Order and Phase Behavior of Thin Film of Diblock Copolymer

Article pubs.acs.org/Macromolecules

Order and Phase Behavior of Thin Film of Diblock CopolymerSelective Nanoparticle Mixtures: A Molecular Dynamics Simulation Study Lenin S. Shagolsem*,†,‡ and Jens-Uwe Sommer†,‡ †

Leibniz Institute of Polymer Research Dresden, 01069 Dresden, Germany Institute of Theoretical Physics, TU Dresden, 01069 Dresden, Germany

‡

ABSTRACT: By means of molecular dynamics simulations, we study AB diblock copolymer and nanoparticle mixtures confined between two identical walls in slit geometry. The nanoparticles are selective to the minority A-block, while the walls are neutral to both copolymer and nanoparticle. We obtained the various structures of the copolymer nanocomposites and are summarized in a phase diagram constructed in diblock composition and nanoparticle concentration space. In comparison to the phase diagram in bulk, we observe a much wider lamellar region with a broad class of lamellar structures, and the phase boundaries are shifted with increasing nanoparticle concentration. We find that both vertically and horizontally oriented lamellar structures are realized. The vertically oriented lamellae are formed by slightly asymmetric and symmetric diblock copolymers at low nanoparticle concentrations and have a very limited region of stability in the phase space, whereas the horizontally oriented lamellae are formed by asymmetric copolymer at large nanoparticle concentrations. In the vertically oriented lamellae, the segregated nanoparticles at the polymer−wall interfaces form nanoparticle monolayer above the A-domains and exclude A-monomers from this region. Consequently, the copolymer interface lines near walls are perturbed; also, the chains close to the walls are overstretched compared to the bulk. For horizontally oriented lamellae there is no overstretching of chains near the walls. The test of stability of the lamellar structures against the different thermodynamic pathways is also performed. Lastly, by considering the horizontal lamellae, we study the effect of nanoparticle concentration on the lamellar layer thickness.

I. INTRODUCTION

copolymer nanocomposites represent an interesting system to explore. Mixtures of diblock copolymer (DBC) and selective nanoparticle (NP) in bulk have been studied by Huh and coworkers using both Monte Carlo simulations and theory based on a scaling model for DBC in the strong segregation limit.12 They observed various morphological transitions by varying the concentration and size of the NP; also, they developed the corresponding phase diagram. Such morphological transitions induced by the filler particles are observed in a recent experimental study by Jang et al.13 Lee and co-workers extended the above theoretical study by considering symmetric-DBC/NP mixtures in confinement where they combined both self-consistent field theory (SCFT) describing polymers and density functional theory (DFT) describing particles.14,15 In their study, they demonstrated that a horizontally oriented lamellae (formed with selective confining walls) could transform to a vertically oriented lamellae upon addition of neutral NPs, whereas with the neutral confining

With the advancement of thin film technologies polymer-based thin films with/without nanoinclusions find applications in modern devices such as sensors, organic electronics/photovoltaics, surface modifications, nanolithography, etc.1−6 Control of morphology and positioning of nanoinclusions at the right place in the polymer matrix is essential to achieve the desired properties.7 Use of block copolymers is favored with respect to homopolymers because of their ability to self-organize into various equilibrium structures at nanoscales, thus allowing better control over spatial distribution of nanoinclusions.8 For example, the performance of bulk heterojunction solar cells can be affected by the nanoscale morphology.9,10 Structural disorder in bulk heterojunctions (due to the random phase separation of donor and acceptor) can be removed by using block copolymers which can act as both active materials and structure directors,2 thus giving a semiconductor block copolymer nanocomposite with well-defined stable internal structure,11 and the device, in principle, can attain higher performance. Therefore, it is necessary to be able to control the internal structures in a desired way to enhance the device performance. Also, viewed from the fundamental aspects, © 2014 American Chemical Society

Received: October 23, 2013 Revised: January 2, 2014 Published: January 13, 2014 830

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induced horizontal orientation of the lamellae, stability of lamellar structures, and density profiles are discussed in sections IIIB, IIIC, and IIID, respectively, and finally we summarize the results in section IV.

walls addition of NPs selective to a component of the DBC leads to the formation of horizontal lamellae with particle selective block close to the walls.15 And in a recent experimental study, Yoo et al. used gold NPs with tuned surface chemistry to control the orientation of block copolymer thin films.16 They observed that the addition of neutral-NPs produce a horizontally oriented microdomains at low NP concentrations, whereas at higher NP concentration (about/ above 5 wt %) it changes to vertical orientation. For the case of selective-NPs they observed a horizontal orientation of the microdomains relative to the substrate regardless of the amount of added NPs. Horizontal orientation of the microdomains observed at low NP concentration for neutral-NP case and independent of NP concentration for the selective-NP case may be due to the preferential interaction between the substrate and one of the component of diblock copolymers. Copolymer nanocomposites have the advantage that the morphologies formed by the composites and even the orientation can be controlled by changing the amount/type of fillers alone.12−17 Thus, morphological/orientational transitions are possible without using external fields or modification of copolymer chains. There is a large amount of experimental and theoretical work covering different aspects for the block copolymer thin films,18−25 but there is still more to be explored for the thin films of copolymer nanocomposites. The current state of mesoscopic modeling for the block copolymer nanocomposites is discussed in ref 26. In our earlier molecular dynamics (MD) simulations study of DBC/neutral-NP mixtures in confinement,27 we showed that the spatial distribution of NPs in the copolymer matrix is very sensitive to the NP−monomer interaction. Athermal NPs lead to the formation of nanoclusters (NP condensates), while thermal NPs give rise to a homogeneous distribution of NPs at the diblock interfaces; however, in both cases there is no effect on the copolymer morphology. In this work, we extend our previous study of the confined nanocomposites by considering NPs selective to the minority component of the DBC. The nanocomposites are confined in a slit geometry by purely repulsive walls. Also, in a recent study by Wu et al.,28 various structures formed by the DBC and selective-NP mixtures confined by neutral walls are explored using Monte Carlo simulations. For different values of film thickness they obtain phase diagrams in the NP concentration and NP/polymer interaction strength phase space while keeping the diblock composition fixed; i.e., they consider symmetric DBC. Similar study in bulk has been performed earlier by Schultz and coworkers. 29 In the present study, by varying the NP concentration and diblock composition (at fixed film thickness and NP/polymer interaction strength), we explore various structures formed by the nanocomposites and discuss the effect of confinement on the phase diagram. Here, we show that both horizontally and vertically oriented lamellar structures can be formed even with the purely repulsive wall. We will also show that the NP induce switching of lamellar orientation from vertical to horizontal with respect to the wall at high NP concentrations is a result of the tendency of the chains to relax overstretching near the wall. We also discuss the stability of lamellar structures and the effect of changing NP concentration on the lamellar layers, e.g., NP localization and individual chain properties. The rest of the article is organized as follows: In section II we describe the model and simulation details. Various ordered structures are discussed and summarized in section IIIA; NP

II. MODEL AND SIMULATION DETAILS A coarse-grained bead−spring model is employed to simulate the polymer chains.30 All the pairwise interactions are simulated using a cut and shifted Lennard-Jones (LJ) potential ⎡ d V (r ) ⎤ ULJ = V (r ) − V (rc) − (r − rc)⎢ ⎥ ⎣ dr ⎦ r = r

c

(2.1)

where ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ V (r ) = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠

(2.2)

is the LJ potential. Here, r and rc are the separation between a pair of particles and cutoff radius respectively, and ε is the interaction strength. Connection between the neighboring beads in a chain is modeled by a finitely extensible nonlinear elastic (FENE) potential31 defined as UFENE

⎧ kr 2 ⎪− 0 ln[1 − (r /r0)2 ], r < r0 =⎨ 2 ⎪ r ≥ r0 ⎩∞ ,

(2.3)

where r is the separation of neighboring monomers in a chain. The spring constant, k, is fixed at 30ε/σ2, while the maximum extension between two consecutive monomers in a chain, r0, is fixed at 1.5σ. The above values of k and r0 ensure that the chains avoid bond crossing and very high frequency modes.30 All the physical quantities are expressed in terms of LJ reduced units where σ and ε are the basic length and energy scales, respectively. The reduced temperature, T*, and time, t*, are defined as T* = kBT0/ε and t* = t/τLJ, where τLJ = σ(m/ ε)1/2 is the LJ time, and kB, T, t, and m are the Boltzmann constant, absolute temperature, real time, and mass, respectively. All the monomers have a diameter of σ = 1 and mass m = 1, while NPs have a diameter of σp = 2σ and mass mp scales as cube of diameter. The interaction strength, ε = 0.5, is same for all the pairwise interactions present in the system. In a film of thickness 50σ, we introduce 2500 A−B diblock copolymer chains each with 48 monomers along with NPs. Periodic boundary conditions are applied along X- and Ydirections, while the Z-direction is nonperiodic due to the presence of walls. The diblock composition, f = NA/N, is varied in the range 0.1 ≤ f ≤ 0.5. Here, NA and N are the number of monomers in the particle selective A-block and the total number of monomers per chain, respectively. The total amount of NPs present in the system is quantified by an overall NP volume fraction, Φp, defined as π Φp = (Npσp3/V0) (2.4) 6 where V0 = (π/6)(Npσp3 + Nmσ3) is the total occupied volume; Np and Nm are respectively the total number of NPs and monomers present in the system. In our simulations, we vary Φp in the range 0 ≤ Φp ≤ 0.4. The A−B, B−NP, A/B−wall, and NP−wall interact via a purely repulsive LJ potential (cutoff at the potential minimum, rc = 21/6σ), while we allow attraction between the A−A, B−B, and A−NP (rc = 2.5σ); see Table 1. 831

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Table 1. Interaction Range among the Species interaction

cutoff radius (rc)

nature of interaction

A−B A−A or B−B A−NP B−NP NP−NP A/B−wall NP−wall

21/6 × 1.0 2.5 × 1.0 2.5 × 1.5 21/6 × 1.5 21/6 × 2.0 21/6 × 0.5 21/6 × 1.0

repulsive attractive attractive repulsive repulsive repulsive repulsive

For the above systems, we perform constant NPT simulations using Nosé−Hoover thermostat and barostat to maintain constant temperature and pressure, respectively. To simulate in melt condition, the pressure along X- and Ydirections is set to P = 5ε/σ3,30 and we choose an MD integration time-step of 0.001τLJ. Initially, randomly generated DBC/NP mixtures are equilibrated (at T* ≈ 4.5) with only the purely repulsive LJ interactions. Then we switch on the interactions between the different pairs (see Table 1) and let the system equilibrate at a sufficiently high temperature (T* ≈ 4.5), where they are in disordered state. And then the systems are cooled down to T* = 1 continuously, where the various morphologies are formed. The addition of NPs can effectively change the order−disorder transition (ODT) temperature of copolymer melts or polymer blends;29,32,33 however, in our present study we will not discuss this effect, but rather focus on the morphologies formed at a given temperature of T* = 1.0, which is well below the ODT for a pure DBC melt.27 Typically, in our study, the systems are relaxed at a desired temperature for about 5 × 103τLJ (which is roughly 100τd, where τd is the average end-to-end correlation decay time), and the samples are cooled at the rate of dT/dt = 10−4ε/kBτLJ. MD simulations are performed using LAMMPS,34 and the configurations are visualized using VMD.35

Figure 1. Simulation snapshots of the typical structures formed (at T* = 1.0) by the confined nanocomposites shown for three different values of NP concentration, Φp, indicated in the figure, and at different values of diblock composition, f. NP is represented in blue, NP selective A-species in red, and B-species in green. Symbols: S = sphere, C = cylinder, Lv/h = vertically/horizontally oriented lamellae, PLh = perforated Lh, and ML = mixed lamellae. The NP repulsive B-block is made transparent in all the figures, except for the perforation made by the B-component in the middle lamellar layer of part c:(i) and in parts c:(iv) and c:(v) to see the structures clearly. In order to see the inverted structures clearly, NP is made transparent in part c:(iv), and in part c:(v) only the B-component is shown.

The morphologies formed at a higher NP concentration of Φp = 0.143 is shown in Figure 1b. Here, the spherical structure formed at f = 0.1 (see Figure 1b:(i)) has much larger core size in comparison to that formed at Φp = 0.077. Now if we compare Figures 1a and 1b, we can see that in the range 0.187 ≲ f ≲ 0.3 cylinders are observed for Φp = 0.077, while at larger value of Φp = 0.143 a horizontally oriented lamellar structure is formed. This indicates a shift in the phases to a lower value of f with higher particle concentration. It is interesting to note that with selective-NP lamellar structure can be realized even for highly asymmetric DBC. In Figure 1b:(v), we see a lamellar structure which orients vertically near the surface while in the bulk it orients horizontally. Such lamellar structure with mixed orientations is called mixed lamellae (ML) in our present study. In Figure 1c, we show a snapshot of the morphologies for a NP concentration of Φp = 0.40. We can see that upon increasing diblock composition f lamellae change to inverted cylinders, i.e., cylinders with majority B-component as core (see Figure 1c:(iv,v)). At this high value of Φp the highly asymmetric DBC (f < 0.23) self-organized into horizontal lamellae where segregated NPs from the A layer form a dense NP layer (NP condensate) next to it. Such morphologies are referred to as self-assembled (SA) structures, and it is an exclusive characteristic of DBC/NP mixtures. Figures 1c:(i−iii) are self-assembled lamellar (SA-L) structures, and likewise shown in Figures 1b:(i,ii) are the SA-S and SA-C structures, respectively. Because of the finite NP uptake capacity for a given f with increasing particle concentration the excessive NPs are segregated forming a dense NP layer/core next to the Ablock.12,36 This avoids the overstretching of the NP selective Ablocks and at the same time gains the enthalpic energy due to the contact between A-monomers and NP cluster. However,

III. RESULTS AND DISCUSSION A. Structures Formed by the Nanocomposites. Various morphologies formed by the DBC/selective-NP mixtures have been studied for bulk systems by Huh and co-workers.12 They show that depending on the diblock composition and NP volume fraction and size, the nanocomposites can realize various self-assembled structures. As discussed in earlier studies considering confined lamellae,14,15 NP can influence/modify the preferred orientation of the lamellae. Here, using MD simulations, we explore a much wider parameter space; i.e., we vary both diblock composition and NP concentration and obtain the various structures formed by the DBC/selective-NP mixtures confined by repulsive walls. Below, we briefly summarize the typical structures formed at T* = 1.0. In Figure 1a, we display snapshot of the various structures formed by the DBC/selective-NP mixtures at NP volume fraction Φp = 0.077 and at different values of diblock composition f as indicated in the figure. As expected for highly asymmetric DBC, f = 0.1, we see spheres. At this NP concentration the spheres appear in a core−shell structure with a very small NP core size. Spherical structure changes to cylindrical structures upon increasing f (see Figures 1a:(ii,iii)). And for nearly symmetric and symmetric DBC (see Figures 1a: (iv,v)), the nanocomposites self-organize into a vertically oriented lamellar structure with NP stripes at the polymer− wall interfaces. 832

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are neutral. In the following, we discuss why the lamellar structure formed by highly asymmetric DBCs at large NP concentrations prefer horizontal orientation even when the confining surfaces are neutral. B. NP-Induced Horizontal Lamellar Structures. In copolymer nanocomposite thin films, NP can induce orientation transition of the copolymer morphologies. For example, the SCFT/DFT combined study by Lee and coworkers shows that in symmetric-DBC/neutral-NP mixtures confined by selective walls the orientation of nanoparticle decorated lamellae can be changed by the segregation of nanoparticles at the polymer−wall interfaces which effectively modify the surface properties.14,15 In the same study, the case of selective-NP and neutral confining walls is briefly considered.15 Again, in the latter case of neutral walls, the segregated NPs at the polymer−wall interfaces effectively modify the surface properties; i.e., it makes the surface selective and drives the transition. However, the above study fails to provide microscopic details, such as the individual chain properties close to the surface and bulk regions. Since only symmetric DBC are considered, several morphologies such as the Lh phase formed at other compositions have not been revealed. As we have seen in the phase diagram (Figure 2), a neat Lh structure is formed by asymmetric DBC at large NP concentrations and an Lv structure is confined to f ≈ 0.5 at small NP concentrations. Therefore, the Lh phase considered in our study is not obtained by the symmetric DBC. In the following, we systematically explore the difference between the two lamellar phases. As shown in Figure 3a, if we replot the phase diagram in the f−ΦA+P plane, where ΦA+P is the volume fraction combining both A and NP, we see that in the region ΦA+P ∼ 0.5 there is a transition from Lh to Lv upon increasing f. Thus, vertically oriented lamellae are formed by nearly symmetric DBC, and at low values of Φp only, and have a very limited region in the phase space. The average inclination angle θ of the end-to-end vectors about Z-axis; i.e., the axis normal to the wall is shown in Figure 3b. And in Figure 3c, the laterally averaged volume fraction profiles for the Lh phase corresponding to the circles marked (i)−(iii) in Figure 3a are shown. In order to calculate the laterally averaged density profiles, we first subdivide the film into thin slices and then calculate the volume fraction in each slice for A, B, and NP. We can see in Figure 3c that at larger values of NP concentration the NPs form several condensed layers at the polymer−wall interfaces; also it fills the A-phase. In the following, we will discuss why the lamellar structures formed at high NP concentrations (and low values of diblock composition f) prefer horizontal orientation. In Figure 4a, we display the shape of the A−B interface lines for the Lv phase formed at the composition: Φp = 0.077 and f = 0.50 (see Figure 1a(v)). As we can see in the figure, the interface lines separating the A + P and pure B phases start to bend as it approaches the wall. The interface lines enclosing the pure-B phase becomes narrower while that enclosing the A + P phase becomes wider when approaching the wall. The regions occupied by the segregated NPs at the polymer−wall interface (forming stripes in the XY-plane; see Figure 1a(iv,v)) exclude A-monomers from this region; see regions below the dashed line in the A + P region in Figure 4a. The empty spaces adjacent to the NP stripes are filled by the B-monomers and consequently distort the interface position near the walls. In Figure 4b, we show the difference of the elastic energy per

the segregated NPs lose their translational degrees of freedom. The balance of stretching of the chains, enthalpic energy gain due to A-monomer/NP contacts, and particle’s contribution (translational entropy and packing) which minimizes the total free energy of the system determines the equilibrium conformation. The readers are also referred to refs 12 and 27 for theoretical calculations and further discussion on the formation of such structures. Unlike bulk systems, due to the confinement, there is influence especially on the orientation of the ordered structures. The phase diagram in the f−Φp plane which summarizes all the structures obtained through simulation is shown in Figure 2. Each point in the phase diagram represents the simulated point.

Figure 2. Phase diagram in f−Φp plane summarizing all the structures obtained through simulations. Color code in different regions is to guide the eye. See the caption of Figure 1 for the meaning of the symbols. Laterally averaged density profile and average chain extension for the dotted region in the phase diagram are shown in Figures 10 and 11a, respectively.

In comparison to the phase diagram predicted for the system in bulk,12 the phase diagram in confinement has a much bigger lamellar region with a broad class of lamellar structures. The phases are shifted toward small values of f with increasing Φp, and thus overall the phase boundaries are tilted toward left. However, it is important to note that the phase boundaries might be affected depending on the thermodynamic pathways chosen to reach the final structures. As we show later in the section IIIC, the lamellar phases that we observe are not in true equilibrium, but rather depend on the thermodynamic pathways. As expected for the neutral confining walls, the vertically oriented lamellar phase is formed by the slightly asymmetric and symmetric DBCs. The phase diagram show that Lv phase is formed for f ≈ 0.5 and at very low values of NP concentration. At larger values of NP concentration, other structures (Lh, PLh, ML) are realized. For f = 0.5, the horizontally oriented (perforated) lamellae is formed at a much higher value of Φp (see Figure 2). It is interesting to note that highly asymmetric DBCs at large NP concentrations form horizontally oriented lamellar structures. Existence of both vertically and horizontally oriented lamellae illustrates that the lamellar orientation can be controlled using selective NPs even when the confining surfaces 833

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Figure 3. (a) Phase diagram plotted in the f−ΦA+P plane. In this representation, for ΦA+P ∼ 0.5, we can see transition between the horizontally (points enclosed by circle) and vertically oriented (points enclosed by square) lamellar structures as a function of f. (b) Average inclination angle θ of the end-to-end vectors about the axis normal to the wall, i.e, Z-axis. The data points correspond to the encircled points in (a). (c) Laterally averaged volume fraction profiles for Lh phase, i.e., circles marked (i)−(iii) in (a).

rapidly as we approach the wall, and thus the chains are more stretched near the surfaces. To calculate Eel, we use eq 2.3. Qualitatively, the behavior of ΔEel as a function of Z for the whole chain as well as its individual blocks is the same for the symmetric and slightly asymmetric chains. If we compare the A and B blocks, we see that the value of ΔEel for the B-block in the region Z < σp is significantly higher than the bulk value. In the same region there is no A-block; however, the value of ΔEel for A-block shows an increasing tendency when approaching the wall, but the value does not differ much from the bulk (see the inset of Figure 4b). In order to compare Lh and Lv phases, the excess stretching energy per bond of the chains very close to the walls, i.e., in the region Z ≤ σp, is plotted as a function of diblock composition in Figure 5a. The data points correspond to the encircled points in Figure 3a. As we can see in the figure, for the horizontally oriented lamellae (which includes SA-Lh phase) the value of ΔEel ≈ 0, whereas we get ΔEel ≈ 0.048 kBT for the vertically oriented lamellae, and the transformation between the two orientations of the lamellae is stepwise. For Lv phase at Φp = 0, we get ΔEel ≈ 0.025 kBT, which is lower than the Lv phase formed by the nanocomposites. In the inset of Figure 5a, we show the behavior for the individual blocks. For the A-blocks, transformation from Lh to Lv upon increasing f is smooth in ΔEel and saturates around 0.02 kBT. Whereas, for the B-block, it is rather stepwise (as for the entire chain AB), and the difference in the value of ΔEel between the two lamellar phases is around 0.06 kBT, which is significantly higher than that of Ablock. Thus, in horizontally oriented lamellar structures, the chains at the polymer−wall interfaces are not overstretched. Also, we calculate the per monomer pairwise interaction energy of A-monomers, EALJ, with the other species in the bulk and surface regions

Figure 4. (a) Shape of the A−B interface lines near the wall for Lv phase (Φp = 0.077 and f = 0.50, see Figure 1a(v)). The data shown is obtained from a slice of thickness 4σ in the X−Z plane, and the wall is located at Z = 0. Dots are the interface positions at different times, and the solid line represents the time averaged interface position which separates the A + P and pure-B phases. Regions where the segregated nanoparticles are located in the polymer−wall interface is shown by the filled circles representing nanoparticles. (b) Excess stretching energy ΔEel as a function of distance Z from the confining wall for the Lv phases formed at Φp = 0.077 by a nearly symmetric and symmetric DBC; see Figure 1a(iv,v). Inset: ΔEel for the individual blocks A and B separately, and the vertical dashed line indicates Z = σp. For Z < σp there is no A-block.

E LJA = (VA − A + VA − B + VA − NP)/nA

bond, ΔEel (excess stretching energy), of the chains from that of the average bulk value as a function of distance Z from the wall. As we can see, the chains stretching energy, Eel, increases

(3.1)

where Vi−j is the LJ interaction potential (see eq 2.2) between species i and j, and nA is the number of A monomers in the respective regions (see Figure 5b). Here, the surface region is 834

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Figure 5. (a) Difference of average elastic energy per bond, ΔEel, between the chains near the walls and that in the bulk as a function of diblock composition f. Inset: ΔEel for the A-block and B-block of the DBC. (b) Per monomer pairwise integration energy of A monomers, EALJ, with the other species in the surface and bulk regions, and their difference, ΔELJ = EALJ(surface) − EALJ(bulk), are shown. The data points displayed here corresponds to the encircled points in Figure 3a.

defined as the region within a distance of roughly 5σ from the wall, where the influence of surface is strong (see Figure 4b), and the region beyond is considered as bulk. In the Lh phase, the interaction energy of the A-monomers in the bulk is lower than that in the surface region, while this behavior is inverted in the Lv phase. In Lh morphology, in addition to the zero excess stretching energy, the horizontal orientation generates more area for the segregated NPs at the polymer−wall interfaces. Therefore, the translational entropy related with the particles is maximized, and consequently the interaction energy is further reduced. In Figure 6, we show the excess stretching energy for all the simulated parameters.

En(t ) =

⎛ e(t ) ·e(0) ⎞ Ln⎜ ⎟ , ⎝ |e(t )||e(0)| ⎠

n = 1, 2, ... (3.2)

where e(t) and |e(t)| are the end-to-end vector of a chain and its magnitude respectively at time t, and Ln represents the Legendre polynomial of nth degree.37 We also calculate the average orientation function

g (t ) = ⟨cos2 θ(t )⟩

(3.3)

where θ(t) is the inclination angle of a chain’s end-to-end vector with respect to the Z-axis at time t. For g(t) = 1.0 (upper bound), the chains are oriented perfectly normal to the confining walls, and for g(t) = 0 (lower bound), the chains are aligned parallel to the walls. 1. Slow Heating−Cooling Cycle. The Lh and Lv lamellar phases at T* = 1.0 are continuously heated to T* = 4.5, where the systems are in disordered state. Then, we cooled down the systems again to T* = 1.0 at the same rate. The rate of temperature change is dT/dt = 10−4ε/kBτLJ. The evolution of E2 and ⟨cos2 θ⟩ during heating and cooling cycles for both lamellar structures is shown in Figure 7. We first consider the Lh phase (see Figure 7a). During heating, initially E2(t) decays rapidly from unity and displays a plateau at about 0.25 in the interval of 2.0 < T* < 3.0. By further increasing the temperature, the order parameter decays to zero. Upon cooling, the correlation function increases again and saturates at the plateau value at around 0.25. Thus, a partial memory of the individual chain orientation has been recovered. This is to be related with the re-formation of the horizontally oriented lamellae where the majority of the chains are oriented perpendicular to the substrate. The formation of lamellar structure is observed in the interval 3.0 < T* < 4.0, where the lamellae are weakly ordered. This is indicated by a hysteresis which may be related to a discontinuous nature of the order− disorder transition. The results for the orientation function of the chains with respect to the walls, ⟨cos2 θ⟩, support the reformation of the Lh phase after the heating−cooling cycle and the hysteresis behavior. We can conclude that the chain orientation at the end of heating−cooling cycle is recovered (within the error bar), and the initial and final structures are the same (see Figure 9). Next, we consider the Lv phase. The results for both order parameters indicate that the morphology Lv is metastable or, at least, leads to metastable variants. The result of the heating−

Figure 6. Excess stretching energy per bond, ΔEel, shown for all the simulated points in the phase diagram.

C. Lv and Lh Phases under Heating−Cooling Cycles. The stabilities of the morphologies obtained have been tested against different thermodynamic pathways. In one case, we perform slow heating−cooling cycle, while in the other case, we perform rapid heating−cooling cycle (quenching). Then the structures before and at the end of the cycle are compared. For vertically oriented lamellae we consider Figure 1a(v), where f = 0.5 and Φp = 0.077, and for the horizontally oriented lamellae we consider Figure 1b(iv), where f = 0.271 and Φp = 0.143. The orientational change during heating and cooling is monitored through the orientational correlation of the end-toend vector defined as 835

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Figure 7. Evolution of the average orientation correlation E2(t) and ⟨cos2 θ(t)⟩ during slow heating−cooling cycle for (a) Lh and (b) Lv phases. Simulation snapshots corresponding to the points labeled (i), (ii), and (iii) in the cooling curve of Lv phase (see part b, lower panel) are shown on the right.

Figure 8. Evolution of the average orientation correlation E2(t) and ⟨cos2 θ(t)⟩ during rapid heating−cooling cycle for (a) Lh phase and (b) Lv phase. Δt (= 5 × 103τLJ) represents the time interval. Here, the reduced time is presented without an asterisk.

Figure 9. Snapshots of the typical configurations at the end of slow and rapid heating−cooling cycles compared with the initial configurations.

heat the system at T* = 1.0 to T* = 4.5 for 5 × 103τLJ, where the systems are in disordered state, and then quench the system again to T* = 1.0. The results for E2(t) and ⟨cos2 θ⟩ during the heating−cooling cycle are displayed in Figure 8 for both lamellar phases. For the Lh phase (see Figure 8a), the average chain orientation returns close to the initial value if we relax the sample at T* = 1.0 for a long time, but the final morphology is quite different from the initial (see Figure 9). For instance, the

cooling cycle is a fusion of vertical lamellae containing components of horizontal orientation; see the snapshots indicated on the right of Figure 7b. The complete loss of memory for the E2 functions is explained by a rotation of the Lv morphology around the Z-axis. The morphologies at the end of heating−cooling cycles are shown in Figure 9 for both Lh and Lv phases. 2. Rapid Heating−Cooling Cycle. We perform rapid heating−cooling cycles of the lamellar phases; i.e, we suddenly 836

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Figure 10. Laterally averaged volume fraction profiles for f = 0.23 and at different values of Φp indicated in the figure. Part a is for the C phase, while parts b−d are for the Lh phase; also see dotted region in the phase diagram in Figure 2. The walls are located at Z/σ = −25 and Z/σ = 25. As we can see in the figure, NPs packing affect the B-block (repulsive with NPs) in such a way that NPs compress the B-blocks, i.e., form the layers of A-blocks by decreasing the width of B-layers.

Figure 11. Average end-to-end distance, R, as a function of Φp for f = 0.23 (dotted region in the phase diagram, see Figure 2) shown for (a) A- and B-blocks separately and (b) the entire chain, A−B. The corresponding average elastic energy per bond, Eel, is shown in (c). The overall chain extension shown in (b) is compared to the systems with neutral NPs, and the dotted line corresponds to R at Φp = 0.0.

where the diblock composition f = 0.23 and the morphological transitions upon increasing NP concentration Φp are C → Lh → Cinverted. Furthermore, Lh phase in the above region exists over a wide range of Φp, and this allow us to investigate directly the effect of increasing NP concentration on the lamellar morphology. In Figure 10, we show the laterally averaged volume fraction profile plotted as a function of film thickness Z at different values of Φp indicated in the figure. Figure 10a corresponds to the volume fraction profile for the cylindrical phase, while that of Figure 10b−d corresponds to the lamellar phase, and we will focus on the lamellar phases. Let us denote DA+P and DB for the thickness of the A + P and B layers of the lamellae, respectively. As indicated in Figure 10b, the value of DB ≈ 2DA+P for the low NP concentration, and thus we see a lamellar structure with high asymmetry in the thickness of A + P and B layers. However, as we gradually increase the value of Φp, the thickness of A + P layer increases as expected, while that of B layer decreases. This increase in the

horizontally oriented lamellar structure formed after the cycle is perforated and the number of lamellae layers decreased. For the Lv phase, the value of ⟨cos2 θ⟩ is far from the initial value even after the long relaxation time (see Figure 8b), and this indicates that the initial and final structures are different (see Figure 9). The above study of the behavior of the systems under slow and rapid heating−cooling cycles clearly shows that morphologies can be controlled by the thermodynamic pathways. As we have seen, under slow heating−cooling cycle, the Lh phase returns to the same initial configuration, suggesting that it can be an equilibrium phase, whereas it is not the case for Lv phase. This observations suggest that in particular the vertically oriented phase are prone to metastable forms. However, a complete switching to the parallel phase is also not observed. Thus, the phase boundaries might be affected depending on the thermodynamic pathways chosen to reach the final structures. D. Density Profile and Chain Extension. We consider the dotted region in the phase diagram shown in Figure 2 837

dx.doi.org/10.1021/ma402184w | Macromolecules 2014, 47, 830−839

Macromolecules

Article

value of DA+P with increasing Φp is related to the increase in the NP uptake by the A-phase until it reaches a saturation point and beyond which the excessive NPs are expelled from the Aphase and form a condensate in the middle of A-layers (see Figure 10d). Moreover, upon increasing the value of Φp, additional NP layers are formed at the polymer−wall interfaces by the segregated NPs, and consequently the available volume for the nanocomposite decreases and thus the thickness of NP repulsive B-layer decreases to conserve the film thickness. The role of selective NPs in forming a commensurable Lh phase, e.g., effect of varying NP concentration and film thickness, in the presence of repulsive confining walls has been studied using a mean-field model, and it is reported in ref 36. From the meanfield model, we could see that frustration due to the mismatch of equilibrium lamellar period and the film thickness can be reduced not only by changing the film thickness but also by tuning the NP−monomer interaction. The density profiles shown here for the NP in lamellar phase are in agreement with an earlier theoretical prediction for the inclusion of relatively small nanoparticles in the symmetric DBC matrix; i.e., the excessive NPs are segregated at the A−B interfaces and at the center of the A-layer.38 The root-mean-square end-to-end distance of the chains denoted by R is plotted as a function of Φp as shown in Figure 11a. Here, we have considered the same dotted region in the phase diagram which we have used for the calculation of density profiles (see Figure 10). The first and the last data point are for the C phase, while those in between are for the Lh phase. Transformation from C to Lh phase upon increasing Φp is associated with a decrease for A-blocks and an increase for Bblocks in the value of R (see Figure 11a). However, within the lamellar region the extension of A-blocks increases, while that of B-blocks decreases with the increase of NP concentration. This agrees with the change in the thickness of individual lamellar layers observed in the density profiles shown in Figure 10. As shown in Figure 11b, the value of R for the entire chain within the Lh phase slightly decreases with increasing Φp, and this behavior is consistent with our previous mean-field study.36 If we now look at the average elastic energy per bond, Eel, for the entire chain (AB) and its individual blocks (A and B), we see that A-block pays the highest stretching penalty except at the point where it first form lamellae from the cylindrical phase (Φp ≈ 0.15) (see Figure 11c). The morphological transition at low NP concentration, i.e., C → Lh, is related with a sudden decrease in the value of Eel for A-blocks and thus reduce the penalty related to overstretching of A-blocks. Whereas for the transition Lh → Cinverted at relatively high NP concentration, we see that the value of Eel does not differ significantly and therefore suggest that the transition is driven mainly by the segregated NPs, i.e., tendency to maximize the translational entropy and packing contributions.

the lamellar layer thickness and individual chain extension. The results are briefly summarized and discussed below. The phase diagram we have obtained shows a much wider lamellar region with a broad class of distinct lamellar structures. Upon increasing NP concentration, the phases are shifted to smaller values of diblock composition. It is known that the presence of neutral NPs do not alter the copolymer morphologies and structures can be retained up to large NP concentrations.27 Lamellar structures in nanocomposites with selective NPs are observed for ΦA+P ∼ 0.5, where ΦA+P is the NP and A-block effective volume fraction. Both horizontally and vertically oriented lamellar phases are observed. An interesting observation is that the lamellae formed by highly asymmetric DBC at larger NP concentrations prefer to orient horizontally with respect to the walls. This is in contrast to previous results obtained for symmetric DBC without nanoparticles where a preference for edge-on orientation of lamellar phases close to neutral substrates have been concluded.39,40 In vertically oriented lamellar structures, we have observed that the copolymer interface lines are distorted while approaching the walls due to the presence of segregated nanoparticles forming stripes at the polymer−wall interfaces. Consequently, the chains near the walls are overstretched compared to the bulk. Our analysis of the excess stretching energy of the chains reveal that by transforming the lamellar orientation from vertical to horizontal the chains can overcome the overstretching penalty. Also, we have compared the behavior of per monomer pairwise interaction energy of Amonomers, EALJ, with the other species in the surface and bulk regions for both the lamellar phases. We have exposed the lamellar morphologies to different thermodynamic pathways by applying slow and rapid heating− cooling cycles. Under the slow heating−cooling cycle, the horizontal phase, Lh, returns to the initial morphology. The vertical phase, Lv, displays morphological changes which up to a pure symmetry operation (rotation) consists in fusing of lamellae and a tendency of chains to order perpendicular to the substrate. This underlines the limited stability of the edge-on phase which are prone to metastable forms. However, a switch toward the Lh phase is not observed. This suggest that some of the structures formed by the nanocomposites are stable, while others are not. Under a rapid heating−cooling cycle (i.e., quenching) both the lamellar structures do not return to the initial morphologies even after a long relaxation runs. These observations show that long-living metastable morphologies can be obtained by selection of thermodynamic pathways. The influence of thermodynamic pathways on the final structures also opens the question of whether the phase boundaries will be affected depending on the pathways chosen to obtain the structures. From the aspect of technological applications, it would be desirable to have a stable Lv phase. Earlier studies have considered using patterned substrate as one way to improve the order. In the same line, one could think of exploring wider parameter, e.g., surface selectivity or patterned substrate, and varying the strength and range of NP interaction to improve the stability and range of order. In the horizontally oriented lamellar structure, adding more NP increases the A-layer thickness as expected, while the thickness of B-layer decreases in order to conserve the film thickness. Apart from the NPs at the polymer−wall interfaces, at large NP concentrations, the excessive NPs are segregated at the A−B interfaces and at the middle of NP selective A-layers forming a condensate. The overall chain extension, however,

IV. SUMMARY By means of molecular dynamics simulations, we have investigated the order and phase behavior of diblock copolymer and selective-NP mixtures confined by neutral walls in slit geometry. We have obtained various morphological phases formed by the nanocomposites and summarized them in a phase diagram. We have focused our attention on the following points: (1) why the lamellae formed by highly asymmetric DBC at larger NP concentrations prefer horizontal orientation, (2) stability of lamellar structures under different thermodynamic pathways, and (3) the effect of nanoparticle concentration on 838

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(28) Wu, X.; Chen, P.; Feng, X.; Xia, R.; Qian, J. Soft Matter 2013, DOI: 10.1039/c3sm50169h. (29) Schultz, A. J.; Hall, C. K.; Genzer, J. Macromolecules 2005, 38, 3007−3016. (30) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990, 92, 5057. (31) Grest, G. S.; Kremer, K. Phys. Rev. A 1986, 33, 3628. (32) Chervanyov, A. I.; Balazs, A. C. J. Chem. Phys. 2003, 119, 3529. (33) Ginzburg, V. V. Macromolecules 2005, 38, 2362. (34) Plimpton, S. J. Comput. Phys. 1995, 117, 1−19. (35) Humphrey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33−38. (36) Shagolsem, L. S.; Sommer, J.-U. Soft Matter 2012, 8, 11328. (37) Bennemann, C.; Paul, W.; Baschnagel, J.; Binder, K. J. Phys.: Condens. Matter 1999, 11, 2179−2192. (38) Pryamitsyn, V.; Ganesan, V. Macromolecules 2006, 39, 8499. (39) Mansky, P.; Liu, Y.; Huang, E.; Russell, T. P.; Hawker, C. J. Science 1997, 275, 1458. (40) Mansky, P.; Russel, T. P.; Hawker, C. J.; Pitsikalis, M.; Mays, J. Macromolecules 1997, 30, 6810.

decreases with increasing NP concentration and agrees with our mean-field calculation,36 and it is related with a decrease in the available volume for the nanocomposite due to the formation of NP layers at the polymer−wall interfaces.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (L.S.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the DFG (under Contract SO-277/3-1) is gratefully acknowledged. The work is also funded by the European Union (ERDF) and the Free State of Saxony via TP A2 (“MolDiagnostik”) of the Cluster of Excellence ”European Center for Emerging Materials and Processes Dresden” (ECEMP). L. S. Shagolsem acknowledges support from the ECEMP-International Graduate School. We also thank the Center for High Performance Computing (ZIH) at TU Dresden for the generous grant of computing time.

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