equal to 2.0; Variables: ζ, number of monomer repeating units in the hydrophobic block, varied between 2.4 and 237.6; η, number of monomer repeating units in the hydrophilic block, varied between 237.6 and 2.4; ι, degree of ionization of the hydrophilic units, varied between 0.001 and 0.1; Independent variables: φ, polymer volume fraction for unimers; φshell A , polymer volume fraction in the hydrophilic nanophase of ordered morphologies; φcore B , polymer volume fraction in the hydrophobic nanophase of ordered morphologies; other symbols: ΔG, Gibbs free energy change; k, Boltzmann’s constant; T, absolute temperature; H , asymmetry ratio for the a

h0

hydrophobic nanophase in cylinders;

H , l0

asymmetry ratio for the hydrophilic nanophase in cylinders;

h , h0

asymmetry ratio for the hydrophobic

nanophase in lamellae; l , asymmetry ratio for the hydrophilic nanophase in lamellae. l0

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ÄÅ ÉÑ1/3 Ä É|1/2 ÅÅl ÑÑ 1/3 1/3Ñ Ä ÉÑÅÅÅi ÑÑo ÅÅo ÑÑ core Å y o i y o ÑÑÅÅjj ζ zz Ñ ÅÅo Ñ η o 1 1 j z 2/3 5/3 6φB Å Å Ñ ÅÅm − 1ÑÑÑ 1 + β φ0 πζχ ÅÅÅ core 2/3 + shell 2/3 ÑÑÑÅÅÅjj core2 zz + jjj shell2 zzz ÑÑÑ} o o ( ) ( ) ζφ ηφ Åo ÑÑ o Å Ñ Å Ñ AB φ φ B A 1/3 Å Ç ÖÅÅk B { k A { ÑÑÖo ÑÑ ij πχ yz ÅÅÅo Ç n ~ ÑÖ H 1 AB z Ç j = 2 5 1/9 jjj core zzz ÄÅ ÉÑ2/3 h0 ÅÅ 1 Ñ (β φ0 ) k 6φB { 1 ÑÑ ÅÅ core 2/3 + shell Ñ ÅÅÇ (ζφB ) (ηφA )2/3 Ñ ÑÖ

whereas the relevant derivation can be found in Appendix B. The asymmetry ratio for the hydrophilic nanophase in cylinders is the ratio of the compressed to uncompressed

(3)

heights of the hexagonal shell, H/l0. The expression for H/l0 is given by

ÄÅ ÉÑ1/3 ÄÅ É|1/2 ÅÅl ÑÑ 1/3 1/3Ñ Ä É Å Ñ ÅÅo ÑÑ o Å Ñ core Å Ñ ij η yz ÑÑo ÑÑÅÅijj ζ yzz ÅÅo ÑÑ o o 1 1 2/3 5/3 6φB Å Å j z Å Ñ Å Ñ ÅÅm ÑÑ + β φ + + − 1 1 } j z j z Å Ñ Å Ñ 2 2 core 2/3 shell 2/3 j j 0 o o Å Ñ core z shell z Å Ñ πζχ ( ) ( ) ζφ ηφ Å ÑÑ o o Å Ñ AB Ç φA { Ñ B A 1/3 Å ÖÅÅÅk φB { o ÑÑÖo k Å ÑÑ i y Å Ç ~ ÑÖ jj πζχAB zz ÅÇn H 1 z = 2 5 1/9 jjj Ä É core z 2/3 z Å Ñ ηφ l0 6 ÅÅ 1 ÑÑ (β φ0 ) k 1 B { ÅÅ core 2/3 + shell ÑÑ ÅÅÇ (ζφB ) (ηφA )2/3 Ñ ÑÖ

and the derivation can also be found in Appendix B. Gibbs Free Energy of Mixing. Equations T1.1 and TTT1.2 give the Gibbs free energy of mixing in oil and water, respectively, for the unimer-like network. These equations originate from the classical Flory−Huggins expression for the Gibbs free energy of mixing of solvent with a polymer network.43a In these equations, the enthalpies from the interaction of the hydrophobic and hydrophilic units with water, in the case of unimers in water (eq T1.2), or that of the hydrophobic and hydrophilic units with oil, in the case of unimers in oil (eq T1.1), are expressed through the first two terms in brackets, ζχB−solvent and ηχA−solvent, where “solvent” corresponds either to water (eq T1.2) or to oil (eq T1.1). For simplicity, our approach considered that all χ-parameters are independent of polymer concentration. The third and final term in brackets is the translational entropy of the solvent. The size of the polymer network is infinite, and, consequently, the term corresponding to the translational entropy of the polymer segments is omitted. The terms β(ζ + η) and β(ζ + η)(1 − φ) , in

(4)

this investigation, our simple description of the electrostatic Gibbs free energy ignored the part of the electrostatic energy due to the electrostatic repulsion among the charged polymer chains. For the ordered networks, the expression for the electrostatic Gibbs free energy (eq T1.6) is based on the same concept as in the unimer-like system described above, i.e., based on the counterion translational entropy, with the only difference that the relevant control volume is not that of the whole unit cell but only a part of it, namely, its hydrophilic shell. Elastic Gibbs Free Energy. Equation T1.7 is the expression for the elastic Gibbs free energy for the unimerlike network, in which the first term in brackets (the one raised to the 2/3 power) represents the contribution from the isotropic linear deformation, whereas the last term in brackets (the logarithmic one) is that due to the incompressibility condition.43b,c The constant 3β in this equation is derived from 4

φ

the original Flory coefficient for one elastic chain of

these two equations, represent the total number of monomer repeating units and the total number of solvent molecules within the unit cell, respectively. For the ordered networks, with the morphologies of spheres, cylinders, and lamellae, the Gibbs free energies of mixing in the core and in the shell are given by eqs T1.3 and T1.4, respectively, for all three morphologies. In these expressions, the first term in brackets is the enthalpic interaction between the hydrophobic units and oil in eq T1.3, χB−oil, and between the hydrophilic units and water in eq T1.4, χA−water. The second and final term in brackets is the solvent translational entropy. In both eqs T1.3 and T1.4, the enthalpic interactions between incompatible solvent−block pairs are omitted, as these interactions are described in the expressions for the interfacial Gibbs free energy (eqs T1.15−T1.17). Electrostatic Gibbs Free Energy. Equation T1.5 provides the electrostatic Gibbs free energy for the unimer-like network, originating from the translational entropy of the counterions to the conetwork charges,44 captured by the logarithmic term whose argument is exactly the volume fraction of these counterions. Due to the low degrees of ionization explored in

3 2

and the

β 2

effective number of chains of lying within one unit cell. For the ordered networks, the elastic Gibbs free energies in the core and in the shell are given by eqs T1.8−T1.13. Equations T1.8 and T1.9 are the result of applying eq T1.7 in the core and in the shell, respectively, for the isotropic spherical morphology. For the anisotropic morphologies, the elastic Gibbs free energy expressions in the core and in the shell, given in eqs T1.10−T1.13, are functions of the respective asymmetry ratios.43d The derivations of the expressions for the asymmetry ratios are given in Appendices A and B. These expressions constitute great improvements compared to the ones in our previous work30 in which the elastic energy from the looped chains was taken equal to zero. In contrast, here, the proper elastic energy values from the incompressibility condition were calculated. Interfacial Gibbs Free Energy. Due to lack of microphase separation, the interfacial Gibbs free energy for the unimer-like network in eq T1.14 is zero. In contrast, the interfacial Gibbs free energies for the ordered networks are nonzero and represent the interfacial interaction between the hydrophilic 4725

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nanophase (hydrophilic blocks plus water) and the hydrophobic nanophase (hydrophobic blocks plus oil). These energies, ΔGordered interfacial, are given in eqs T1.15−T1.17 and were calculated as the product of the interfacial tension between the hydrophilic and hydrophobic nanophases, γ, times the interfacial area in the unit cell, Sordered interfacial ordered ΔGinterfacial

=

ordered γSinterfacial

cylinders cylinders cylinders ΔGtotal = ΔGmixing _core + ΔGmixing_shell cylinders cylinders + ΔGelectrostatic _shell + ΔGelastic_core cylinders cylinders + ΔGelastic _shell + ΔGinterfacial

For the lamellar morphology, the total Gibbs free energy, ΔGlamellae total , is

(5)

lamellae lamellae lamellae ΔGtotal = ΔGmixing _core + ΔGmixing_shell

The interfacial tension, γ, is related to χAB through the following equation45 kT ij χAB yz jj zz A2 k 6 {

lamellae lamellae + ΔGelectrostatic _shell + ΔGelastic_core

1/2

γ=

lamellae lamellae + ΔGelastic _shell + ΔGinterfacial

(6)

DS =

Subsequently, R can be related to the composition (ζ) and swelling (φcore B ) characteristics of the hydrophobic nanophase through volume balance. Volume balance dictates that the volume of the spherical hydrophobic nanophase, Vspheres core , can be calculated both from its radius R and also from the volume occupied by its (dry) polymer content, βζA3, in combination with its polymer volume fraction, φcore B spheres V core

DSshell =

i 3βζ zy jj core zzz jφ z k B {

1/3 j j

= (4π )

(15)

volume of water + volume of (dry) hydrophilic blocks volume of (dry) hydrophilic blocks (16)

Similarly, the degree of swelling in oil for the hydrophobic nanophase, DScore, is DScore =

(8)

volume of oil + volume of (dry) hydrophobic blocks volume of (dry) hydrophobic blocks (17)

For the calculation of the global degree of swelling, DSglobal, the following definition is used

Equation 8 can be used to obtain an expression for R, which can be substituted into eq 7, giving spheres Sinterfacial

volume of solvent(s) + volume of (dry) polymer volume of (dry) polymer

Consequently, the degree of swelling in water for the hydrophilic nanophase, DSshell, is

(7)

βζ 4π 3 = R = core A3 φB 3

(14)

Equilibrium Degree of Swelling (DS). In this investigation, we adopt the following definition for the (volumetric) degree of swelling, DS

where A2 is the area corresponding to one monomer repeating unit. For the spherical morphology, the interfacial area, Sspheres interfacial, can be first related to the radius of the spherical oil-swollen hydrophobic nanophase, R spheres Sinterfacial = 4πR2

(13)

DSglobal =

2/3 2

Α

(volumes of water and oil + volumes of (dry) hydrophilic and (9)

hydrophobic blocks)/(volumes of (dry) hydrophilic

Substituting eqs 6 and 9 into eq 5 leads to spheres 1/2 ΔGinterfacial i χ y ij 3βζ yz = (4π )1/3 jjj AB zzz jjjj core zzzz kT k 6 { k φB {

and hydrophobic blocks)

2/3

As already noted, the present model requires (at least) two independent variables for minimization for the three ordered morphologies, and these variables were chosen to be the polymer volume fractions in the hydrophilic and hydrophobic nanophases. Thus, the expression for the total Gibbs free energy for each morphology was minimized with respect to both φshell and φcore A B . In the case of unimers, either in water or in oil, only one minimization variable was required, and this was the polymer volume fraction, φ. The total Gibbs free energy versus φ curve for unimers in water usually presented two minima.1,6,46 The low-φ minimum corresponded to swollen unimers in water, whereas the higher-φ minimum corresponded to shrunk unimers in water. Subsequently, the values for the minimized total Gibbs free energies for the six (or five, unimers in water may be either in a swollen or in a shrunk state) possible conetwork structures, swollen unimers and shrunk unimers in water and spheres, cylinders, lamellae, and unimers in oil, were compared to each other, the one possessing the lowest value was determined, and the corresponding structure was identified as the prevailing one. When the prevailing structure was an ordered morphology, the two polymer volume fractions corresponding to its total Gibbs free energy minimum were noted and used to calculate

(10)

The calculations for the interfacial areas of the lamellar and cylinders cylindrical oil-swollen cores, Slamellae interfacial and Sinterfacial, are more complicated, and the derivations are given in Appendices A and B, respectively. Total Gibbs Free Energy. The total Gibbs free energies for the conetworks are equal to the sum of all their Gibbs free energy components. In particular, for unimers (either in oil or water), the total Gibbs free energy, ΔGunimers total , is given by the expression unimers unimers unimers unimers ΔGtotal = ΔGmixing + ΔGelectrostatic + ΔGelastic

(11)

For the spherical morphology, the total Gibbs free energy, ΔGspheres total , is spheres spheres spheres spheres ΔGtotal = ΔGmixing _core + ΔGmixing_shell + ΔGelectrostatic_shell spheres spheres spheres + ΔGelastic _core + ΔGelastic_shell + ΔGinterfacial

(18)

(12)

For the cylindrical morphology, the total Gibbs free energy, ΔGcylinders , is given by the expression total 4726

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Assumptions Made and Range of Variables Investigated. The system was considered to have an ideal structure, comprising ABA triblock copolymers of well-defined size and composition, end-linked at cores of exact functionality. The A end-blocks were assumed to be hydrophobic, whereas the B mid-block, hydrophilic and ionizable, as shown in Figure 1. This assignment of hydrophilic and hydrophobic blocks would lead to self-assembled and shrunk end-blocks next to the crosslinks and swollen and expanded mid-blocks, whereas the reverse assignment would have resulted in a more frustrated microphase-separated structure, with self-assembled and shrunk mid-blocks and swollen and expanded end-blocks next to the cross-links. Furthermore, the counterions were assumed to be restricted only within the hydrophilic nanophase, being excluded from the hydrophobic one where they would be insoluble. Constant values were assigned to the quantities φ0, β, χA−water, χB−oil, χA−oil, χB−water, and χAB throughout this investigation, with, as mentioned before, all χ-parameters being treated as polymer-concentration-independent. The choice of these values is justified as follows. φ0, the polymer volume fraction in the unperturbed state, was taken equal to 0.2, the same as the volume fraction upon synthesis in our experimental system.12 β, the number of arms per cross-link, was fixed to 20, close to the values determined by static light scattering for star polymers prepared by analogous methods.10 The Flory−Huggins interaction parameters between water and the hydrophilic units and between oil and the hydrophobic units, χA−water and χB−oil, respectively, were set equal to 0.45, consistent with the experimentally determined χ values for the interaction between water and 2(dimethylaminoethyl)methacrylate31 and between toluene and polystyrene.48,49 The Flory−Huggins interaction parameters between the incompatible solvent−block pairs, χA−oil, χB−water, and χAB, were assigned the value of 2.0, typical for the value for the propylene oxide−water pair.50,51 In this investigation, we systematically varied the values of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in the conetworks at constant (ζ + η = 240) overall degree of polymerization of a chain within the unit cell. The fractional degrees of ionization were varied from 0.001 to 0.10 (0.1−10%), whereas the mol fraction of the hydrophobic units was varied in the range from 0.01 to 0.99 (1−99%).

the equilibrium degrees of swelling in the two nanophases as their inverse 1

DSshell =

φA shell

(19)

and DScore =

1 φBcore

(20)

Then, the global degree of swelling, DSglobal, was also calculated from the polymer volume fractions in the hydrophilic and hydrophobic nanophases and conetwork composition DSglobal =

ζ φBcore

+

η φA shell

ζ+η

=

η ζ DScore + DSshell ζ+η ζ+η (21)

When the prevailing structure was the unimer-like one, the single equilibrium DS was simply calculated as the inverse value of the polymer volume fraction, φ, at the total Gibbs free energy minimum DSunimer =

1 φ

(22)

Solubilization Capacity. The ability of APCNs to solubilize organic solvents was quantified through the solubilization capacity, SC, defined as SC =

volume of oil solubilized volume of water solubilized

(23)

It can be easily shown that the SC can be related to the polymer volume fractions in the hydrophilic and hydrophobic nanophases and the copolymer composition as SC =

shell 1 − φBcore ζ φA η φBcore 1 − φA shell

(24)

SC has a meaning only for the three ordered morphologies, which take on both water and oil. In contrast, the SC cannot be calculated for the unimer-like structures because these structures lack one of the two liquids, the oil or the water. Minimization Procedure. The Newton−Raphson method47 was employed to numerically minimize the total Gibbs free energies. This necessitated the prior analytical calculation of the first and second derivatives of the total Gibbs free energies with respect to φ in the case of unimers and with respect to φshell and φcore for the three ordered morphologies. A B For the anisotropic morphologies, cylinders and lamellae, the minimization problem has four unknowns, the asymmetry ratios in the hydrophilic and hydrophobic nanophases and the polymer volume fractions in these two nanophases. However, as mentioned before, this problem was simplified by first analytically determining the asymmetry ratios as a function of the polymer volume fractions, φshell and φcore A B , via a rather simple analytical minimization procedure. This was possible because the asymmetry ratios appear only in the elastic Gibbs free energies in the core and in the shell and the interfacial Gibbs free energy components (Appendices A and B). The obtained expressions for the four asymmetry ratios are given in eqs 1−4. Thus, the problem was simplified to one with only two, rather than four, unknowns and could be solved as that for the spherical morphology.

■

RESULTS AND DISCUSSION This section presents all results of this study, which include a phase diagram, the equilibrium degrees of swelling in water and oil, the overall (global) equilibrium degree of swelling, the asymmetry ratios in water and oil for the anisotropic morphologies (i.e., cylinders and lamellae), and the oil solubilization capacity, all as a function of the conetwork composition and the degree of ionization of the hydrophilic units. The degrees of swelling, the asymmetry ratios, and the oil solubilization capacity are given in three-dimensional (3-D) plots, whose base (x−y plane) is the phase diagram, with axes the conetwork composition (x axis), and the hydrophile ionization degree (y axis), and also in the two possible twodimensional (2-D) projections. Phase Diagram. Figure 3 is the phase diagram for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers in oil and water as a function of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units. This phase diagram is in 4727

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and their high degree of ionization. In contrast, when the mol fraction of the hydrophobic units was high and the degree of ionization of the hydrophilic units was low, lamellae and unimers in oil were the prevailing structures in the conetworks. One may observe from Figure 3 that as the mol fraction of the hydrophobic units spanned the range from 0.01 to 0.99, three (at two constant values of the degree of ionization, 0.001 and 0.01) or four (nine cases, 0.02−0.10 degrees of ionization) different structures were encountered at constant degree of ionization. In contrast, as the degree of ionization increased from 0.001 to 0.10, one (at six constant values of the mol fraction of the hydrophobic units, 0.01, 0.10, 0.70, 0.80, 0.90, and 0.99), two (four cases, 0.30, 0.40, 0.50, and 0.60 hydrophobe mol fractions), or three (one case, 0.20 hydrophobe mol fraction) different structures were encountered at constant mol fraction of the hydrophobic units. Thus, it appears that, for the selected range of values for the degree of ionization and the mol fraction of the hydrophobic units, the latter quantity affected more strongly the prevailing structures. Gibbs Free Energy Components at the Energy Minima. Figures 4 and 5 provide the minimized total Gibbs free energies for the various structures, from which the lowest minimum and the prevailing structure were obtained. Figure 4 focuses on the effect of the conetwork composition at constant degree of ionization, whereas Figure 5 focuses on the effect of the degree of ionization at constant conetwork composition. The particular values at which the degree of ionization (0.02) in Figure 4 and the hydrophobe mol fraction (0.20) in Figure 5 were kept constant were chosen such that a maximum number of equilibrium structures be encountered as conetwork composition and degree of ionization were varied (see the phase diagram in Figure 3). Figure 4 illustrates the effect of the mol fraction of the hydrophobic units, at a constant degree of ionization of the hydrophilic units of 0.02, on the minimized total Gibbs free energy for each of the six structures considered for the conetworks. Figure 4a includes the whole range of values of the hydrophobe mol fraction, whereas Figure 4b focuses on the lower range of values where most of the transitions took place. A first inspection of Figure 4a reveals that lamellae presented

Figure 3. Phase diagram for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers in oil and water as a function of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units. The values used for the parameters were φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA−oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240.

qualitative agreement with previous works on APCNs in the melt25−28 or swollen only in water29,30 but with important differences arising from the inclusion of oil. Four of the five structures considered in this study appeared in the present phase diagram: spheres, cylinders, lamellae, and unimers in oil. The structure not encountered in the phase diagram was that of unimers in water, as its minimum total Gibbs free energy was always somewhat greater than that for spheres (Figure 4a,b). The main observations from the phase diagram in Figure 3 were the following. When the degree of ionization of the hydrophilic units was high and the mol fraction of the hydrophobic units was low, spheres and cylinders were the dominant structures in the conetworks. This may be attributed to the fact that these ordered morphologies could more efficiently accommodate the high curvature52−54 induced by their highly swollen hydrophilic nanophases (mid-blocks B are the hydrophilic ones), arising from the high conetwork content in hydrophilic units

Figure 4. (a) Effect of the mol fraction of the hydrophobic units on the minimized total Gibbs free energy per chain, in kT units, for the six possible structures considered for ionizable APCNs with a 0.02 degree of ionization. (b) Magnification of the low-hydrophobicity region of (a). The values used for the parameters were the same as those used in Figure 3, i.e., φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA‑oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240. 4728

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Figure 5. (a) Effect of the degree of ionization of the hydrophilic units on the minimized total Gibbs free energy per chain, in kT units, for each of the six possible structures considered for ionizable APCNs with a 0.20 mol fraction of the hydrophobic units. (b) Magnification of the low degree of ionization region of (a). The values used for the parameters were also the same as those used in Figure 3, i.e., φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA‑oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240.

Figure 6. Degrees of swelling of the hydrophilic nanophase in water and of the hydrophobic nanophase in oil for ionizable APCNs based on endlinked ionizable amphiphilic ABA triblock copolymers in an oil−water mixture, as a function of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units. Dependence of the degree of swelling of the hydrophilic nanophase in water (a) on both the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in a 3-D plot, (b) on the mol fraction of the hydrophobic units for different degrees of ionization (0.001−0.10 = 0.1−10%) of the hydrophilic units in a 2-D plot, and (c) on the degree of ionization of the hydrophilic units for different mol fractions of the hydrophobic units (0.01−0.99 = 1−99%) in a 2-D plot. Dependence of the degree of swelling of the hydrophobic nanophase in oil (d) on both the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in a 3D plot, (e) on the mol fraction of the hydrophobic units for different degrees of ionization (0.001−0.10 = 0.1−10%) of the hydrophilic units in a 2D plot, and (f) on the degree of ionization of the hydrophilic units for different mol fractions of the hydrophobic units (0.01−0.99 = 1−99%) in a 2-D plot. The values of the parameters used were the same as those used in Figure 3, and, in particular, φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA‑oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240.

the lowest minimized total Gibbs free energy for a large compositional range, corresponding to a mol fraction of the

hydrophobic units from 0.30 to 0.90. At the composition extremes, spheres and unimers in oil displayed the lowest 4729

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polymeric components are parts (blocks, segments) of the same conetwork, which leads to the simultaneous transitions. Regarding the latter, these nanophases can still swell relatively independently of each other as the solvents they absorb are mutually immiscible, permitting the occurrence of one volume phase transition in each nanophase. Another key observation is the significant difference (∼2 orders of magnitude) in the degrees of swelling in water and oil for the two polymeric nanophases. In particular, the degrees of swelling in water for the hydrophilic nanophase were very high, especially where the conditions greatly favor swelling in water, i.e., high degrees of ionization and low hydrophobe mol fractions, leading to a maximum calculated value for the degree of swelling in water of about 1700, whereas the degrees of swelling in oil for the hydrophobic nanophase were relatively low, with a maximum calculated value of about 20. This large difference was a consequence of the presence of the charged groups in the hydrophilic nanophase, building up an osmotic pressure through the translational entropy of the counterions to these charges. A third observation is that these 3-D plots interestingly displayed a complementarity in shape, i.e., when the degrees of swelling in water for the hydrophilic nanophase were high, the degrees of swelling in oil for the hydrophobic nanophase were low, whereas when the degrees of swelling in oil for the hydrophobic nanophase were relatively high, the degrees of swelling in water for the hydrophilic nanophase were low. Moreover, the 3-D plots in Figure 6a,d indicated that the degrees of swelling in water and oil for the two polymeric nanophases were highly dependent on structure. In particular, the degrees of swelling in water for the hydrophilic nanophase decreased when the most energetically favored structure changed from spherical to cylindrical, lamellar, and unimers in oil, whereas for the same sequence of structure change, the degrees of swelling in oil for the hydrophobic nanophase increased. Figure 6b,e illustrates the effect of the mol fraction of the hydrophobic units on the degrees of swelling in water and oil for the hydrophilic and hydrophobic nanophases, respectively, for different degrees of ionization of the hydrophilic units in the conetworks. The figures show simultaneous discontinuous changes in the degrees of swelling in water and oil arising from the transitions between different structures (morphologies). For example, at a degree of ionization of the hydrophilic units of 0.02 (2%), an increase in the mol fraction of the hydrophobic units in the conetworks from 0.01 to 0.99 caused three discontinuous changes in the degrees of swelling in water and oil, corresponding to transitions from spheres to cylinders (at a 0.101 mol fraction of the hydrophobic units), from cylinders to lamellae (at a 0.22 mol fraction of the hydrophobic units), and from lamellae to unimers in oil (at a 0.955 mol fraction of the hydrophobic units), consistent with Figures 3 and 4a. For the spherical and cylindrical morphologies, at a given degree of ionization, an increase in the mol fraction of the hydrophobic units in the conetworks from 0.01 to 0.99 expectedly led to a systematic reduction in the degrees of swelling in water for the hydrophilic nanophase (Figure 6b) and a systematic increase in the degrees of swelling in oil for the hydrophobic nanophase (Figure 6e). In contrast, in the regions where lamellae prevailed, a more complex swelling behavior was observed. First, it is worth mentioning that the swelling in the two lamellar nanophases was restricted between 4 and 8. Furthermore, the (small) variation of the degrees of

minimized total Gibbs free energies at low and high hydrophobe contents, respectively. The highest minimized total Gibbs free energies were exhibited by unimers in oil and shrunk unimers in water at low and high hydrophobe contents, respectively. Examining now Figure 4b, which is a magnification of the low-hydrophobicity region (hydrophobe mol fractions of 0.01−0.30) in Figure 4a where most transitions took place, the lowest minimized total Gibbs free energies at 0.01−0.10, 0.20, and 0.30 hydrophobe mol fractions corresponded to spheres, cylinders, and lamellae, respectively. Furthermore, it is noteworthy that the difference between the lowest minimized total Gibbs free energy and the next one was very small. Figure 5 shows the effect of the degree of ionization of the hydrophilic units, at a constant mol fraction of the hydrophobic units of 0.20, on the minimized total Gibbs free energy for each of the six possible structures considered for the conetworks. Figure 5a includes the whole range of values of the degrees of ionization, whereas Figure 5b focuses on the lower range of values where the transitions between different structures took place. A first observation from Figure 5a is that spheres and cylinders had similar values of their minimized total Gibbs free energies for the whole range of values of the degrees of ionization considered. This observation was confirmed by examining Figure 5b too. However, a closer look, especially at Figure 5b, reveals that cylinders prevailed for degrees of ionization from 0.02 to 0.04 and spheres were the dominant structure for degrees of ionization of 0.05 and higher. At the lowest degrees of ionization (0.001−0.01), lamellae displayed the lowest total minimized Gibbs free energies. The highest minimized total Gibbs free energies were exhibited by unimers in oil (Figure 5a). Appendix C in the Supporting Information analyzes, with the aid of tables, the Gibbs free energy components whose change is mainly responsible for several of the transitions shown in Figures 4 and 5. Degrees of Swelling in Water and Oil. Figure 6a−f shows the degrees of swelling in water and oil for the hydrophilic and hydrophobic polymeric nanophases, respectively, for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers, as a function of the mol fraction of the hydrophobic units and the degree of ionization of the hydrophilic units. Figure 6a,d shows 3-D histograms for the swelling degrees in water and oil, respectively, in which the degrees of swelling are presented as bars of different colors according to the structure, whereas the rest of the graphs in the figure are 2-D projections of these two 3-D plots. We wish to reiterate here that the x−y base of Figure 6a,d is the phase diagram of Figure 3. Furthermore, we note that simplified versions of the degrees of swelling in water and oil, together with the asymmetry ratios and oil solubilization capacities, are plotted against conetwork composition and degree of ionization in Appendix D in the Supporting Information. A significant observation from the 3-D plots in Figure 6a,d is the discontinuous volume phase transitions, taking place simultaneously in the aqueous and organic nanophases at the points where a structural change was observed. This is the first time this simultaneous (dual) discontinuous volume phase transition is reported, and it is due to the fact that the two nanophases are inter-related but, at the same time, they keep a fair degree of independence from each other. Regarding the former, the two nanophases depend on each other as their 4730

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Figure 7. Global degrees of swelling for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers in an oil−water mixture, as a function of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units. Dependence of the global degree of swelling (a) on both the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in a 3-D plot, (b) on the mol fraction of the hydrophobic units for different degrees of ionization (0.001−0.10 = 0.1−10%) of the hydrophilic units in a 2-D plot, and (c) on the degree of ionization of the hydrophilic units for different mol fractions of the hydrophobic units (0.01−0.99 = 1−99%) in a 2-D plot. The values of the parameters used were the same as those used in Figure 3, and, in particular, φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA‑oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240.

Figure 6c,f displays the effect of the degree of ionization of the hydrophilic units on the degrees of swelling in water and oil for the hydrophilic and hydrophobic nanophases, respectively, for different mol fractions of the hydrophobic units in the conetworks. The figure showed simultaneous discontinuous increases in the degrees of swelling in water and reductions in the degrees of swelling in oil, which were again due to transitions between different structures. For example, at a 0.20 mol fraction (20%) of the hydrophobic units, an increase in the degree of ionization in the conetworks from 0.001 to 0.100 caused two discontinuous changes in the degrees of swelling in water and oil, corresponding to transitions from lamellae to cylinders (at a degree of ionization of 0.019) and from cylinders to spheres (at a degree of ionization of 0.048), consistent with Figures 3 and 5a. For most mol fractions of the hydrophobic units, an increase in the degree of ionization in the conetworks from 0.001 to 0.100 led to a systematic increase in the degrees of swelling in water for the hydrophilic nanophase (Figure 6c), except for the 0.99 mol fraction of the hydrophobic units, where unimers in oil exclusively prevailed, a structure that did not swell in water at all. This increase in the degrees of swelling in water was to be expected as a higher degree of ionization of the hydrophilic units led to more charged groups in the hydrophilic nanophase, which favored aqueous swelling. Despite the absence of such charged groups in the hydrophobic nanophase, an increase in the degree of ionization in the conetworks from 0.001 to 0.100 caused a slight increase in the degrees of swelling in oil in the areas where cylinders, lamellae, and unimers in oil prevailed (Figure 6f). This slight increase was attributed to the asymmetry ratios of the anisotropic morphologies and the disordered state of unimers in oil. The degrees of swelling in oil for the hydrophobic nanophase of the more symmetrical spherical morphology were totally independent of the degree of ionization. All above trends in swelling are explained in more detail in Appendices D and E in the Supporting Information. Careful examination of Figure 6c led to the observation of linear increases of the degrees of swelling in water for the hydrophilic nanophase with the degree of ionization for all three ordered morphologies, above a certain degree of

swelling in both lamellar nanophases exhibited, in some cases, unexpected dependence on hydrophobe content. In particular, for degrees of ionization from 0.001 to 0.03, the degrees of swelling in water for the hydrophilic nanophase unexpectedly slightly increased with the mol fraction of the hydrophobic units, whereas for degrees of ionization of 0.04 and higher, the degrees of swelling in water for the hydrophilic nanophase presented a shallow minimum as the content in hydrophobic units increases. On the other hand, the degrees of swelling in oil for the hydrophobic nanophase unexpectedly slightly decreased with the mol fraction of the hydrophobic units for all degrees of ionization considered. This counterintuitive behavior in lamellae was due to the dependence of their degrees of swelling not only on polymer composition and degree of ionization but also on the asymmetry ratios (see relevant subsequent section in the article), which were usually large and varied, sometimes, greatly with polymer composition and, more importantly, in a way opposite to the one by which polymer composition directly influenced the degree of swelling. This dependence of the lamellar degrees of swelling on both polymer composition and asymmetry ratios is clearly shown in eq E43 in Appendix E whose two terms on the right-hand side separately indicate the effects of the asymmetry ratio (first term), and polymer composition and degree of ionization (second term) on the degree of swelling in water for the hydrophilic nanophase. Further discussion on this is provided in Appendices D and E in the Supporting Information, where the explanation of the counterintuitive behavior in the swelling of the lamellar hydrophobic nanophase in oil is also given. The non-counterintuitive behavior in the degrees of swelling of the other anisotropic morphology, the cylinders, was due to the much smaller asymmetry ratios observed, whose variation with polymer composition could not be large enough to counteract the effect of polymer composition and degree of ionization on the degree of swelling. Furthermore, the hydrophilic nanophase in cylinders usually swelled much more than that in lamellae, with this great swelling in cylinders being due to the relatively high content in hydrophilic units and high degree of ionization. 4731

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Figure 8. Asymmetry ratios within the hydrophilic nanophase in water and within the hydrophobic nanophase in oil for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers in an oil−water mixture, as a function of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units. Dependence of the asymmetry ratio within the hydrophilic nanophase in water (a) on both the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in a 3-D plot, (b) on the mol fraction of the hydrophobic units for different degrees of ionization (0.001−0.10 = 0.1−10%) of the hydrophilic units in a 2-D plot, and (c) on the degree of ionization of the hydrophilic units for different mol fractions of the hydrophobic units (0.01−0.99 = 1−99%) in a 2-D plot. Dependence of the asymmetry ratio within the hydrophobic nanophase in oil (d) on both the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in a 3-D plot, (e) on the mol fraction of the hydrophobic units for different degrees of ionization (0.001−0.10 = 0.1−10%) of the hydrophilic units in a 2-D plot, and (f) on the degree of ionization of the hydrophilic units for different mol fractions of the hydrophobic units (0.01−0.99 = 1−99%) in a 2-D plot. The values of the parameters used were the same as those used in Figure 3, and, in particular, φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA‑oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240.

energy of mixing in the shell, the elastic Gibbs free energy in the core, and the interfacial Gibbs free energy, in addition to the dominant first derivatives (Table S3). Finally, the significant deviation in the case of lamellae was due to the fact that, apart from the first derivatives of the electrostatic Gibbs free energy in the shell and the elastic Gibbs free energy in the shell, the first derivatives of the Gibbs free energy of mixing in the shell, the elastic Gibbs free energy in the core, and the interfacial Gibbs free energy also contributed significantly (Table S3). Global Degree of Swelling. Figure 7a,c shows the global degree of swelling for the ionizable APCNs based on endlinked ionizable amphiphilic ABA triblock copolymers as a function of the mol fraction of the hydrophobic units and the degree of ionization of the hydrophilic units. Figure 7a is a 3-D histogram for the global degree of swelling, in which the global degrees of swelling are presented as bars of different colors according to the structure, whereas the rest of the graphs in the figure are 2-D projections of the 3-D plot. According to eq 21, the global degree of swelling is the polymer-composition-weighted average of the degrees of swelling in the two nanophases. Thus, the global degree of

ionization (above 0.01 for spheres and above 0.04 for cylinders and lamellae) in the double-logarithmic plot. This observation implied power-law dependencies of the degrees of swelling in water on the degree of ionization. The power-law exponents were calculated for certain values of the mol fraction of the hydrophobic units, 0.10, 0.40, and 0.70, plotted separately in Figures S6a, S7a, and S8a in Appendix D in the Supporting Information. These power-law exponents were found to be 1.40, 1.27, and 0.48, for spheres, cylinders, and lamellae, respectively. These values should be compared with the value of 1.50 derived from the approximate analytical solutions (based on the balancing of the two dominant first derivatives of the Gibbs free energy components, which, in this region of the phase diagram, were those of the electrostatic Gibbs free energy in the shell and the elastic Gibbs free energy in the shell) provided in Appendix E. The slight deviation in the case of spheres could be attributed to the small, but finite, contribution of the first derivative of the Gibbs free energy of mixing in the shell to the hydrophilic nanophase of spheres (Table S3 in Appendix E) as well. The slightly greater deviation in the case of cylinders was due to the contribution of the first derivatives of the Gibbs free 4732

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fraction of the hydrophobic units in the conetworks from 0.01 to 0.99 caused two discontinuous changes in the asymmetry ratios within the hydrophilic and hydrophobic nanophases, corresponding to transitions from spheres to cylinders (at a 0.101 mol fraction of the hydrophobic units) and from cylinders to lamellae (at a 0.22 mol fraction of the hydrophobic units), consistent with Figures 3, 4a, and 6b,e. By examining parts (b) and (e) of Figure 8, one may observe how the asymmetry ratios within the hydrophilic and hydrophobic nanophases, respectively, changed as the conetwork hydrophobe content increased. Focusing first on lamellae, for all degrees of ionization where this anisotropic morphology prevailed, the asymmetry ratios in the two nanophases exhibited opposite dependencies on conetwork composition. In particular, an increase in the hydrophobe content in the conetworks resulted in a reduction in the asymmetry ratio within the hydrophilic nanophase in water and in an increase in the asymmetry ratio within the hydrophobic nanophase in oil. The increase in the asymmetry ratio within the hydrophobic nanophase in oil for lamellae (h/ h0) (Figure 8e) was mainly due to the increase of the height of the hydrophobic nanophase in the unit cell at swelling equilibrium, h, with the number of units in the hydrophobic end-block, ζ (eqs A1 and A2 in Appendix A), in combination with the relative constancies of φcore (eq A1, for its constancy, B see Figure 6e), and the length of the edge of the square corresponding to the interfacial surface area per unit cell, w (eq A2, for its constancy, see Figure S9 in Appendix D in the Supporting Information). The dependence of the height of the hydrophobic nanophase in the unit cell at the state of conetwork preparation, h0, on ζ (eq A14) was much weaker than that of h on ζ. The latter dependence was stronger due to the elongated state of the polymer chains in lamellae. Analogous explanations may be given for the opposite dependence of the asymmetry ratio within the hydrophilic nanophase in water for lamellae (l/l0) (Figure 8b) on conetwork composition. In particular, this asymmetry ratio decreased as hydrophobe content increased, owing to the reduction of the overall height of the hydrophilic nanophase in the unit cell at swelling equilibrium, l, with the number of units in the hydrophobic end-block, ζ (eqs A20 and A21; increase of l with η, which varied inversely with ζ), in combination with the relative constancies of φshell (eq A21, for its constancy, see A Figure 6b), and the length of the edge of the square corresponding to the interfacial surface area per unit cell, w (eq A20, for its constancy, see Figure S9). The dependence of the height of the hydrophilic nanophase in the unit cell at the state of conetwork preparation, l0, on η (eq A19) was much weaker than that of l on η. The latter dependence was again stronger due to the elongated state of the polymer chains in lamellae. Focusing now on cylinders, for all degrees of ionization where this anisotropic morphology prevailed, the asymmetry ratios in the two nanophases also exhibited opposite dependencies on conetwork composition, but, in this case, an increase in the hydrophobe content in the conetworks resulted in an increase in the asymmetry ratio within the hydrophilic nanophase in water and in a reduction in the asymmetry ratio within the hydrophobic nanophase in oil. As mentioned before, unlike those in lamellae, the polymer chains in cylinders are compressed. The reduction in the asymmetry ratio within the hydrophobic nanophase in oil for cylinders (H/h0) (Figure 8e) was mainly due to the increase of the

swelling, the calculated values of which are plotted in Figure 7, is very important for the overall appearance of an APCN, absorbing both water and water-immiscible organic solvent. Figure 7a is very similar to Figure 6a both qualitatively and quantitatively because the APCN water absorption behavior dominated in the overall absorption behavior as oil absorption was much less than water absorption for most of the composition−degree of ionization space. The only region where the APCN oil absorption behavior prevailed over the water absorption behavior was where unimers in oil prevailed (highest hydrophobe content over whole degree of ionization range). The global degrees of swelling for unimers in oil were higher than those of the neighboring lamellae, which, in combination with the fact that cylinders and spheres had much higher global (and aqueous) degrees of swelling than those in lamellae, gave rise to minima for each APCN global degree of swelling vs. conetwork composition curve. Figure 7b,c is similar to Figure 6b,c, respectively, in the same way as Figure 7a was similar to Figure 6a. Furthermore, Figure 7b,c displayed discontinuous volume phase transitions at the same positions as Figure 6b,c, respectively. Asymmetry Ratios in Water and Oil. Figure 8 plots the asymmetry ratios within the hydrophilic nanophase in water and within the hydrophobic nanophase in oil as a function of the mol fraction of the hydrophobic units and the degree of ionization of the hydrophilic units in ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers. This information is presented both in 3-D histograms, in which the asymmetry ratios are presented as bars of different colors according to the structure (parts (a) and (d)), and in the two possible 2-D projections (parts (b), (c), (e), and (f)) of the 3D plots. A first observation from the 3-D plots in Figure 8a,d is that the asymmetry ratios for spheres within both the hydrophilic and hydrophobic nanophases were equal to 1, set to that value because of the high symmetry of spheres. In contrast, the asymmetry ratios for the anisotropic ordered morphologies, cylinders and lamellae, within either of the two polymeric nanophases were different from 1 and spanned a range of values from 0.65 to 4.26. In particular, cylinders had asymmetry ratios within the hydrophilic and hydrophobic nanophases lower than or slightly higher than 1 (ranging from 0.65 to 1.16) because cylinders are a “compressed” (essentially to disks) morphology (see Figure S2 in Appendix B), whereas lamellae had asymmetry ratios within the hydrophilic and hydrophobic nanophases greater than 1 (ranging from 1.28 to 4.26) because lamellae are an “elongated” morphology (see Figure S1 in Appendix A). The higher asymmetry ratios in lamellae were facilitated by the low degrees of swelling in both nanophases of this morphology, which kept the polymer chains near their unperturbed state, subsequently allowing them to be relatively easily deformed to an elongated shape to minimize the interfacial area between the nanophases. Figure 8b,e displays the effect of the mol fraction of the hydrophobic units on the asymmetry ratios within the hydrophilic nanophase in water and within the hydrophobic nanophase in oil, respectively, for different degrees of ionization of the hydrophilic units in the conetworks. As in the case of the degrees of swelling in water and oil (Figure 6b,e), these figures also showed simultaneous discontinuous changes in the asymmetry ratios in water and oil arising from the transitions between different structures. For example, at a degree of ionization of 0.02 (2%), an increase in the mol 4733

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Figure 9. Oil solubilization capacities for ionizable APCNs based on end-linked ionizable amphiphilic ABA triblock copolymers in an oil−water mixture, as a function of the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units. Dependence of the oil solubilization capacity (a) on both the degree of ionization of the hydrophilic units and the mol fraction of the hydrophobic units in a 3-D plot, (b) on the mol fraction of the hydrophobic units for different degrees of ionization (0.001−0.10 = 0.1−10%) of the hydrophilic units in a 2-D plot, and (c) on the degree of ionization of the hydrophilic units for different mol fractions of the hydrophobic units (0.01−0.99 = 1−99%) in a 2-D plot. The values of the parameters used were the same as those used in Figure 3, and, in particular, φ0 = 0.2, β = 20, χA−water = 0.45, χB−oil = 0.45, χA‑oil = 2.0, χB−water = 2.0, χAB = 2.0, and η + ζ = 240.

By examining parts (c) and (f) of Figure 8, one may observe how the asymmetry ratios within the hydrophilic and hydrophobic nanophases, respectively, changed as the degree of ionization of the hydrophilic units increased. Focusing first on lamellae, for all hydrophobe mol fractions where this anisotropic morphology prevailed, the asymmetry ratios in the two nanophases exhibited opposite dependencies on ionization. In particular, an increase in the degree of ionization in the conetworks resulted in an increase in the asymmetry ratio within the hydrophilic nanophase in water and in a reduction in the asymmetry ratio within the hydrophobic nanophase in oil. The increase in the asymmetry ratio within the hydrophilic nanophase in water for lamellae (l/l0) (Figure 8c) was due to the increase of the overall height of the hydrophilic nanophase in the unit cell at swelling equilibrium, l, with the aqueous degree of swelling in lamellae, DSlamella (eqs A20 and A21 in shell Appendix A; DSlamella shell increased with the degree of ionization; for its trend, see Figure 6c), in combination with the slight increase in the length of the edge of the square corresponding to the interfacial surface area per unit cell, w, with the degree of ionization (eq A20, for its trend see Figure S11 in Appendix D). The height of the hydrophilic nanophase in the unit cell at the state of conetwork preparation, l0, was totally independent of the degree of ionization (eq A19). Analogous explanations may be given for the opposite dependence of the asymmetry ratio within the hydrophobic nanophase in oil for lamellae (h/ h0) (Figure 8f) on the ionization degree. In particular, this asymmetry ratio decreased as the degree of ionization increased, owing to the reduction of the height of the hydrophobic nanophase in the unit cell at swelling equilibrium, h, with the degree of ionization of the hydrophilic units (Figure S11). This latter trend was despite the increase in the degree of swelling in oil for the hydrophobic nanophase of lamellae, DSlamella core , with the degree of ionization (for this trend, see Figure 6f); this increase was not significant and was only reflected in the length of the edge of the square corresponding to the interfacial surface area per unit cell, w (eq A2, for its trend, also see Figure S11), allowing the reduction of h. The height of the hydrophobic nanophase in the unit cell at the state of conetwork preparation, h 0 , was duly totally independent of the degree of ionization (eq A14).

height of the hydrophobic nanophase in the unit cell at the state of conetwork preparation, h0, with the number of units in the hydrophobic end-block, ζ (eq B14 in Appendix B). The dependence of the height of the hydrophobic nanophase in the unit cell at swelling equilibrium, H, on ζ was much weaker than that of h0 on ζ (eqs B1 and B2) because the increase in the hydrophobe content was mainly reflected in the radius of the cylinder, r (eqs B2 and B3, for its trend, see Figure S10 in Appendix D). This fact, in combination with the relative constancy of φcore (eq B1, for its constancy, see Figure 6e), B made the reduction of H ζ almost negligible. Analogous explanations may be given for the opposite dependence of the asymmetry ratio within the hydrophilic nanophase in water for cylinders (H/l0) (Figure 8b) on conetwork composition. In particular, this asymmetry ratio increased as the hydrophobe content increased, owing to the reduction of the height of the hydrophilic nanophase in the unit cell at the state of conetwork preparation, l0, with the number of units in the hydrophobic end-block, ζ (eq B19; increase of l0 with η, which varied inversely with ζ). The dependence of the height of the hydrophilic nanophase in the unit cell at swelling equilibrium, equal to H, on ζ (eqs B1 and B2) was much weaker than that of l0 on ζ. The latter dependence was stronger due to the compressed state of the polymer chains in cylinders. Figure 8c,f illustrates the effect of the degree of ionization of the hydrophilic units on the asymmetry ratios within the hydrophilic nanophase in water and within the hydrophobic nanophase in oil, respectively, for different mol fractions of the hydrophobic units in the conetworks. As in the case of the degrees of swelling in water and oil (Figure 6c,f), these figures also showed simultaneous discontinuous changes in the asymmetry ratios in water and oil arising from the transitions between different structures. For example, at a 0.20 mol fraction (20%) of the hydrophobic units, an increase in the degree of ionization in the conetworks from 0.001 to 0.10 caused two discontinuous changes in the asymmetry ratios within the hydrophilic and the hydrophobic nanophases, corresponding to transitions from lamellae to cylinders (at a degree of ionization of 0.019) and from cylinders to spheres (at a degree of ionization of 0.048), consistent with Figures 3, 5a, and 6c,f. 4734

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sponding conetwork more compatible with the organic solvent and, therefore, more capable to solubilize it. The discontinuous changes in the oil solubilization capacity were due to the transitions between different structures, as discussed before. Figure 9c depicts the effect of the degree of ionization of the hydrophilic units on the oil solubilization capacity, for different mol fractions of the hydrophobic units in the conetworks. For low degrees of ionization, the oil solubilization capacity was high. An increase in the degree of ionization in the conetworks resulted in a reduction in the oil solubilization capacity for all hydrophobe contents. This reduction was a consequence of the presence of more charged groups in the hydrophilic nanophase of the conetwork at higher degrees of ionization. These charged groups made the corresponding conetwork more compatible with the aqueous solvent and hence less capable to solubilize the incompatible organic solvent. The reduction in the oil solubilization capacity was not significant for compositions in hydrophobic units of 0.70 and higher, but it was considerable for lower hydrophobe contents. The discontinuous changes in the oil solubilization capacity were again due to the previously mentioned transitions between different structures. Careful examination of part (c) of Figure 9 leads to the observation of linear decreases of the oil solubilization capacities with the degree of ionization for all three ordered morphologies, above a certain degree of ionization (above 0.01 for spheres and above 0.04 for cylinders and lamellae) in the double-logarithmic plot. This observation implied power-law dependencies of the oil solubilization capacities on the degree of ionization. The power-law exponent for spheres was calculated for a 0.10 mol fraction of the hydrophobic units and was found to be −1.41 (Figure S16 in Appendix D). This exponent was consistent with the +1.40 power-law exponent for the degree of swelling in water versus degree of ionization for the same hydrophobe content (Figure S6a), noting that the oil solubilization capacity and the degree of swelling in water for the hydrophilic nanophase were inversely proportional (DScore nearly independent of the degree of ionization). The inverse dependence of the oil solubilization capacity and the degree of swelling in water for the hydrophilic nanophase is shown in detail in Appendix D (eq D2). The power-law exponents of the oil solubilization capacities for the anisotropic morphologies were also calculated for certain values of the mol fraction of the hydrophobic units, 0.40 for cylinders and 0.70 for lamellae, and are plotted separately in Figures S17 and S18 in the Supporting Information. These power-law exponents were found to be −1.23 and −0.37 for cylinders and lamellae, respectively. These exponents also indicated that the oil solubilization capacity and the degree of swelling in water for the hydrophilic nanophase were inversely proportional because the corresponding degree of swelling in water versus degree of ionization power-law exponents were 1.27 and 0.47 for cylinders and lamellae, respectively. The deviations from the expected value (−1.50) were explained in detail in the discussion of the aqueous swelling degrees. Comparison with Experimental Solubilization Capacities. To examine the usefulness of the oil solubilization capacities predicted by our model, in this paragraph, we compare them with those determined in a rare experimental study by Kennedy and co-workers.56 The experimental study utilized four APCNs based on end-linked poly(dimethylsiloxane) (PDMS) and poly(ethylene glycol) (PEG) segments (four different compositions, spanning a

Focusing now on cylinders, for all hydrophobe mol fractions where this anisotropic morphology prevailed, an increase in the degree of ionization in the conetworks resulted in reductions in the asymmetry ratios within both nanophases, the aqueous and the organic, H/l0 and H/h0, respectively (Figure 8c,f). Since the heights of both the hydrophilic and hydrophobic nanophases in the unit cell at the state of conetwork preparation, l0 and h0, respectively, were expectedly totally independent of the degree of ionization (eqs B14 and B19 in Appendix B, respectively), the reductions in the asymmetry ratios within both nanophases were due to the slight reduction in the common (for the two nanophases) height in the unit cell at swelling equilibrium, H, with the degree of ionization of the hydrophilic units (see Figure S12 in Appendix D). Again, this latter trend was despite the increases in the degrees of swelling in water and oil for the hydrophilic and hydrophobic nanophases, DScylinder and DScylinder shell core , respectively, for cylinders with the degree of ionization (for these trends, see Figure 6c,f); these increases were only reflected in the radius of the cylinder, r (eqs B2 and B3; for its trend, see Figure S12), allowing the reduction of H. Oil Solubilization Capacity. Figure 9 illustrates the oil solubilization capacity for ionizable APCNs based on endlinked ionizable amphiphilic ABA triblock copolymers as a function of the mol fraction of the hydrophobic units and the degree of ionization of the hydrophilic units. Figure 9a is a 3-D histogram for the oil solubilization capacity, in which the oil solubilization capacities are presented as bars of different colors according to the structure, whereas Figure 9b,c shows 2D projections of the 3-D plot. Figure 9a shows that the oil solubilization capacity expectedly increased with the mol fraction of the hydrophobic units and decreased with the degree of ionization of the hydrophilic units. Another very important observation is that the oil solubilization capacity was highly dependent on the structure, with the transitions between the different structures being accompanied by discontinuities in the oil solubilization capacity, arising from the corresponding discontinuities in the degrees of swelling in water and oil (Figure 6a,d). The highest oil solubilization capacity was presented when the APCN system was organized into lamellae, in which the values of the oil solubilization capacity ranged between 0.18 and 6.38. The important experimental implication of this observation is that an APCN is more capable to selectively solubilize organic solvents when it forms lamellae.55 Furthermore, given that the oil solubilization capacities in the other ordered morphologies, i.e., spheres and cylinders, were much lower than those in lamellae, the large change in oil solubilization capacity effected upon a transition between the ordered morphologies would result in a large oil absorption or expulsion. The latter may be exploited in applications such as delivery of hydrophobic drugs21 or regeneration of an APCN system saturated with hydrophobic pollutants.39,41 Figure 9b displays the effect of the mol fraction of the hydrophobic units on the oil solubilization capacity, for different degrees of ionization of the hydrophilic units in the conetworks. For low hydrophobe contents, the oil solubilization capacity was very low. An increase in the hydrophobe content of the conetworks resulted in an increase in the oil solubilization capacity for all degrees of ionization. This observation was expected as the increase in hydrophobe content, thereby the reduction in hydrophilic content at a constant overall degree of polymerization, made the corre4735

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high conetwork contents in hydrophobic units and low degrees of ionization, lamellae and unimers in oil were found to be the prevailing structures in the conetworks. Thus, the variation of the conetwork composition or/and the degree of ionization of the hydrophilic units in the conetworks caused transitions between different structures. These transitions were always accompanied by discontinuous changes in the degrees of swelling in water and oil. It is noteworthy that these discontinuous changes took place simultaneously in the aqueous and organic nanophases. This is the first time this simultaneous (dual) discontinuous volume phase transition is reported, and it is due to the fact that the two nanophases are inter-related but, at the same time, they keep a fair degree of independence from each other. The anisotropic morphologies, cylinders and lamellae, were characterized by asymmetries in each nanophase, calculated by the computational model as asymmetry ratios, some of which acquired rather high values (430% for some of the lamellae). Upon each transition, the model predicted simultaneous discontinuous changes in the asymmetry ratios in water and oil, similar to the degrees of swelling. The high values of asymmetry ratios in lamellae significantly affected the degrees of swelling in water and oil for lamellae and could explain the counterintuitive, in some cases, lamellar swelling behavior. In contrast, the much smaller asymmetry ratios in water and oil observed in the case of cylinders (cylindrical asymmetry ratio of down to 65%) could not strongly affect the cylindrical swelling behavior. The overall appearance of the APCN, absorbing both water and water-immiscible organic solvent, was captured by the global degree of swelling of the conetwork, which was a composition-averaged degree of swelling. The global degree of swelling also displayed discontinuous changes at the transitions due to the corresponding transitions in the two individual nanophases. One of the most useful properties of APCNs is that of oil solubilization arising from their ability to also absorb oil, in addition to water. The APCN oil solubilization capacities were predicted using the developed computational model and were found to agree well with experimental literature data. Our model predictions indicated that the oil solubilization capacities were highest when the APCNs were organized into lamellae. This observation could be exploited experimentally via the use of APCNs as efficient solubilizers of organic derivatives. The calculated oil solubilization capacities exhibited a discontinuous change upon a volume phase transition, similar to the degrees of swelling and the asymmetry ratios. These discontinuous changes, as well as the much greater capacity of the lamellar APCNs for oil solubilization compared to that of the spherical and cylindrical ones, could be exploited for the use of APCNs in a variety of applications.

range from 60.0 to 84.0 wt % in hydrophobic content, i.e., PDMS plus a silane cross-linker), simultaneously swollen in nheptane and water. Figure 10 plots the experimentally

Figure 10. Comparison between the predicted oil solubilization capacities and experimental oil solubilization capacities determined by Kennedy and co-workers on a PDMS−PEG system simultaneously swollen in n-heptane and water.56

measured values of the oil solubilization capacity against the corresponding predictions of our model. The experimentally determined oil solubilization capacities varied between 0.61 and 10.67, whereas the corresponding predicted values ranged from 1.43 to 4.18. The comparison of these values is done in the figure, with the aid of the diagonal. In the figure, two points lie below the diagonal, one on it, and the fourth above it. Although the highest experimentally measured oil solubilization capacity in the figure was higher than the corresponding predicted value by a factor of 2.55, there was rather good quantitative agreement between experiment and theory for the other three points. This latter observation is very satisfactory given the differences in the structure of the experimental and model systems. The underestimation by our model by a factor of 2.55 of the oil solubilization capacity of the strongest solubilizer might be attributed to the greater affinity between the PDMS units and n-heptane than our model considered, which assumed an almost θ-behavior between the two (χB−oil = 0.45), although a range of values, from 0.39 to 0.51, have been mentioned in the literature for the PDMS−n-heptane χ-parameter.57

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CONCLUSIONS A computational model was developed to predict the phase behavior and swelling properties of perfect ionizable amphiphilic polymer conetworks (APCNs) based on endlinked ABA triblock copolymers of different hydrophilic/ hydrophobic compositions and different degrees of ionization, equilibrated in an oil−water mixture. Depending on composition and ionization, the APCNs were found either to organize into ordered morphologies (spheres, cylinders, or lamellae) and absorb both oil and water (usually more water than oil) or to remain in a disordered state absorbing only oil (unimers in oil). In particular, under conditions that greatly favored aqueous swelling, i.e., high conetwork contents in hydrophilic units and high degrees of ionization, spheres and cylinders prevailed because these ordered morphologies were more appropriate for water-swollen structures. In contrast, for

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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/acsomega.8b03658. Derivation of the analytical expressions for the asymmetry ratios for the lamellar morphology (Appendix A); the corresponding derivation for cylinders (Appendix B); tables disclosing the main Gibbs free energy components responsible for some of the 4736

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transitions (Appendix C); extra figures better illustrating the connection among the degrees of swelling, the asymmetry ratios, and the oil solubilization capacity (Appendix D); and approximate analytical expressions that can explain some of the swelling trends displayed in the figures in the Results and Discussion section (Appendix E) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Costas S. Patrickios: 0000-0001-8855-0370 Notes

The authors declare no competing financial interest.

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DEDICATION This work is dedicated to the pioneers of the volume phase transition in polymer networks, Professor Karel Dušek (Czech Academy of Sciences) and the late Professor Toyoichi Tanaka (formerly of the Massachusetts Institute of Technology).

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DOI: 10.1021/acsomega.8b03658 ACS Omega 2019, 4, 4721−4738