The Effects of Nanoparticles on the Lamellar Phase Separation of

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J. Phys. Chem. B 2008, 112, 4194-4203

The Effects of Nanoparticles on the Lamellar Phase Separation of Diblock Copolymers Linli He,† Linxi Zhang,*,‡ and Haojun Liang‡,§ Department of Physics, Zhejiang UniVersity, Hangzhou 310027, P R China, Department of Physics, Wenzhou UniVersity, Wenzhou 325027, P R China, and Department of Polymer Science and Engineering, UniVersity of Science and Technology of China, Hefei, 230026, P R China ReceiVed: July 22, 2007; In Final Form: January 25, 2008

The phase separation of diblock copolymers containing some energetically neutral/biased nanoparticles is studied by means of large-scale dissipative particle dynamics (DPD) simulations. The effects of the volume fraction of nanoparticles, the size of nanoparticles, and the interaction strength between nanoparticles and blocks on the lamellar phase separation of diblock copolymers are investigated. When these effects are up to a critical value, the diblock copolymer nanocomposites can form a new bicontinuous morphology, which is well consistent with the experimental results. It is also found that the degree of order of phase separation for a given system increases slightly and then decreases abruptly until the bicontinuous morphology is formed as the volume fraction of nanoparticles increases. Furthermore, we discuss the microphase transition through the position distributions of nanoparticles and present a phase diagram in terms of the nanoparticle volume fraction, size, and surface interaction strength.

1. Introduction Phase-separation phenomena are widely observed in various kinds of condensed matter including metals, simple liquids, and complex fluids such as polymers, surfactants, and liquid crystals.1 Diblock copolymers can form a variety of microphase separation structures on nanometer length scales, the shapes of which depend on the length ratio and the interaction strength of blocks.2-4 Theory and computer simulation have also been used to study the phase behaviors of diblock copolymers. Theories are capable of predicting the locations of the orderdisorder transition (ODT),5-7 the point where an ordered structure dissolves into a disordered one, and the order-order transitions(OOT),5,6,8,9 the spacing between periodic repeat units such as lamellae, the energy, and the entropy,8-13 all as functions of the copolymers chain length (N), the volume fraction of each component (f), the interaction strength (χ), and the volume fraction of copolymers in the solution (φc).14 In the past decade, computer simulation methods have been proven to be valuable tools to study the phase behaviors of polymers. Computational research has focused on the location of the ODT,13,15 the structural spacing of lamellae,13,16 and the transitions between some ordered structures, including lamellae, perforated lamellae, cylinders, and BCC spheres.14,17 In recent years, polymer nanocomposites are receiving great attentions for potential applications including magnetic storage media, high-surface-area catalysts, selective membranes, and photonic band gap materials, etc.18 This combination of polymers and nanoparticles can lead to the enhancement of mechanical, optical, thermal, fire-retardant, and ablative properties as well as gas-transport properties compared to either of the individual components.18,19 Furthermore, due to the ability to microphase separatr into a variety of ordered structures on nanometer length scales, diblock copolymers can be used as * Corresponding author. Electronic mail: [email protected]. † Zhejiang University. ‡ Wenzhou University. § University of Science and Technology of China.

ordered template to afford opportunities for controlling the spatial and orientational distribution of nanoparticles to form new materials with tailored microstructure-dependent properties.18,20-22 The phase behaviors of nanoparticles in block copolymers have been extensively studied both theoretically14,23-28 and experimentally.21,23,29 Thompson et al. studied block copolymer/ nanoparticle composites using a theory based on a self-consistent field theory (SCFT) for the block copolymers and a density functional theory for the nanoparticles.27 They found that the larger nanoparticles concentrated in the center of the preferred domain while the smaller nanoparticles concentrated near the interface, which is consistent with some of the experimental observations.24-25 In lamellae, the nanoparticles formed ordered nanosheets (monolayers or bilayers), while in cylinders, they formed helical or ring structures. Although nanoparticles also concentrated within the BCC spherical domains, they do not order within the spheres.27,29 Kim et al. studied a system containing diblock copolymers PS-b-P2VP and Au nanoparticles. They demonstrated that nanoparticles could act as surfactants for block copolymers and lead to form stable bicontinuous morphologies. The increasing of Au nanoparticle surfactants volume fraction first caused a decrease in the lamellar period and then beyond a critical volume fraction gave rise to stable bicontinuous block polymer microstructures with characteristic dimensions well below 100 nm.30 Recent studies have also used a variety of simulation techniques to investigate the effects of particles dispersed in complex fluids such as block copolymers.28,31-33 Wang et al. performed lattice Monte Carlo simulations to examine the position of a single spherical or cubic nanoparticle within a lamellae, forming symmetric block copolymer of length 24. They found that both large and small neutral nanoparticles were preferentially located at the copolymer interface, while a small nanoparticle that preferred component A of the copolymer was preferentially located within domain A. They also found that larger nanoparticles which preferred component A had little

10.1021/jp0757412 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/19/2008

Effects of Nanoparticles on Phase Separation

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preference to stay in any one place, but they attributed this to the simulation method and the system size, which was larger than that of their simulations with smaller nanoparticles.28 Their work mainly focused on the distribution of different nanoparticles sizes in lamellar diblock copolymers. Liu et al. investigated the self-assembly of a single kind of nanoparticles and binary nanoparticles mixtures in lamellar diblock copolymers A10B10 using dissipative particle dynamics simulations.31 They also mainly discussed the distribution of the nanoparticles with different sizes, shapes, and interaction strength in lamellar diblock copolymers. However, the effects of energetically biased nanoparticles on the morphologies of phase separation for diblock copolymers have not been studied in detail. Therefore, in this work, a systematic three-dimensional dissipative particle dynamics (DPD) simulation is performed, and our aim is to discuss the effects of the nanoparticles volume fraction, the nanoparticles size, and the interaction strength between nanoparticles and fluid particles on the morphologies of phase separation for diblock copolymers. We take nanoparticles as hard spheres by introducing a new interactive potential between blocks and nanoparticles in order to simulate real nanoparticles. Our study is motivated by the fact that some impurities unfortunately drop into complex fluids in industry, which will affect the structures and some properties of fluids, thus providing, furthermore, useful information on the synthesis of new polymer nanocomposites with tailored properties. Our results show that, as the volume fraction of nanoparticles increases, it first causes a tiny improvement in the average degree of order of microphase structures and then forms the stable bicontinuous diblock copolymer microstructures. Bicontinuous microstructures with a certain dimension are ideally suited for many of the emerging applications such as photovoltaic films,34,35 catalysts,36 and chemical/biological sensors.37 2. Simulation Method and Models 2.1. DPD Simulation Method. In 1992, Hoogerbrugge and Koelman38 proposed a new simulation technique referred to as dissipative particle dynamics (DPD), which is very similar to molecular dynamics(MD) but more appropriate for the investigation of the generic properties of macromolecular systems. The particles or “beads” in the DPD simulation represent “fluids packages” or groups of particles. This method combines soft conservative interactions between particles with random and dissipative interactions, which are interrelated through the fluctuation-dissipation theorem. Furthermore, in contrast to the molecular dynamics approach with noise,39 the dissipative and random forces obey Newton’s third law in the DPD approach, which can preserve local momentum conservation. This is important for the correct description of long-range and longtime correlations in the velocity field. For a system composed of N DPD particles, the force acting on the ith particle Bfi contains three parts: a conservative force B FCij , a dissipative force B FDij , and a random force B FRij .

Bf i )

(F BCij + B FDij + B FRij ) ∑ i*j

(1)

where the sum is over all other particles within a certain cutoff radius rc. As this is the only length scale in the system, the cutoff radius is used as a unit of length, rc ) 1, so that all lengths are measured relative to the DPD particle radius.

The conservative force B FCij is a soft repulsive force and given by

B FCij )

{

aij(1 - rij)rîj (rij < rc) (rij g rc) 0

(2)

where rij is the magnitude of the particle-particle vector and b rij is the unit vector joining particles i and j, b rij ) b ri - b rj, rij ) |r bij|, rîj ) b rij/rij. The repulsion parameter aij is often related to the Flory-Huggins interaction parameter χij. The dissipative force B FDij is a hydrodynamic drag force and given by

B FDij )

{

-γωD(rij)(rîj‚νij)rîj (rij < rc) (rij g rc) 0

(3)

where γ is a friction parameter and ωD(rij) is a r-dependent νij ) b νi - b νj. weight function vanishing for rij g rc and b The random force B FRij corresponds to the thermal noise and has the form of

B FRij )

{

σωR(rij)θijrîj (rij < rc) (rij g rc) 0

(4)

where σ is a parameter, ωR(rij) is also a weight function, and θij is a randomly fluctuating variable satisfying

〈θij(t)〉 ) 0

(5)

〈θij(t)θkl(t′)〉 ) (δikδjl + δilδjk)δ(t - t′)

(6)

with i * j and k * l. Note that these two forces B FDij and B FRij also act along the line of centers and conserve linear and angular momentums. There is an independent random function for each pair of particles. Also, there is a relation between both constants γ and σ, which is as follows2

σ2 ) 2γkBT

(7)

In our simulation, γ ) 6.57 and the temperature kBT ) 1. Therefore, σ ) 3.62 according to eq 7. In order for the steady-state solution to the equation of motion to be the Gibbs ensemble and for the fluctuation-dissipation theorem to be satisfied, it has been shown14 that only one of FRij can be chosen arbitrarily: the two weight functions B FDij and B

ωD(r) ) [ωR(r)]2 )

{

(rc - rij)2 (rij < rc) (rij g rc) 0

(8)

Finally, the spring force BfSi , which acts between the connected beads in a chain, has the form of

Bf Si )

∑j Cbr ij

(9)

where C is a harmonic type spring constant for the connecting pairs of beads in a polymer chain, chosen to be equal to 4 here (in terms of kBT).40 2.2. Models. In our simulation, we consider diblock copolymers, which are represented by six DPD beads of A3B3. For simplicity, the nanoparticles are modeled as single spheres. So, the simulation system contains a kind of nanospheres and two types of DPD particles (i.e., A and B). A and B polymer beads are connected by springs. Initially, DPD particles are arranged in a face-centered-cubic (fcc) lattice via spring forces, and the

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Figure 1. Snapshots, densities, order parameter profiles, and position distributions of nanoparticles for 9 × 9 × 9 DPD units systems with values of φN ) 0 (a1-a3), 0.0775% (b1-b4), 0.155% (c1-c4), 0.217% (d1-d4), 0.233% (e1-e4), and 0.248% (f1-f4), respectively. A and B blocks are presented in white and red, and nanoparticles are shown in yellow, respectively. When φN g φCN ) 0.233% (see e1), the bicontinuous morphology is formed. Solid lines in (a3-f3) represent the average order parameter. Here, ωAN ) 5.0, ωBN ) 1.0, and Rnano ) 0.3.


J. Phys. Chem. B, Vol. 112, No. 14, 2008 4197 separated morphology. The box size of the simulation system is 9 × 9 × 9 DPD units, containing 3645 DPD particles for the density of F ) 5. The snapshot and density profile of A3B3 lamellar structures are given in Figure 1a1,a2, which are in accord with previous studies.2 For the energetically biased nanoparticles, in order to study the effects of nanoparticles on the lamellar phase separation of diblock copolymers, we perform a series of simulations by adding nanoparticles into the fluids one by one. Some snapshots, density profiles of the microphase separation, and the distributions of nanoparticles with φN ) 0.0775, 0.155, 0.217, 0.233 and 0.248% are shown in Figure 1. Here, the nanoparticles volume fraction φN is defined as

Figure 2. Average order parameter φ as a function of nanoparticles volume fraction φN for different DPD systems with 9 × 9 × 9, 12 × 12 × 12, 15 × 15 × 15, and 17 × 17 × 17 units. Here, ωAN ) 5.0, ωBN ) 1.0, and Rnano ) 0.3.

mobile nanoparticles are randomly dispersed in the fluids. Periodic boundary conditions are applied in all three directions. The Newton equations for all particles’ positions and velocities including nanoparticles’ positions and velocities are solved by a modified version of the velocity Verlet algorithm.2 In previous investigations,31,41,42 the nanoparticles were also composed of DPD particles. All DPD particles interacted with each other via the same conservative, dissipative, and random forces given by eqs 2-4. These forces are all soft interactions, which may lead to some overlappings of nanoparticles. For the present study, we take nanoparticles as hard spheres, which can represent real nanoparticles. The additional interactions between nanoparticles and DPD particles as well as between nanoparticles and nanoparticles have the same form and they are defined as43

[( ) ( ) ]

n

Eij )

∑ j)1

ωiN

i∈A,B,N

δiN

9

rij

-

δiN

3

rij

(10)

which can avoid the undesirable penetration of DPD particles into the nanoparticles. Here, rij represents the distance between the center position of DPD particle i and the surface of nanoparticle j, ωiN is the potential strength, and δiN is the range of interaction around each particle. For convenience, the cutoff radius rc, the particle mass m, and the energy kBT are all taken as unity. The parameters of our model are as follows:

aAA ) aBB ) 15, aAB ) aBA ) 28

(11)

ωNN ) 10.0

(12)

δAN ) δBN ) 1.2, δNN ) 0.5

(13)

Here, N, A, and B represent nanoparticle, A and B DPD particles, respectively. ωNN and δNN are interaction parameters between nanoparticle and nanoparticle. The mass of the nanoparticles mnano is 1, and the radius of nanoparticle is Rnano. 3. Results and Discussion 3.1. Effect of Nanoparticles Number. We perform the DPD simulations on the phase separation of diblock copolymers A3B3 without any nanoparticles, which have the lamellar microphase-

4 Nnano × πRnano3 3 φN ) D3

(14)

Here, Nnano represents the number of nanoparticles, and Nnano ) 5, 10, 14, 15, and 16 in Figure 1b-f, respectively. However, in our study, due to the strong effective potential energy and the excluded volume effect of nanoparticles, the effective radius of nanoparticles is larger than Rnano and a few nanoparticles can be accepted in the diblock copolymer systems. The effective nanoparticles volume fraction is larger than φN, and therefore our value of φN is far less than the experimental one in ref 30. In Figure 1, these snapshots indicate that the nanoparticles preferentially aggregate to domain A because the adsorption interaction between A DPD particles and nanoparticles ωAN (ωAN ) 5.0) is greater than that between B DPD particles and nanoparticles ωBN (ωBN ) 1.0). In order to discuss the reason why the nanoparticles can destroy the lamellar phase, we investigate the position distribution of nanoparticles shown in Figure 1b4-f4. Here, the total number of nanoparticles is 10 times that of Nnano because 10 times adds up at intervals of 3000 steps after reaching the equilibrium state, and the 2D coordinates are the projection of 3D coordinates. The transition process from ordered lamellar structures to disordered ones is clearly shown in Figure 1e1. Interestingly, when φN is small, the nanoparticles are entirely concentrated on domain A and the lamellar structures can be preserved well in Figure 1b1-b4. However, the nanoparticles tend to mainly centralize in the A-B interfaces, and the distinct curve of the lamellar structures is observed as the nanoparticles volume fraction φN further increases in Figure 1d1-d4, though there are lamellar structures as a whole yet. When φN becomes 0.233% (Nnano ) 15), about 20% of the nanoparticles are distributed in domain B, as shown in Figure 1e4, and the ordered mesostructures are destroyed, corresponding to the domains signed “disordered” in Figure 1e1-e2. We define the critical nanoparticles volume fraction as φCN (≡(NC × 4/3πRnano3)/D3), indicating the phase transition from the ordered mesostructure to the disordered. Here, φCN is equal to 0.233% for the system with the box size of 9 × 9 × 9 DPD units. This means that when φN g φCN, more and more nanoparticles infiltrate into domain B because of the excluded volume effect of nanoparticles, consequently, leading to the destruction of ordered lamellar structures and the formation of bicontinuous morphologies, as shown in Figure 1f1-f4, which is in good agreement with the experimental results.30 Usually, the degree of order of phase separation is described by the average order parameter, which has many equivalent


He et al.

Figure 3. Snapshots, densities, and order parameter profiles for 9 × 9 × 9 DPD units systems with φN ) 0.140% (a1-a3), 0.295% (b1-b3), 0.310% (c1-c3), and 0.326% (d1-d3), respectively. Here, ωAN ) 3.0, ωBN ) 1.0, and Rnano ) 0.3.

definitions. In our simulation, in order to present a quantitative analysis, we define the average order parameter φ as

φ)

(

N

)

|φAi - φBi| /N ∑ i)1

(15)

where φAi or φBi is the density of A or B block in the ith layer and N is the total layers of the simulation box on a certain direction. Here, we give a value of N ) 100. Combined with the microphase structures discussed above, the average order parameter φ can be used to describe the phase separation behaviors quantitatively in Figure 1a3-f3. In Figure 2, the average order parameters φ for 9 × 9 × 9, 12 × 12 × 12, 15 × 15 × 15, and 17 × 17 × 17 DPD units systems are shown as a function of nanoparticles volume fraction φN, with ωAN ) 5.0 and Rnano ) 0.3. It is clear from this figure that the average

order parameter φ increases slightly at first as the nanoparticles volume fraction φN increases. Subsequently, with φN further increasing, the average order parameter φ decreases dramatically near the critical point φCN, corresponding to the dashed line in Figure 2. This figure implies that when the nanoparticles volume fraction φN is small the degree of order of the lamellar structure of symmetrical diblock copolymers can be improved slightly. This tendency may depend on the simulation box size. If our simulation system is large enough, an accurate result can be obtained. However, computational restrictions may prohibit an accurate plot of this system. For example, it took about 25 days to calculate the phase separation of φN ) 0.0230% using our PC cluster for the 17 × 17 × 17 DPD units systems. The computer time for each system increases exponentially with the increase of the box size. At last, the microphase separated structures all get disrupted as φN becomes greater than or equal



Figure 4. Average order parameter φ as a function of nanoparticles volume fraction φN for the 9 × 9 × 9 DPD units systems with different interaction strengths of ωAN and ωBN: (a) the nanoparticles are energetically biased to A block with ωAN ) 3.0, 5.0, 7.0, and ωBN ) 1.0; (b) the neutral nanoparticles are considered with ωAN ) ωBN ) 0.1, 0.5, and 1.0, respectively, and the upper/lower snapshots correspond to φN ) 1.32% (Nnano ) 85) with ωAN ) ωBN ) 0.1 and φN ) 0.635% (Nnano ) 41) with ωAN ) ωBN ) 0.5, respectively. Here, Rnano ) 0.3.

Figure 5. The critical nanoparticles volume fraction φCN as a function of the interaction strength between A blocks and nanoparticles ωAN for 9 × 9 × 9 DPD units systems. Here, ωBN ) 1.0 and Rnano ) 0.3.

Figure 6. The value of φN,A as a function of φN for the 9 × 9 × 9 DPD units systems with different interaction strengths of ωAN. Here, ωBN ) 1.0 and Rnano ) 0.3.

to the critical nanoparticles volume fraction φCN for the four systems. Figure 2 shows that the variations of the average order parameter φ with the nanoparticles volume fraction φN for these four systems are the same. 3.2. Effect of Interaction Strength. In order to study the effects of the interaction strength between nanoparticles and the blocks of the copolymers on the lamellar phase separation, we perform another series of simulations by changing the interaction strength. The snapshots, densities, and order parameter profiles for 9 × 9 × 9 DPD units simulation systems with the interaction strength of ωAN ) 3.0 are shown in Figure 3. Here the volume fraction of the nanoparticles φN are 0.140, 0.295, 0.310, and 0.326% and Rnano ) 0.3. When φN increases from 0 to 0.140%, the average order parameter φ increases from 4.50 to 4.70. When φN increases from 0.155 to 0.326%, however, the average order parameter φ decreases from 4.68 to 2.85. The corresponding microphase structures are clearly observed from the ordered structures to the disordered structures in Figure 3. Similarly,

the “disordered” domains have been displayed in Figure 3c1c3, corresponding to the critical nanoparticles volume fraction φCN ) 0.310%. In Figure 4a, the average order parameter φ for 9 × 9 × 9 DPD units systems is shown as a function of the nanoparticles volume fraction φN for different interaction strengths of ωAN ) 3.0, 5.0, and 7.0. Here ωBN is fixed. The tendency of the average order parameter φ with respect to φN is consistent with the results mentioned above. Meanwhile, we note that the critical nanoparticles volume fraction φCN for ωAN ) 3.0 is greater than that for ωAN ) 7.0, but the maximum value of the average order parameter φ for ωAN ) 3.0 is less than that for ωAN ) 7.0. The explanation for this result is as follows. On one hand, the attractive interaction between the nanoparticles and A blocks is stronger than that between the nanoparticles and B blocks, which plays a driving role in the forming of lamellar phase and leads to a definite extent of improvement for the order of order


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Figure 7. Snapshots, densities, and order parameter profiles for 9 × 9 × 9 DPD units systems with nanoparticle radius of Rnano ) 0.2 corresponding to φN ) 0.0368% (a1-a3), 0.0735% (b1-b3), 0.0781% (c1-c3), and 0.0873% (d1-d3) and Rnano ) 0.4 corresponding to φN ) 0.110% (e1-e3), 0.184% (f1-f3), 0.221% (g1-g3), and 0.257% (h1-h3), respectively. Here, ωAN ) 5.0 and ωBN ) 1.0.

of microphase structures. On the other hand, the uneven distribution of the interaction strength may lead to the imbalance of system energy and structure, so it is easier for the organized structures of the system with a stronger interaction ωAN ) 7.0 to be destructed by the nanoparticles than that of the system with a weaker interaction ωAN ) 3.0. As a result, it is easy to understand that the final morphology and degree of order for a given system are mainly controlled by the competing interplays between interaction strength and systematic stability. In order to know whether there are different effects of neutral and energetically biased nanoparticles on the phase separation of diblock copolymers or not, we also investigated another three systems containing the neutral nanoparticles with ωAN ) ωBN ) 0.1, ωAN ) ωBN ) 0.5, and ωAN ) ωBN ) 1.0. The relationship between the average order parameter φ and the nanoparticles volume fraction φN is shown in Figure 4b. It is indicated that the effects of the neutral nanoparticles on the lamellar phase separation of diblock copolymers are similar with the biased nanoparticles except for very weak interaction of ωAN ) ωBN ) 0.1. In order to further investigate the effects of the interaction strength between neutral/biased nanoparticles and blocks on the phase separation of diblock copolymers, we calculate the total interaction energy Enano between nanoparticles and blocks for the whole system. When the nanoparticles volume fraction is fixed as φN ) 0.552%, the average order parameter φ is 4.56, 4.20, and 3.17 and the total interaction energy is Enano ) -8.5, -89.3, and -168.8 for the three systems with ωAN ) ωBN ) 0.1, ωAN ) ωBN ) 0.5, and ωAN ) ωBN ) 1.0, respectively. For the two systems with ωAN ) ωBN ) 1.0 and ωAN ) ωBN ) 0.5, the dramatic changes from the lamellar morphology to the bicontinuous morphology occur at φCN ) 0.387% (Enano ) -126.6) and 0.543% (Enano ) -89.7), respectively. The lower snapshot in Figure 4b represents a stable bicontinuous structure for the system ωAN ) ωBN ) 0.5 with φN ) 0.635% and Enano ) -105.5 (Nnano ) 41). However, for the system with ωAN ) ωBN ) 0.1, the average order parameter φ keeps unchanged and the lamellar structures remain surprisingly unaltered in the whole process with increasing volume fraction of nanoparticles, even at φN ) 1.32% (Nnano ) 85), which is the maximum volume fraction in our simulation,

corresponding to the upper snapshot in Figure 4b. If the nanoparticle fraction volume φN is greater than 1.32%, the system cannot reach the equilibrium state after a very long time because there exists a very strong excluded volume effect among the nanoparticles. For the system with ωAN ) ωBN ) 0.1, the total attractive interaction energy Enano between nanoparticles and blocks is very small, for example, Enano ) -8.5 for φN ) 0.552%, which is only about 5% of Enano for the system with ωAN ) ωBN ) 1.0. These two snapshots in Figure 4b also show the different effects of the neutral nanoparticles with different interaction strength on the final phase structures of copolymers. If the total attractive interaction energy Enano is very small, the ordered phase structure is always preserved, such as in the system with ωAN ) ωBN ) 0.1, which is consistent with the experimental results.30 We also perform simulations for another two values of ωAN, i.e., ωAN ) 9.0, and 11.0. Here the value of ωBN is fixed, i.e., ωBN ) 1.0. The critical nanoparticles volume fraction φCN as a function of ωAN is given in Figure 5. This figure clearly shows that the increase of the interaction strength between the nanoparticles and A block ωAN leads to a decrease in the critical nanoparticles volume fraction φCN. This result implies that the stronger the interaction strength between nanoparticles and blocks is the more easily the lamellar structures of diblock copolymers are destroyed. To further provide more physical insights into the phase separation, we study the transition process from ordered lamellar structures to bicontinuous ones through the nanoparticle distribution. In Figure 6, we present the value of φN,A as a function of nanoparticles volume fractfion φN for 9 × 9 × 9 DPD units systems with different interaction strengths of ωAN ) 3.0, 5.0, and 7.0, where φN,A represents the average volume proportion of the nanoparticles locating in domain A to the total nanoparticles, i.e.,

φN,A )

Nnano,A Nnano

(16)

Here Nnano,A represents the number of nanoparticles which are located in the domain A. The results shown in Figure 6


Figure 8. Average order parameter φ as a function of the nanoparticles volume fraction φN for 9 × 9 × 9 DPD units systems with different nanoparticle radii, Rnano. Here, ωAN ) 5.0, and ωBN ) 1.0.

areaveraged over 1000 times at intervals of 1000 steps for the equilibrium system. When the nanoparticles volume fraction φN is small, all nanoparticles (φN,A ) 100%) concentrate on domain A and the lamellar structures remains unchanged. However, with the increase of nanoparticles volume fraction φN, more and more nanoparticles infiltrate into domain B because of the excluded volume effect of nanoparticles. When values of φN reach the critical values φCN, the average volume proportions of nanoparticles locating in domain A to the total nanoparticles φN,A becomes 77%, 80%, and 83% respectively, corresponding to the systems with ωAN ) 3.0, 5.0, and 7.0. All above results imply that when φN is small the phase separation between A and B blocks takes a key effect on the final morphology of copolymer nanocomposites and nanoparticles mainly concentrate on domain A. As φN increases, some of nanoparticles have to be distributed in domain B because of the excluded volume effect of nanoparticles. The nanoparticles located in domain B will induce phase separation of diblock copolymers and can promote the formation of the bicontinuous structures to some extent. This investigation can provide some insights into the effects of nanoparticles on the lamellar phase separation of diblock copolymers. 3.3 Effects of Nanoparticles Size. We now investigate the effects of nanoparticles size on the phase behaviors of diblock copolymers. In Figure 7, the snapshots, densities, and order parameter profiles for two systems with Rnano ) 0.2 and 0.4 are shown. The “ordered” and “disordered” domains have also been displayed in panels c1-c3 and panels g1-g3 of Figure 7, corresponding to φCN ) 0.0781% (Nnano ) 17) and φCN ) 0.221% (Nnano ) 6). As shown in Figure 8, for these systems with different nanoparticles radii, the average order parameter φ also increases slightly first and then decreases with loading the nanoparticles one by one. The critical nanoparticles volume fraction φCN depends on the radius of nanopartcles. Additionally, the critical nanoparticles volume fractions φCN for five different values of nanoparticles radius with the box size of 9 × 9 × 9 DPD units are plotted in Figure 9. Here, the neutral nanoparticles with ωAN ) ωBN ) 1.0 and ωAN ) ωBN ) 0.5 as well as the energetically biased nanoparticles with ωAN ) 5.0, ωBN ) 1.0 are considered. The curves show that the critical nanoparticles volume fraction φCN increases as the nanoparticles radius Rnano increases, especially for the neutral nanoparticles with ωAN ) ωBN ) 0.5. For the same number of nanoparticles, the larger the nanoparticle size is, the more obvious the effect of nanoparticles on the phase separation is. The reason may be that the excluded volume effect is more obvious for the large nanoparticle size. For the same nanoparticle size, the smaller

He et al.

Figure 9. The critical nanoparticles volume fraction φCN as a function of the nanoparticles radius Rnano for 9 × 9 × 9 DPD units systems with different interaction strengths of ωAN and ωBN.

Figure 10. The phase diagram of copolymer nanocomposites in terms of the nanoparticles volume fraction φN, nanoparticles radius Rnano, and interaction strength between nanoparticles and A block ωAN for 9 × 9 × 9 DPD units systems with ωBN ) 1.0.

the interaction strength is, the larger is the critical nanoparticles volume fraction φCN and the more is the number of nanoparticles. 3.4. The Phase Transition from the Lamellar to the Bicontinuous. From the above discussion, we arrive at the phase diagram of the copolymer nanocomposites in terms of the interaction strength ωAN, nanoparticles radius Rnano, and the nanoparticles volume fraction φN for 9 × 9 × 9 DPD units systems shown in Figure 10. The curved surface in Figure 10 represents the transition points from lamellar structures to bicontinuous structures for every system. The domains below and above this surface, respectively, represent the ordered lamellar structures and the stable bicontinuous morphology. This phase diagram can serves as a guideline to know more about the phase behavior of the copolymer nanocomposites. Of course, some new morphology may be obtained if we change the volume fraction of A block fA or the interaction expression between the nanoparticles and blocks. 4. Conclusion In summary, the effects of nanoparticles on the lamellar phase separation of diblock copolymers are investigated by large-scale

Effects of Nanoparticles on Phase Separation dissipative particle dynamics. In order to avoid the undesirable penetration of fluid particles into the nanoparticles, we introduce new interactive energies between nanoparticles and DPD particles as well as between nanoparticles and nanoparticles. Two types of nanoparticles, i.e., energetically biased and neutral nanoparticles, are considered. Our simulations show that loading the spherical nanoparticles, increasing the interaction strength between nanoparticles and blocks, and enlarging the nanoparticles radius do largely affect the lamellar phase of diblock copolymers, which is in qualitative agreement with the experimental results. In particular, we find that the degree of order of phase separated structures increases slightly and then decreases abruptly until a bicontinuous morphology is formed as the nanoparticle volume fraction increases. In fact, the position distribution of nanoparticles plays a decisive role in the formation of the final bicontinuous morphology, because there exists a strong excluded volume effect among these nanoparticles. Finally, the morphologies of symmetric diblock copolymers A3B3 with nanoparticles are mainly controlled by the cooperative effects of the number of nanoparticles, the radius of nanoparticles, and the interaction strength between nanoparticles and the blocks. Acknowledgment. This research was financially supported by the National Natural Science Foundation of China (Nos. 20574052, 20774066, 90403022, 20525416), the Program for New Century Excellent Talents in University (NCET-05-0538), and the Natural Science Foundation of Zhejiang Province (No. R404047). The authors thank the referees for their critical reading of the manuscript and their very good ideas. References and Notes (1) Onuki, A. Phase Transition Dynamics; Cambridge University Press: Cambridge, 2002. (2) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423. (3) Matsen, M. W.; Schick, M. Curr. Opin. Colloid Interface Sci. 1996, 1, 329. (4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091. (5) Leibler, L. Macromolecules 1980, 13, 1602. (6) Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87, 697. (7) Barrat, J.; Fredrickson, G. H. J. Chem. Phys. 1991, 95, 1281. (8) Semenov, A. N. SoV. Phys. JETP 1985, 61, 733. (9) Matsen, M. W.; Schick, M. Phys. ReV. Lett. 1994, 72, 2660. (10) Whitmore, M. D.; Noolandi, J. J. Chem. Phys. 1990, 93, 2946. (11) Matsen, M. W.; Bates, F. S. J. Chem. Phys. 1997, 106, 2436. (12) Fredrickson, G. H.; Ganesan, V.; Drolet, F. Macromolecules 2002, 35, 16.

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The Effects of Nanoparticles on the Lamellar Phase Separation of

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