Phase Behavior of Diblock CopolymerâHomopolymer Ternary Blends

Article pubs.acs.org/Macromolecules

Phase Behavior of Diblock Copolymer−Homopolymer Ternary Blends: Congruent First-Order Lamellar−Disorder Transition Robert J. Hickey,†,‡ Timothy M. Gillard,† Matthew T. Irwin,† David C. Morse,† Timothy P. Lodge,*,†,‡ and Frank S. Bates*,† †

Department of Chemical Engineering and Materials Science and ‡Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, United States S Supporting Information *

ABSTRACT: We have established the existence of a line of congruent firstorder lamellar-to-disorder (LAM−DIS) transitions when appropriate amounts of poly(cyclohexylethylene) (C) and poly(ethylene) (E) homopolymers are mixed with a corresponding compositionally symmetric CE diblock copolymer. The line of congruent transitions, or the congruent isopleth, terminates at the bicontinuous microemulsion (BμE) channel, and its trajectory appears to be influenced by the critical composition of the C/E binary homopolymer blend. Blends satisfying congruency undergo a direct LAM−DIS transition without passing through a two-phase region. We present complementary optical transmission, small-angle X-ray scattering (SAXS), transmission electron microscopy (TEM), and dynamic mechanical spectroscopy (DMS) results that establish the phase behavior at constant copolymer volume fraction and varying C/E homopolymer volume ratios. Adjacent to the congruent composition at constant copolymer volume fraction, the lamellar and disordered phases are separated by two-phase coexistence windows, which converge, along with the line of congruent transitions, at an overall composition in the phase prism coincident with the BμE channel. Hexagonal and cubic (double gyroid) phases occur at higher diblock copolymer concentrations for asymmetric amounts of C and E homopolymers. These results establish a quantitative method for identifying the detailed phase behavior of ternary diblock copolymer− homopolymer blends, especially in the vicinity of the BμE.

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critical solution (UCST) behavior.18,19 Additionally, convenient melting and glass transition temperatures (Tm,E ≅ Tg,C ≅ 100 °C) allow real space imaging by transmission electron microscopy (TEM) on rapidly quenched specimens.18,19 Melt density differences between E and C, and access to deuterium labeling with both polymers, provides access to reciprocal space analysis by small-angle X-ray and neutron scattering (SAXS and SANS) techniques as shown previously.18,19 Mixtures of A/B/AB polymers occupy a phase space most effectively represented by a ternary phase prism, where the vertices are associated with the pure components and temperature forms the vertical axis, as shown in Figure 1a (a constant pressure of 1 atm is usually assumed unless indicated otherwise). The phase space is located between the limits of two distinctly different universality classes. Volumetrically symmetric AB diblock copolymers of finite molecular weight exhibit an ordered lamellar morphology (LAM) at high values of the segment− segment interaction (χ) parameter (i.e., at low temperatures for UCST systems), and a homogeneous disordered phase (DIS) at high temperatures, separated by a fluctuation-induced weak firstorder phase transition of the Brazovskii class.25,26 On the

INTRODUCTION Ternary polymer blends composed of immiscible A and B homopolymers stabilized with AB diblock copolymers are high molecular weight analogues of oil/water/surfactant mixtures. Such ternary systems exhibit a range of morphologies and display a variety of structural and transport properties.1−9 As a result, ternary mixtures have found uses in applications ranging from detergents and pharmaceuticals to the creation of nanoporous templates and organic photovoltaics.3,4,10−13 One morphology in particular, the bicontinuous microemulsion (BμE), has motivated extensive research in efforts to understand when and why this state of matter forms.7,14−24 The BμE is a disordered, cocontinuous structure with interpenetrating A (oil) and B (water) domains in which the amphiphile (AB diblock copolymer or surfactant) resides at the interface, preventing coarsening and macrophase separation. Ternary polymeric mixtures are ideal model systems for studying the intricate phase behavior and structural dynamics of this class of blends owing to the precise synthetic control over the molecular weight and dispersity of the building blocks afforded by controlled polymerization. Saturated hydrocarbon polymers such as poly(ethylene) (E) and poly(cyclohexylethylene) (C) are particularly attractive, as we have shown in recent publications, due to the simple (van der Waals) interactions that govern the mixing thermodynamics, which result in predictable upper © XXXX American Chemical Society

Received: August 26, 2016 Revised: October 3, 2016

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DOI: 10.1021/acs.macromol.6b01872 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) A/B/AB ternary polymer phase prism showing a constant homopolymer volume fraction (ϕH) slice in the ordered region near the polymeric BμE channel. At one limit, where ϕH = 0, volumetrically symmetric AB diblock copolymers undergo a lamellar-to-disorder transition at TODT. On the opposite end of the ternary phase prism, where ϕH = 1, A/B homopolymer blends at the critical composition exhibit Ising-like critical behavior through Tc. (b) Representative constant ϕH phase diagram as a function of temperature and volume fraction of A (ϕA) with respect to ϕH near the polymeric BμE channel. The work presented here demonstrates that for constant ϕH phase diagrams near the polymeric BμE channel there is a LAM and DIS coexistence window (blue) when the ϕH,A/ϕH ratio deviates from the congruent composition.

blends, with ϕH,A = ϕH,B,7 while Schwahn and co-workers investigated a set of slightly asymmetric mixtures of homopolymers coincident with the calculated critical blend composition.41,42 To the best of our knowledge, no one has established the actual phase behavior as a function of ϕH,A/ϕH,B for 0 ≤ ϕH < ϕH,BμE. We do so here by studying slices in the phase prism at constant overall homopolymer content and variable ϕH,A/ϕH,B, as illustrated in Figure 1b. Polymeric systems are ideal for studying the phase behavior and dynamics of ternary mixtures. Judicious choice of the components and the associated molecular weights provides precise control over compositions and temperatures of the LAM−DIS and macroscopic phase transitions. Also, unlike with oil/water/surfactant mixtures, which are often governed by complex molecular interactions that can produce both UCST and LCST characteristics, the phase behavior of A/B/AB mixtures is controlled by a single, relatively simple segment− segment interaction (χ) parameter. Moreover, the ternary phase diagram for most water/oil/surfactant mixtures is skewed due to the combined effects of polar and nonpolar (e.g., hydrogen bonding) interactions and the asymmetric molar volumes between oil and water.1,2,4 In contrast, the symmetry of the phase diagram for polymers can be controlled through the degrees of polymerizations of the components, as anticipated by the mean-field theoretical work of Janert and Schick.43 Here, we report a relatively comprehensive experimental study on the phase behavior of the ternary phase prism for a model saturated hydrocarbon system comprising poly(cyclohexylethylene) (C) and poly(ethylene) (E) homopolymers and a corresponding CE diblock copolymer, using light transmission, small-angle X-ray scattering (SAXS), transmission electron microscopy (TEM), and dynamic mechanical spectroscopy (DMS). We find that at constant ϕH = ϕH,C + ϕH,E there is a specific C/E homopolymer ratio, ϕH,C/ϕH,E, where the sample undergoes a congruent phase transition in which a lamellar phase transitions directly to a disorderd phase with the same value of ϕH,C/ϕH,E without traversing a measurable two-phase window. For compositions off the congruent condition, lamellar and disordered phases with different value of ϕH,C/ϕH,E are separated by a measurable coexistence window (Figure 1b). We demonstrate that the line of congruent first-order LAM-to-DIS transitions terminates at an overall composition coincident with the occurrence of the BμE channel. Prior investigations involving

opposite end of the prism, at a volume fraction of homopolymer ϕH = 1, A/B binary homopolymer blends undergo macroscopic phase separation at temperatures (χ values) below (above) the second-order critical temperature (χ > χc) associated with the Ising universality class.27−29 Within mean-field theory these two universality classes converge at an unbinding transition that terminates at finite temperature at an isotropic Lifshitz critical point (L) when NAB > 3N, where NAB and N are the degrees of polymerization of the diblock copolymer and homopolymer, respectively, and 0 < ϕH,L < 1.30,31 Over the past 20 years, several groups have experimentally explored such ternary polymer blends leading to the identification of a BμE channel in the vicinity of the predicted unbinding transition and Lifshitz point.7−10,15,18,19,32−38 Existence of the BμE has been associated with fluctuation effects, leading to the speculation that the unbinding line and Lifshitz point are destroyed in the non-meanfield (N < ∞) limit.7,39 Since the discovery of the BμE, some researchers have focused on developing a better understanding of this morphology for technological purposes,10 while several groups have explored the fundamental issues associated with these ternary polymer mixtures.15,19,20,40 Schwahn and co-workers have extensively investigated the critical behavior along the Scott line, ϕH,BμE < ϕH ≤ 1,10,34−36,41,42 in the homopolymer-rich region of the ternary phase prism, addressing the crossover behavior from Ising to isotropic Lifshitz critical phenomena.34−36,41,42 In the ordered region of the ternary phase prism, 0 ≤ ϕH < ϕH,BμE, a complete understanding of the LAM-to-BμE transition is lacking. In a recent publication, we reported experimental evidence showing a change in the character of the order−disorder transition (ODT) for LAM-forming blends along the volumetrically symmetric isopleth with increasing ϕH.18 Increased light scattering occurred in a temperature window that corresponded to the ODT for LAM-forming blends near the BμE. This scattering behavior was hypothesized to result from either a change in the order of the transition (first vs second order) or a LAM and DIS coexistence window. In addition to this recent report, several previous works have documented coexistence of different ordered phases at blend compositions near the BμE along the volumetrically symmetric isopleth. 8,9,38 To date, each group that has investigated blends of diblocks and homopolymers has restricted attention to a single isopleth, i.e, a fixed value of ϕH,A/ϕH,B. Bates and co-workers have focused on volumetrically symmetric B


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Figure 2. Isothermal slice (T = 120 °C) of the C/E/CE ternary polymer phase prism. All data points indicate where samples were characterized using a combination of optical transmission, SAXS, and TEM. The existence of lamellar (blue), hexagonal (red), gyroid (green), single-phase disordered (orange), and macrophase-separated (maroon) regions were confirmed by the methods stated above. The yellow star indicates the critical composition (ϕH,C = 0.63) of the C/E binary blend. Lines separating colored regions are to guide the eye. liquid nitrogen, and freeze-drying under dynamic vacuum for 24 h to remove the solvent. Small-Angle X-ray Scattering (SAXS). Synchrotron SAXS experiments were conducted at both DND-CAT 5-ID-D and 12-ID-B beamlines at the Advanced Photon Source (Argonne National Laboratory, Argonne, IL). The DND-CAT 5-ID-D beamline was configured with a 8.5 m sample-to-detector distance (SDD) and a Rayonix CCD area detector. The 12-ID-B beamline was fitted with a PILATUS 2M area detector (Dectris Ltd.) and was operated with a 4 m SDD. For brevity, we will refer to all such experiments simply as SAXS. All scattering patterns were azimuthally isotropic and were integrated to obtain 1D plots of scattered intensity (in arbitrary units) versus scattering wave vector q = 4πλ−1 sin(θ/2). Samples were contained in a 1.5 mm nominal diameter quartz capillary (Charles Supper Company), and the temperature was controlled during scattering experiments at both beamlines with a modified Linkam HFS91 heating stage. Prior to the SAXS experiments, all samples were heated to 200 °C to facilitate equilibration and to remove any processing history, slowly cooled (ca. 1 °C/min) to the desired temperature and annealed under vacuum for 15 h, and then rapidly quenched to room temperature by immersion in a room temperature water bath to kinetically trap the structure. At the beamline, the samples were quickly (ca. ∼10 s) reheated to the annealing temperature, and data were acquired. Samples were subsequently heated or cooled to desired temperatures and annealed for 2 min before collecting scattering patterns. Transmission Electron Microscopy (TEM). TEM specimens were prepared by heating the sample to 200 °C for 30 min and then cooling to the temperature of interest and annealing (cooling ramps and annealing times were sample dependent; see text for details). Samples were then plunged into a mixture of dry ice and 2-propanol to rapidly cool and solidify the material (Tg,C ≅ Tm,E ≅ 100 °C), fixing the blend morphology present at the annealing temperature. We estimate that the specimens are cooled below the solidification temperature in less than 5 s. All samples were stained with ruthenium tetroxide vapor for 4 h to enhance contrast between microphases. Microtoming was performed at room temperature using a Leica EM UC6 ultramicrotome and a MicroStar diamond knife. Sample sections were microtomed to a thickness of approximately 70 nm, collected on a tabbed copper grid (PELCO, 300 mesh), and then imaged using a FEI Tecnai G2 Spirit BioTWIN transmission electron microscope operating at a 120 kV accelerating voltage.

individual isopleths associated with several different polymer systems report BμE channels that span 0.02 < ΔϕH < 0.05, where ΔϕH represents the difference between the highest and lowest homopolymer content associated with the Brazovskii (LAM− DIS) and Ising (one−two phase) regimes, respectively (see Figure 1). We find that this BμE channel shrinks in size in the vicinity of the congruent composition to ΔϕH < 0.01.

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EXPERIMENTAL SECTION

Materials, Molecular Characterization, and Blend Preparation. Detailed synthetic procedures and molecular characterization methods employed in this work are described elsewhere.18 Briefly, poly(cyclohexylethylene) (C, Mn = 2.6 kg/mol, Đ = 1.06) and poly(ethylene) (E, Mn = 2.1 kg/mol, Đ = 1.05) homopolymers and a symmetric poly(cyclohexylethylene)-block-poly(ethylene) diblock copolymer (CE, Mn = 13.6 kg/mol, Đ = 1.08, and the volume fraction of the C block f C = 0.52 ± 0.01) were prepared by catalytic hydrogenation of the corresponding unsaturated precursors (poly(1,4-butadiene), poly(styrene), and poly(styrene)-block-poly(1,4-butadiene), respectively) synthesized by anionic polymerization. Number-average molecular weights and molecular weight dispersities for the unsaturated precursors were determined by size-exclusion chromatography (SEC) using three 5 mm Phenomenex Phenogel columns, a Waters 410 differential refractometer, and tetrahydrofuran at 25 °C. The poly(styrene) and poly(1,4-butadiene) homopolymer molecular weights were determined by direct comparison with polystyrene standards (EasiCal PS-2, Polymer Laboratories) and Mark−Houwink parameters, respectively. Volumetric degrees of polymerization are based on a 118 Å3 reference volume and published densities at 140 °C.44 The C volume fraction in the CE diblock copolymer was calculated based on 1H NMR spectroscopy results obtained on the unsaturated precursors and on melt densities at 140 °C.44 High temperature SEC (Polymer Laboratories PL-GPC 220 held at 135 °C with 1,2,4-trichlorobenzene as the mobile phase) was used to confirm that chain degradation did not occur during hydrogenation of polymers containing poly(ethylene). All blends in this article were prepared by codissolving the appropriate quantity of each polymer in hot benzene (just below the boiling temperature ∼80 °C), followed by quenching the solution in C


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Figure 3. Isothermal SAXS patterns for samples where (a) ϕH = 0.30 (T = 140 °C), (b) ϕH = 0.60 (T = 120 °C), and (c) ϕH = 0.80 (T = 140 °C), with changing ϕH,C/ϕH. Dynamic Mechanical Spectroscopy (DMS). All measurements were conducted using 25 mm diameter, stainless steel parallel plates on a Rheometrics Scientific ARES strain-controlled rheometer equipped with a forced convection oven (nitrogen atmosphere). Dynamic temperature ramps were used to determine the ODT of polymer blends. Samples were loaded to give a gap of approximately 1 mm, heated above any thermal transitions (e.g., glass, melting, or order− disorder transitions), cooled at 1 °C/min to specific temperatures, and annealed for 30 min. After annealing, the storage (G′) and loss (G″) moduli were recorded at constant frequency (1 rad/s) and strain amplitude (0.5%) while increasing the temperature at 1 °C/min. The TODT is associated with an abrupt decline in the storage modulus. Optical Transmission. Transmission measurements were made using a custom-built setup. A 50 mW HeNe laser (wavelength 633 nm) emits a beam that is conditioned with a neutral density filter before passing through the sample, which is contained in a cylindrical ampule under dynamic vacuum and held in a large temperature-controlled (Omega CN3251) copper heating block. A lens located beyond the sample focuses the transmitted beam onto a photodiode detector. Automated temperature control and data collection are accomplished with home-built Labview software (National Instruments). Raw intensity data are normalized to a transmittance scale (0−100%) based on the light collected on the detector relative to the incident intensity. Reduction in transmittance is attributed to light scattered to angles wider than the collection optics; no changes in absorption are anticipated for the polymers in the temperature range of the experiments. In a typical experiment, a specimen was heated above any thermal transitions (e.g., glass transitions, order−disorder transitions, or phase separation temperatures), held until a steady intensity was recorded, and then cooled at a controlled rate (1 °C/min) to a temperature well below TODT or phase separation, followed by heating at a controlled rate (typically 1 °C/min) through the phase transition, while continuously monitoring the transmitted intensity.

distribution of morphology as a function of composition, should not be interpreted in detail; most notably, we have not included two-phase regions in the ordered portion of the phase diagram, required by the Gibbs phase rule, in order to avoid cluttering this illustration. As seen in the magnified region of Figure 2, LAM and DIS coexistence is documented. In this section we describe how these phase assignments were determined. Macroscopic phase separation at high homopolymer content (ϕH > 0.88) was deduced based on the observation of turbidity by eye and through transmission experiments as discussed below. A state of disorder was established based on the combined properties of optical clarity and the presence of a single broad peak in the associated SAXS pattern. Of particular interest is the behavior of mixtures in the vicinity of the region separating the ordered and phase-separated portions of the phase diagram shown in Figure 2, centered at ϕH ≈ 0.88 and ϕH,C/ϕH ≈ 0.60. SAXS data obtained in this portion of the phase prism are presented in the following sections.

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RESULTS AND ANALYSIS Morphology and Phase Behavior. All ternary polymer blends contain a mixture of C and E homopolymers with nearly equal molecular volumes (NC = 39 and NE = 38 are the numberaverage degrees of polymerization based on a 118 Å3 reference volume and published densities)44 and a volumetrically symmetric CE diblock copolymer (f C = 0.52 ± 0.01, NCE = 230). Figure 2 summarizes the C/E/CE ternary phase behavior at 120 °C. Three distinct regions are identified in this isothermal slice through the phase prism: (i) ordered phases, including lamellar (LAM), hexagonal (HEX), and gyroid (GYR) morphologies; (ii) macrophase separation; and (iii) disorder (DIS). The shaded regions associated with the colored points, which are intended to convey a qualitative image of the

Figure 4. TEM image for a HEX sample (ϕH = 0.60 and ϕH,C/ϕH = 0.20). The image represents a quenched state after the sample was annealed at 120 °C for 15 h, stained with ruthenium tetroxide vapors, and microtomed (ca. 70 nm).

The ordered morphologies in the diblock-rich region of Figure 2 were determined using isothermal SAXS experiments. Representative SAXS patterns are shown in Figure 3 for three values of ϕH and various C/E homopolymer ratios expressed as ϕH,C/ϕH. At ϕH = 0.30 (Figure 3a) three distinct SAXS patterns are seen corresponding to LAM (ϕH,C/ϕH = 0.55; Bragg peaks at D


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Figure 5. Determination of TODT using a combination of DMS and light scattering for samples in the ordered region of the ternary phase prism with ϕH = 0.30. (a) LAM (ϕH,C/ϕH = 0.55), (b) GYR (ϕH,C/ϕH = 0.80), and (c) HEX (ϕH,C/ϕH = 1.0).

Figure 6. Experimentally determined constant homopolymer volume fraction phase diagrams for (a) ϕH = 0.30 and (b) ϕH = 0.60. Black squares are ODT values determined from a combination of optical transmission and DMS experiments. Blue, green, red, and orange cross-marks signify LAM, GYR, HEX, and disordered morphologies, respectively, and were determined from isothermal SAXS measurements. All lines are to guide the eye, and phase diagrams are a function of ϕH,C/ϕH.

q/q* = 1, 2, 3 where q* is the first reflection), GYR (ϕH,C/ϕH = 0.80; q/q* = √6, √8, √14, √16, ...), and HEX (ϕH,C/ϕH = 1.0; q/q* = √1, √4, √7) phases. Increasing the homopolymer content to ϕH = 0.60 resulted in the sequence HEX (ϕH,C/ϕH = 0.20; q/q* = √1, √3, √4, √7), LAM (ϕH,C/ϕH = 0.63; q/q* = 1, 2, 3), and HEX (ϕH,C/ϕH = 0.80; q/q* = √1, √3, √4, √7), while at ϕH = 0.80 only the LAM phase was found (data shown for ϕH,C/ϕH = 0.50, 0.57, and 0.65). TEM images indicate that the HEX phases contain a cylindrical morphology as shown in Figure 4 for the ϕH = 0.60, ϕH,C/ϕH = 0.20 sample; see Figure S1 for additional micrographs. As seen in Figure 3, all samples at constant ϕH have virtually the same domain spacing, d = 2π/q*. In addition to characterizing the isothermal morphology of the ternary blends using SAXS, we used a combination of DMS and optical transmission to determine TODT for all the samples in the ordered region of the ternary phase prism. These results are presented in two parts. At lower homopolymer content, ϕH = 0.30 and ϕH = 0.60, we identified multiple ordered phases, each exhibiting a dynamic elastic modulus (G′) that drops precipitously at the ODT on heating (see Figure 5). We also employed a previously described optical transmission technique to corroborate the transition from the ordered to disordered states.18 For ϕH ≤ 0.60 the ODT associated with the LAM and HEX phases is accompanied by a step change in the transmission

attributable to the loss of light depolarization due to locally anisotropic and birefringent morphologies (see Figures 5a and 5c).45−47 This optical step at TODT was reported earlier for the undiluted CE diblock copolymer used in this study.18 For samples with isotropic (e.g., cubic) morphologies like the double gyroid, there should be no optical step when heating the sample through the ODT, consistent with the data in Figure 5b. Interestingly, similar to experimentally determined phase portraits for single component poly(isoprene)-block-poly(styrene) diblock copolymers, TODT for the GYR (TODT = 153 °C, Figure 5b) phase is depressed relative to the LAM (TODT = 171 °C, Figure 5a) and HEX (TODT = 170 °C, Figure 5c) phases at ϕH = 0.30.48,49 At higher homopolymer content, we found only the LAM and DIS phases in the range of experimentally accessible temperatures (T > Tm, Tg > 100 °C). The results of SAXS, DMS, and optical transmission measurements are presented in Figure 7 for ϕH = 0.80 and ϕH = 0.83. These phase diagrams contain an additional feature not resolved in Figure 6: two-phase regions that separate the LAM and DIS phases on either side of the composition associated with the highest TODT, corresponding to a congruent first-order transition. This phase behavior is closely analogous to the vapor−liquid azeotrope in nonideal binary fluid mixtures such as water and ethanol50 and the melting behavior of E


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Figure 7. Constant homopolymer volume fraction phase diagrams: (a) ϕH = 0.8 and (b) ϕH = 0.83. The coexistence window (blue) was determined from the width of the local minimum of the optical transmission experiment (see Figure 8). A congruent transition from LAM to DIS phases occurs when ϕH,C/ϕH = 0.57 and ϕH,C/ϕH = 0.60 for constant homopolymer volume fractions of ϕH = 0.80 and ϕH = 0.83, respectively. Blue cross-marks indicates LAM morphology determined from isothermal SAXS measurements. All lines are to guide the eye, and phase diagrams are a function of ϕH,C/ ϕH.

Figure 8. Optical transmission and G′ as a function of temperature for LAM samples of compositions: (a) ϕH,C/ϕH = 0.50, (b) ϕH,C/ϕH = 0.57, and (c) ϕH,C/ϕH = 0.65 when ϕH = 0.80.

binary inorganic compounds such as γ-TiNi that display a congruent melting temperature.51 In the system of interest here, the value of ϕH,C/ϕH at which TODT is maximum separates a twophase window in which an E-rich disordered phase coexists with a C-rich lamellar phase from a two-phase window in which a Crich disordered phase coexists with a more E-rich lamellar phase. In a binary system, the composition at which the two coexisting phases have the same composition must correspond to an extremum in both phase boundaries and must be a point at which the low temperature phase transforms directly into a high temperature phase without passing through a two-phase window. Though the phase behavior found here is similar to that found in these binary systems, the analogy is complicated by the fact that we are actually studying a ternary system, where the rules of thermodynamics would allow coexistence between a disordered and lamellar phase with the same value of ϕH,C/ϕH but different values of copolymer concentration. If this occurred, it would be expected to lead to a two-phase region that persists even at the temperature at which TODT is maximum, leading to a small gap between the temperature at which the lamellar phase melts and

that at which the disordered phase orders. In fact, our experiments show no evidence of a two-phase window at this composition. The absence of evidence of a two-phase composition at this value of ϕH,C/ϕH indicates that the system is behaving as a pseudobinary system at the congruent point and undergoes a direct transition between two phases with the same surfactant concentration as well as the same value of ϕH,C/ϕH. We describe the evidence leading to the congruent phase diagrams shown in Figure 7 in the following paragraphs. In Figure 8 we present select DMS and optical transmission data obtained for ϕH = 0.80 and ϕH,C/ϕH = 0.50, 0.57, and 0.65. Two distinctly different optical transmission signatures are evident in these results. At ϕH,C/ϕH = 0.57, the transmission increases (steps) discontinuously (i.e., within 1 °C, the temperature resolution of the experiment) upon heating at 1 °C/min, consistent with a direct first-order LAM-DIS transition (see Figure 8). A sharp discontinuity in G′(T) at the congruent TODT is consistent with this interpretation. For ϕH,C/ϕH = 0.50 and 0.65, however, there is a pronounced dip or minimum in the transmission spanning about 4 deg in F


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respective compositions dictated by constant temperature tielines.51 Such composition differences produce subtle refractive index contrast (nC ≅ 1.5152 and nE ≅ 1.48 in the melt state53), which combined with finite particle sizes results in the scattering of light, hence a reduction in transmission. In the ϕH = 0.60 slice, there is a LAM region (0.35 ≤ ϕH,C/ϕH ≤ 0.47) in which a narrow two-phase window (≈ 4 °C) exists as indicated by the appearance of an optical “dip” (see Supporting Information). We have validated this argument with three additional sets of experiments. First, the drop in G′ shown in Figure 8a (ϕH,C/ϕH = 0.50) and Figure 8c (ϕH,C/ϕH = 0.65) appears to be more gradual than at the congruent composition (Figure 8b), consistent with the notion that the transition from LAM to DIS involves the continuous melting of the ordered state over a range of temperatures (see Figure 9). Second, Figure 10 shows SAXS measurements that coincide with the results presented in Figure 8. Differences in the peak position (q*) and peak width make it possible to discern the presence of LAM or DIS, or both phases, by SAXS. Within the 1 deg temperature precision afforded by the technique the LAM phase transitions directly to the DIS phase at ϕH,C/ϕH = 0.57 with no evidence of a two-phase region. In contrast, coexistence of LAM and DIS phases is readily apparent in the ϕH,C/ϕH = 0.50 and 0.65 mixtures around TODT. Third, to further assess the morphology of the mixtures at the local minimum of the transmittance, we used TEM to obtain real space images. Previously we reported that the minimum is stable over time and that it is possible to halt either heating or cooling ramps at temperatures corresponding to the local minimum in transmission.18 Additionally, because the samples are amenable to TEM imaging, we were able to rapidly quench the samples into a mixture of dry ice and 2-propanol (T = −78 °C) from the annealed state near the minimum (Figure 11). Figure 11a shows the cooling profile for a sample displaying an optical dip (ϕH = 0.80 and ϕH,C/ϕH = 0.50). In this experiment, the sample was heated above TODT and slowly cooled at a rate of 0.1 °C/min and annealed for 1 h at 148 °C. From the annealed state, the sample was rapidly quenched below the glass and melting temperatures (Tg,C ≅ Tm,E ≅ 100 °C). The TEM image in Figure 11b reveals LAM and DIS phases coexisting at the local minimum in transmission. The single LAM grain seen in the TEM image is well ordered, and the DIS phase appears to be strongly fluctuating. The globally isotropic DIS phase in Figure 11b

temperature, superimposed on a small step change. We associate the onset and termination of the dip with the limits in temperature of the two-phase windows on either side of the congruent composition shown in Figure 7a. We first reported this “dip” feature in a recent publication18 and attributed it to either phase separation or the development of composition fluctuations. Here we show definitively that it is due to passing through a two-phase window. Figure 9 illustrates our explanation

Figure 9. Representation of the development of lamellar microstructure in the coexistence window for sample ϕH = 0.80 and ϕH,C/ϕH = 0.50. At high temperature, the ternary blend is homogeneously mixed. As the sample is cooled into the coexistence region, lamellar grains begin to grow, and the relative amount of each phase (LAM or DIS) can be determined using the lever rule (constant temperature dashed lines). Finally, at the lowest temperature, a polycrystalline lamellar phase is developed.

for this effect. Cooling the one-phase DIS state or heating the LAM phase into the two-phase window leads to the nucleation and growth of grains of lamellae or disordered melt material, with

Figure 10. SAXS patterns for ϕH = 0.80 samples near TODT: (a) ϕH,C/ϕH = 0.50, (b) ϕH,C/ϕH = 0.57, and (c) ϕH,C/ϕH = 0.65. Blue, red, and gold SAXS patterns indicate LAM, DIS, and coexisting phases, respectively. G


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Figure 11. TEM image of a quenched sample (ϕH = 0.80 and ϕH,C/ϕH = 0.50) after annealing in the local minimum of the transmittance. (a) Cooling ramp and annealing profile for the prepared sample. The sample was cooled at a rate of 0.1 °C/min and annealed for 1 h at 148 °C. (b) The TEM image was obtained after rapidly quenching the sample to a solid state, staining with ruthenium tetroxide vapors and microtoming (ca. 70 nm).

morphologies were verified using TEM images obtained from stained (RuO4) samples that had been annealed at 120 °C and then rapidly solidified and microtomed into thin (ca. 70 nm) sections. Figure 14 shows selected micrographs from samples at three compositions: ϕH = 0.83, 0.86, and 0.875. These remarkable images reveal extraordinary long-range order along with smectic-like focal conic textures surrounding dislocation and disclination defects commonly seen in liquid crystal systems.54,55 The increase in layer spacing with increasing homopolymer context is plainly evident (see below). Increasing the homopolymer content from ϕH = 0.875 to ϕH = 0.885 results in a dramatic change in the optical transmission behavior as shown in Figure 13c. This sample was heated to and held at 180 °C for several minutes producing a nearly transparent state, then cooled at 1 °C/min to 130 °C, and reheated at the same rate to 180 °C, resulting in a complete loss and subsequent recovery of transmission with hysteresis, symptomatic of phase separation, and remixing, respectively. The equilibrium phase boundary lies between the onset of recovery of transmission on heating (169 °C) and the initial drop in transmission on cooling (166 °C); we have assumed the former since remixing occurs spontaneously. The kinetics of spinodal decomposition (or nucleation and growth if the congruent condition does not precisely coincide with the line of critical points; see Discussion section) upon cooling, and diffusion limited mixing during heating, account for hysteresis in this phase transition. The locus of points defining the line of (nominally) second-order transitions for ϕH > 0.88 in Figure 12, terminating near the critical point for the binary homopolymer blend (ϕH > 0.88) at Tc = 245 °C,18 was determined using this “cloud point” technique. A third type of optical transmission behavior was recorded at ϕH = 0.88 as shown in Figure 13b, where heating produces neither a step change in transmission nor any evidence of phase separation. However, there is a distinct change in the slope of the transmission with temperature at about 148 °C, nearly coincident with the line of congruent LAM−DIS transitions in Figure 12. Real-space TEM images obtained from this mixture annealed at 120, 140, and 160 °C and shown in Figure 15 provide clear evidence that a distinctly different morphology is present, which we interpret as the BμE state. The TEM images in Figure

resembles previously reported TEM images in which the sample exhibits locally correlated smectic-like fluctuations.18 Congruent Isopleth. We now focus on the isopleth defined by the ratio of homopolymer compositions ϕH,C/ϕH = 0.60 as a function of overall homopolymer content 0 ≤ ϕH ≤ 1. Figure 12

Figure 12. Phase diagram of the congruent isopleth (ϕH,C/ϕH = 0.60) for C/E/CE blends where ϕH represents the total homopolymer volume fraction. Blue triangles and maroon circles identify TODT and phase separation temperatures, respectively. The BμE channel is restricted to ΔϕH < 0.01. All lines connecting data points are to guide the eye.

summarizes the results along what we refer to as the congruent isopleth. This phase diagram includes the line of congruent firstorder LAM−DIS transitions described in the previous section and the line of (nominally) second-order transitions delineating the single-phase and macroscopic two-phase regions associated with the Ising-like behavior at high hompolymer contents. Figure 13 illustrates how these phase boundaries have been identified using the optical transmission technique in the vicinity of the transition between the two universality classes. LAM samples were annealed at 120 °C for several hours and then heated at 1 °C/min while recording transmission data. Similar to Figure 8b, a discontinuous step in the transmission signals the LAM−DIS transition, shown in Figure 13a for ϕH = 0.87. Lamellar H


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Figure 13. Optical transmission measurements obtained as a function of temperature along the congruent isopleth ϕC/ϕH = 0.60. (a) Heating ϕH = 0.87 through the LAM−DIS transition. (b) Heating ϕH = 0.88 in the BμE state (see Figure 15). (c) Heating and cooling ϕH = 0.885 through the macrophase separation transition.

Figure 14. TEM images of blends forming LAM morphology along the congruent isopleth (ϕH,C/ϕH = 0.60). Sample compositions are (a) ϕH = 0.83, (b) ϕH = 0.86, and (c) ϕH = 0.875. The images were obtained after the samples were annealed at 140 °C for 15 h, rapidly solidified, and then stained with ruthenium tetroxide vapors and microtomed (ca. 70 nm). Scale bars are 500 nm.

Figure 15. TEM images showing the temperature dependence of the BμE-forming blend where ϕH = 0.88 and ϕH,C/ϕH = 0.60. Temperatures are (a) 120, (b) 140, and (c) 160 °C. The images represent a quenched state after the samples were annealed at the specified temperatures for 15 h, stained with ruthenium tetroxide vapors, and microtomed (ca. 70 nm). Scale bars are 500 nm.

established using transmission measurements during heating of specimens in the LAM state (see Supporting Information). In order to prepare the SAXS specimens in the LAM phase, we annealed samples in the capillary tubes for several hours at 120 °C after slowly cooling (ca. 1 °C/min) from the disordered state, rapidly solidified the mixture by immersion in a room temperature water bath, then quickly reheated the mixture to 120 °C on the SAXS beamline. Finally, optical transmission experiments with ϕH ≥ 0.885 and varying ϕH,C/ϕH have revealed the size of the multiphase window adjoining the region where the bicontinuous microemulsion exists. Representative data obtained while cooling mixtures from

15 indicate that no qualitative change in the BμE structure occurs with respect to temperature, consistent with the optical transmission experiment in Figure 13b. Increasing the fraction of homopolymer along the congruent isopleth swells the lamellae as documented by the SAXS patterns shown in Figure 16a, where the domain period d = 2π/q* is determined by the principal peak position q*. As illustrated in Figure 16b, d appears to diverge at 140 °C just above ϕH = 0.875. Here we note that at compositions 0.85 < ϕH < 0.88 we encountered considerable hysteresis in the LAM−DIS transition, i.e., the disordered state remained in a supercooled state for extended times (>15 min) when cooled below TODT, which was I


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Figure 16. Isothermal SAXS patterns and associated d-spacing for LAM samples near the BμE channel along the congruent isopleth. (a) Background subtracted SAXS patterns obtained at 140 °C for the LAM-forming samples and (b) the calculated d-spacing from the corresponding SAXS patterns in (a). The orange point corresponds to the BμE sample where ϕH = 0.88. TEM images confirming the LAM and BμE states are shown in Figures 14 and 15, respectively.

Figure 17. Optical transmission measurements and constant homopolymer volume fraction phase diagram. (a) “Cloud point” measurements for ternary blends where ϕH = 0.90 through the macrophase separation transition. All samples were cooled at a rate of 0.1 °C/min. (b) Constant homopolymer volume fraction phase diagram determined from the “cloud point” measurements in (a). All lines are to guide the eye, and the phase diagram is a function of ϕH,C/ϕH.

the one-phase state at 0.1 °C/min are shown in Figure 17a; additional data acquired while heating and cooling at different rates are presented in the Supporting Information. We have estimated the binodal temperatures associated with the onephase to multiphase transition as the point where the transmission drops steeply and plot the associated phase diagram in Figure 17b.

lishes the importance of mapping the full 3-dimensional phase prism and provides a methodology for doing so. We have demonstrated that the boundaries of the ordered lamellar (LAM) phase near the disordered phase forms a ridge in the phase prism, along the top of which runs a line of virtually congruent melting transitions. Studies of phase behavior within planes of constant copolymer fraction exhibit a phase diagram similar to that found near a congruent melting point of a binary solid, in which the value of ϕH,C/ϕH at which the melting temperature is maximum coincides with a point at which the lamellar crystal melts into a disordered phase of almost exactly the same composition. This coincidence of a maximum of the melting temperature and congruent melting is required in a binary system but not in a ternary system, such as that studied here, and occurs in the ternary system only if, for whatever reason, the ordered phase coexists with a disordered phase with the same copolymer volume fraction. This line of virtually congruent first-order ODTs appears to terminate at a point in the phase prism coincident with the presence of a BμE state. Notably, the BμE structure is only observed in this system over a range of less than 1% copolymer volume fraction, which is a significantly narrower range than that found in previous studies in which a

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DISCUSSION The results and analyses presented in the previous section reveal how the phase behavior of a model ternary system comprising a compositionally symmetric poly(cyclohexylethylene)-b-poly(ethylene) (CE) diblock copolymer and the corresponding C and E homopolymers with α = 0.18 (α = NA/NAB) varies as a function of composition and temperature. A host of prior experimental studies have dealt with related ternary mixtures but have not provided the comprehensive treatment of the order− disorder transition presented in this article. Specific subsystems addressed in earlier publications include individual volumetrically symmetric and asymmetric isopleths, focusing in particular on regions near the microemulsion channel.7,8,10,15,18−20,33−36,38,40−42,56 The work reported here estabJ


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Macromolecules microemulsion channel was identified by studying a specific volumetric isopleth. In what follows, we briefly summarize what is known about the phase behavior of A/B/AB ternary homopolymer blends on the basis of theory, simulation and previous experiments, and its relationship to the present work. Most theoretical work on this type of ternary blend has focused on symmetric systems, containing diblock copolymers with fA = 1/2, homopolymers with NA = NB, and equal statistical segment length, bA = bB. We assume that the phase behavior of real systems with slight asymmetry arising, e.g., from unequal statistical segment lengths, is qualitatively similar to that of a hypothetical symmetric system, except for the lack of perfect symmetry with respect to the volumetrically symmetric isopleth. Self-Consistent Field Theory. Self-consistent field calculations have been used extensively to predict the phase behavior of A/B/AB ternary polymer blends.30,31,43,57−59 To the best of our knowledge, the effects of conformational asymmetry (bA ≠ bB) have not been addressed theoretically. We focus here on the phase diagram of a symmetric system within the volumetrically symmetric isopleth (ϕH,A/ϕH,B = 1). The SCFT phase diagram of such a system with sufficiently large values of α = NA/NAB contains a region of lamellar (LAM) order at high values of χN and high copolymer volume fraction, a region of two phase separation between homogeneous phases rich in different copolymers at high χN and low copolymer volume fraction, and a disordered phase at lower values of χN. The lamellar and disordered regions are separated by a line of order−disorder transitions. The two-phase and disordered regions are separated in this isopleth by a line of critical points that terminates at the critical point of the binary homopolymer blend. The lamellar phase can exist at temperatures below TODT (high values of χN) over a range of concentrations 0 ≤ ϕH < ϕH,UT, where ϕH,UT is the homopolymer concentration at an unbinding transition (UT). Decreasing the amount of copolymer within the lamellar phase swells the lamellae, leading to a divergence in the lamellar period d at ϕH,UT. Any further decrease in the amount of copolymer leads to macroscopic phase separation into A-rich and B-rich homogeneous phases. Within SCFT, the unbinding transition, the order−disorder transition, and the line of critical points of a symmetric system meet within the volumetrically symmetric isopleth (ϕH,A/ϕH,B = 1) at a Lifshitz critical point (L), at homopolymer volume fraction ϕH,L = 1/(1 + 2α2) and segregation strength (χN)L = 2(1 + 2α2)/α, where α = NA/NAB.30 Figure 18 reproduces the mean-field symmetric isopleth reported by Düchs et al. for α = 0.2 (red curves), along with results of field theoretic simulations, which are discussed below.31 We note that the SCFT phase diagram shown in Figure 18 includes a narrow three-phase window located between the UT and the two-phase region, which also terminates at the tricritical Lifshitz point. Prior to 2003, most calculated symmetric isopleth phase diagrams with α < 1 did not include this threephase region.30,43 The three-phase region predicted by SCFT is a region of coexistence of a lamellar phase with two disordered phases rich in A and B homopolymers. The appearance of this three-phase region is a result of an extremely weak attraction between monolayers with slightly overlapping brushes.60−62 This attraction is opposed by the entropy associated with monolayer bending fluctuations,63 which gives rise to an effective pressure between monolayers in the lamellar phase that can prevent the formation of this type of phase separation, but which is not captured by SCFT.

Figure 18. Mean-field and fluctuation-corrected (C = 50) phase diagrams for A/B/AB ternary polymer mixture along the volumetrically symmetric isopleth for α = 0.2.31 Red lines indicate the mean-field results, and circles and squares are results from Monte Carlo and complex Langevin simulations, respectively. The C parameter acts as the Ginzburg parameter and determines the relative influence of fluctuations.31 2ϕ and 3ϕ signify two- and three-phase regions. The dashed lines are to guide the eye. Reproduced with permission from ref 31.

Fluctuation Effects. Fluctuations influence each region of the symmetric ternary phase diagram differently. The main effect of fluctuations upon the ODT of finite molecular weight (N < ∞) diblock copolymers is to destroy the second-order character of the transition predicted by SCFT, inducing a first-order ODT that in systems with a positive mixing entropy occurs at a temperature below the predicted SCFT critical temperature.64,65 This behavior, first identified in neat block polymers by Fredrickson and Helfand based on concepts developed by Brazovskii,66 has been confirmed by numerous experiments.65,67−86 In general, the existence of a first-order ODT is expected to lead to a region of phase coexistence between the lamellar and disordered phase. In a rigorously symmetric system, this would lead to coexistence between lamellar and disordered phases of slightly different copolymer concentration within the volumetrically symmetric isopleth. In a slightly asymmetric system, one can instead define an analogous line along the boundary of the lamellar phase along which the lamellar phase coexists with a disordered phase with the same value of the ratio ϕH,A/ϕH,B of the two types of homopolymer as that found in the lamellar phase. Even along such a line, the coexisting phases could exhibit different copolymer concentrations. In the system studied here, however, we find that coexisting phases along this line appear to have virtually the same volume fractions of all three components, leading to congruent melting. In the limit of A/B binary blends (ϕH = 1), composition fluctuations lead to a crossover from mean-field to Ising-like critical behavior as the temperature of the mixture approaches Tc.87−95 For most experimental homopolymer blends, fluctuations renormalize the critical point to a temperature a few degrees below the apparent mean-field critical point, as defined by extrapolating scattering intensity data from temperatures outside the region of Ising critical behavior. The most profound effect of fluctuations on the ternary phase diagram is the appearance of a bicontinuous microemulsion phase along a channel close to the expected unbinding transition. Numerous experimental studies have shown that the disordered phase is continuously connected to a channel of bicontinuous microemulsion.7−9,18,32,33 In analogous model systems of small molecule surfactants with strongly immiscible polar and nonpolar components (e.g., oil and water), the bicontinuous microemulsion phase is known to exhibit two-phase coexistence K


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the components in the E/P/EP investigation were more than 8 times larger than those used here, NEP = 1930 versus NCE = 230 (calculated using a common segment volume of 118 Å3). Schwann and co-workers have explored two other model systems using SANS, PEE/PDMS/PEE−PDMS (α = 0.18),35,36 and PB/PS/PB−PS (α = 0.16),41,42 both with comparable degrees of polymerization to what we describe in this report. In both systems they identified a BμE state between LAM and twophase regions along single isopleths (ϕPEE/ϕH = 0.52 and ϕPB/ ϕH = 0.42, respectively) and provided a detailed assessment of the critical scattering around the nominal Lifshitz composition. The Ginzburg number was determined as a function of diblock copolymer content along the Ising portion of the second-order phase transition in PB/PS/PB−PS. Schwann et al. report BμE windows that are wider than what we find in the present study: ΔϕH ≅ 0.05 for PEE/PDMS/PEE−PDMS and ΔϕH ≅ 0.02 for PB/PS/PB−PS. Experimental studies on A/B/AB ternary polymer blends are generally carried out on systems with some degree of asymmetry between A and B monomers, even for volumetrically symmetric systems. The remaining asymmetry arises in part from the difference in statistical segment lengths between chemically different monomers, as discussed in more detail below. The present study is the first one to systematically study the nature of the order−disorder transition in the phase prism, rather than focusing on single isopleth, and the first to identify a congruent melting line. The upper bound of ΔϕH < 0.01 obtained here for the width of the microemulsion channel is substantially narrower than that reported in any previous study. This suggests that the apparent microemulsion channel may be particularly narrow near the congruent line. The widths of the BμE channel identified along the volumetrically symmetric isopleth in the E/P/EP and PEE/PDMS/PEE−PDMS, and the asymmetric one for PB/PS/ PB−PS, may thus reflect the consequences of being slightly off the condition of congruency (see below). Nevertheless, the SANS results obtained with all three systems reinforce the notion that the microemulsion appears only over a very narrow range of values of ϕH in the phase prism. Sources and Consequences of A/B Asymmetry. While theory for A/B/AB ternary systems has focused on an idealized case of systems that are completely symmetric with respect to exchange of the identity of A and B monomers, real chemical systems are never symmetric in this sense. Even in volumetrically symmetric systems, with fA = 1/2 and NA = NB, there are at least two sources of chemically asymmetry between the two types of monomer. The first is the difference in the statistical segment lengths of A and B monomers of equal volume, which we will refer to as conformational asymmetry. The second is asymmetry in the excess free energy of mixing per monomer gex of the binary homopolymer system, which is approximated in the classical Flory−Huggins theory by a symmetric function gex = kTχϕAϕB with a composition-independent χ parameter. (Excess free energy is defined here as the difference between the true free energy of mixing of a binary homopolymer mixtures and the Flory−Huggins combinatorial contribution.) The latter asymmetry, which we will refer to as thermodynamic asymmetry, can be accounted for within a generalized variant of Flory−Huggins theory by allowing for the possibility of a compositiondependent χ parameter. Notation aside, this is the same asymmetry that causes the excess free energy of mixing of mixtures of small molecules to be asymmetric with respect to exchange of the identities of the two components even when expressed in terms of volume fraction. This thermodynamic

with the lamellar phase and regions of two- and three-phase coexistence with homogeneous phases rich in oil or water.96,97 The bicontinuous microemulsion phase is known to appear in analogous small molecules surfactant/oil/water systems only over a relatively narrow range of values of the oil/water ratio centered on the value at which the microemulsion exhibits threephase coexistence with excess phases of oil and water. In A/B/AB ternary polymer mixtures, the composition of the microemulsion phase that exhibits three phase coexistence with homogeneous phases rich in A or B homopolymers defines a unique point in an isothermal phase triangle, or a line in the compositiontemperature phase prism. The possible coexistence of a microemulsion with excess homopolymer-rich phases is a different phenomenon than the predicted coexistence of a lamellar phase with homopolymer-rich phases shown in Figure 18. A bicontinuous microemulsion can actually appear only if monolayer fluctuations stabilize the lamellar phase sufficiently to overcome the weak attraction between monolayers predicted by SCFT and thereby prevent the appearance of a microemulsion from being preempted by coexistence of the lamellar phase and homopolymer-rich phases. Experiments on systems with small values of α have thus far found evidence for existence of a microemulsion channel, rather than for two- and three-phase coexistence of the lamellar phase with homogeneous homopolymer rich phases. Düchs et al. have investigated how fluctuations influence the mean-field ternary phase diagram along the volumetric isopleth of a symmetric system using 2-dimensional Monte Carlo and complex Langevin field theoretic simulations.31 A large but finite invariant degree of polymerization, characterized by the parameter C = Rg3/V = 50 or N̅ = 63C2 ≈ 5 × 105, was employed in this simulation study.31 Their results, shown in Figure 18, reveal suppression of the entire LAM−DIS transition and development of a cusp in the ODT near the region of convergence of the ODT and Ising critical point lines. However, owing to finite size limitations, they were not able to extend these calculations as close to the nominal unbinding transition as is possible in experiments. Another important conclusion drawn by Düchs et al. is that a two-phase window at the ODT, if present in the system they studied, is narrower than about 0.001 in ϕH. This is consistent with our experimental identification of congruency, but quantitative comparison is complicated by the fact that these are two-dimensional simulations carried out at a much larger invariant degree of polymerization than that characteristic of these experiments. Relation to Prior Experiments. The first SANS experiments that exposed Lifshitz-like behavior in compositionally symmetric poly(ethylene)-b-poly(ethylene-propylene) (PE-PEP or EP) diblock copolymers and the corresponding PE (E) and PEP (P) homopolymers (α = 0.21)98 suggested that fluctuation effects are restricted to a surprisingly small region in the phase prism around the predicted mean-field Lifshitz point. SANS measurements taken with a specimen located within about 1% in ϕH of the composition associated with the BμE on the Ising side of the ϕE/ ϕH = 0.5 isopleth produced mean-field exponents (γ = 1 and ν = 1/4) within several degrees of an apparent critical temperature. The width of the BμE along the compositionally symmetric isopleth was established to be ΔϕH ≅ 0.02 based on SANS, SAXS, and transmission electron microscopy.7 Two experimental features distinguish that system from the one under consideration in this study: (i) PEP and PE are considerably less conformationally asymmetric than C and E (see below), and (ii) L


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rationalize. In a melt of volumetrically symmetric diblock copolymers, with fA = 1/2, the block on the interior of a spherical or cylindrical region must stretch more than the block on the exterior in order to maintain an equal average volume per block in an incompressible liquid. Because the free energy cost of stretching a block is lower for the block with larger statistical segment length, the system tends to put the block with larger statistical segment length on the concave or interior side of the interface in volumetrically balanced one-component melts. In the present system, this effect is expected and observed48 to shift the phase diagram of the one-component CE diblock copolymer toward f C < 1/2 because of the larger statistical segment of poly(ethylene) monomers when defined using monomers of equal volume. Adding a thermodynamically neutral diluent (χ = 0) to both domains of the conformationally asymmetric lamellar diblock copolymer will swell the morphology, separating the diblock monolayers. Because blocks with larger random-walk coil size stretch more easily, this tends to cause the block with larger coil size to swell more, even in a lamellar phase in which the interfaces remain flat, i.e., a system with volumetrically balanced A and B blocks. The greater swelling of the block with larger ideal coil radius in a highly swollen system creates a torque that tends to make the interface bend away from this larger block and toward the block with the smaller coil size. Because this effect of swelling by diluent is opposite in sign to that obtained in a neat system, the relative importance of the two tendencies will depend upon the degree of swelling. The degree of swelling in turn depends upon the degree polymerization of the added homopolymer “solvent” relative to the block sizes. Figure 19 illustrates how changes in α =

asymmetry often requires the introduction of two- and threeparameter empirical models to fit the composition dependence of the excess free energy. Each of the boundaries between lamellar, disordered, and twophase regions of an A/B/AB ternary system is characterized by a special line in the phase prism. These are (1) the congruent line within the boundary between the lamellar and disordered phase, (2) the line of critical points within the boundary of the disordered phase with respect to macroscopic two-phase separation in the homopolymer-rich part of the phase prism, and (3) the line along which a balanced microemulsion exists in three phase coexistence with excess phases rich in A and B homopolymers. These three lines are expected to almost meet near the opening of the microemulsion channel, if the region of microemulsion three-phase coexistence extends to near the top of the microemulsion channel (which is not yet known). All three of these lines would lie within the volumetric isopleth in an idealized symmetric system, and all are expected to lie somewhat outside of this isopleth in a real system as a result of the combined effects of conformational and thermodynamic asymmetry. It is thus far unclear if the shift of the congruent melting line from the volumetric isopleth is related to the corresponding shift of either of these other lines in the phase prism. The most important effect of thermodynamic asymmetry is to cause the line of critical points of a volumetrically symmetric mixture to move off the volumetric isopleth. For the system studied here, this line of critical points begins with the pure binary homopolymer mixture (ϕH = 1) at a composition ϕC = 0.63 ± 0.0318 and descends in temperature as ϕH approaches the point of convergence at the BμE. In the system studied here, the value of ϕH,C/ϕH near the point of termination of the congruent melting line with increasing ϕH seems to correspond at least roughly with its value at the end of the line of critical points. We hasten to point out that we have not yet examined the multiphase region of the C/E/CE phase prism in any detail, which would require further characterization of all relevant phase boundaries as functions of ϕH,C/ϕH. Nevertheless, the data obtained for this system are consistent with the idea that the congruent melting line and the line of Ising critical points are coupled in the vicinity of the BμE. Conformational asymmetry affects the tendency of diblock copolymer monolayers to curve toward A- or B-rich domains, which in turn controls where the microemulsion can exhibit three-phase coexistence with excess phases of A and B homopolymers. The composition at which the BμE exhibits three-phase coexistence with excess A- and B-rich phases is expected to move toward higher volume fractions of A in systems in which monolayers exhibit a spontaneous tendency to curve around A domains and away from B domains. As discussed below, however, the relationship between spontaneous curvature and conformational asymmetry may be different in the case of highly swollen ternary systems than in the better-studied case of diblock copolymer melts. Conformational asymmetry is known to play an important role in the phase behavior of neat diblock copolymers. Matsen has shown by self-consistent field theory (SCFT) that the overall phase portrait is skewed toward fA > 1/2 when bA/bB > 1.99 This implies that the equilibrium interfacial curvature favors placing the block with the larger statistical segment length on the concave side of the domain boundary; i.e., the range of compositions that support A-domains with a spherical or cylindrical morphology is expanded in the phase portrait of χN versus fA. This seemingly counterintuitive result is actually easy to

Figure 19. Schematic illustration showing the influence of homopolymer on the interfacial curvature of a conformationally asymmetric (bA > bB) diblock copolymer in the limit of strong segregation. In the “dry brush” regime (α ≫ 1) the monolayer curves toward the block with the larger statistical segment length (A, red). Swelling of the blocks in the “wet brush” regime (α ≪ 1) inverts the curvature toward the more compact block (B, blue).

N/NAB influences the interfacial curvature in a system with wellseparated monolayers, near the unbinding transition, in the limit of relatively strong segregation. When α ≫ 1, the layers will separate without swelling of the blocks (often referred to as the “dry brush” limit),100−102 leading to an interface that curves toward the block with the larger statistical segment length for the reasons outlined above regarding neat diblock copolymers. For α ≪ 1 the diluent will instead swell the blocks, which can reverse the preferred interfacial curvature. Here we restrict our attention M


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Macromolecules to the case of volumetrically symmetric polymers, with fA = 1/2 and NA = NB = N, but note that one could also vary NA/NB or fA to tune interfacial curvature. In the system of interest, if the effect of swelling of both blocks by homopolymer dominates, we would expect to see spontaneous curvature away from the E domains and around the C domains because of the larger statistical segment length of poly(ethylene) (E) monomers when defined using effective monomers of equal volume. This in turn would be expected to shift the region of stability of the bicontinuous microemulsion to ϕH,C/ϕH > 1/2. The fact that we observe a congruent melting line at compositions ϕH,C/ϕH > 1/2 is thus consistent with either of two simplifying assumptions, namely that the value of ϕH,C/ϕH where the congruent melting line terminates near the BμE channel tends to lie either (i) near its value at the termination of the line of Ising critical points or (ii) near its value along the termination of the line along which the BμE coexists with two homopolymer-rich phases. The latter assumption is consistent only if one assumes that the sign of the preferred monolayer curvature is dominated by the effects of swelling by homopolymer. One experiment that could add to our understanding of the relationship among different features of the phase diagram, and guide the development of theory, would be to compare studies of two A/B/AB ternary systems that contain the same volumetrically balanced homopolymers but diblock copolymers with somewhat different values of fA. Changes in the relative volume of the two blocks of the copolymer would have no effect on the location of the critical point of the binary blend, and little effect on the location of the line of critical points that terminates at the binary critical point, but could be used to move the preferred location of the microemulsion channel to either side of the volumetric isopleth by varying the preferred curvature of copolymer monolayers. General Comments on Ternary Mixtures. The phase behavior for C/E/CE bears remarkable similarities to the thermodynamic properties of oil/water/surfactant mixtures, which are frequently depicted using an isothermal 2-dimensional slice, a constant surfactant concentration (“chi cut”) slice, or an isopleth (“fish cut”) through the 3-dimensional phase prism,2,4,103 similar to what is shown in Figures 2, 7, and 12, respectively. Surfactant-based systems are sensitive to the detailed molecular structure of the amphiphiles, including the precise molecular geometry and the interactions with water and oil. Furthermore, owing to the combined effects of van der Waals interactions and hydrogen bonding, many surfactant/water/oil mixtures exhibit both LCST and UCST behavior, which drives the bicontinuous microemulsion into a relatively small volume in phase space that spans a narrow range of temperatures; see for example the phase behavior of C14H30/H2O/C12E5.104 Nevertheless, the qualitative features found in this class of mixtures, including two- and three-phase regions and ordered lamellar and bicontinuous microemulsion phases, are closely related to the phase behavior of A/B/AB ternary polymer blends. In many respects, the polymer-based ternary blends are more experimentally controllable and more amenable to statistical mechanical theoretical treatment than the low molecular weight analogues. For example, by adjusting NA, NB, and NAB (at constant diblock copolymer composition), the composition associated with the bicontinuous microemulsion can be precisely tuned, and the TODT and critical temperatures of the diblock, binary homopolymer blend, and ternary mixtures can be adjusted to desired values. A single segment−segment interaction

parameter (χ) governs the diblock and homopolymers, greatly simplifying theoretical modeling. We believe these considerations make A/B/AB polymer systems ideal platforms for exploring the fundamental consequences of fluctuation effects on this fascinating class of condensed matter materials at the point of coincidence between the Brazovskii and Ising universality classes.

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CONCLUSIONS This report describes the detailed phase behavior of a C/E/CE ternary polymer blend in the ordered/disordered and macroscopically phase-separated regions of the phase prism. We establish the importance of mapping the full 4-dimensional (ϕCE−ϕC−ϕE−T) phase prism by using a number of characterization methods. Central to this finding is the development of a light transmission technique that is able to distinguish between direct (congruent) and indirect (i.e., mediated by a two-phase window) lamellar-to-disorder transitions with variations in temperature. The assignment of morphology and quantification of phase boundaries were further established using SAXS and TEM experiments. Through the combination of these characterization techniques, we have established the existence of a line of congruent first-order lamellar-to-disorder transitions, called the congruent isopleth, which terminates at the BμE channel. The trajectory of the congruent isopleth appears to be influenced by the critical composition of the C/E binary homopolymer blend (ϕH = 1, ϕC = 0.63). We propose that the line of congruent transitions identified at ϕC/ϕH ≈ 0.6 balances the interfacial curvature of a conformationally asymmetric diblock copolymer as the homopolymer content increases toward the limit at which the BμE forms. Along the congruent isopleth, the BμE window, separating the ordered/disordered and macroscopically phaseseparated regions of the phase prism, is significantly reduced to ΔϕH < 0.01, which is the narrowest reported to date.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01872. Figures S1−S8 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected]. *E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the National Science Foundation under Awards DMR-1104368 (F.S.B.) and DMR-01206459 (T.P.L.). Portions of this work were performed at both Sector 12ID-B and the DuPont−Northwestern−Dow Collaborative Access Team (DND-CAT) located at Sector 5 of the Advanced Photon Source (APS). DND-CAT is supported by E.I. DuPont de Nemours & Co., The Dow Chemical Company, and Northwestern University. Use of the APS, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract DE-AC0206CH11357. Parts of this work were carried out in the Characterization Facility, University of Minnesota, which N


Article

Macromolecules

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receives partial support from NSF through the MRSEC program. We thank James Lettow for helping with SAXS and DMS sample preparation and characterization.

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