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Thermodynamic and Morphological Behavior of Block Copolymer Blends with Thermal Polymer Additives Daniel F. Sunday,*,† Adam F. Hannon,† Summer Tein,‡ and R. Joseph Kline† †

Materials Science and Engineering Division, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, United States ‡ McKetta Department of Chemical Engineering, University of Texas, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: Block copolymer (BCP) blends offer a facile route toward customizable nanomaterials. To better understand these systems, the thermodynamics of polystyrene-bpoly(methyl methacrylate) (PS-b-PMMA) and poly(vinylphenol) (PVPH) blends were examined using scattering measurements and self-consistent field theory (SCFT). PVPH hydrogen bonds to the PMMA block, resulting in the selective infusion into the PMMA layer. Measurements on initially disordered blends show that this interaction can induce an order− disorder transition (ODT). The ODT was observed to be a continuous transition, unlike the first-order thermal ODT typically observed in BCPs. Free energy curves extracted from SCFT simulations also observed a lack of a discontinuity in the first-order derivative. Lamellar systems underwent a greater increase in BCP period compared to identical athermal systems due to the extension of the PMMA chains away from the interface. Comparison with the SCFT models finds good agreement in the predicted behavior of the blends using a negative χ parameter to model interactions between PVPH and PMMA, including the predicted distribution of the PVPH throughout the PMMA layer.

■

INTRODUCTION

Entropic forces primarily dictate the behavior of AB/B blends. The key parameters are η, which is the ratio of the homopolymer degree of polymerization (NHB ) to the degree of polymerization of the corresponding block in the BCP (NBCP B ), and the homopolymer volume fraction (f H).10−12 When η ≥ 1, there is a significant entropic penalty for interpenetration of the homopolymer into the BCP chains, and the homopolymer segregates to the interior of its corresponding block. The pitch of the system expands in proportion to the amount of homopolymer added until a critical concentration (Φc) is reached and macrophase separation occurs.13 When η < 1, the entropic penalty for mixing with the BCP chains is reduced, and the homopolymer becomes uniformly distributed throughout its corresponding block. The interdigitation of the homopolymer into the BCP expands the area per junction (aj) at the interface between the two blocks and swells the chains of the resident block perpendicular to the interface. At very low volume fractions of homopolymer and η ≪ 1 this can result in a reduction in the overall pitch.10 For example, blends of PS with PS-b-polyisoprene (PS-b-PI) resulted in contraction of the PI block that exceeded the expansion of the PS block for f H
1 results in limited homopolymer solubility, the PAA/Pluronic blend had no observable macrophase separation at f H = 0.5 and η > 10. The molecular mass of the additive did have an impact on the change in pitch, with higher molecular mass resulting in a greater increase compared to lower molecular mass at equivalent f H.8 Similar results were observed for the addition of PVPH to poly(methyl methacrylate-b-vinylpyrrolidone) (PMMA-b-PVP).16 In this system PVPH hydrogen bonds to both the PMMA and PVP blocks, but it bonds more strongly to PVP. This results in a closed loop phase diagram, where at low or high f H there is a disordered phase and for 0.16 < f H < 0.67 there was an emergence of an ordered microphase. For the AB/C blend there is some evidence that η plays a role in determining the f H at which the system undergoes an order−order transition (OOT). PVPH with a range of molecular mass was added to asymmetric polystyrene-bpoly(2-vinylpyridine) (PS-b-P2VP).17,18 For three different values of η the system was seen to undergo a transition from spheres → lamella → cylinder → spheres, and the volume fraction at which the transition occurred depended on η. The phase window for the lamellar morphology was expanded for η > 1 compared to η < 1, which resulted in a wider window for the cylindrical morphology. It is unclear how much this behavior depends on η compared to the strength of the hydrogen-bonding interactions between the homopolymer and its corresponding block. It is well-known that when BCPs are placed in a selective solvent for one of the blocks, the chains in that block stretch.19 The degree of stretching and distribution of the added component affect the degree of curvature at the interface, potentially shifting the location of the morphological transitions. This behavior was taken advantage of to design a highly asymmetric BCP blend with a lamellar morphology.20−22 Blending PS-b-P2VP with PS-b-PVPH (where for both materials the PS layer was the majority block) resulted in a lamellar morphology even though the native BCPs were

cylindrical The chain stretching in the blended layer induced by the strong hydrogen bonding between the PVPH and P2VP blocks suppressed the interfacial curvature and resulted in the asymmetric lamellar structure. Several counterintuitive results were observed when PS-bPVPH was blended with either P2VP or PMMA.23 The addition of P2VP resulted in the observation of the usual morphologies, but the addition of PMMA expanded the lamellar phase window significantly compared to P2VP. Instead of transitioning from lamellae to cylinders (or gyroid), the PSb-PVPH/PMMA transitioned from lamellae to distorted lamellae at f H = 0.4 and remained distorted lamellae up to f H = 0.7, the highest concentration examined. Even more unusual was that the pitch was observed to shrink compared to the native BCP, up to f H = 0.7, which resulted in L/L0 = 0.77, where L is the perturbed pitch of the BCP/homopolymer blend and L0 is the pitch of the native BCP. Some of this reduction was explained by the decrease in LPS (length of the PS layer), resulting from the increase in aj, but after an initial increase at f H = 0.1, LBl (length of the blended layer) also decreases, even at high f H. This system was examined using self-consistent field theory (SCFT) simulations with two different model considerations: one with negative χ parameters and one using polymer−polymer complexes to model hydrogen bonding.24 The negative χ model predicted that L would increase with the addition of homopolymer, whereas the complexation model predicted that L would decrease. This study examines PS-b-PMMA/PVPH systems to better understand how the addition of a selectively associating additive affects both the ODT and the morphological behavior in the system. Both small-angle neutron (SANS) and X-ray (SAXS) scattering measurements were conducted on PS-b-PMMA/ PVPH blends over a range of f H and η. This enables characterization of both the morphology and spacing as a function of homopolymer concentration. Coupling these results with SCFT calculations provides an improved understanding of how added homopolymer can impact blend morphology and pitch.

■

MATERIALS AND METHODS

Experimental Methods. All polymers were acquired from Polymer Source25 and used as received; the properties of all materials used in this study are shown in Table 1. Propyl glycol methyl ether acetate (PGMEA) was acquired from Sigma-Aldrich. The blends were

Table 1. Labeled Names of Polymers Used, Corresponding Molecular Masses, Volume Fraction of PS f PS for BCPs, and PDI (Polydispersity Index)

B

sample name BCP

molecular mass[kg/mol] PS-b-PMMA

f PS volume fraction PS

PDI

dPS11-bPMMA8 dPS37-bPMMA46 PS10-bPMMA10 PS25-bPMMA26 PS47-bPMMA40 homopolymer

11-b-8.5

0.59

1.07

37-b-46

0.47

1.06

10-b-10

0.53

1.09

25-b-26

0.52

1.08

47-b-40

0.57

1.08

molecular mass [kg/mol]

f PS volume fraction PS

PDI

PVPH7 PMMA7

7 7

not applicable not applicable

1.12 1.05

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Macromolecules prepared by dissolving the desired amount of PS-b-PMMA and PVPH into PGMEA at a mass fraction of ΦM ≅ 0.04. The solution was then drop-cast and allowed to dry for 48 h, then placed in a 160 °C vacuum oven for 5 days, and finally annealed at 200 °C under vacuum for 30 min to ensure the removal of any residual solvent. SANS. Measurements were conducted at the National Institute of Standards and Technology Center for Neutron Research (NCNR) on the NGB 10 m SANS instrument. Neutron wavelengths λ = 10 and 5 Å were used at sample−detector distances of 5.2 and 4.6 m, respectively. This resulted in an available q range of 0.01−0.64 Å−1 (q = 4π sin(θ/2)/λ). The sample was placed on a heating stage and raised to the desired temperature, followed by equilibrating for 30 min, after which the SANS pattern was acquired. SAXS. Measurements were conducted at 5ID-D at the Advanced Photon Source at Argonne National Laboratory.26 The beam had an incident energy of 17 keV (λ = 0.729 Å) and a spot size of 70 μm × 120 μm; images were collected on a MAR-CCD detector with a pixel size of 79.14 μm and a sample−detector distance of 4.51 m. Collection times ranged from 10 to 60 s. SCFT Simulations. In order to better explore the trends observed for changes in periodicity and morphology with increased homopolymer volume fraction, SCFT simulations were performed.27−33 In SCFT, the polymers are modeled in terms of local density values and their conjugate chemical potential fields. By varying thermodynamic variables, SCFT can predict the equilibrium phase behavior of polymer blends under a variety of boundary conditions. In our study, we consider a model with three species among two chains system with an incompressible blend of an AB copolymer with a C homopolymer (here A is the model designation for PS, B for PMMA, and C for PVPH). In the model, the total partition function Z of the system is written as Z=

∫ e−H[ρ,Ω]/kT DρDΩ

For a three species system with an AB diblock and C homopolymer, the Hamiltonian is given as

∫ dr(⃗ χAB N(ϕA − fA )(ϕB − fB ) + χBC N(ϕB − fB )

H[ϕ , Ω] = G(

× (ϕC − fC ) + χAC N (ϕA − fA )(ϕC − fC ) − ΩA ϕA − Ω BϕB − ΩCϕC − Ω P(1 − ϕ+)) − V ln(Q AB[ΩA , Ω B]) − V ln(Q C[ΩC]))

where ϕX is the normalized density for species X (i.e., ϕX = ρX/ρ0 where ρ0 is the average monomer density), N is a reference degree of polymerization for the system (chosen as the AB diblock N in this study), G is a dimensional proportionality constant, G = kTρ0Rg3/N with Rg being the radius of gyration of the AB block, χXY is the Flory− Huggins interaction parameter between the respective XY species pair, f X is the volume fraction of species X, ΩP is the pressure field which is independent of Ω A , Ω B , and Ω C and is used to enforce incompressibility, ϕ+ is the sum of all normalized densities (i.e., ϕ+ = ϕA + ϕB + ϕC), V is the grid volume of the system (i.e., V = NxNyNz), QAB is the single chain partition function for the AB diblock, and QC is the single chain partition function of the C homopolymer. These partition functions are solved for by solving for their corresponding propagator functions qAB and qC such that

δH = 0 and δΩ

1 V

Q AB[ΩA , Ω B] = Q C[ΩC] =

1 V

∫ dr ⃗ qAB(r ⃗, s = NS,AB , Ω)

∫ dr ⃗ qC(r ⃗, s = NS,C , Ω)

(4)

where s is a chain position variable that varies from 0 to NS,Z along a given chain where s = 0 corresponds to the first statistical segment of the chain and NS,Z to the last statistical segment of component Z, where NS,Z is the number of statistical chain segments used to partition chain Z in the simulation. The chains are assumed to be Gaussian coils for the polymers used in experiment, and thus the propagators are solved from the following Fokker−Planck modified diffusion equations

(1)

where ρ is the set of density fields for each species (i.e., ρ ≡ {ρA, ρB, ρC}), Ω is the set of chemical potential fields for each species (i.e., Ω ≡ {ΩA, ΩB, ΩC}), k is the Boltzmann constant, T is the temperature, D is the functional integral differential operator, and H is the Hamiltonian of the system as a functional of ρ and Ω. ρ and Ω are themselves functions of position r⃗ = {x, y, z} which corresponds to grid points on an Nx by Ny by Nz lattice in the simulations. The goal of SCFT is to find sets of ρ = ρ* and Ω = Ω* that satisfy the saddle point equations

δH δρ

(3)

∂qAB( r ⃗, s , Ω) ∂s ∂qC( r ⃗, s , Ω)

=0

∂s

=

=

2 R g,AB

NAB 2 R g,C

NCH

∇2 qAB( r ⃗, s , Ω) − Ω(ϕ( r ⃗, s , Ω))qAB( r ⃗, s , Ω) ∇2 qC( r ⃗, s , Ω) − Ω(ϕ( r ⃗, s , Ω))qC( r ⃗, s , Ω)

(2)

(5)

When ρ and Ω correspond to a solution of these equations, H corresponds to the normalized free energy F of the system. This free energy is calculated with the reference state of a fully disordered homogeneous system having F = 0 and written as ΔF (positive energies are disordered and negative energies are ordered). The SCFT solutions can be either equilibrium (global minimum) or metastable (local minima, saddle point, or maxima). Verification of what kind of solution is found is confirmed by comparing free energies from multiple simulations, and the lowest free energy solution is chosen as the candidate for the global equilibrium structure.

with initial conditions qAB(r⃗, 0, Ω) = 1 and qC(r⃗, 0, Ω) = 1. The chemical potential field Ω here corresponds to the current chain type such that

ρ*

Ω*

⎧ ⎪ ϕ ( r )⃗ = ⎪ A ⎪ ⎪ ϕ( r )⃗ = ⎨ ϕB( r )⃗ = ⎪ ⎪ ⎪ ⎪ ϕC( r )⃗ = ⎩

ρA ( r )⃗ ρ0 ρB ( r )⃗ ρ0 ρC ( r )⃗ ρ0

=

=

=

⎧ s < fA ⎪ ΩA Ω=⎨ ⎪ ⎩ Ω B s ≥ fA

for the diblock and simply Ω = ΩC for the homopolymer. The normalized densities ϕ are

fAB

∫0

Q AB fAB Q AB fC QC

(6)

∫f

∫0 C

1

fA

1

† ds qAB ( r ⃗, 1 − s , Ω)qAB( r ⃗, s , Ω)

† ds qAB ( r ⃗, 1 − s , Ω)qAB( r ⃗, s , Ω)

A

ds qC†( r ⃗, 1 − s , Ω)qC( r ⃗, s , Ω)

(7)

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Table 2. Tabulated L0 Values for the Four Different NAB with Respective f B AB Diblocks Used in Simulation Calculated Either by the Simulated Free Energy Curves or by Extrapolation of the Data to f C = 0a

a

exptl BCP

dPS11-b-PMMA8

PS10-b-PMMA10

PS25-b-PMMA26

PS47-b-PMMA40

NAB fB free energy L0 normalization L0

183 0.42 undefined (3.02 ± 0.01)Rg

196 0.47 undefined (3.02 ± 0.01)Rg

500 0.48 (3.64 ± 0.02)Rg (3.64 ± 0.01)Rg

811 0.41 (4.17 ± 0.01)Rg (4.17 ± 0.01)Rg

Uncertainty reported is from performing on fits within the standard deviation of the free energy calculations.

where the backward propagators q†AB and q†C are found by solving the Fokker−Planck modified diffusion equations starting from the opposite chain end (i.e., s = NS,Z) and fAB is the total fraction of AB diblock in the system such that fAB + f C = 1. These equations can be found noting that the saddle point equation δH/δΩ = 0 is in general −ϕZ − δ(GV ln(QZ))/δΩ = 0 → ϕZ = −(GV/QZ)(δQZ/δΩ) and that −GV(δQZ/δΩ) = f Z∫ q†ZqZ ds. A pseudospectral method is used to solve the Fokker−Planck equations for convenience as no a priori assumptions on the morphology solutions are necessary.28,30 Simulations are initiated using random field values assigned to the Ω fields. Solutions are converged toward using a combination of steepest descent and complex Langevin dynamics. The first half of the simulation relaxes the fields with complex Langevin noise that helps kick the fields out of local minimum states and progress toward the global minimum. The second half of the simulation takes the lowest energy field solution found and lets the fields finish relaxing with the steepest descent scheme to ensure the saddle point conditions are satisfied (i.e., finds the actual mean field solution). The simulation time step schemes for this steepest descent algorithm is accomplished by taking the SCFT condition (δH/δΩ)|Ω* = 0 and relating a linear combination of the Hamiltonian field derivatives to time derivatives such that

δ ΩX = dt

⎧ X if j = 1 ⎪ δH and Wj = ⎨ Y if j = 2 ∑ λj δϕW ⎪ j=1 j ⎩ Z if j = 3

using the equation ΩP = κ(ϕ+ − 1) where κ is a compressibility constant that in the limit κ → ∞ would result in a theoretical completely incompressible system. However, because of numerical instability issues, κ was fixed to a value of 100 where the observed field values are approximately incompressible but such that no divergence in the fields is observed. Larger values of κ could be used by changing the values of λj, but since the set found works well no further optimization of these parameters were performed as changing λj further would reduce convergence speed. Two sets of simulations were performed to elucidate two different aspects of the copolymer and homopolymer blend physics. 1D simulations were performed to extract the theoretical lamellae periodicity behavior with increasing homopolymer volume fraction. 3D simulations were preformed to examine the phase behavior of the blended systems. Unit cell calculations in this study are performed on either 1D or 3D grids of size Nx = 30 or Nx = Ny = Nz = 16. All simulations assumed periodic boundary conditions. As a check, simulations were conducted examining an AB/B system, and the results were compared with those of prior simulations.34 The results of these simulations are included in the Supporting Information (see SI 1). For comparing the experimental system, the following set of parameters were chosen: χAB = 0.03 based on experimental ranges of 0.02−0.04 for PS and PMMA,35 χAC = 0.12 based on experimental ranges of 0.11−0.13 for PVPH and PS,35 four different NAB values corresponding to four molecular masses tested in the experiment (NAB = 183, 196, 500, and 811 with respective corresponding volume fractions f B = 0.42, 0.47, 0.48, and 0.41, which then correspond to the dPS11-b-PMMA8, PS10-b-PMMA10, PS25-b-PMMA26, and PS47-bPMMA40, two different NHB for the B or PMMA homopolymer of 6.5 kg/mol with the first corresponding to the actual molecular mass NHB = 65 and the second NHB = 200 for comparison, and similarly two different NC for the C or PVPH homopolymer of 6.5 kg/mol with the first corresponding to the molecular mass NC = 54 and the second NC = 200 for comparison. For χBC, four different values were tested to see what best modeled the expected hydrogen-bonding interactions. Previous studies suggest that negative χ values model hydrogen bonding well36−40 while still other studies suggest more complicated complexation models are necessary to capture the behavior of the hydrogen-bonding interactions between the blocks.24 Here we test values of χBC = 0, −0.01, −0.03, and −0.05 and only consider the simple negative χ value model based on the observed findings agreeing relatively well with the experimental observations. The results of all the 1D simulations with the different χBC, NAB, and NC or NHB are detailed in the Supporting Information (see SI 2). For normalization purposes, L0 values for the bulk AB diblocks were calculated as follows. Simulations varying Λ, the assigned unit cell length, around the expected L0 from strong segregation theory were performed and free energy curves found and fitted (see SI 1 for details on these fits). These resulted in values of L0 ≅ (3.64 ± 0.02)Rg for NAB = 500 and L0 ≅ (4.17 ± 0.01)Rg for NAB = 811. The NAB = 183 and 196 polymers have χN values of 5.5 and 5.9, respectively, lower than the ODT of χN = 10.5 and did not form ordered structures, so values calculated for those structures are reported as undefined. To have normalization values for the two pure cases below the ODT, an extrapolation to f C = 0 for simulation results with f C > 0 was performed and is detailed in the Supporting Information (see SI 3). These fit values are reported for those two diblocks in the normalization L0 column. Ideally just the free energy L0 values

3

(8)

where the set of constants λj are chosen to be large enough to make the system converge fast in as few iterations as possible but small enough to ensure the simulation is numerically stable. Wj is used to demark the species whose field is being updated, where X represents that species and Y and Z are the other species. Additionally, the relative values of the constants are chosen to keep the simulation numerically stable (usually such that λ1 > λ2 = λ3). Every step in time t is one iteration forward in the simulation. For the study, the time constant values were set as λ1 = 0.014 with λ2 = λ3 = λ1/5. The analytical form of the functional derivatives of H with respect to ϕX is

δH = χXY N (ϕY − fY ) + χXZ N (ϕZ − fZ ) − ΩX + Ω P δ ΩX

(9)

Inserting these into the time derivative equation and rewriting things as a numerical iteration update scheme yields 3

ΩX , J + 1 = ΩX , J +

∑ λj(χWjWkj N(ϕWkj − fWkj ) j=1

+ χW W N (ϕW − fW ) − ΩWj + Ω P) j mj

mj

mj

with

⎧ 3 if j = 1 ⎧ 2 if j = 1 ⎪ ⎪ kj = ⎨ 3 if j = 2 and mj = ⎨1 if j = 2 ⎪ ⎪ ⎩ 2 if j = 3 ⎩1 if j = 3

(10)

where J denotes the Jth iteration. A term ξ(r⃗) is added to each field which is Gaussian distributed random noise when the complex Langevin dynamics is implemented. The pressure field ΩP is updated D

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Figure 1. (A) SANS for PVPH7/dPS11-b-PMMA8 blends as a function of mass fraction PVPH7; curves are shifted vertically for clarity. (B) SANS for dPS11-b-PMMA8 as a function of mass fraction PMMA7; curves are shifted vertically for clarity. (C) fwhm for dPS11-b-PMMA8 blended with PVPH7 (●) or PMMA7 (■) as a function of the mass fraction of the homopolymer additive.

PVPH7 blends show a steady decrease in fwhm until ΦH > 0.05, at which point it begins to level off and shows only a small decrease throughout the remainder of the lamellar concentration range. For the thermal ODT the fwhm will show a discontinuity at the transition temperature, a signature of a firstorder transition. In this case the fwhm gradually decreases with no indication of a discontinuity, indicating that the PVPHinduced ODT is a continuous higher order transition. Additionally temperature-dependent SANS was conducted on the PVPH7 blend series (shown in the Supporting Information). Below ΦH = 0.05 the intensity of the scattering increased with decreasing temperature; at and above that concentration there is no longer a temperature dependence on the scattering within the measured range.

would have been used, but since the lower molecular mass values were undefined, this method was chosen.

■

RESULTS AND DISCUSSION In order to examine the behavior of an AB/C type blend in the vicinity of the ODT, dPS11-b-PMMA8 was blended with PVPH7, and the resulting blends were examined using SANS. dPS11-b-PMMA8 has a low enough molecular mass to be disordered in its native state; χN at 160 °C was calculated to be 5.4 (assuming χ ≅ 0.03).41 This is well below the ODT, which was found to be (χN)ODT = 13.2 once the fluctuation corrections and the finite chain length were accounted for (eq 11) (where N̅ = ρ02b6N).42 SANS measurements were conducted on PVPH7/dPS11-b-PMMA8 blends with the homopolymer mass fraction (ΦH,) ranging from 0 to 0.32, the results of which are shown in Figure 1A. The native BCP shows the broad correlation hole scattering associated with a disordered state. As PVPH7 is added, the peak begins to narrow and shift to lower q. At ΦH = 0.095 there is the emergence of a peak at 3q*, indicating that the system was now in the lamellar phase. Lamella with equal volume fractions in each phase result in a minima in the form factor at 2q*. A shift away from equivalent volume fractions results in the emergence of the second-order scattering peak as observed in the measurements. As homopolymer continues to be added, there eventually emerges a peak at 2q*, indicating the presence of an asymmetric lamellar morphology, consistent with the blended phase having a larger width than the PS phase. At ΦH = 0.32 peaks are now observed at √3q* and √7q*, indicating a transition to a cylindrical morphology. In order to verify that the ODT observed in this system results from interactions between PVPH and PMMA, and not simply from the presence of homopolymer, we also examined blends of dPS11-b-PMMA8 with PMMA7; Figure 1B shows the results of SANS measurements on these blends. For all concentrations in the PMMA7 blends the correlation hole scattering observed for the native BCP is present. Figure 1C shows the fwhm of the firstorder peak as a function of concentration for blends with both PMMA7 and PVPH7. The blends with PMMA7 show no change in fwhm over the concentration range measured, while the

(χN )ODT = 10.495 + 41.0N̅ −1/3 + 123.0N̅ −0.56

(11)

The 3D SCFT simulations also show a continuous change in free energy with the addition of the homopolymer. To examine the nature of the ODT induced by adding C homopolymer to the low degree of polymerization AB BCPs, plots of the free energy difference and its derivative with respect to f C were made for the NAB = 196 case. These plots are shown in Figure 2. Both ΔF and the first derivative of ΔF are continuous across the region the phase transition from disorder to order was observed (between f C = 0.05 and f C = 0.09). These plots give further credence to the idea the homopolymer added induced ODT is continuous in nature rather first order. In order to explore the phase behavior of AB/C type blends, measurements were conducted on samples with ΦH ranging from 0 to 0.7 and η ranging from 0.15 to 0.74. Representative SAXS curves from concentration series prepared with BCPs having three different molecular masses are shown in Figure 3. Figure 3A shows SAXS images from blends of PS10-b-PMMA10 with PVPH7, where PS10-b-PMMA10 was initially disordered. Similarly to the blend series investigated with SANS, the PS10b-PMMA10/PVPH7 blend transitions from a disordered to ordered state at ΦH ≅ 0.05, maintaining a lamellar composition until ΦH ≅ 0.25, at which point a transition to a cylindrical morphology was observed. For ΦH ≥ 0.5 the blend exhibits a E

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as the result of either a narrow phase window or due to the polydispersity of the materials. For the lower molecular mass polymers (η = 0.64 and 0.76), the morphology starts out disordered, transitions to disordered PMMA-rich spherical micelles, PMMA-rich cylinders, lamellae, perforated lamellae/ gyroid, PS-rich cylinders, and a metastable region of PS-rich cylinders and spheres. The general trends observed are similar to experiment, although PMMA-rich spheres and cylinders predicted near the ODT of the lower molecular mass BCP were not directly observed in experiment. Since the SCFT calculations did not model fluctuation effects explicitly, these phases observed are likely transitional phases seen normally going from the disordered phase space region into the ordered region for asymmetrical systems in the canonical mean field phase diagram. More discussion of this possibility is included in the Supporting Information (see SI 4). For blends exhibiting a lamellar morphology the pitch was extracted and normalized by the pitch of the native BCP; these changes in L/L0 as a function of ΦH can be seen in Figure 5A. In order to place these results in context with AB/B blends, measurements were also conducted on PMMA7/PS47-bPMMA40 blends; the pitch of these blends was found to expand slowly with the addition of homopolymer, at a rate of 0.27(L/L0)/ΦH. This result was consistent with previous studies of AB/B blends.10 L/L0 expands considerably faster for the PVPH blends, at rates of 1.06(L/L0)/ΦH for η = 0.15 and 0.25. The rate of expansion was slightly faster for η = 0.65, where the slope was found to be 1.23(L/L0)/ΦH. (This slope does not include volume fractions below 0.05 for η = 0.65 as the material is disordered in this region.) More insight can be gained into this system by examining the change in aj as a function of added homopolymer. From the pitch the relative change in aj compared to the native aj0 can be calculated, using eq 12a to calculate aj and eq 12b to calculate aj/aj0.10

Figure 2. Plots of the normalized free energy difference ΔF/kT and its first two derivatives with respect to f C for the simulation conditions of NAB = 196 and χBC = −0.05. Red dots are the calculated points from the simulation, and blue lines are to guide the eye.

spherical morphology. The same trends are observed for blends of PVPH7 with PS25-b-PMMA26 and PS47-b-PMMA40, except that the native state for both of these systems was ordered lamella. The results of the SAXS measurements are summarized in Figure 4A, which shows the phase behavior of the blends as a function of ΦH. For all values of η examined the system transitions from lamellar to cylindral around ΦH = 0.25 and cylindrical to spherical around ΦH = 0.45. The experimental results are compared to a phase diagram developed from SCFT calculations, which is shown in Figure 4B. The results shown here are for χBC = −0.05; a number of different values for χBC were tested to determine what values best matched the experimental results (see SI S2). For the higher molecular mass BCPs (η = 0.16 and 0.23), as ΦH increases the morphologies transition from lamellae to PS phase perforated lamellae (likely gyroid at larger unit cell calculations) to PS-rich cylinders to PS-rich spheres. The SCFT results predict the morphology will remain lamellar at higher homopolymer concentrations than is observed in the experimental results. Additionally, neither the perforated lamella nor gyroid phases were experimentally observed, likely

aj =

2MPS NAvρPS (1 − fBl )L

(12a)

Figure 3. Representative SAXS curves for blends of PVPH7 with PS10-b-PMMA10 (A), PS25-b-PMMA26 (B), and PS47-b-PMMA40 (C) as a function of concentration. The morphology and PVPH7 mass fraction are indicated in the labels at the right end of each curve (D = disordered, L = lamellar, C = cylindrical, S = BCC spherical). F

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Figure 4. (A) Phase diagram for PS−PMMA/PVPH blends as a function of η and PVPH mass fraction ΦH. (B) Phase diagram of 3D morphologies observed in SCFT simulations. Morphologies that were somewhat metastable (very close in free energy) are shown with the lower free energy structure as the corresponding symbol and the color as the corresponding metastable structure. These results are thus metastable PS spheres with stable PS cylinders as green triangles, metastable perforated lamellae with stable lamellae as red ×, and metastable perforated lamellae with stable PS cylinders as yellow ×. (C) Example morphologies for each type found in the unit cell calculations showing the PS regions in cyan and the PMMA and PVPH regions in magenta. Many of these morphologies were slightly metastable (i.e., similar free energies) with a neighboring competing morphology in the simulations; as such, the morphology with the lower free energy is designated on top of that morphology in the phase diagram legend.

Figure 5. L/L0 (A) and aj/aj0 (B) for PMMA7/PS46-b-PMMA40 (blue ▲), PVPH7/PS47-b-PMMA40 (red ◆), PVPH7/PS25-b-PMMA26 (red ●), and PVPH7/PS10-b-PMMA10 (red ■). Simulation results for L/L0 (C) and aj/aj0 (D) results for NHA = 65/NAB = 811/η = 0.20 (blue ■), NHA = 175/NAB = 811/η = 0.53 (blue ▲), NC = 54/NAB = 811/η = 0.16 (red ●), NC = 54/NAB = 500/η = 0.23 (red ◆), and NC = 54/NAB = 196/η = 0.59 ( red ■).

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DOI: 10.1021/acs.macromol.6b00651 Macromolecules XXXX, XXX, XXX−XXX

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Figure 6. 1D density plots for simulated blends of C homopolymers with AB diblocks for f C = 0.01−0.29, NAB = 196 or 811, NC = 54 or 200, and χBC = −0.01 or 0.05. Blue dotted are the A block, red dashed lines the B block, green solid lines the C homopolymer, and magenta dotted/dashed lines the sum of the B block and C homopolymer densities. The sum of the densities is approximately ϕ+ ≅ 1 from incompressibility.

aj a j0

=

fractions).43 As a result, there will be PVPH chains near the interface, which should result in an expansion in the distance between the junction points. This suggests that the PMMA chains stretch away from the interface, reducing their crosssectional area and counterbalancing the presence of the PVPH. The stretching of the PMMA chains perpendicular to the interface also explains the larger L/L0 scaling relative to the PMMA additive. Figures 5C and 5D show the SCFT simulation results of L/ L0 and aj/aj0, respectively (aj/aj0 was calculated from eq 12b). Two different parameter sets were examined for the AB/B reference system. The first had an equivalent η to the PMMA7/ PS47-b-PMMA40 blend (η = 0.20) and resulted in a decrease in L/L0 over the entire range of concentrations examined. In order to obtain comparable results to the experimental system, η was increased to 0.53 (NHB = 175). Preliminary calibration results of the homopolymer blends agreed with previous results by Matsen34 which are shown in the Supporting Information (see SI 1). This resulted in a difference between the simulated and experimental results, potentially due to the proximity to the crossover in stretching behavior in the SCFT results (shown in Figure S2) or the mild polydispersity on the experimental systems. For the AB/C system NHC = 54 provided an excellent match to the experimental result.

L0(1 − fPMMA ) L(1 − fBl )

(12b)

Here MPS is the molecular mass of PS, NAv is Avogadro’s constant, and ρPS is the density of PS. Figure 5B show the results of aj/aj0 for both the BCPs blended with PVPH7 and PMMA7. The PMMA7/PS47-b-PMMA40 blends show a gradual increase in aj/aj0, which is consistent with an expansion in the distance between junction points due to complete intermixing of the homopolymer and the BCP. The expansion in aj is attributed to the presence of homopolymer chains at the interface, which increase the average distance between junction points. In contrast, for the η = 0.15 and 0.25 PVPH blends aj/ aj0 shows a slight reduction at low ΦH before slowly trending upward. A similar trend is observed for η = 0.65 where after the immediate drop a slow decrease is observed up until ΦH ≅ 0.1, after which it begins to drift back upward. While this would be consistent with a segregation of the homopolymer to the interior of the block (leaving the interface undisturbed) previous results have shown that at least for ΦH < 0.1 the ratio of PVPH/PMMA will be constant throughout the PMMA block (SCFT calculations, discussed later in the paper, also indicate that the PVPH volume fraction will remain proportional to that of the PMMA block at low additive volume H

DOI: 10.1021/acs.macromol.6b00651 Macromolecules XXXX, XXX, XXX−XXX

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larger η (η > 1 for both systems examined).44 The SCFT results suggest that χPMMA−PVPH ≅ −0.05; for comparison, a similar process was used to estimate χP2VP−PVPH and χP4VP−PVPH, which were found to be ≈−0.075 and ≈−0.17, respectively.22 The estimated χ value of the hydrogen-bonding pairs becomes increasing negative with the formation of stronger hydrogen bonds, as PMMA is considered the weakest hydrogen bond acceptor of the three, followed by P2VP and P4VP. The results shown here are starkly different from those observed in the studies by Chen et al., where the addition of PMMA or P2VP to PS-b-PVPH was shown to result in a decrease in the overall pitch.23 Their system was an inverted case of the system we examined, where the polymers with the most unfavorable interaction parameters were chemically connected and a hydrogen-bonding acceptor was added to the blend (whereas in this study a hydrogen-bonding donor was added). They observed a decrease in the pitch as a function of the added homopolymer, which corresponds to increase in aj. At the highest volume fractions aj becomes almost 80% larger than the native BCP. This situation should be thermodynamically unfavorable, as the increasing aj leads to greater interfacial area between the two phases and significant distortion of the polymer chains from their native conformations, both factors which will lead to increase the free energy of the system. This suggests that the homopolymer was localized to the interface in their system, resulting in the increase in aj and decrease in L/L0. This suggests that the chain architecture in this type of system may impact how the homopolymer is distributed throughout the blocks. PS-b-PVPH has a much higher χ than PS-b-PMMA (≈0.1 and 0.03, respectively).6,45 The homopolymer additive may prefer to segregate to the interface to mediate that unfavorable interaction between PS and PVPH. In the PS-bPMMA/PVPH system the estimated χ for PMMA/PVPH was ≈−0.05, which has a greater magnitude than the unfavorable interactions between PS and PMMA. This difference in the relative magnitude of the interaction parameters for the three components in the system could explain the different distribution of the homopolymer additive.

For the AB/C systems, the parameters used for the 1D simulations were equivalent to those in the 3D simulations used to examine the morphology (χBC = −0.05, NAB = 196, 500, 811, and NC = 54). All conditions examined show a sharp initial increase in L/L0 for f C < 0.05. For both η = 0.16 and 0.23 L/L0 then levels off and shows a more gradual increase throughout the remainder of the concentration range examined. For η = 0.59 L/L0 continues to increase more quickly until f C approaches 0.2, at which point it also levels off. Both simulated systems diverge from the experimental results around this concentration, where the experimental pitch continues to increase at a faster rate. Similarly the values of aj/aj0 extracted from the simulations qualitatively agree for lower volume fractions, although they diverge sharply as the concentration increases. This could be due to the system approaching the lamellar−cylindrical transition, which the 1D simulation cannot account for. The other potential divergence between the simulations and the experiment is the assumption in the SCFT calculation that the chains are Gaussian coils. The experimental results suggest that the PMMA/PVPH block becomes increasingly stretched with increasing PVPH concentration and can likely no longer be approximated by a Gaussian coil. In fact, the PVPH and PMMA hydrogen-bonded segments may be better modeled as a wormlike chain. A more in-depth exploration of the simulation results for different parameters is found in the Supporting Information (see SI S2). Overall, the trends predicted by the SCFT calculations for both the morphologies and pitches predict similar results to what was observed in the experimental systems. This reinforces the idea that the use of a negative interaction parameter in SCFT calculations can qualitatively predict the behavior of hydrogenbonding blends. The 1D simulation results allowed for exploration of the density profiles as shown in Figure 6. The parameter combinations includes NAB = 193 and 811, NC = 54 and 200, χBC = −0.01 and −0.05, and f C = 0.01−0.29. For all parameter combinations NAB = 196 was seen to be disordered at the lowest volume fractions. For χBC = −0.01 and η = 0.59 the system transitions to an ordered phase around f C = 0.09. Throughout this concentration series the homopolymer can be seen to be slightly enriched at the center of the blended layer, with the degree of segregation increasing as the concentration increases. For the same homopolymer at the same interaction strength in a higher NAB BCP (NAB = 811, η = 0.16) the homopolymer is predicted to be uniformly distributed; even at the highest concentration examined there is only a very small amount of localization of the homopolymer to the center of the blended layer. When these two series are compared to the results with a higher molecular mass homopolymer (identical BCP NAB and χBC), it is clear that the higher NC drives much greater segregation of the homopolymer to the center of its resident block. The shift to stronger interactions between the B and C blocks (χBC = −0.05) results in a more uniformly distributed homopolymer under all molecular mass combinations. In this case the only series where the homopolymer is predicted to be significantly localized within the blocks is for NAB = 196 and η = 2.17, and equivalent conditions were not explored in the experimental measurements. While no localization was observed in previous measurements or directly indicated in these measurements, there have been experimental results which show this occurring. SANS measurements of PAA in PEO−PPO−PEO show localization of the PAA in the center of the PEO phase, with the degree of localization increasing for

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CONCLUSIONS A combination of scattering measurements and SCFT calculations was used to investigate the behavior of PS-bPMMA blended with PVPH. This is a model system for investigating the thermodynamics of block copolymers blended with selectively associating polymer additives. Measurements on initially disordered PS−PMMA revealed that the addition of PVPH induces a continuous ODT. The continuous nature of this transition was also observed in the SCFT calculations. SAXS measurements on a series of blends with different molecular mass BCPs showed the morphological transitions typically observed for native diBCPs. The SCFT morphologies and phase transitions as a function of homopolymer mass fractions agreed qualitatively well with the experiment. The shift in L/L0 and aj for samples in the lamellar region were examined and compared to results from AB/B systems. The pitch of the PS−PMMA/PVPH blend increased much faster than the PS−PMMA/PMMA blend. The aj of the PS−PMMA/ PVPH blend was found to show little change below f C = 0.2 and in some cases decreased. These results indicate that the PMMA chains stretch away from the BCP interface upon the addition of PVPH. The changes in periodicity predicted by the SCFT calculations in the lamellar region agreed quantitatively well up to around f C = 0.2. The agreement between the SCFT I

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Macromolecules calcualtions and experiments suggests that a negative χBC parameter is an appropriate model for the hydrogen-bonding interactions with further model refinement needed for better quantitative agreement between the model and experiment.

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(6) Sweat, D. P.; Kim, M.; Larson, S. R.; Choi, J. W.; Choo, Y.; Osuji, C. O.; Gopalan, P. Rational Design of a Block Copolymer with a High Interaction Parameter. Macromolecules 2014, 47, 6687−6696. (7) Tirumala, V. R.; Romang, A.; Agarwal, S.; Lin, E. K.; Watkins, J. J. Well Ordered Polymer Melts from Blends of Disordered Triblock Copolymer Surfactants and Functional Homopolymers. Adv. Mater. 2008, 20, 1603−1608. (8) Tirumala, V. R.; Daga, V.; Bosse, A. W.; Romang, A.; Ilavsky, J.; Lin, E. K.; Watkins, J. J. Well-Ordered Polymer Melts with 5 Nm Lamellar Domains from Blends of a Disordered Block Copolymer and a Selectively Associating Homopolymer of Low or High Molar Mass. Macromolecules 2008, 41, 7978−7985. (9) Daga, V. K.; Watkins, J. J. Hydrogen-Bond-Mediated Phase Behavior of Complexes of Small Molecule Additives with Poly(ethylene Oxide- b -Propylene Oxide- b -Ethylene Oxide) Triblock Copolymer Surfactants. Macromolecules 2010, 43, 9990−9997. (10) Winey, K. I.; Thomas, E. L.; Fetters, L. J. Swelling of Lamellar Diblock Copolymer by Homopolymer: Influences of Homopolymer Concentration and Molecular Weight. Macromolecules 1991, 24, 6182−6188. (11) Shull, K. R.; Winey, K. I. Homopolymer Distributions in Lamellar Copolymer/homopolymer Blends. Macromolecules 1992, 25, 2637−2644. (12) Hashimoto, T.; Tanaka, H.; Hasegawa, H. Ordered Structure in Mixtures of a Block Copolymer and Homopolymers. 2. Effects of Molecular Weights of Homopolymers. Macromolecules 1990, 23, 4378−4386. (13) Koizumi, S.; Hasegawa, H.; Hashimoto, T. Ordered Structures of Block Copolymer/Homopolymer Mixtures. 5. Interplay of Macroand Microphase Transitions. Macromolecules 1994, 27, 6532−6540. (14) Winey, K. I.; Thomas, E. L.; Fetters, L. J. Isothermal Morphology Diagrams for Binary Blends of Diblock Copolymer and Homopolymer. Macromolecules 1992, 25, 2645−2650. (15) Bodycomb, J.; Yamaguchi, D.; Hashimoto, T. A Small-Angle XRay Scattering Study of the Phase Behavior of Diblock Copolymer/ Homopolymer Blends. Macromolecules 2000, 33, 5187−5197. (16) Chen, W.-C.; Kuo, S.-W.; Jeng, U.-S.; Chang, F.-C. SelfAssembly through Competitive Interactions of Miscible Diblock Copolymer/Homopolymer Blends: Poly(vinylphenol- b -Methyl methacrylate)/Poly(vinylpyrrolidone) Blend. Macromolecules 2008, 41, 1401−1410. (17) Dobrosielska, K.; Wakao, S.; Suzuki, J.; Noda, K.; Takano, A.; Matsushita, Y. Effect of Homopolymer Molecular Weight on Nanophase-Separated Structures of AB Block Copolymer/C Homopolymer Blends with Hydrogen-Bonding Interactions. Macromolecules 2009, 42, 7098−7102. (18) Dobrosielska, K.; Wakao, S.; Takano, A.; Matsushita, Y. Nanophase-Separated Structures of AB Block Copolymer/C Homopolymer Blends with Complementary Hydrogen-Bonding Interactions. Macromolecules 2008, 41, 7695−7698. (19) Hanley, K. J.; Lodge, T. P.; Huang, C.-I. Phase Behavior of a Block Copolymer in Solvents of Varying Selectivity. Macromolecules 2000, 33, 5918−5931. (20) Han, S. H.; Pryamitsyn, V.; Bae, D.; Kwak, J.; Ganesan, V.; Kim, J. K. Highly Asymmetric Lamellar Nanopatterns via Block Copolymer Blends Capable of Hydrogen Bonding. ACS Nano 2012, 6, 7966− 7972. (21) Kwak, J.; Han, S. H.; Moon, H. C.; Kim, J. K.; Koo, J.; Lee, J.-S.; Pryamitsyn, V.; Ganesan, V. Phase Behavior of Binary Blend Consisting of Asymmetric Polystyrene- block -poly(2-Vinylpyridine) Copolymer and Asymmetric Deuterated Polystyrene- block -poly(4Hydroxystyrene) Copolymer. Macromolecules 2015, 48, 1262−1266. (22) Kwak, J.; Han, S. H.; Moon, H. C.; Kim, J. K.; Pryamitsyn, V.; Ganesan, V. Effect of the Degree of Hydrogen Bonding on Asymmetric Lamellar Microdomains in Binary Block Copolymer Blends. Macromolecules 2015, 48, 6347−6352. (23) Chen, S.-C.; Kuo, S.-W.; Jeng, U.-S.; Su, C.-J.; Chang, F.-C. On Modulating the Phase Behavior of Block Copolymer/Homopolymer Blends via Hydrogen Bonding. Macromolecules 2010, 43, 1083−1092.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00651. Simulation testing/calibration (S1), simulation results (S2), normalization extrapolation (S3), rationalization of phases observed only in simulations (S4), chain end distribution comparison (S5), and additional SANS data (S6) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (D.F.S.). Author Contributions

D.F.S. and A.F.H. contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Portions of this work were performed at the DuPontNorthwestern-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-AC02-06CH11357. We thank Steven Weigand and Denis Keane for assistance at sector 5-ID-D. The use of the 10 m small-angle neutron scattering beamline at the NIST Center for Neutron Research was provided in support of the nSoft consortium, and we thank Ronald Jones for assistance conducting the SANS measurements. Thanks to Karim Gadelrab for assistance in finding appropriate relaxation constants for the three species SCFT formulism.

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