Modeling Hydrogen Bonding in Diblock Copolymer ... - ACS Publications

Jul 9, 2013 - Cheng-Chang TsaiZhihua GanTao ChenShiao-Wei Kuo. Macromolecules ... Shuang Yang , Zhen Lei , Nan Hu , Er-Qiang Chen , An-Chang Shi...
0 downloads 0 Views 888KB Size
Download PDF
Article pubs.acs.org/Macromolecules

Modeling Hydrogen Bonding in Diblock Copolymer/Homopolymer Blends Ashkan Dehghan and An-Chang Shi* Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada ABSTRACT: Blends of AB-diblock copolymers and C-homopolymers (AB/C), in which A and C are capable of forming hydrogen bonds, are used as a model system to examine and validate two approaches for modeling the effects of hydrogen bonding on the phase behavior of polymeric systems. The first model assumes a negative Flory−Huggins parameter between the A/C monomers. This attractive-interaction model reproduces qualitatively correct phase transition sequences as compared with experiments, but it fails to correctly describe the change of lamellar spacing induced by the addition of the C-homopolymers. The second model assumes that hydrogen bonding leads to A/C complexation. The interpolymercomplexation model predicts correctly the order−order phase transition sequences and the decrease of lamellar spacing at strong hydrogen bonding. The results demonstrate that hydrogen bonding of polymers should be modeled by the interpolymer-complexation model.

■

INTRODUCTION Scientists and engineers are constantly searching for methods to diversify and enhance the properties of polymeric materials. One method to enrich the functionality of polymeric materials is to blend polymers of different chemical structure and architecture.1,2 In the case of polymeric systems containing block copolymers, a simple example is provided by mixtures of AB-diblock copolymers and C-homopolymers (AB/C).3−5 In recent years, the phase behavior of AB/C blends has attracted considerable attention both theoretically and experimentally.3−21 One of the attractive feature of the AB/C blends is the opportunity to control the self-assembled morphologies using blending. In a polymer blend containing block copolymers, the size and morphology of the self-assembled structures can be controlled by adjusting the chemical and physical properties of the polymeric components. Some potential applications of these nanostructures include nanolithography, photonics, data storage, and drug delivery.6,22 Although blending is a promising and efficient route to engineer polymeric materials, most polymer chains are immiscible due to their high degree of polymerization. The mixing entropy is reduced for polymers with large molecular weight, thus the miscibility depends mostly on the monomer− monomer interactions. This fact dictates that polymer blends often tend to phase separate macroscopically.23 To induce the formation of a homogeneous phase, favorable intermolecular interactions such as ion−dipole, π−π interactions, or hydrogen bonding are often utilized.4,5,11,18,24,25 In this paper, we focus on the utilization of hydrogen bonding to enhance miscibility in AB/C blends. More specifically, we are interested in examining and validating theoretical models for the study of hydrogen bonding in polymeric systems. Such a study will be very useful for the © 2013 American Chemical Society

understanding of the effects of hydrogen-bonding strength and homopolymer composition on the phase behavior of the AB/C blends. The effect of hydrogen bonding on the phase behavior of diblock copolymer/homopolymer blends has been investigated recently in a number of experiments.5,16,18,26 For example, Chen et al.25 examined the effect of hydrogen-bonding strength on the phase transitions of AB/C mixtures. They demonstrated that the phase behavior of the blend is affected by the strength of the hydrogen bonds formed between the A and C monomers. Specifically, they observed that increasing the homopolymer concentration resulted in order−order phase transitions and macroscopic phase separation for strong and weak hydrogen-bonding blends, respectively. In similar studies, Dobrosielska et al.11,12 reported on the formation of micro and macrophase separated regions in hydrogen-bonded AB/C blends. They further investigated the effect of homopolymer molecular weight and blend composition on the formation of nanophase separated structures and concluded that strong hydrogen bonding dramatically enhances the miscibility of the C-homopolymers in the AB blend. Lee et al.18 studied the effect of homopolymer concentration on the phase behavior of AB/C blends, where hydrogen bonding occurs between the A/C and B/C monomers. It was demonstrated that the stronger hydrogen bonding between the A/C monomers results in a phase separated state, with the formation of mixed A/C domains in the B-rich matrix. These experiments demonstrated the richness of the phase behavior due to hydrogen bonding. Therefor it is important to understand the effect of hydrogenReceived: April 26, 2013 Revised: June 20, 2013 Published: July 9, 2013 5796

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

homopolymer blend. The hydrogen bonds are assumed to form between the A and C monomers, as shown schematically in Figure 1. The equilibrium phase behavior of the system is

bonding strength, the concentration and the molecular weight of the additives on the phase behavior of the AB/C blends. Although experimental studies provide a solid foundation for investigating hydrogen bonding in AB/C blends, it is useful and essential to develop theoretical models for exploring the large parameter space of these complex systems.9,20,27−30 A commonly used model to mimic the hydrogen bonding between polymers assumes a large negative (attractive) Flory−Huggins parameter (χ < 0) between the hydrogenbonding monomers.24,29 This attractive-interaction model (AIM) provides a simple method to study the enhanced miscibility created by the hydrogen bonds. The AIM has been used extensively in the literature to investigate the phase behavior of hydrogen-bonded polymer blends. For example, Han et al.24 studied the qualitative phase behavior of binary (AB/AC) blends of diblock copolymers using a negative interaction parameter to model the formation of hydrogen bonds between the C and B monomers. Using the selfconsistent mean field theory (SCFT), they investigated the phase behavior of the system as a function of AB molecular weight and volume fraction. They obtained a qualitative agreement between the phase behavior calculated using the negative interaction SCFT model and experiments. Similar approaches have been used to study hydrogen bonding in diblock copolymer/homopolymer blends.16,21 Löwenhaupt et al.21 studied the phase behavior of AB/A and AB/C blends using the random phase approximation (RPA) analysis. They modeled the hydrogen bonding between the A and C monomers using a negative interaction parameter, and reported on a micro/macro phase transition upon increase in the ABdiblock concentration. Most recently, Pryamitsyn et al.29 investigated the effect of hydrogen bonding on the phase behavior of AB/AC blends using the strong stretching theory. By modeling strong hydrogen bonding using a large negative χBC and assuming that the B and C blocks form B + C mixed domains, they predicated that the driving force behind the order−order phase transitions is the bending force at the BC− AA interface.29 These theoretical studies indicate that the AIM can produce qualitatively correct phase transition sequences in hydrogen bonded blends. Although the success of the AIM provides a compelling argument for using this model for studying hydrogen bonding in polymers, hydrogen bonding is fundamentally different from the attractive interaction, in that the bonding leads to interpolymer complexation. The proper, albeit more complicated, approach for modeling hydrogen bonding is to use interpolymer complexation, where the hydrogen bonds between the monomers lead to the formation of supramolecular structures. This approach has been used recently to investigate hydrogen bonding in homopolymer/homopolymer28,31,32 and diblock copolymer/homopolymer blends.10 Feng et al.32 developed a field-theoretic model for studying diblock copolymer complexes. In their work, they investigated the properties of supramolecular diblock chains formed via reversible bonding of two chemically distinct homopolymer chains. Similar work by Nakamura and Shi28 examined the formation of ladder like structures formed by complete and symmetric polymer−polymer complexation. These studies suggest that hydrogen bonding should, and could, be modeled by assuming a donor−acceptor relationship between monomers with hydrogen-bonding capability. In this paper, a detailed comparison between the attractiveinteraction model (AIM) and the interpolymer-complexation model (ICM) is carried out for an AB-diblock copolymer/C-

Figure 1. Schematic diagram showing hydrogen bonding in the AB/C system. Hydrogen bonds are assume to form between the A and C segments.

obtained using the self-consistent field theory (SCFT), formulated for both of the AIM and ICM. In the AIM, hydrogen bonding is represented by assuming a negative Flory−Huggins parameter χAC. Thus, the strength of hydrogen bonds is modeled by the magnitude of the χAC in the AIM. In contrast, the ICM describes hydrogen bonding by a complete and symmetric complexation between the A and C segments. In this approach, the hydrogen-bonding strength is modeled by the energy gain (ε) per hydrogen bond. A detailed analysis of the phase behavior of the system is performed using the SCFT and RPA. We will demonstrate that both the AIM and ICM predict similar phase transition sequences. However, these two models predict opposite change of the lamellar spacing as a function of the homopolymer concentration at the strong hydrogen-bonding regime. By comparing our theoretical predictions with available experiments we conclude that the AIM is not adequate for modeling hydrogen bonding especially in the strong hydrogen-bonding regime, and the correct method for modeling hydrogen bonding in polymers is the ICM.

■

THEORETICAL FRAMEWORK In this section, we present a brief description of the selfconsistent field model for the complexation between the ABdiblock copolymer and C-homopolymer chains. This process is denoted as AB + C ⇄ DB′, where the D-blocks are produced by the complexation of the A and C blocks. Specifically, the model system contains n1 AB-diblock copolymers, n2 Chomopolymers and n3 DB′-complexed chains in a box of volume V. The degree of polymerization of each block is specified by Nα, where α = A,B,C,D and B′. The monomers are assumed to occupy a volume vα, where the ratio between the volume occupied by α-monomer and a reference volume is represented by ξα = vα/v0. This parameter will be used to describe the conformational dissimilarity between the single stranded chains (A, B, C, B′) and the complexed chains (D).28,33 In what follows, we will describe the key aspects of the theory and leave the details of the SCFT formalism to Appendix A. The thermodynamic properties of our model is most conveniently described by the grand canonical ensemble at a fixed temperature, volume and activity. Starting with the Gaussian model of the polymers and applying the SCFT formalism, the partition function can be written in the form Ξ=

∫ +[ϕ(r)]+[ω(r)] ∏ δ[∑ ϕα(r) − 1]e−G[ϕ,ω] r

α

(1) 5797

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

where G[ϕ,ω] is the grand potential functional or the grand canonical free-energy functional. In many of the calculations, it is convenient to write the free-energy density per volume in the form, G[ϕ,ω]N1/ρ0V, where the length of the polymer chains are scaled with respect to the length of the AB-diblock copolymers. For simplicity, we assume ξα = 1 for α = A, B, C, and B′. The relative volume occupied by the complexed monomers (ξD) is a varying parameter, which is used to describe the conformational dissimilarity between the single stranded and the duplexed chains. In the ICM, the DB′ chains are formed from the complexation between the AB and C chains. The equilibrium of the complexation reaction is reached when μ3 = μ1 + μ2, where μ1, μ2, and μ3 are the chemical potentials of the diblock copolymers, homopolymers, and complexed diblock copolymers, respectively. At equilibrium, using the incompressibility condition and the fact that μ3 = μ1 + μ2, we can write down the activity of the complexed chains in terms of the activity of the diblock copolymers. For a symmetric diblock chain (fA = f B), we obtain z 3 = z1e−NAε − ln(NA)

homogeneous or disordered phase. Using the homogeneous mean field equations and the free-energy density functional eq 3, we can obtain the free-energy density of the disordered phase: ⎛ ϕ̅ ⎞ ϕ̅ ⎛ ϕ̅ ⎞ ϕ̅ g[ϕ ̅ , ω̅ , η ̅ ] = ϕ1̅ ln⎜ 1 ⎟ + 2 ln(ϕ2̅ ) + 3 ln⎜⎜ 3 ⎟⎟ fA ψ ⎝ z1z′ψ ⎠ ⎝ z1 ⎠ − ϕ1̅ −

∫

−

∑

ϕ3̅

(2)

Vξα

α

+

1 2V

∑

ϕα(r)ϕβ (r)

α≠β

ξαξβ

ϕ1̅ ϕ2̅

(3)

Within the mean-field approximation, the density profiles ϕα(r) and their conjugate fields ω α (r) are determined by minimization of the SCFT free-energy functional (eq 3) with respect to the ϕ, ω, and η fields, leading to the following SCFT equations: ϕα(r) ξα

ϕC (r) ξC ϕβ (r) ξβ

ωγ (r) ξγ

= z1

=

∫0

1 fA

fα

∫0

=

1 2

∑ γ≠γ′

ϕγ ′(r) ξγξγ ′

fβ

Q3 Q 1Q 2

(6)

= ψz′exp[fA χAB (ϕ1̅ + τB ′ϕ3̅ ) + fA χCD τDϕ3̅ ]

ds qC (r, s)qC†(r, fA − s)

∫0

(5)

[1 − ϕH + (τD/ξD)ϕ3̅ − ϕ3̅ ][ϕH − (τD/ξD)ϕ3̅ ]

τD ϕ̅ ξD 3

ϕ1̅ = 1 − ϕ2̅ − ϕ2̅

(7)

The first expression is a transcendental equation with ϕ̅ 3 written as a function of ϕH. It is important to note that determining ϕH using eq 6 is only valid for the homogeneous phase. To determine the concentration of the complexed chains in the system for a given set of parameters, we use the general form of the mass action law, eq 2. This equation allows us to calculate the concentration of the complexed chains ϕ3 for the homogeneous and in-homogeneous phases. After determining ϕ3, the concentration of the diblock copolymers and homopolymers can be calculated. Using ϕH and ε, we can explore the phase behavior of the system with respect to the concentration of the added homopolymers and the hydrogenbonding strength.

ds qβ (r, s)qβ†(r, fβ − s)

χγγ + η(r) ′

∑ ϕγ (r) = 1 γ

= ψ e−NAε − ln(NA)

ϕ2̅ = ϕH −

= z1e−NAε − ln(NA)

+ W (ϕ ̅ )

ϕ3̅

ds qα(r, s)qα†(r, fα − s) fA

ψ

where Q1, Q2, and Q3 are the single chain partition functions of the diblock copolymers, homopolymers and complex chains in the homogeneous phase. In the current study, we are interested in the effect of the homopolymer concentration on the phase behavior of the AB/C blend. Therefore, we will use the total concentration of the homopolymers (ϕH) as the controlling parameter. It is important to note that ϕH is not always equal to ϕ̅ 2. Instead, ϕH is the sum of the concentrations of the free and complexed homopolymer chains. The total concentration of the added homopolymers in the system for a given hydrogenbonding strength (ε) can be written as, ϕH = ϕ̅ 2+(τD/ξD)ϕ̅ 3, where τD = (ξD f D)/(ξD f D + f B′). Let us first consider a system where NA = NB = NC, χAC = χAD = χCD and χBC = χBD. Using the mass action law eq 6 and the incompressibility condition, ϕ̅ 1 + ϕ̅ 2 + ϕ̅ 3 = 1, we can write,

⎤ χαβ N1⎥ ⎥⎦

1 Q − z1e−NAε − ln(NA)Q 3 fA 2

− z1Q 1 −

ϕ3̅

Here, z′ = e and ψ = f B′ + f DξD. In the homogeneous phase, ϕ̅ 1, ϕ̅ 2, and ϕ̅ 3 are the volume fraction of the diblock copolymers, homopolymers and complexed copolymers, respectively. In the above equation, W(ϕ̅ ) accounts for the interaction energy contribution to the free-energy. In the homogeneous phase, the equilibrium volume fraction for each species can be determined by applying the mass action law.10,32 For the specific case described here, the mass action law can be written as follows:

⎡ dr ⎢η(r)[∑ ϕα(r) − 1] ⎢⎣ α

N1ωα(r)ϕα(r)

fA

−

−NAε−ln(NA)

where ε represents the energy gained when a single hydrogen bond is formed.32 Using eq 2 and a Lagrangian multiplier to ensure the incompressibility of the system, the free-energy density can be written as g[ϕ , ω , η] =

ϕ2̅

(4)

where α = {A,B}, β = {D,B′} and γ, γ′ = {A,B,C,D,B′}. A trivial but exact solution of the SCFT equations is obtained when ϕ, ω and η fields are constants, corresponding to the 5798

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

■

Article

RESULTS AND DISCUSSION We now turn to the results on the phase behavior of the AB/C blends, calculated using the RPA and SCFT. The solutions to the modified diffusion equations were obtained using the spectral method19,27,34,35 for the classical lamellar (Lam), cylindrical (Hex) and spherical (BCC) phases. For simplicity, the gyroid phase is omitted in the current study. Instead we focus on the comparison between the AIM and ICM. In this section, we will investigate the effect of hydrogen-bonding strength and the concentration of the homopolymer additives on the phase behavior of the AB/C blend. Our attention is focused on the phase behavior of the system in the microphase separated regions of the phase space predicted by our RPA calculations.30,36−38 The results are summarized in terms of phase diagrams, containing phase boundaries obtained from the RPA and SCFT. The RPA results are shown using dotted and dashed lines. Here, the dotted lines correspond to disordered → macrophase and dashed lines correspond to disordered → ordered phase transitions. These two lines meet at a Lifshitz point, represented in the phase diagrams by a solid circle. The SCFT phase boundaries are shown using the solid lines in the phase diagrams. We will first focus on the AIM and examine the effect of the homopolymer concentration on the phase behavior of the system. Figure 2 represents phase diagram calculated for the

effect as increasing fA in a neat AB system. Furthermore, the results of Figure 2 are similar to the theoretical results calculated previously for the AB/C blends with negative χAC values.7,9 We mentioned that the strength of hydrogen bonds can be mimicked by the relative magnitude of the χAC parameter. It is interesting to investigate the effect of hydrogen-bonding strength on the phase behavior of the AB/C system by exploring the phase diagram in the χAC−ϕH plane. Figure 3

Figure 3. SCFT phase diagram for the AB/C blend with χAB = 12, χBC = 15, fA = 0.5, and κ = 0.5. The RPA phase diagram is shown using the dotted and dashed lines, which meet at the Lifshitz point, represented with the solid circle.

shows the SCFT phase diagram for the AB/C blend with χAB = 12, χBC = 15, fA = 0.5, and κ = 0.5. The strength of hydrogen bonding is modeled by the magnitude of the χAC, shown on the y-axis in Figure 3. In the strong hydrogen-bonding region, χAC ≲ −2, the attractive interaction between the A and C monomers results in the miscibility of the homopolymers in the diblock blend. Increasing the concentration of the homopolymer additive in the system changes the packing structure, which results in the order−order phase transitions from Lam → Hex → BCC → Dis. At the weak hydrogenbonding region, χAC ≳ −2, the positive B/C interaction parameter (χBC = 15) results in the immiscibility of the homopolymers in the system. Increase in the ϕH drives the system from the lamellar structure into a two-phase region. The phase behavior for the AB/C blend predicted by the AIM (Figures 2 and 3) is consistent with those observed experimentally. Specifically, starting with a pure lamellar phase separated AB blend, Chen et al.25 observed that increasing the homopolymer concentration resulted in Lam → Hex → BCC → Dis phase transitions for strong hydrogen bonding. They also observed a Lam → two-phase phase transition with increase in ϕH for weak hydrogen bonding. The phase transition sequences observed by Chen et al.25 is consistent with those shown in Figure 3. The order−order phase transitions shown for χAC ≲ −2 is consistent with those observed for strong hydrogen bonding. Similarly, the order-totwo-phase phase transition for χAC ≳ −2 is consistent with the phase transition observed experimentally.25 These results indicate that the phase behavior predicted by the AIM is in qualitative agreement with experiments. We now turn to the results from the ICM. In this model, we assume a donor−acceptor relationship between the A and C

Figure 2. Phase diagram for the AB/C blend with χAC = −10, χBC = 15, χAB = 12, and κ = 0.5. Disordered, microphase-separated and twophase regions are observed. In the microphase region, lamellar (Lam), cylindrical (Hex) and spherical (BCC) phases are shown. The RPA phase diagram is shown using the dotted and dashed lines, which meet at the Lifshitz point, represented with the solid circle.

AB/C system with χAB = 12, χAC = −10, χBC = 15, and κ = 0.5 in the fA−ϕH plane. Here, κ is the ratio between the degree of polymerization of the C chains to that of the AB-diblock chains κ = NC/NAB. As shown in Figure 2, for blends with fA ≲ 0.25 the system is driven from the disorder phase into a two-phase region when ϕH is increased. In the case where 0.25 ≲ fA ≲ 0.3, increasing the homopolymer concentration results in disorder→ order → disorder phase transitions. The microphase region exhibits order−order phase transitions from BCCA → HexA → Lam → HexB → BCCB. Here, the subscript A indicates an A-rich core in a A-rich matrix and vice versa for subscript B. The phase behavior predicted by the theory resembles that of a neat AB-diblock melt.7 This indicates that the attractive interaction between the A and C monomers has the same 5799

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

to, at ϕH ∼ 0.3 the hydrogen-bonding sites are saturated. After this point, increase in ϕH results in an increase in the free homopolymer concentration in the system. The immiscibility of the C-homopolymers in the DB′ blend (χBD = 15) results in a disorder-to-two-phase phase transition. It is important to note that the saturation effect observed here is unique to the ICM and is lacking in the AIM. The saturation effect is caused by the fact that there is a one to one ratio in the A/C complexation and there are only a finite amount of A segments available. The phase transition sequences obtained from the SCFT for both weak and strong hydrogen bonding are consistent with those observed experimentally.25 Our results indicate that both the AIM and the ICM provide qualitatively correct phase transition sequences as compared with experiments. To investigate the difference between these two models, we need to go beyond the overall phase behavior of the system by examining the mechanisms of the phase transitions in each model. The driving force of the phase transitions in the ICM is the increase of the effective volume and the interfacial area per chain of the hydrogen-bonded chains. This change is caused by the complexation between the A and C monomers, resulting in the formation of the duplexed chains. The increase in the effective volume and the interfacial area per chain drives the phase transitions of the system. This mechanism is different from that of the AIM, where the most important change is the localization of the C chains in the A-rich domains, resulting in an increase of the effective volume of the A domains but very little change in the A−B interfaces. Because of the different effects on the interfacial area per chain, the difference between these two models leads to different prediction of the effect of the homopolymer concentration on the lamellar spacing. Figure 5 shows the normalized lamellar spacing for two blends, as predicted by the AIM. Here, the increase in the

monomers. The complexation strength is characterized by ε, which is the energy gain of a single hydrogen bond. The complexation between the AB and C chains is assumed to result in the formation of the DB′ supramolecular diblock chains. Here, the D monomers are formed when the A and C segments complex. The volume occupied by the complexed monomers is assumed to be twice of that of the single-stranded chains. In the ICM, the concentration of the complexed chains in the system is determined by the mass action equation given by eq 2. To investigate the effect of hydrogen-bonding strength on the phase behavior of the AB/C blends, we will explore the phase diagram in the ε−ϕH plane. It is important to notice that ϕH is the total concentration of the homopolymers in the system. This means that, depending on the strength of hydrogen bonds and interaction parameters, the C chains could be in the complexed and/or free form. Figure 4 shows

Figure 4. SCFT phase diagram calculated using the interpolymer complexation model, for the symmetric AB/B′D/C system, where χAB = 12, χBC = χBD = 15, χAC = χAD = χCD = 2, ξD = 2, and NA = 100 with ε and ϕH being the hydrogen-bonding strength and the concentration of the added homopolymers, respectively. The RPA phase diagram is shown using the dotted and dashed lines, which meet at the Lifshitz point, represented with the solid circle.

the SCFT phase diagram for the AB/C blends, where χAB = 12, χBC = χBD = 15, χAC = χAD = χCD = 2, ξD = 2, and NA = 100. First we will focus on the phase behavior of the system in the weak hydrogen-bonding regime (ε ≲ 0.04). In this case, the complexation between the A and C chains results in an increase in the overall free-energy. This can be understood since the entropy loss caused by the complexation process is greater than the energy gained during hydrogen bonding. For this reason, it is energetically more favorable for the C chains to stay free. The immiscibility between the C-homopolymers and the diblock copolymers (χAC, χBC > 0) results in a macrophase separation upon increase in ϕH. The phase behavior in Figure 4 is similar to that observed in the AIM, Figure 3, for χAC ≳ −2. By increasing ε, it is energetically more favorable for the A− C complexation to take place. This corresponds to the strong hydrogen-bonding regime, characterized by χAC ≲ −2, in the AIM. For the AB/C blends with ε ≳ 0.05, all the additive homopolymers complex to form supramolecular DB′ chains. Starting with a pure AB blend, increase in the DB′ concentration results in order−order phase transitions with the sequence of Lam → Hex → BCC → Dis. Because there are only a finite number of A segments for the C chains to complex

Figure 5. Normalized lamellar spacing for the AB/C system with χAB = 12, χBC = 15, fA = 0.5, κ = 0.5, and χAC = −10 (dotted line), and χAC = −1 (solid line).

homopolymer concentration results in an increase in the lamellar spacing for both strong and weak hydrogen bonding. This increase of lamellar spacing is due to the localization of the C-homopolymers in the middle of the A-domains as illustrated in Figure 6, which shows the density profiles for blends with χAB = 12, χBC = 15, fA = 0.5, κ = 0.5, and χAC = −10 (dotted line) and χAC = −1 (solid line) for ϕH of 20%. The attractive interaction between the A and C monomers results in the localization of the C segments in the A-rich domains. Naively, 5800

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

Figure 7. Normalized lamellar spacing for the symmetric AB/C system (interpolymer complexation model) with χAB = 12, χBC = χBD = 15, χAC = χCD = χAD = 2, ξD = 2, and N = 100 for ε = 0.15 (dotted line) and ε = 0 (solid line).

Figure 6. Density profile for the AB/C system, with χAB = 12, χBC = 15, fA = 0.5, κ = 0.5 and χAC = −10 (dotted line), and χAC = −1 (solid line). ϕH is 20% for both blends. The x axis is the axis perpendicular to the lamellar interface, normalized with respect to the lamellar domain spacing.

in ϕH results in an increase in the lamellar spacing. In this case (ε = 0), it is energetically unfavorable for the homopolymers to complex to the diblock chains. This means that homopolymers are free and thus localize at the A-rich domains, increasing the lamellar spacing. The shrinking in the lamellar spacing predicated by the ICM at the strong hydrogen-bonding regime can be related to the thickness asymmetry in the chains. The complexation between the A and C monomers results in the formation of duplexed segments (D) with twice the volume of the single stranded A, B, C, and B′ monomers. The complexation increases the effective volume and the interfacial area of the hydrogenbonded A chains, driving the system into phase transitions. Similar argument was made by Pryamitsyn et al.29 to explain phase transitions in AB/AC blends, where B and C segments form strong hydrogen bonds. Using strong stretching theory, they assumed that the large negative interaction parameter (χBC) between the B and C monomers results in the localization of the AC chains at the AB interface. Using this argument, the authors demonstrated that the change in the effective volume, caused by the localization of the AC chains at the interface, results in order−order phase transitions. To further investigate the mechanism of the shrinking of the lamellar spacing, we consider the effect of chain thickness asymmetry on the lamellar spacing in an AB/A′B′ blend. The AB and A′B′ chains are chemically identical and thus χAA′ and χBB′ are set to zero. The A′ monomers are chosen to be larger than other species, to isolate for the effect of thickness asymmetry on the lamellar spacing. Figure 8 shows the change in lamellar spacing with respect to ϕH (ϕA′B′) for an AB/A′B′ system with χAB = χA′B′ = 12 and ξA′ = 1.25. As shown, increase in ϕA′B′ results in a decrease in the lamellar spacing. The fact that the AB and A′B′ chains are chemically identical means that the added A′B′ chains localize at the AB interface. This increases the effective volume and the interfacial area of the A chains. Increase in the interfacial area between the A and B chains and the fact that the volume of the B chains must stay the same means that the B chains must shrink toward the interface. This can be shown explicitly by calculating the A, A′, B, and B′ segment lengths individually. Figure 9 shows the result of this analysis, where the segment lengths are normalized with respect to their initial values. These results

one would expect that it is energetically favorable for the C chains to completely penetrate the lamellar structure, since there is an attractive interaction between the A and C segments. However, penetrating the lamellar structure requires a reduction of the entropy for both diblock and homopolymer chains. For this reason, the homopolymers prefer to localize in the region between the lamellar sheets, thus increasing the lamellar spacing. In contrast to the increase in lamellar spacing predicted by the AIM for strong hydrogen bonding, Chen et al.25 observed a decrease in the lamellar spacing upon increase in ϕH. The inconsistency between the lamellar spacing predicted by the AIM and those given by experiments indicates that the mechanism behind the change in the effective volume caused by hydrogen bonding is fundamentally different from that produced by the negative interaction parameter. This leads us to conclude that the AIM can only be an approximate model for studying hydrogen bonding in polymer blends. We now turn to our analysis of the effect of ϕH on the lamellar spacing for both strong and weak hydrogen bonding using the ICM. Figure 7 shows the normalized lamellar spacing for the AB/C system with χAB = 12, χBC = χBD = 15, χAC = χCD = χAD = 2, ξD = 2, and N = 100 for ε = 0.15 (dotted line) and ε = 0 (solid line). Here, ε = 0.15 and ε = 0 correspond to the system in the strong and weak hydrogen-bonding regimes, respectively. At the strong hydrogen-bonding regime, Figure 7 demonstrates that increase in the homopolymer concentration results in a decrease in the lamellar spacing. This is in contrast to the increase in the lamellar spacing predicated by the AIM. In the strong hydrogen-bonding regime (ε ≳ 0.07), it is energetically favorable for the added homopolymers to complex to the A blocks. The complexation increases the effective volume and the interfacial area of the hydrogen-bonded A chains. The increase in the interfacial area per chain between the AB chains and the fact that the B chains must occupy the same volume as before the complexation, dictates that the B domains must shrink. This results in the decrease in the lamellar spacing seen in Figure 7, which is consistent with experimental predictions at the strong hydrogen-bonding regime.25 For weak hydrogen bonding, as observed in experiments25 and the negative interaction model, an increase 5801

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

shown to be consistent with experiments,25 where increase in the homopolymer concentration resulted in order−order and order-to-two-phase phase transitions for strong and weak hydrogen bonding, respectively. The difference between these two models was shown to be in the mechanism of the phase transitions. We demonstrated that the increase in the effective volume of the A-domains in the AIM was due to the negative χAC, resulting in the localization of the C chains at the A-rich domains, driving the system into phase transitions. In contrast, we showed that phase transitions in the ICM is caused by the increase in the effective volume and the interfacial area per chain of the hydrogen-bonded A chains. The difference in the mechanism causing phase transitions in the AIM and ICM can be tested experimentally and theoretically. This was done by investigating the changes to the lamellar spacing as a function of the homopolymer concentration. We demonstrated that changes to the lamellar spacing predicted by the AIM and ICM were consistent with experiments in the weak hydrogen-bonding regime. In contrast to the increase in the lamellar spacing predicted by the AIM at the strong hydrogen-bonding regime, the ICM gives a decrease in the lamellar spacing. The results from the ICM were in agreement with experimental measurements. Another unique property of the ICM is the saturation effect, caused by the saturation of the hydrogen-bonding sites available in the blend. We showed that this effect is only observed in the ICM and the phase behavior of the blends beyond the saturation point is dependent on the interaction between the C-homopolymers and AB-diblock chains. The consistency of our results with experiments indicate that the correct method for modeling hydrogen bonding in polymers is by using the ICM. Despite the approximate nature of the AIM, it provides a simple and fast method to obtain a qualitative understanding of the phase behavior of polymer blends with hydrogen bonding. The accuracy of the predictions from the AIM is sensitive to the type of the system studied. For example, in AB/BC blends, where A and C segments hydrogen bond, the attractiveinteraction between the A and C segments and the architecture of the chains results in the localization of the BC chains at the AB interface. This implies that in contrast to the AB/C system, the AIM can provide an accurate description of hydrogen bonding in the AB/BC blends. The ICM considered in this work assumes complete and linear complexation between the A and C blocks. The supramolecular architecture considered here is only one of the many possible ways the A and C chains can complex. A more complete theory must allow for an ensemble of structures, which could depend on parameters such as the hydrogenbonding strength and the degree of polymerization of the hydrogen-bonding blocks. In particular, the association of one C-polymers with different A-blocks can lead to random crosslinked configurations of the A blocks, in which the C-polymers act as the cross-linking agents. The phase behavior of crosslinked block copolymers has been studied experimentally39,40 and theoretically.41,42 The general conclusion from these studies is that, when the density of cross-links is low, the system undergoes order−disorder transitions similar to neat block copolymers. On the other hand, when the density of the cross-links is high enough, the system will assume a gel state.39,40 In the current study, we were focusing on the comparison of the two commonly used models, thus we have adapted the simplest assumption about the polymer complexation.

Figure 8. Normalized lamellar spacing for the AB/A′B system with χAB = 12 and ξA′ = 1.25.

Figure 9. Normalized segment length for the AB/A′B′ system, where λ represents the length of each segment, normalized to its initial value.

indicate that the thick A′ chains are longer than the chemically identical A chains. It is also interesting to point out that the increase in the A′B′ concentration results in an increase in the A and A′ segment lengths. In contrast, B and B′ segment lengths decrease with increase in the A′B′ concentration. The analysis of the relative segment changes for all species firmly establishes that the overall decrease in the lamellar spacing is caused by shrinking of the B and B′ chains toward the lamellar interface.

■

CONCLUSION In this paper, we examined and validated two methods for the modeling of hydrogen bonding of polymers. The first approach is the widely used attractive-interaction model (AIM), where hydrogen bonding is represented using a negative Flory− Huggins parameter. The second approach models hydrogen bonding using the interpolymer-complexation model (ICM). In the ICM, segments with hydrogen-bonding capability complex to form supramolecular structures. Using random phase approximation (RPA) and self-consistent field theory (SCFT), we examined the effect of the homopolymer concentration on the phase behavior of the system and analyzed the mechanism behind the phase transitions. We demonstrated that the qualitative phase behavior of the blends were captured by both the AIM and ICM. These results were 5802

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

■

Article

n3!)−1 takes into account the indistinguishability of the polymer chains. n Using the mathematical relationship ex = ∑∞ 0 (x /n!), the partition function can be written in the form

APPENDIX A

Self-Consistent Field Theory of Interpolymer−Complexation Model

In this Appendix, we provide a detailed description of the SCFT for the interpolymer−complexation model. The system contains n1 AB-diblock copolymers, n2 C-homopolymers, and n3 DB′-complexed chains in a box of volume V. The degree of polymerization for each chain is represented by Nα, where α = A,B,C,B′ and D. The monomers are assumed to occupy a volume vα, where the ratio between the volume occupied by α monomer and a reference is represented by ξα, such that ξα = vα/v0. The conformation of the polymer chains is described by the Gaussian model. In this model, chains are considered flexible with a stretching energy given by43 H(0)[R iα(s)]

3 = 2 2bα

∫0

Nα

Ξ=

ρ0 2

(11)

where G[ϕ,ω] is the functional form of the grand potential. Using this information, eq 10 can be written as ∞

∞

n1= 0 n2 = 0 n3 = 0

∑ ∑ ∑

Ξ=

n1= 0 n2 = 0 n3 = 0

+

−

α

r

+

⎢⎣

ρ0 2

∑ ∫ dr α≠β

ϕα(r)ϕβ (r) ξαξβ

ξαξβ

α≠β

e μ2Q 2(0)Q 2V

−

λT3N2

α

χαβ −

ξα

e μ1Q 1(0)Q 1V λT3N1

e(μ3 − NDε)Q 3(0)Q 3V ⎤ ⎥ ⎥⎦ λT3N3

⎡

1 + 2V −

(12)

∫ dr ⎢⎢−∑

∑

⎣

α

ϕα(r)ϕβ (r) ξαξβ

α≠β

N1e μ2Q 2(0)Q 2 ρo λT3N2

−

N1ωα(r)ϕα(r) Vξα ⎤ N1e μ1Q 1(0)Q 1 ⎥ χαβ N1 − ⎥⎦ ρo λT3N1

N1e(μ3 − NDε)Q 3(0)Q 3 ρo λT3N3

(13)

In this model, DB′ chains are formed from the complexation between the AB and C chains. The equilibrium of the complexation reaction is reached when μ3 = μ1 + μ2. Here, the chemical potentials of the diblock copolymer and homopolymer chains can be written as, μα = G(0) α + ln(zα), where G(0) α and zα are the standard-state Gibbs free-energy and activity of the α-type chains.32 The standard-state Gibbs freeenergy can be written as, Gα(0) = −ln(Qα(0)/λTα3N). At equilibrium, the condition μ3 = μ1 + μ2 leads to

ξα

⎤⎡ e μ1Q (0)Q V ⎤n1 1 1 ⎥ χαβ ⎥⎢ ⎥⎦⎢⎣ ⎥⎦ λT3N1

⎡ e μ2Q (0)Q V ⎤n2 ⎡ e(μ3 − NDε)Q (0)Q V ⎤n3 3 3 ⎥ 2 2 ⎥ ⎢ ×⎢ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ λT3N2 λT3N3

2

ϕα(r)ϕβ (r)

∑ ∫ dr

g[ϕ , ω] =

ρo ωα(r)ϕα(r)

α

ρo

ρ0 ωα(r)ϕα(r)

The free-energy density functional can be determined by multiplying G[ϕ,ω] by N1/ρ0V. Here, the length of polymer chains are scaled with respect to the length of the AB-diblock copolymers. The free-energy density (g[ϕ,ω] = (G[ϕ,ω]N1)/ (ρ0V)) can be written as,

∫ +[ϕ(r)]+[ω(r)]

⎡

∫ +[ϕ(r)]+[ω(r)]

⎢⎣

α

r

(8)

(9)

∏ δ[∑ ϕα(r) − 1] exp⎢−∑ ∫ dr

1 n1! n2 ! n3! ⎡

∑ ∫ dr ϕα̂ (r)ϕβ̂ (r)χαβ

1 n1! n2 ! n3!

∞

∏ δ[∑ ϕα(r) − 1] exp⎢−∑ ∫ dr

⎛ dR α(s) ⎞2 ds ⎜ i ⎟ ⎝ ds ⎠

α≠β

∞

∞

∑ ∑ ∑

Ξ=

where ϕ̂ α(r) is the volume fraction of the α segment and ρ0 is a reference monomer density defined as the number monomers per unit volume. The thermodynamic properties of our model is most conveniently described using the grand canonical ensemble at a fixed temperature, volume and activity. Given that the probability distribution of the the blocks follows the standard Wiener form43 and that the interaction between the chains is described by eq 9, the grand canonical partition function of the system can be written as ∞

α

r

The space curve Rαi (s) specifies the position of a given segment s, with i running over the number chains for given species. Here, bα represents the Kuhn length of a given monomer. The type and strength of the interactions between segments is modelled by the Flory−Huggins parameters.44 The interaction between segments of two different species is represented by χαβ, where α, β = A, B, C, D, B′. Using this notation, the interaction energy of the system is given by W [ϕ(̂ r)] =

∫ +[ϕ(r)]+[ω(r)] ∏ δ[∑ ϕα(r) − 1]e−G[ϕ,ω]

z3 = exp[−G3(0) + G1(0) + G2(0)] z1z 2

(10)

(14)

where the chemical potential of the homopolymer chains (μ2) can be adjusted so that z2 = 1. The dependency of the chemical potentials is the result of the incompressibility condition. Given the definition for the standard-state Gibbs free-energy and by setting z2 = 1, we can write,

where μ1, μ2, and μ3 are the chemical potentials of the diblock copolymer, homopolymer and complexed chains, respectively. In the above equation, ε measures the energy gained when a single hydrogen bond is formed. Here, ND measures the degree of polymerization of the complexed chains. Therefore, the strength of hydrogen bonding can be modelled by the energy (ε) and number (ND) of hydrogen bonds. In eq 10, λT is the thermal wave length, and Q(0) b (b = 1,2,3) are the un-normalized single chain partition functions in zero fields. Here, (n1!n2!

⎡ ⎤ ⎛ Q (0) ⎞ z3 3 3NA ⎥ ⎢ ⎜ ⎟ = exp −NDε + ln⎜ (0) (0) ⎟ + ln(λT ) ⎢ ⎥ z1 ⎝ Q1 Q2 ⎠ ⎣ ⎦ 5803

(15)

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

The un-normalized single chain partition functions in zero 3N 2 3/2 fields can be written as Q(0) α = VαgMα−1 , where gM = (2πb /3) with b being the Kuhn length.32,43 The volume V in the zero field single chain partition function is the volume occupied by one chain and can be written as, Vα = Nαvα. Given this information, we can write

∂qγ (r, s) ∂s ∂qγ†(r, s) ∂s

■

−3NA v0NA −NAε λT

z 3 = z1e−NAε − ln(NA)

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The computation was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET).

(17)

■

−2λT−3N A ν0

where is absorbed into the definition of the hydrogen-bonding energy ε. Equation 17 defines the activity of the DB′ complexed chains (z3) in terms of the activity of the AB-diblock copolymers (z1). Using a Lagrangian multiplier to ensure the incompressibility in the system, the free-energy density functional eq 13 can be written as,

∫ dr ⎢⎢η(r)[∑ ϕα(r) − 1] ⎣

N1ωα(r)ϕα(r) Vξα

α

α

+

1 2V

∑

ϕα(r)ϕβ (r)

α≠β

ξαξβ

⎤ χαβ N1⎥ ⎥⎦

1 Q − z1e−NAε − ln(NA)Q 3 fA 2

− z1Q 1 −

(18)

The saddle point solutions to the free-energy functional can be determined by minimizing eq 18 with respect to the ϕ, ω and η fields. This results in the self-consistent field equations ϕα(r) ξα

ϕC (r) ξC ϕβ (r) ξβ

ωγ (r) ξγ

= z1

=

∫0

1 fA

fα

∫0

ds qα(r, s)qα†(r, fα − s) fA

ds qC (r, s)qC†(r, fA − s)

= z1e−NAε − ln(NA)

=

1 2

∑ γ≠γ′

ϕγ ′(r) ξγξγ ′

∫0

fβ

ds qβ (r, s)qβ†(r, fβ − s)

χγγ + η(r) ′

∑ ϕγ (r) = 1 γ

REFERENCES

(1) Paul, D. R., Ed. Polymer blends; Academic Press: New York, 1978. (2) Utracki, L. Polymer Alloys and Blends: Thermodynamics and Rheology; Hanser Pub: New York, 1989. (3) Jiang, M.; Xie, H. Prog. Polym. Sci. 1991, 16, 977−1026. (4) Jiang, M.; Huang, T.; Xie, J. Macromol. Chem. Phys. 1995, 196, 803−813. (5) Zhao, J. Q.; Pearce, E. M.; Kwei, T. K. Macromolecules 1997, 30, 7119−7126. (6) Hamley, I. Developments in block copolymer science and technology; Wiley: New York, 2004. (7) Matsen, M. W. Macromolecules 1995, 28, 5765−5773. (8) Semenov, A. N. Macromolecules 1993, 26, 2273−2281. (9) Zhou, J.; Shi, A.-C. J. Chem. Phys. 2009, 130, 19. (10) Zhuang, Y.; Wang, L.; Lin, J. J. Phys. Chem. B 2011, 115, 7550− 60. (11) Dobrosielska, K.; Wakao, S.; Takano, A.; Matsushita, Y. Macromolecules 2008, 41, 7695−7698. (12) Dobrosielska, K.; Wakao, S.; Suzuki, J.; Noda, K.; Takano, A.; Matsushita, Y. Macromolecules 2009, 42, 7098−7102. (13) Hashimoto, T.; Tanaka, H.; Hasegawa, H. Macromolecules 1990, 23, 4378−4386. (14) Winey, K. I.; Thomas, E. L.; Fetters, L. J. Macromolecules 1991, 24, 6182−6188. (15) Kuo, S.-W. Polym. Int. 2009, 58, 455−464. (16) Hameed, N.; Salim, N. V.; Guo, Q. J. Chem. Phys. 2009, 131, 214905. (17) Janert, P. K.; Schick, M. Macromolecules 1998, 31, 1109−1113. (18) Lee, H.-F.; Kuo, S.-W.; Huang, C.-F.; Lu, J.-S.; Chan, S.-C.; Wang, C.-F.; Chang, F.-C. Macromolecules 2006, 39, 5458−5465. (19) Matsen, M. W. Macromolecules 2003, 36, 9647−9657. (20) Matsen, M. W. Phys. Rev. Lett. 1995, 74, 4225−4228. (21) Löwenhaupt, B.; Steurer, A.; Hellmann, G. P.; Gallot, Y. Macromolecules 1994, 27, 908−916. (22) Park, M.; Harrison, C.; Chaikin, P. M.; Register, R. A.; Adamson, D. H. Science 1997, 276, 1401−1404. (23) Kuo, S.-W. J. Polym. Res. 2008, 15, 459−486. (24) Han, S. H.; Kim, J. K.; Pryamitsyn, V.; Ganesan, V. Macromolecules 2011, 44, 4970−4976. (25) Chen, S.; Kuo, S.; Jeng, U.-S.; Su, C.; Chang, F. Macromolecules 2009, 43, 1083−1092. (26) Han, Y.-K.; Pearce, E. M.; Kwei, T. K. Macromolecules 2000, 33, 1321−1329. (27) Matsen, M. W.; Schick, M. Phys. Rev. Lett. 1994, 72, 2660−2663.

⎡

∑

AUTHOR INFORMATION

*E-mail: (A.C.S.) [email protected].

(16)

where we have assumed NA = ND = NC and ξD = 2. This condition corresponds to the complete and symmetric complexation. Here, the complexed segments occupy twice the volume of the single-stranded chains. If we consider a symmetric diblock chain (fA = f B = 0.5), then eq 16 can be written as,

−

(20)

Corresponding Author

fA − 1

g[ϕ , ω , η] =

= −∇2 qγ†(r, s) + ω(r)qγ†(r, s)

where the solutions to the modified diffusion equations are solved for using the spectral method.19,27,34,35

⎡ ⎤ ⎛ vN ⎞ z3 = exp⎢ −NDε − ln⎜⎜ 0 A ⎟⎟ − ln(λT−3NA )⎥ ⎢⎣ ⎥⎦ z1 ⎝ fA − 1 ⎠ =e

= ∇2 qγ (r, s) − ω(r)qγ (r, s)

(19)

where α = {A,B}, β = {D,B′} and γ, γ′ = {A,B,C,D,B′}. In the above set of equations, we rescale χγγ′N1 → χγγ′ and ωγ(r)N1 → ωγ(r). In eq 19, q and q† are the forward and complementary chain propagators.43 These propagators satisfy the modified diffusion equations 5804

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805

Macromolecules

Article

(28) Nakamura, I.; Shi, A.-C. J. Phys. Chem. B 2011, 115, 2783−2790. (29) Pryamitsyn, V.; Han, S. H.; Kim, J. K.; Ganesan, V. Macromolecules 2012, 45, 8729−8742. (30) Leibler, L. Macromolecules 1980, 13, 1602−1617. (31) Lee, W. B.; Elliott, R.; Katsov, K.; Fredrickson, G. H. Macromolecules 2007, 40, 8445−8454. (32) Feng, E. H.; Lee, W. B.; Fredrickson, G. H. Macromolecules 2007, 40, 693−702. (33) Lefèvre, N.; Daoulas, K. C.; Müller, M.; Gohy, J.-F.; Fustin, C.A. Macromolecules 2010, 43, 7734−7743. (34) Fredrickson, G. H.; Ganesan, V.; Drolet, F. Macromolecules 2002, 35, 16−39. (35) Laradji, M.; Shi, A.-C.; Desai, R. C.; Noolandi, J. Phys. Rev. Lett. 1997, 78, 2577−2580. (36) Kawakatsu, T. Statistical Physics of Polymers: An Introduction; Springer: New York, 2004. (37) Bosse, A. W.; Tirumala, V. R.; Lin, E. K. J. Polym. Sci., Part B: Polym. Phys 2009, 47, 2083−2090. (38) Kim, J. K.; Kimishima, K.; Hashimoto, T. Macromolecules 1993, 26, 125−136. (39) Hahn, H.; Eitouni, H. B.; Balsara, N. P.; Pople, J. A. Phys. Rev. Lett. 2003, 90, 155505. (40) Hahn, H.; Chakraborty, A. K.; Das, J.; Pople, J. A.; Balsara, N. P. Macromolecules 2005, 38, 1277−1285. (41) Gomez, E. D.; Das, J.; Chakraborty, A. K.; Pople, J. A.; Balsara, N. P. Macromolecules 2006, 39, 4848−4859. (42) Li, D.; Gruhn, T.; Emmerich, H. J. Chem. Phys. 2012, 137, 024906. (43) Fredrickson, G. The Equilibrium Theory of Inhomogeneous Polymers; Oxford University Press: Oxford, U.K., 2006. (44) De Gennes, P. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.

5805

dx.doi.org/10.1021/ma4008576 | Macromolecules 2013, 46, 5796−5805