Phase Behavior of Binary Blends of Diblock Copolymers - American

Nov 15, 2010 - where the end-integrated propagators qpR(r,s) (p ) l, s; R ) A,. B) satisfy the modified .... achieved by varying χN from 0 up to 100,...
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J. Phys. Chem. B 2010, 114, 15789–15798

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Phase Behavior of Binary Blends of Diblock Copolymers Zhiqiang Wu, Baohui Li,* Qinghua Jin, and Datong Ding* School of Physics, Nankai UniVersity, Tianjin, 300071, China

An-Chang Shi* Department of Physics and Astronomy, McMaster UniVersity, Hamilton, Ontario L8S 4M1, Canada ReceiVed: August 20, 2010; ReVised Manuscript ReceiVed: October 17, 2010

The phase behavior of binary blends of a long symmetric AB diblock copolymer and a short asymmetric AB diblock copolymer is studied using the self-consistent mean-field theory. The investigation focuses on blends with different short diblocks by constructing phase diagrams over the whole blending compositions and a large segregation regime. The influences of the chain length ratio (R) of the long and short diblock copolymers on their miscibility and on the stability of various ordered structures are explored. The theoretical results reveal that the blends have a much more complex phase behavior than each constituent copolymer. With the increase of the volume fraction of the short diblocks in the blends, multiple transitions from a long-period lamellar phase to phases with nonzero interfacial curvatures including cylindrical and spherical phases, and finally to a short-period lamellar phase or disordered phase, are predicted. In particular, consistent with experiments, the theory predicts that the cylindrical phase is stabilized over a wide blending compositions region in the strong segregation region, even though the two constituent diblock copolymers are both lamellaforming. When the ratio R is large enough, macrophase separation occurs over a wide range of blending compositions in a relatively strong segregation regime. Various coexisting phases, including those of lamellar and disorder, lamellar and cylindrical, cylindrical and cylindrical, cylindrical and disorder, spherical and disorder, and cylindrical and spherical, are predicted. In addition, the density profiles of the typical ordered structures are presented in order to understand the self-organization of the different copolymer chains. Introduction The phase behavior of block copolymers has sparked considerable interest due to the fact that these macromolecules can spontaneously order into various nanostructures, which have potential applications in the nanotechnology.1-3 Particular attention has been paid to diblock copolymers which have the simplest possible architecture. To a good approximation,4 the morphology of an AB diblock copolymer depends on only two parameters, χN and f, where χ is the Flory-Huggins interaction parameter between the A- and B-monomers, N is the total degree of polymerization and f the volume fraction of the A-diblock. By varying f and χN, morphologies ranging from lamellae, hexagonal-packed cylinders, and body-centered-cubic spheres to a complex bicontinuous gyroid phase can be obtained from diblock copolymers.5 One goal in the study of block copolymers is to develop methods to control the properties of the material and to produce superior materials. A straightforward approach is the development of precisely tailored block copolymers. However, the necessity of accurately synthesizing complex molecules remains a serious obstacle impeding their commercial exploitation. On the other hand, polymer blending provides an economical alternative in engineering materials. Especially in the past several decades, binary mixtures of two AB diblock copolymers have received considerable experimental and theoretical attention. Studies show that blends of two AB diblock copolymers exhibit a much richer phase behavior than each neat constituent diblock * To whom correspondence should be addressed. E-mail: baohui@ nankai.edu.cn (B.L.); [email protected] (A-C.S.).

copolymer.6-16 This behavior is attributed to the large number of controlling parameters, including those for each constituent copolymers and the blending composition, as well as to the possibility of macrophase separation which is not encountered in the neat system. Theoretical exploration plays an important role in the understanding of these blending systems. By comparing the free energies of different morphologies, one can predict the phase diagrams which may provide useful guidelines for obtaining a specific phase experimentally. In the past years, a number of theoretical methods based on mean-field approximation have been developed to understand the underlying principle of the phase behavior of diblock copolymer blends.17-23 Among these, the self-consistent mean-field theory (SCFT)5 provides very accurate predictions. Using SCFT in the canonical ensemble, Shi and Noolandi constructed phase diagrams of the binary blends of diblock copolymers with similar degrees of polymerization in terms of compositions of the two constituent copolymers.24 They found that phases of the blends can be qualitatively predicted by the one-component approximation if the difference of the compositions of the two constituent copolymers is not too large. In the one-component approximation, the phase behavior of the blends is assumed to be governed by two parameters, the incompatibility χN and the total volume fraction of the A-monomers in the blends, and the phase diagram follows that of a neat diblock copolymer. However, if the compositions of the constituent diblock copolymers differ significantly, the blends tend to undergo macrophase separation into two disordered phases. Matsen and Bates obtained similar results using SCFT formulated in the grand-canonical en-

10.1021/jp107907s  2010 American Chemical Society Published on Web 11/15/2010

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semble.25 Furthermore, they also predicted that, with increasing χN, the two diblocks of very different compositions at first undergo macrophase separation, resulting in two disordered phases, and then they mix to form a single microstructure.25 These results agree well with previous experiment12,26 except that morphology with a long-range order is replaced by the one with local order only, due to the limited annealing time.12 When the degrees of polymerization of the two diblock copolymers are different, the phase behavior of the blends is even more complicated. The first issue is the immiscibility of the two diblock copolymers which arises from the mismatch of their degrees of polymerization. Matsen27 examined the phase behavior of binary blends of a long and a short symmetric AB diblock copolymers. His study showed that the long and the short diblocks are completely miscible when the ratio of their molecular weights is smaller than ∼5. When the ratio exceeds 5, the blend tends to phase separate into two distinct lamellar phases, or a lamellar and a disordered phase. This behavior agrees well with the available experimental results.6,7,28 Another issue that has attracted much attention is the large effect of the short diblocks on the interfacial curvature of the well-separated ordered structures formed by the long diblocks. Shi and Noolandi23 investigated the phase behavior of blends of small amounts of short diblocks blended with chemically identical long diblocks, when the long diblocks are in the strongly segregated regime. Their studies showed that, when the short diblocks are highly asymmetric, they tend to localize in the corresponding bulk domains and act like homopolymer fillers. Therefore, when the short diblocks are highly asymmetric, the phase behavior of the blends of two diblock copolymers is expected to be similar to that of blends of a diblock copolymer and a homopolymer. On the other hand, if the short diblocks are symmetric or nearly symmetric, a large amount of short diblocks tend to segregate to the interfaces and behave as compatibilizers to largely affect the local interfacial curvature of the ordered structures, which is the so-called cosurfactant effect.13 Experimentally, Hashimoto and co-workers have carried out a series of studies on phase transitions and morphologies of binary blends of polystyrene-block-polyisoprene (PS-PI) diblock copolymers,6-16 mainly focusing on three series of blends. In the first series, the constituent diblock copolymers of the blend both have nearly symmetric compositions but rather different molecular weights.7,9-11 In the second series, the constituent diblock copolymers have almost the same molecular weights but complementary compositions.12 In the third series, the constituent diblock copolymers were composed of a long asymmetric copolymer having a spherical (or cylindrical) morphology and a short symmetric copolymer.8,13-16 They systematically studied the phase behavior of these blends and constructed phase diagrams in the parameter space of temperature, blending composition and molecular weight ratio of the two constituent diblock copolymers. They found that the phase behavior of these blends is much richer and shows some fascinating features. For example, in the first series of blends, they observed an interesting and puzzling transition from lamellar to cylindrical phase, even though the two constituent copolymers are both lamella-forming by themselves.9-11 Moreover, they observed various coexisting phases, including those of lamellae and disorder, lamellae and cylinders, and two lamellae with different domain sizes.10 The possibility of the formation of cylindrical phase in the mixture of two lamellaforming diblock copolymers has been predicted by Lyatskaya et al. using the strong segregation theory (SST).19 These authors

Wu et al. also studied mixtures of two cylinder-forming block copolymers and cylinder- and lamella-forming block copolymers using SST. Their predicted dependence of domain width on the composition of the block copolymer mixture agrees well with the available experimental data. Motivated by these previous experimental and theoretical studies, especially by the interesting observation that blending two lamella-forming diblocks may lead to the formation of nonlamellar phases, we have carried out a series of studies of the phase behavior of binary blends of a long diblock copolymer and a short diblock copolymer using SCFT. One of the advantages of SCFT is that this theory is not restricted by the degree of segregation. In two previous reports, we have presented SCFT results of these blends in the relatively weak, or intermediate, segregation regime with χN e 4029 or χN e 45.30 In our first report,29 the diblock copolymer blends consist of a long symmetric diblock copolymer (with volume fraction flA ) 0.5) and a short nearly symmetric diblock copolymer (fsA ) 0.4). In the second report,30 the blends consist of two nearly symmetric diblock copolymers with flA ) 0.47 and fsA ) 0.4-0.49. The occurrence of order-order transition, orderdisorder transition (ODT), and regions of order-disorder coexistence has been examined. These studies provide some useful information about the phase behavior of binary block copolymer blends. However, due to the limitation to intermediate segregation (χN e 45), some important features from previous experiments were not examined in these previous studies. Examples are the observation that blending two lamella-forming diblocks can lead to nonlamellar phases and various coexisting phases, which usually occur in the strong segregation regime.10,11 In the present paper, we present a more detailed study of the phase behavior of binary blends of diblock copolymers using SCFT. The blends consist of a long symmetric diblock copolymer (flA ) 0.5) and a short asymmetric diblock copolymer. In contrast to our previous studies,29,30 the current study extends the calculations to the strong segregation regime with χN e 100. Furthermore, the composition of the short diblocks is extended to a more asymmetric one with fsA ) 0.35, besides the nearly symmetric one with fsA ) 0.4. SCFT phase diagrams over the whole blending compositions and from weak to strong segregation regimes are constructed by solving the SCFT equations using a reciprocal-space method.5,31 The influences of the compositions of the short diblocks and the length ratio of the two diblock copolymers on their miscibility and on the stability of different ordered structures are explored. In addition, the density profiles of the ordered structures are presented in order to understand the distribution of different copolymer chains. Theory In this section, we outline the SCFT formulated in the grand canonical ensemble. The grand-canonical ensemble is convenient for the study of phase behavior of blends when a macrophase separation may occur. We consider incompressible blends of two types of AB diblock copolymers with different degrees of polymerization contained in a volume V. The polymer chains are modeled as flexible Gaussian chains. The degree of polymerization of the long AB diblock copolymer is N and the volume fraction of the A-block in each polymer chain is flA. The degree of polymerization of the short diblock copolymer is N/R (R denotes the length ratio of the two diblock copolymers) and the volume fraction of the A-block in each chain is fsA. The monomer density F0 and the statistical Kuhn length b are assumed to be the same for the two species, A and B. We scale

Binary Blends of Diblock Copolymers

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distances by the Gaussian radius of gyration of a long diblock copolymer chain. Within the self-consistent formulation, the grand partition function of a blend is given by22

∫ DΦADΦBDΦADΦBDΞ exp{-F/kBT}

Z∝

(1)

where kB is the Boltzmann constant, ΦR is the monomer density, and WR is the corresponding conjugate fields for monomer R with R ) A, B. The function Ξ is a Lagrange multiplier to ensure the incompressibility condition. The free energy functional is given by the expression

NF 1 ) kBTVF0 V

∫ dr [χNΦAΦB - WAΦA - WBΦB Ξ(1 - ΦA - ΦB)] - Ql - eµQs

φA(r) ) -V

δQl δQs - Veµ δωA δωA

(3)

φB(r) ) -V

δQl δQs - Veµ δωB δωB

(4)

ωA(r) ) χNφB(r) + ξ(r)

(5)

ωB(r) ) χNφA(r) + ξ(r)

(6)

φA(r) + φB(r) ) 1

(7)

It is convenient to express the single-chain partition functions in terms of the end-integrated propagators

1 V

Qs )

∫ dr qlA†(r, flA)

(8)

∫ dr qsA†(r, fsA/R)

(9)

1 V

And sequentially, eqs 3 and 4 can be rewritten as

φA(r) )

∫0f

lA

† ds qlA(r, s)qlA (r, flA - s) +

eµ

∫0f

sA/R

∫0f

lB

† ds qlB(r, s)qlB (r, flB - s) +

eµ

∫0f

sB/R

† ds qsB(r, s)qsB (r, fsB /R - s) (11)

where the end-integrated propagators qpR(r,s) (p ) l, s; R ) A, B) satisfy the modified diffusion equation

∂qpR(r, s) ) ∇2qpR(r, s) - ωR(r)qpR(r, s) ∂s

(12)

with the initial conditions qpR(r,0) ) 1. q†pR(r,s) satisfy the same † † (r,0) ) qlβ(r,flβ),qsR (r,0) equations but with initial conditions qlR ) qsβ(r,fsβ/R), where β ) B if R ) A and vice versa. Once solution of the SCFT eqs 3-7 is obtained, the SCFT free energy density of the blend is given by

(2)

In eq 2, µ corresponds to the chemical potential for the short diblocks and the chemical potential for the long diblocks has been set to be zero without loss of generality since the blend is incompressible. Ql and Qs are the single-chain partition functions of the long and short diblock copolymers, respectively. Because exact evaluation of the partition function is in general not possible, the free energy is usually approximated by F[φA,ωA,φB,ωB,ξ], where φA, φB, ωA, ωB, and ξ are functions for which F attains its minimum. The minimization condition leads to the following set of SCFT equations

Ql )

φB(r) )

† ds qsA(r, s)qsA (r, fsA /R - s) (10)

NF 1 ) kBTVF0 V

∫ dr [χNφA(r)φB(r) - ωA(r)φA(r) ωB(r)φB(r)] - Ql - eµQs

(13)

Generally speaking, there may be more than one solution to the SCFT equations, corresponding to different phases of the system. At a given point in the parameter space, the equilibrium phase corresponds to the phase that has the lowest free energy. By comparing the free energy of the different possible phases, phase diagrams can then be constructed. Except for the homogeneous phase, an analytical solution of the SCFT equations is hard to obtain. Therefore, a numerical method is required. One method is to solve the equations directly in real space by using a combinatorial screening algorithm proposed by Drolet and Fredrickson,32,33 which is useful in exploring new morphologies of complex copolymers.34-37 Another choice, the one we employed in this work, is to solve the SCFT equations using the reciprocal-space method,5,31 which is numerically efficient and allows high-precision calculations of free energies and phase diagrams of the systems. The reciprocal-space method involves selecting a set of basis functions which are consistent with the symmetry of the ordered phase under investigation, and reformulating the theory in the reciprocal-space spanned by the basis functions. The basis functions are chosen as the eigenfunctions of the Laplacian operator ∇2fn(r) ) -(λn/D2)fn(r), with appropriate boundary conditions, where D is the period of the ordered structure. In particular, we take λ1 ) 0 and f1(r) ) 1. Thus, an arbitrary function g(r) with the symmetry of the ∞ gnfn(r). In terms ordered phase can be expanded as g(r) ) ∑n)1 of the expansion coefficients, the SCFT equations become a set of nonlinear algebraic equations, and they can be solved numerically. In our calculations, the Broyden method38 is used to iterate the equations. For an ordered structure, the free energy density is minimized with respect to the period. Although the calculations are performed in the grand-canonical ensemble, we plot our results in terms of the canonical coordinates χN, R, flA, fsA, and φl, where φl is the average volume fraction of the long diblocks in the blend. φl is conjugating to µ, and can be calculated by

φl ) Ql ) 1 - eµQs /R

(14)

Results and Discussion Phase Diagrams. We now present our theoretical results of the phase behavior of blends of two AB diblock copolymers.

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Figure 1. Phase diagrams for binary blends of a long symmetric diblock copolymer and a short asymmetric diblock copolymer with fsA ) 0.4. Different length ratios are considered: (a) R ) 10, (b) R ) 5, (c) R ) 2.5, and (d) R ) 1.67. L, H, S, and D represent lamellar, cylindrical, spherical, and disordered phases, respectively.

Theoretically, in a blend system, there are five independent controlling parameters, including the interaction strength, χN, the blending composition in terms of the volume fraction of the long diblocks in the blend, φl, the chain length ratio, R, and the volume fractions of the two constituent copolymers, flA and fsA. We are primarily interested in the case where the long diblocks are symmetric with flA ) 0.5, and the short diblocks are asymmetric. This choice of composition parameters complements the previous SCFT study on the blends of two symmetric diblock copolymers with mismatch lengths by Matsen.27 For the neat long diblocks, we noted that, when χN is small, they are in a disordered state. With increasing χN, they microphase separate into lamellar phase via a second-order phase transition at χN ) 10.495. We studied two series of blends where the compositions of the short diblocks are fsA ) 0.4 and fsA ) 0.35. In each series, the phase behavior of the blends is determined by the remaining three parameters, χN, φl, and R. To study their effects on the phase behavior of the blends, we extend our calculations to a stronger segregation regime, to the whole φl range, and to a wide R range. This enlarged parameter space is achieved by varying χN from 0 up to 100, φl from 0 to 1, and R in the whole range of 1 < R < 10. Since the reciprocal space method described in the last section requires the symmetry of the potential equilibrium phases, we need to select the candidate phases. Based on the available experimental results10,11 and for simplicity, we included all the classical phases, the lamellae (L), hexagonally packed cylinders (H), body-centered cubic spheres (S), and homogeneous disordered phase (D), in our calculations. This choice of phases excludes the bicontinuous gyroid phase, which is expected to occur in a small region at the lamellar-cylindrical phase boundary. Phase diagrams in different parameter space are

constructed by comparing the free energies of these phases. For all the cases, up to 200 basis functions are used and the concentration and field accuracy is calculated to 10-6. By convention, in all the plots of the phase diagrams shown below, all the pure single-phase regions have been labeled while the biphasic regions separating two neighboring pure single phases are, generally, not labeled. However, in cases where the pure single-phase regions are too small to be visible on the scale of the plots, we labeled such two-phase regions for the convenience of identifying the related regions. The phase diagrams for the blends with fsA ) 0.4 are summarized in Figure 1, where parts a-d correspond to the blends with R ) 10, 5, 2.5, and 1.67, respectively. For each value of R, the phase diagram is presented in the φl and χN space. In the phase diagram shown in Figure 1a with R ) 10, the neat short diblock copolymers are not able to form an ordered phase by themselves over the entire range of χN we considered, mainly due to their relatively short length. As shown in the phase diagram, blending these short diblocks with the long ones leads to a complex phase behavior. For small values of χN, the two constituent copolymers of the blend are completely miscible at all blending compositions, forming a single disordered phase. Above the critical point of the long diblocks at χN ) 10.495, lamellar phase is the equilibrium structure at large φl. Several important effects induced by the addition of the short diblocks can be clearly identified. First of all, the addition of the short diblocks has an effect on the domain sizes and stability of the ordered structures. As we can see in Figure 1a, when the volume fraction of the short diblocks in the blend is small, the two diblock copolymers are completely miscible and form a single lamellar phase. We noticed (and will show later) that, in this lamellar phase region, the

Binary Blends of Diblock Copolymers equilibrium domain size decreases with the increase of the volume fraction of the short diblocks. The reason for this is that, due to enthalpy gain, a considerable amount of the short diblocks segregates to the interface of the lamellae, resulting in a decrease of the interfacial energy and an increase of the stretching energy.23 The competition of these two factors results in a decrease of the equilibrium domain size without changing the lamellar structure. In this case, the long diblocks can absorb about 25-50% short ones, depending on the χN value. Then, with a further increase of the volume fraction of the short diblocks to about 30-50%, i.e., φl ≈ 0.5-0.7, a first-order phase transition from the lamellar phase to a cylindrical phase occurs in the range of 20 < χN < 74.3. This order-order transition is due to the competition between the interfacial and stretching energy. Furthermore, in the range of 20 < χN < 51.3, phase transition from cylindrical phase to spherical phase occurs in a narrow φl range of 0.3-0.4. Such order-order phase transitions also reveal the fact that in a blend even a slight asymmetry of the short diblocks can be greatly enhanced, thus leading to the formation of morphologies with curved interfaces. Second, when the incompatibility is relative larger (χN > 30), at high concentrations of the short diblocks, the two diblock copolymers become partially miscible. This is because when the ordered phase becomes saturated with the absorbed short chains, extra short chains are expelled from the ordered structures and the blends separate into a coexisting phase of an ordered (spherical or cylindrical) phase formed by both diblock copolymers and a disordered phase rich in the short diblocks. With the increase of χN, the φl region of the coexisting phase enlarges, which causes the shrinking and eventually disappearing of the φl region of the spherical phase at χN ) 51.3, and then the cylindrical phase at χN ) 74.3. Above χN ) 74.3, the coexistence region of the cylindrical and disordered phases is replaced by that of a lamellar phase rich in the long diblocks and a disordered phase rich in the short diblocks. The reason for this is that the phase transition from a lamellar phase to a cylindrical phase needs the addition of a sufficient amount of short chains; however, as χN increases, the long diblocks can absorb less short chains, and therefore, the blends phase separate into coexistence of lamellar and disordered phases. Figure 1b shows the phase diagram for the blends with R ) 5. Due to the relatively larger length, as χN increases, the neat short diblocks exhibit a sequence of first-order phase transitions from a disordered phase to a spherical phase, to a cylindrical phase, and finally to a lamellar phase. As a consequence of this added factor, the phase diagram of Figure 1b exhibits many new features when compared with Figure 1a. The most important feature is that, as a sufficient amount of short diblocks are added, the phase transition from lamellar to cylindrical phases always exists even for the largest χN under consideration. This behavior does not depend on the morphology of the neat short diblocks. Especially, in the strong separation regime, blending two lamella-forming diblock copolymers can also lead to the formation of a cylindrical morphology over a wide range of φl ) 0.4-0.6, and the phase boundary is almost independent of χN, at φl ≈ 0.7. Spherical phase can still be found in the relatively weak segregation regime. However, comparing with Figure 1a, the region of spherical phase is much smaller and shifts toward smaller χN. Furthermore, the miscibility of the constituent copolymers of the blends also exhibits some new features. The size of the biphasic region is largely dependent on the phase of the neat short diblocks. When χN > 20, the two-phase coexisting region of a spherical phase formed by the long and the short diblock copolymers and a disordered phase

J. Phys. Chem. B, Vol. 114, No. 48, 2010 15793 rich in the short diblocks appears, just like that in Figure 1a. As χN increases, this region is replaced by the one of a cylindrical phase and a disordered phase. Above the value of χN ≈ 56, the neat short diblocks order into a spherical phase and then a cylindrical phase. As a consequence, the orderdisorder coexistence region is replaced by the two-phase coexistence region of a spherical phase and a cylindrical phase and then the one of a long-period cylindrical phase formed by the long and the short diblock copolymers and a short-period cylindrical phase rich in the short diblocks. With further increase of χN, a coexistence region of a lamellar phase and a cylindrical phase appears. It is interesting to notice that there is a small range of incompatibility, 70.5 < χN < 73.3, in which the lamellar phase formed by the short diblocks can transform into a cylindrical phase as a small amount of the long diblocks is entered. And as a result, in such a range of χN, with increase of the volume fraction of the short diblocks, transitions from a long-period lamellar phase to a long-period cylindrical phase, to a short-period cylindrical phase, and eventually to a shortperiod lamellar phase take place sequentially. The above results can be compared with previous studies. Hashimoto et al. have constructed a phase diagram experimentally for the blends of two nearly symmetric PS-PI diblock copolymers in the parameter space of temperature-blending composition.11 Although the theoretical χN-φ phase diagram is analogous to the temperature-composition diagram typically measured in experiments, a comparison of the theoretical phase diagram with the experimental one is not straightforward, mainly because of the lack of knowledge of the relationship between the Flory-Huggins parameter χ and temperature. Under such a consideration, we will only simply distinguish between strong, intermediate, and weak segregation regimes. In the experiments of Hashimoto et al.,11 the range of temperature can be considered as corresponding to regimes between weak segregation to strong segregation. The length ratio of the two diblock copolymers used in their experiments is R ) 4.8 and the compositions of the two PS-PI diblocks are flPS ) 0.47 and fsPS ) 0.4, which are very similar to the blends shown in Figure 1b. Similar to the theoretical phase diagram, the long diblocks remain a lamellar phase over the whole segregation regime due to the relatively high molecular weight and nearly symmetric composition while the short ones show a lamellar phase in strong and intermediate segregation regimes and disorders in weak segregation regime.11 Their experimental phase diagram showed a good agreement with our theoretical one. For the blend with 30% of the short diblocks, only a lamellar phase could be found in the experimental system. As the volume fraction of the short diblocks increases to 60%, over the whole segregation regime, only a cylindrical phase was observed,11 indicating that the phase transition between lamellar and cylindrical phases occurs within the range of 0.4 < φl < 0.7. As more short diblocks (70-80%) are added, in the weak segregation regime, the blends were disordered while in the strong segregation regime, macroscopically phase separated, coexisting phase of lamellar and cylindrical phases were observed. For the blend with 10% of the long diblocks, in the strong segregation regime, a lamellar phase was observed. All these observations are in general agreement with our theoretical prediction. Despite the above agreements, there are some important differences between the theoretical and experimental results. One difference is that the experimental study found that, as the volume fraction of the long diblocks in the blend increases, the ODT temperature is lowered (i.e., larger χN), which is contrary to our theoretical results. This could be attributed to thermal fluctuation near the ODT of the short

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Figure 2. Phase diagrams for binary blends of a long symmetric diblock copolymer and a short asymmetric diblock copolymer with fsA ) 0.35: (a) R ) 10, (b) R ) 5, (c) R ) 2.5, and (d) R ) 1.67. In panel b, a region of H + S is labeled for the convenience of identifying the related regions.

diblocks. Another difference is that the coexistence region of ordered phase and disordered phase was not found in the experiments. This may be attributed to the fact that only the dominant phase is observed in experiments even though the blend is in the two-phase coexistence region. In addition, the spherical phase was not observed experimentally, which is probably due to its narrow parameter window. In addition to the experimental studies, the possibility of the formation of cylindrical phase in the mixture of two lamella-forming diblock copolymers was predicted in the strong segregation limit by Lyatskaya et al.19 Phase diagrams with R ) 2.5 and 1.67 are shown in Figure 1, c and d, respectively. These two phase diagrams reveal that, when the length ratio becomes smaller, the phase behavior of the blends becomes simple and exhibits similar topological characters. The long and short diblocks are completely miscible over most of the blending composition in the χN range considered, so that corresponding to a given (φl, χN) the blend only forms a single structure. As χN increases, regions of disordered, spherical, cylindrical, and lamellar phases appear in succession. The addition of the short diblocks still has the effect to induce a phase transition from a lamellar phase to a cylindrical phase and eventually spherical phase. However, as χN increases, such an effect becomes weaker and it requires more short chains before the phase transition takes place. As χN continues to increase, the region of spherical phase, and then cylindrical phase eventually disappears, indicating that these phases become unstable and the only ordered structure is lamellar phase at all the blending compositions. A comparison of panels c and d of Figure 1 reveals that, with the decrease of

R, the region of the spherical and cylindrical phase shrinks and shifts toward smaller χN while that of the lamellar phase enlarges. In Figure 1, we noted that all the neat short diblocks with R e 5 can microphase separate into a lamellar phase by themselves at large values of χN. However, only in the phase diagram with R ) 5 the interesting feature that a cylindrical phase can be stabilized by blending two lamella-forming diblock copolymers occurs. This observation indicates that the occurrence of this interesting feature depends sensitively on R. Besides, as R decreases, the region of two-phase coexistence shrinks and eventually disappears. This means that, as the mismatch of the lengths of the two diblock copolymers becomes smaller, the two diblock copolymers tend to be more miscible. Figure 2 shows phase diagrams in the χN-φl plane for the blends with fsA ) 0.35. Parts a-d of Figure 2 correspond to the blends with R ) 10, 5, 2.5, and 1.67, respectively. Although the composition of the short diblocks is only slightly varied, the phase diagrams represent several important differences when compared with those shown in Figure 1 for the case of fsA ) 0.4. The most obvious difference is that the region of spherical phase is much larger. This is because the effect of changing the interfacial curvature is enhanced due to more asymmetric in composition of the short diblocks. In Figure 2a-d, phase transitions from lamellar to cylindrical, and then to spherical phases are observed with the addition of the short diblocks in the weak segregation regime. On the other hand, the size of the spherical region shrinks with the decrease of R, and in the larger χN regimes, the spherical phase is replaced by the cylindrical phase or a coexistence phase of cylindrical and disordered phases.

Binary Blends of Diblock Copolymers In the phase diagram shown in Figure 2a with R ) 10, the neat short diblocks are not able to form an ordered phase. The long diblocks need to absorb at least 30% of the short chains to induce a phase transition from a lamellar to a cylindrical phase. Such a phase transition always occurs and the region of the cylindrical phase can extend to the largest χN value considered in the calculations. Therefore, the coexistence region of lamellar and disordered phases, occurring in Figure 1a, never occurs in Figure 2a, while a coexistence region of cylindrical and disordered phases occurs instead. Figure 2b shows the phase diagram for the blends with R ) 5. In this case, the neat short diblocks exhibit a sequence of first-order transitions from disordered phase to a spherical phase and then to a cylindrical phase. Since the neat short diblocks are more asymmetric, they are not able to form a lamellar phase in the parameter space considered. As χN increases, the biphasic regions of spherical and disorder (S + D) phases, cylindrical and disorder (H + D) phases, cylindrical and spherical (H + S) phases, and cylindrical and cylindrical (H + H) phases appear in succession, where H + H denotes a two-phase coexistence of two cylindrical phases differing in the short-chain concentration and in the lattice parameter. It is interesting to note that, when χN is close to 100, the region of coexisting phases ends at a critical point above which the two diblock copolymers become miscible and form a single cylindrical phase. In the phase diagram shown in Figure 2c for the blends with R ) 2.5, the two constituent copolymers are miscible in the most part of the phase diagram, which is similar to that in Figure 1c. However, different from the latter, a cylindrical phase formed by blending two lamella-forming diblock copolymers can be found in Figure 2c in a relatively wide φl range. Hence, in the strong segregation regime, phase transitions from a long-period lamellar phase to a cylindrical phase, and eventually to a shortperiod lamellar phase can be expected with increasing the volume fraction of the short diblocks in the blends. Comparing the phase diagram shown in Figure 2d for the blends with R ) 1.67 with that shown in Figure 1d, we noted that the overall tendency of phase transitions is similar in these two cases. Up to now, phase diagrams have been presented in the χN-φl plane which are analogous to the temperature-composition diagrams typically measured in experiments. Through the above discussions, we noted that some interesting features, such as the macrophase separation of the two constituent diblock copolymers of a blend and the occurrence of a cylindrical phase by blending two lamella-forming diblock copolymers, are sensitively dependent on the length ratio R. To examine the effects of R on the phase behavior of the blends more thoroughly, we have constructed phase diagrams in terms of blending composition, φl and R, which are shown in Figure 3 and Figure 4 for the two blends with fsA ) 0.4 and fsA ) 0.35, respectively. The particular choice of the range for R (in the range of 2-10) is for the convenience of comparing with experiment,10 and the range is sufficient for us to see the characteristic features. Since the above-mentioned interesting features usually occur in the relatively strong segregation regime, we choose the interaction strength at χN ) 100 in the following calculations. Figure 3 shows the phase diagram for the blends with fsA ) 0.4. We noted that when φl > 0.75 the blends possess a lamellar phase because of the dominance of the long diblocks. As more short diblocks are added, several interesting features appear. The following discussions mainly focus on the region φl < 0.75. When R < 4, the long and the short diblock copolymers are completely miscible at all the blending compositions, forming

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Figure 3. Phase diagram as functions of φl and R for the binary blends of a long symmetric diblock copolymer and a short asymmetric diblock copolymer with fsA ) 0.4. The interaction strength is fixed at χN ) 100. Some regions of two-phase coexistence are labeled for the convenience of identifying them.

Figure 4. Phase diagram as functions of φl and R for the binary blends of a long symmetric diblock copolymer and a short asymmetric diblock copolymer with fsA ) 0.35. The interaction strength is fixed at χN ) 100. The coexistence regions of H + S and H + D are labeled for the convenience of identifying them.

a single lamellar phase. At R ≈ 4, a critical point (at φl ≈ 0.46) is encountered, above which an order-order phase transition from lamellar to cylindrical phases can be found. Such a transition occurs when about 25-50% of the short diblocks are added, depending on the R value. When more of the short diblocks are added, the two diblock copolymers phase separate into coexisting lamellar and cylindrical (L + H) phases, which also occurs just above the critical point. As R increases to ∼7.1, the neat short diblocks undergo a phase transition from a lamellar phase to a cylindrical phase and, as a consequence, in the diagram shown in Figure 3, the L + H region is replaced by a coexistence region of two cylindrical phases (H + H) with different lattice periods. As R increases further, the short-period cylindrical phase phase-transits into spherical phase first and eventually disorders and hence the H + H region is replaced by the H + S region and then H + D region. Finally, when R > 9.1, the cylindrical phase region eventually disappears and a coexistence region of L + D, a lamellar phase rich in the long diblocks and a disordered phase of almost pure short diblocks, is observed. In the phase diagram, the cylindrical phase formed by blending two lamella-forming diblock copolymers can be found over the length ratio range of 4 < R< 7.1, and the particular case with R ) 5 shown in Figure 1b is in this range. The miscibility criterion on the length ratio can also be read off from the phase diagram. The long and short diblock copolymers are completely miscible when the ratio R is smaller than 4.

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Earlier theoretical work of Masten focused on the miscibility of two symmetric diblock copolymers with different lengths using SCFT.27 In his work, ordered structures with nonzero interfacial curvatures are not stable due to the symmetry of the diblock copolymers. A critical value of R ≈ 5 was found from his results, while the two symmetric diblocks are completely miscible if their length ratio is smaller than this value. As this ratio is exceeded, the blend can phase separate into two distinct lamellar phases or coexistence lamellar and disordered phases. Comparing our results with those of Matsen’s, we can easily identify the effects induced by the slightly asymmetric short diblocks. First, an ordered structure with a nonzero interfacial curvature, i.e., the cylindrical phase, is stabilized in the range of 4 < R < 9.1. More interestingly, the cylindrical morphology can be obtained even if the two constituent copolymers of the blend are both lamella-forming by themselves in the range of 4 < R < 7.1. Second, the critical value of R (Rc) above which the two copolymers become immiscible is shifted to a smaller one, from Rc ≈ 5 to Rc ≈ 4. Such a shift is correlated with the stability of the cylindrical phase. With the increase of R, the appearance of two-phase coexistence and the cylindrical phase occurs simultaneously. It is also useful to compare our phase diagram with previous experimental results of Yamaguchi and Hashimoto10 on the blends of two nearly symmetric PS-PI diblock copolymers. Their experiments were performed at a sufficiently low temperature (ambient temperature), corresponding to a strong segregation regime and so a comparison with our theoretical results is feasible. Although the specific values of parameters in the experiments are different from those we used in our studies, it is expected that such differences may not bring qualitative errors in the comparison. In their experiments, the length ratio R ranges from 4.8 to 8.3, fsPS from 0.40 to 0.49, and flPS is fixed at 0.47. For all the blends, over the range of φl > 0.7, a lamellar morphology was retained in their experiments,10 which is in qualitative agreement with our theoretical result. For the blend with fsPS ) 0.4 and R ) 4.8, a cylindrical phase was observed in wide blending compositions of 0.3 e φl e 0.6 and the blend separates into coexisting phase of lamellar and cylindrical phase over the range of 0.2 e φl e 0.3. For the blend with fsPS ) 0.45 and R ) 6.9, a cylindrical phase was also observed but in a much narrow blending composition range and the coexistence region of lamellar and cylindrical phases was found within the blending composition range of 0.02 e φl e 0.5.10 All these observations agree very well with our theoretical predictions. On the other hand, for the blends with fsPS ) 0.49 and R ) 7.4, and with fsPS ) 0.49 and R ) 8.3, cylindrical phases were not observed,10 which is obviously because the short PS-PI chains are not asymmetric enough. Although the experimental work did not give a miscibility criterion on the length ratio, it was noted that, as R decreases, the blending composition range where the constituent copolymers macrophase separate becomes small. This tendency accords with our theoretical prediction on the miscibility criterion. Comparing Figure 4 with Figure 3, we notice several important differences in the topology of the phase diagram in the region with φl < 0.75, even though they have similarities in the region with φl > 0.75. The differences are due to the fact that the composition of the short diblocks becomes more asymmetric (fsA ) 0.35). As shown in Figure 4, the critical point is shifted toward an even smaller value of R ≈ 2.4 above which a phase transition between lamellar and cylindrical phases is observed when sufficient short diblocks are added. When 2.4 < R < 4.1, the cylindrical phase is stabilized by blending two

Wu et al. lamella-forming diblock copolymers in a wide range of blending composition. When 4.1 < R < 5.3, the cylindrical phase remains stable and its composition range increases with the increase of R since that the short diblocks are forming cylinder by themselves. A critical point is encountered at Rc ) 5.3, above which a region of two-phase coexistence of two cylindrical phases (H + H) with different microdomain sizes appears. The H + H region is stable until R ≈ 7.9 above which it is replaced by the coexistence of H + S and H + D in succession as R increases further. When R > 5.3, the composition range of the coexisting region enlarges with the increase of R, which results in the shrinking in the region of the cylindrical phase. In spite of this, the cylindrical phase remains stable in the largest R value we investigated, which is quite different from that in Figure 3 where the cylindrical phase eventually disappears when R > 9.1 and a coexistence region of L+D occurs. Density Profiles of the Ordered Structures. In order to understand the self-organization of the polymer chains in the binary blends, it is helpful to examine the density profiles and to relate them to the equilibrium structures. We are primarily interested in the evolution of the spatial distribution of each segment as the volume fraction of short chains increases, especially in the cylindrical phase formed by blending two lamella-forming diblock copolymers. A series of blends are investigated, and in all cases, the parameters χN, flA, fsA, and R are fixed at 100, 0.5, 0.4 and 5, respectively. As an example, the density profiles for blends with φl ) 1.0, 0.8, 0.1, and 0.5 are shown in Figure 5 parts a-d, respectively. According to the phase diagram shown in Figure 1b, all the blends form a lamellar phase except that for the case with φl ) 0.5 where the blend forms a cylindrical phase. In Figure 5a-c, one-dimensional density profiles are shown for the lamellar phases, where the contributions due to the long chains and due to the short chains are plotted as solid lines and dashed lines, respectively. Moreover, each density profile spans a complete phase period from the middle of one domain to the middle of the adjacent domain, and the spatial distance is scaled by the Gaussian radius of gyration of the long diblocks. For the case with φl ) 1.0, the system corresponds to the neat long symmetric diblock copolymer melts. Since the system is in the strong segregation regime, the melts self-assemble into a wellordered lamellar structure and the interfaces between the adjacent domains of A-monomers and B-monomers are extremely narrow as shown in Figure 5a. For the case with φl ) 0.8, 20% of the short diblocks have been blended. As shown in Figure 5b, almost all of the short chains are localized at the A/B interfaces. To avoid the contact between the unlike monomers, the A- and B-monomers of the short chains are distributed in their corresponding domains formed by the long diblocks. Such a distribution leads to a reduction in the interfacial energy but an increase in the stretching energy, resulting in shrinkage of the domain size of the lamellar structure.23 When φl ) 0.1, the short chains are the dominant constituent copolymers. As we can see in Figure 5c, the long and short copolymers share a common interface. In this case, the domain size is largely determined by the short chains, which causes a further reduction in the period of lamellar phase. When φl ) 0.5, a cylindrical phase is stabilized since enough of short and long chains have been added. For a cylindrical phase, the density profile is two-dimensional (2-D) in nature, so we need to use 2-D plot to show the complete polymer distribution for the cylindrical phase. It is found that the variation of the densities is not circularly symmetric in such a cylindrical phase, as shown in the 2-D plots in Figure 5d. It is observed

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Figure 5. Density profiles for the blends of a long symmetric diblock copolymer and a short asymmetric diblock copolymer with fsA ) 0.4 when χN ) 100 and R ) 5. Parts a-d correspond to the blends with φl ) 1.0, 0.8, 0.1, and 0.5, respectively. In panels a-c, the profiles of long chains are plotted as solid lines and those of short ones are plotted as dashed lines. In panel d, the density profiles are shown in 2-D for the cylindrical phase when φl ) 0.5; (i)-(iv) correspond to φlA, φlB, φsA, and φsB, respectively. The red and blue colors correspond to the maxima and minima, respectively.

that the A-monomers of the long chains form the cores of A-rich domains, while most of the short chains segregate at the interfaces, forming shells surrounding the A-rich cores. The B-monomers of the long chains stretch out to fill the space between the A-rich cores, and their maximum density appears at the center of three adjacent A-rich domains. Hence, the B

blocks of the long chain are largely stretched along the direction from the interface to the center of the three adjacent A-rich domains. However, along the direction from one A-rich domain to an adjacent one, little B-blocks of the long chain can be found. This may be because the space between two adjacent A-rich domains is too narrow to allow the stretching of B blocks of

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the long chain. On the other hand, the B-blocks of the short chain are largely stretched along the direction from one A-rich domain to an adjacent one. Therefore, the stretching directions are different for different copolymer chains in such a cylindrical phase. The SST provided an intuitive understanding on the formation of a cylindrical phase in the mixture of two lamellaforming diblock copolymers.19 However, due to the narrow interface approximation and the approximation of using a circular cylinder instead of the unit cell of cylindrical phase, the structure of the cylindrical phase obtained in that theory is an approximation. Figure 5d gives more reasonable density profiles of the cylindrical phase. Through the analysis of the density profiles, it can be seen that the asymmetric short diblocks have a very different mechanism in changing the interfacial curvature of a well-ordered lamellar structure when compared with that of homopolymers. In the latter case, chains tend to be the filling of the corresponding domain and changing the stability of ordered structures by swelling the corresponding domain.39 Conclusions The phase behavior of binary blends of diblock copolymers has been studied using the reciprocal-space self-consistent mean filed theory, focusing on the cases where the long diblocks have a symmetric composition (flA ) 0.5) and the short ones have an asymmetric composition (fsA ) 0.4 and 0.35). The relative stability of various phases, including lamellar, cylindrical, spherical, and disordered phases, is examined. Phase diagrams are constructed in both (χN-φl) and (R-φl) spaces. Density profiles are presented in order to understand the self-organization of the different polymer chains. Several conclusions can be drawn from the theoretical studies. First of all, ordered structures with nonzero interfacial curvature can be stabilized in blends containing enough short chains. In both series of blends with fsA ) 0.4 and 0.35, cylindrical phase is found to be stable over wide φl and χN ranges. Especially, the cylindrical phase remains stable even when the two constituent copolymers are both lamella-forming by themselves. In addition, spherical phase is found to be stable in the relatively weak segregation regime. Therefore, multiple phase transitions can be expected with the change of the blending composition of the blends. Especially for the series of blends with fsA ) 0.4, narrow regions of spherical phase are also found. The total volume fractions of the A monomers corresponding to these equilibrium spherical structures are all large than φA > 0.4. Therefore, such a behavior cannot be explained by the one-component approximation,24,25 which is drawn from the binary blends of diblock copolymers with similar lengths. Second, the presence of a second component may lead to macrophase separation. The length ratio of the two constituent diblock copolymers plays an important role in control the miscibility between them. When R is small, the constituent copolymers are completely miscible over the whole blending compositions and they together form a single phase. When R is large enough, macrophase separation occurs in relatively strong segregation regime. The critical R value is correlated with the composition of the short diblocks. In the blends with fsA ) 0.4, Rc ) 4, while in the blends with fsA ) 0.35, Rc ) 5.3. Above these critical R values, various coexisting phases are predicted. This study complements the previous one of Matsen on the miscibility of two symmetric diblock copolymers.27 Careful comparisons show that our predictions are in good agreement with available experiments and theories for the most part, especially in the strong segregation regime.

Wu et al. Acknowledgment. This research is supported by the National Natural Science Foundation of China (Grants No. 20474034, 20774052, and 20990234), by the National Science Fund for Distinguished Young Scholars of China (No. 20925414), by the Chinese Ministry of Education with the Program of New Century Excellent Talents in Universities (Grants No. ncet-050221), and by Nankai University ISC. A.-C.S. gratefully acknowledges the support from the Natural Sciences and Engineering Research Council (NSERC) of Canada. References and Notes (1) Abetz, V.; Simon, F. W. AdV. Polym. Sci. 2005, 189, 125–212. (2) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52 (2), 32–38. (3) Hamley, I. W., The physics of block copolymers; Oxford University Press: Oxford, UK, 1998; p viii, 424 pp. (4) Helfand, E. J. Chem. Phys. 1975, 62, 999. (5) Matsen, M. W.; Schick, M. Phys. ReV. Lett. 1994, 18, 2660–2663. (6) Hashimoto, T.; Yamasaki, K.; Koizumi, S.; Hasegawa, H. Macromolecules 1993, 26, 2895–2904. (7) Hashimoto, T.; Koizumi, S.; Hasegawa, H. Macromolecules 1994, 27, 1562–1570. (8) Yamaguchi, D.; Bodycomb, J.; Koizumi, S.; Hashimoto, T. Macromolecules 1999, 32, 5884–5894. (9) Yamaguchi, D.; Shiratake, S.; Hashimoto, T. Macromolecules 2000, 33, 8258–8268. (10) Yamaguchi, D.; Hashimoto, T. Macromolecules 2001, 34, 6495– 6505. (11) Yamaguchi, D.; Hasegawa, H.; Hashimoto, T. Macromolecules 2001, 34, 6506–6518. (12) Yamaguchi, D.; Takenaka, M.; Hasegawa, H.; Hashimoto, T. Macromolecules 2001, 34, 1707–1719. (13) Court, F.; Hashimoto, T. Macromolecules 2001, 34, 2536–2545. (14) Court, F.; Hashimoto, T. Macromolecules 2002, 35, 2566–2575. (15) Court, F.; Yamaguchi, D.; Hashimoto, T. Macromolecules 2006, 39, 2596–2605. (16) Court, F.; Yamaguchi, D.; Hashimoto, T. Macromolecules 2008, 41, 4828–4837. (17) Birshtein, T. M.; Liatskaya, Y. V.; Zhulina, E. B. Polymer 1990, 31, 2185–2196. (18) Birshtein, T. M.; Lyatskaya, Y. V.; Zhulina, E. B. Polymer 1992, 33, 2750–2756. (19) Lyatskaya, Y. V.; Zhulina, E. B.; Birshtein, T. M. Polymer 1992, 33, 343–351. (20) Zhulina, E. B.; Birshtein, T. M. Polymer 1989, 32, 1299–1308. (21) Zhulina, E. B.; Lyatskaya, Y. V.; Birshtein, T. M. Polymer 1990, 33, 332–342. (22) Matsen, M. W. Phys. ReV. Lett. 1995, 74, 4425–4428. (23) Shi, A. C.; Noolandi, J. Macromolecules 1994, 27, 2936–2944. (24) Shi, A. C.; Noolandi, J. Macromolecules 1995, 28, 3103–3109. (25) Matsen, M. W.; Bates, F. S. Macromolecules 1995, 28, 7298–7300. (26) Vilesov, A. D.; Floudas, G.; Pakula, T. Macromol. Chem. Phys. 1994, 195, 2317–2326. (27) Matsen, M. W. J. Chem. Phys. 1995, 103, 3268–3271. (28) Papadakis, C. M.; Mortensen, K.; Posselt, D. Eur. Phys. J. B 1998, 4, 325–332. (29) Wu, Z. Q.; Li, B. H.; Jin, Q. H.; Ding, D. T.; Shi, A. C. Acta Polym. Sin. 2007, 11, 1035–1039. (30) Ge, H.; Wang, Z.; Li, B. H.; Ding, D. T.; Shi, A. C. Acta Polym. Sin. 2007, 2, 119–122. (31) Matsen, M. W. Eur. Phys. J. E 2009, 30, 361–369. (32) Drolet, F.; Fredrickson, G. H. Macromolecules 2001, 34, 5317. (33) Drolet, F.; Fredrickson, G. H. Phys. ReV. Lett. 1999, 83, 4317. (34) Jiang, Z.; Wang, R.; Xue, G. J. Phys. Chem. B 2009, 113, 7462– 7467. (35) Tang, P.; Feng, Q.; Zhang, H.; Yang, Y. Phys. ReV. E 2004, 69, 031803. (36) Tang, P.; Qiu, F.; Zhang, H.; Yang, Y. J. Phys. Chem. B 2004, 108, 8434–8438. (37) Wang, R.; Xu, T. Polymer 2007, 48, 4601–4608. (38) Press, W. H.; S. A. Teukolsky, W. T. V.; Flannery, B. P. Numerical Recipes in C: The Art of Scientific Computing; Cambridge University Press: Cambridge, UK, 2002. (39) Matsen, M. W. Macromolecules 2003, 36, 9647–9657.

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