Order and Phase Behavior of a Cylinder Forming Diblock Copolymers and Nano-Particles Mixture in Confinement: A Molecular Dynamics Study

Full Paper Order and Phase Behavior of a Cylinder Forming Diblock Copolymers and Nano-Particles Mixture in Confinement: A Molecular Dynamics Study Lenin S. Shagolsem,* Jens-Uwe Sommer We study a coarse grained model of cylinder forming diblock copolymers and nano-particles (NPs) mixture confined between Lennard–Jones hard walls. Two models for non-selective interactions between monomers and NPs are applied. In the case of purely repulsive interactions between NPs and monomers (athermal case) strong segregation of NPs at the film surfaces and the formation of droplets of particles inside the copolymer film can be observed. For weakly attractive interactions between NPs and monomers (thermal case) formation of droplets of particles disappears and segregation on the film surfaces depend on temperature. The uptake of NPs by the copolymer film in the thermal case displays a non-monotonic dependence on temperature which can be qualitatively explained by a mean-field model. In both cases of nonselective interactions NPs are preferentially localized at the interface between the microphase domains. Introduction For the past few decades, mixtures of polymers and nanoparticles have been a subject of intense research. For L. S. Shagolsem Leibniz Institute of Polymer Research Dresden, 01069 Dresden, Germany E-mail: shagolsem@ipfdd.de J.-U. Sommer Leibniz Institute of Polymer Research Dresden, 01069 Dresden, Germany and Institute of Theoretical Physics, Technische Universität Dresden, 01069 Dresden, Germany Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim example, mixtures of diblock-copolymers (DBCs) and nanoparticles (NPs) are shown to have improved the quality of polymer materials by producing highly ordered and complex composite structures which can serve as a next generation catalysts, selective membranes, and photonic band gap materials.[1–3] Block-copolymer nano-templates are also used in producing ordered arrays of metal dots or nanowires.[4] In addition to the well known ordered phases of a pure DBC melt, DBC-NP mixtures show new selfassembled morphologies.[5] A pure DBC melt can be completely characterized by the product xN and block wileyonlinelibrary.com DOI: 10.1002/mats.201000095 329 L. S. Shagolsem, J.-U. Sommer www.mts-journal.de ratio, f. Here, x and N are the Flory–Huggins interaction parameter and the degree of polymerization of the polymer chains, respectively. In polymer nano-composites the equilibrium structures additionally depend on the overall volume fraction, selectivity, and size of the added NPs. Copolymer thin films confined between parallel walls, unlike in bulk, display interesting features caused by the ratio of the characteristic length scales (microphase period and film thickness) and by surface forces. The conformational statistics of polymer chains are anisotropic when close to a wall and this give rise to an entropic surface field as discussed by Sommer et al.[6] for DBC melt confined between neutral walls. Thus, there is a favored orientation of the micro domains. For example, lamallae formed by symmetric DBCs are oriented perpendicular with respect to neutral walls. However, for selective walls there is a transition between perpendicular and parallel orientations of the lamallae depending on the wall separation. Such transitions in thin films have been studied experimentally[7–9] and theoretically.[10–13] Geisinger et al. investigated the phase behavior of symmetric DBC thin films using a self-consistent field technique (SCFT) and Monte Carlo (MC) simulation.[13] Considering the confining walls to be selective they obtained a phase diagram as a function of incompatibility xN and the film thickness. In their phase diagram, there exists stable regions of parallel/perpendicular orientations of the lamallae. Naturally, an extension to such systems would be to consider DBC-NP thin films. Recent studies on DBC-NP thin films show new selfassembled structures induced by NPs.[14–18] Segregation of non-selective NPs at the diblock interfaces as well as at the wall–polymer interfaces are observed. NPs are localized within a domain in the case of selective NPs. Chiu et al.[19] illustrated experimentally that location of NPs in a DBC-NP thin film can be controlled by properly tuning the surface affinity of NPs. Surface affinities of NPs are tuned experimentally by either coating them with a mixture of ligands or random copolymers. Random copolymer coated NPs behaves like surfactants i.e., non-selective and localize at the diblock interfaces.[19] Lee, Shou, and Balazs have studied DBC-NP thin films using a method which combines SCFT and density functional theory (DFT).[14] They found that in such thin films a polymer-induced depletion attraction drives the non-selective NPs to the walls. Also localization of the NPs at diblock interfaces forming a NP decorated structures (lamellae) are observed. On the other hand, a recent DFT based calculation of a confined homopolymer/NP mixtures under athermal condition showed that formation of NP layer in the wall–polymer interface corresponds to a first order transition.[20] But, for DBC-NP thin films we find no systematic study to determine the order of NP layer formation transition. Moreover, direct simulation studies of mixtures of diblock-copolymers (DBCs) and NPs are rare. We will discuss two types of 330 non-selectivity which are described in the Section Model and Simulation Details. The aim of this work is to explore the consequences of inclusion of non-selective NPs in DBC thin films using MD simulations within the range of stability of the cylindrical phase in a slit-like geometry. In particular we are interested in the distribution of NPs within the copolymer morphology, segregation of NPs at the surfaces, and the dependence of these properties on temperature and NP concentration. The rest of the paper is organized as follows: In the Section Model and Simulation Details, we describe details of our simulations; results and discussions are presented in the Section Results and Discussion, and we give our conclusions in the Section Conclusions. Model and Simulation Details We used a coarse-grained bead-spring model for polymer chains. Neighboring beads in a chain are connected by a finitely extensible nonlinear elastic (FENE) potential[21] defined as "
 # 2 kR0 r 2 UFENE ðrÞ¼
ln 1
; r < R0 R0 2 (2.1) ¼ 1; r  R0 ; where r is the separation of neighboring monomers in a chain. The spring constant, k, is fixed at 30e/s2, while the maximum extension between two consecutive monomers in a chain, R0 is fixed at 1.5s. The above values of k and R0 ensure that the chains avoid bond crossing and very high frequency modes.[22] All the physical quantities are expressed in terms of Lennard–Jones(LJ) reduced units where s and e are the basic length and energy scales, respectively. All the non-bonded pairwise interactions are simulated using a cut and shifted LJ potential   dVðrÞ ULJ ðrÞ ¼ V ðrÞ
V ðrc Þ
ðr
rc Þ (2.2) dr r¼rc    s 12 s 6 with V ðrÞ ¼ 4"
. Here r and rc are the r r pairwise separation of particles and cut-off radius, respectively. The different cut-off radii for LJ interaction among the different species are summarized in Table 1. The reduced temperature,p T,ffiffiffiffiffiffiffiffiffi and timestep, Dt, are defined  as T ¼ kB T0/e and Dt ¼ t s m=" where kB, T0, and m are the Boltzmann constant, absolute temperature, and mass, respectively. All the monomers of type A and B have same sizes with a diameter of s ¼ 1 and mass m ¼ 1. NPs have a diameter of sp ¼ 2s and mass mp scales as cube of diameter. The interaction strength, e ¼ 0.5 is identical for all LJ interactions. We modeled A–B interaction by a purely repulsive LJ potential (cut-off at the potential minimum, rc ¼ 21/6s) Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.MaterialsViews.com Order and Phase Behavior of a Cylinder Forming Diblock Copolymers . . . www.mts-journal.de Table 1. Interaction range among the species. Interaction between A–B A–A or B–B A–NP or B–NP Cut-off radius (rc) in units of s 21/6  1.0 2.5  1.0 2.5  1.5 (thermal NP) 21/6  1.5 (athermal NP) NP–NP A or B–Wall 21/6  2.0 21/6  0.5 21/6  1.0 NP–Wall while allowing some attraction between the like monomers (rc ¼ 2.5s). For non-selective NPs, we studied the following two cases: (i) purely repulsive and (ii) slightly attractive. One could imagine the first and second situation as effective poor and good solvent conditions, respectively, for NPs. In both cases, it is obvious that there is no energetic preference for NPs to either species of the diblock. We refer to the first case as athermal and to the second case as thermal. Because of the longer range of NP interactions, the thermal case corresponds to an effective attraction between monomers and NPs. We performed a constant NPT-ensemble MD simulation with periodic boundary conditions along X and Y directions, while the Z direction is non-periodic due to the presence of walls. Nose–Hoover thermostat and Bernstend barostat were employed to maintain a constant temperature and pressure of the system, respectively. To simulate a thin film, we fixed the pressure in X and Y directions at P ¼ 5e/s3[22] and the MD integration time step used is 0.001. Chains with a polymerization index of N ¼ 48 are used to study symmetric/asymmetric DBC matrices and the number of chains ranging from 2 500 to 5 000. In an asymmetric DBC matrix, we have chosen the fraction of one species of DBC to be f ’ 0:23 which form cylinders in the microphase separated state. The LJ hard walls are kept at a distance of 50s, and they interact with the monomers and the NPs by a purely repulsive 12–6 LJ wall potential. The cutoff radius for a purely repulsive LJ interaction depends on the particle size as rc ¼ 21/6seff, where seff is the distance of closest approach between particles. Therefore, in order to simulate a LJ-hard wall, we provide different cut-off radii for the interactions at the interface between the monomers/ NPs and the wall, see Table 1. In this way, the monomers and NPs irrespective of having different sizes feel a repulsive interaction only when they touch the walls. The total amount of NPs present in the system is quantified by an overall NP volume fraction defined as Fp ¼ .  p Np s 3p V0 ; 6 www.MaterialsViews.com (2.3)   where V0 ¼ p6 Np s 3p þ Nm s 3 is the total occupied volume; Np and Nm are the total numbers of NPs and monomers present in the system, respectively. In our simulations, we vary Fp in the range F ¼ 0.0–0.4. A pure DBC thin film (Fp ¼ 0) of thickness 50s is considered as our reference system. For cylinder forming DBC thin films without NPs, we estimated an order disorder transition (ODT) temperature of TODT ’ 3:125.[23] In studying systems with athermal NPs, we considered both cylinder and lamellae forming DBC matrices, while only cylinder forming DBC matrices are considered for thermal NPs. For all the systems (with/without NPs) a disordered state was first prepared considering only a purely repulsive interaction among the different species. Then introducing attractive interactions (according to Table 1) we relaxed the systems at a sufficiently high temperature (T ¼ 3.5) which is above TODT of our reference systems. Relaxed samples are then cooled continuously to a temperature (T ¼ 1) well below TODT. Typically, the systems are relaxed at a desired temperature for about 3–5  106 MD steps which roughly corresponds to 25–45t (t is the relaxation time of a single chain in melt). Cooling of the samples by 1 unit of temperature was performed using 1  107 MD steps. We used LAMMPS[24] to perform MD simulations and the configurations are visualized using VMD.[25] The simulation box is subdivided into regions according to Figure 1 for analysis. We define a surface region which extends to a distance of sp from the walls. A bulk region is defined by a layer with a minimal distance of 5sp from the walls. An intermediate region extends for 4sp from the first monolayer. The distance used to define bulk is estimated Figure 1. Sketch of the surface(S), intermediate(I), and bulk(B) regions. Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 331 L. S. Shagolsem, J.-U. Sommer www.mts-journal.de from our reference systems at T ’ 1:0. It corresponds to the distance from a wall at which the wall’s influence on the normal component of the radius of gyration of polymer chains vanishes. Results and Discussion Athermal NP/Polymer Interaction: Segregation and Clustering We simulated athermal NPs in cylinder/ lamellae forming DBC matrices for a fixed NP volume fraction of Fp ¼ 0.143. Our systems consist of 5 000 chains and 5 000 NPs confined between two LJ hard walls. Snapshots of relaxed samples at T ¼ 1.5 are displayed in Figure 2. It has been shown theoretically and in previous simulations that DBC Figure 2. Athermal NPs in lamellae (a) and cylinder (c) forming DBC matrices under chains in microphase separated state are confinement at T ¼ 1.5 and Fp ¼ 0.143. In (c) the minority components of DBC and NPs [6,26–28] stretched. This stretching contriare shown. In (b) and (d) only NPs are displayed to see nano-droplets in the polymer butes to an entropic force pushing NPs matrices. out of the copolymer phases. Interfaces in the phase-segregated state attract non(‘‘nano-droplets’’) are also observed in both the symmetric selective NPs (as non-selective solvent) both enthalpically and asymmetric DBC cases. (compatibalizing effect) and entropically (less excluded We show 3D radial distribution function, g3(r), for NPs in volume). For the non-selective case the interaction between the bulk region in Figure 3a. For NPs in the asymmetric DBC the monomers and the NPs as well as the entropy of mixing matrices, f ¼ 0.23, the 2nd and 3rd order peaks of g3(r) are further contributes to the free energy balance. For our DBCwell developed. In comparison, the 3rd order peak is almost model, the athermal case corresponds to a positive effective absent for NPs in the symmetric DBC matrices, f ¼ 0.5. Thus, interaction between NPs and monomers. Thus, the polymer from the inspection of g3(r) we can conclude that the matrix can be considered as a poor solvent for NPs and at a average radius of nano-droplets is 3sp for f ¼ 0.23. The certain value of control parameters (T and Fp), NPs are average size of nano-droplets in a symmetric DBC matrices expected to segregate from the matrix. As displayed in is slightly smaller. This size difference is consistent with our Figure 2, NPs are preferentially located at AB-interfaces and visual inspection also (see Figure 2). Moreover, the presence segregated at the film surfaces. Formation of NP droplets Figure 3. Comparison of NP radial distribution function for NPs in the asymmetric ( f ¼ 0.23) and symmetric ( f ¼ 0.50) DBC matrices at (p ¼ 0.143 and T ¼ 1.5. (a) 3D pair correlation functions, g3(r), in the bulk region; (b) 2D pair correlation functions, g2(r), in the surface region. 332 Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.MaterialsViews.com Order and Phase Behavior of a Cylinder Forming Diblock Copolymers . . . www.mts-journal.de of a peak at r/sp  13 for f ¼ 0.23 implies an average separation of nano-droplets. The peak is absent for NPs in symmetric DBC matrix indicating a broad distribution of distances between the nanodroplets (see inset of Figure 3a). Further, for NPs in the asymmetric DBC matrix, a direct calculation yields value of the averaged radius of gyration of nanodroplets of Rg ’ 3:25s p and an averaged center of mass separation between the nearest neighbor droplets of 16.33sp. Thus, the direct calculated values of Rg and center of mass separation of the nanodroplets are consistent with that shown by g3(r). Therefore, we conclude an increased tendency for formation of nano-droplets in asymmetric DBC Figure 4. Monomer index histogram of NP contacts at T ¼ 1.5 and Fp ¼ 0.143 for f ¼ 0.23 matrices as compared to the symmetric (a) and f ¼ 0.50 (b). Region enclosed by dotted lines belongs to the diblock interface and case. Nc is the number of direct contacts between monomers and NPs. In Figure 3b, we display the 2D radial distribution function, g2(r) for NPs in the length, and film thickness remain unchanged. Since sp ¼ 2s, surface region. The regular peaks at r/sp ¼ 1, 2, 3, etc are commonly observed in hard sphere fluid simulations and NP–NP interactions are purely repulsive, the monoindicating a fluid-like phase formed by the segregated mer–NP interactions have the largest interaction range NPs.[29] However, there is no difference in g2(r) for NPs in (see Table 1). Thus, the effective pairwise interaction between monomers and NPs is slightly negative. Therefore, symmetric and asymmetric DBC matrices. the polymer matrix can be considered as an effective good Nano-particles (NPs) are observed to be localized solvent for the NPs. preferentially in the AB-interfaces. The role of diblock interfaces for the localization of NPs has been analyzed as follows: We construct a histogram of the number of direct contacts, Nc, of a given monomer index with NPs. In Figure 4, we display the monomer index histogram for NP contacts in the cylinder/lamellae forming DBC matrices. In both the cases, the histogram has a prominent peak in the region where the monomer indices belong to the AB-interface. This is a clear indication that the nano-droplets preferantially grow at the AB-interfaces. Thermal NP/Polymer Interaction and Role of Interfaces In simulating systems with thermal NPs, we considered cylinder forming DBC chains only. In total, 2 500 chains were simulated, and we varied the amount of added NPs. Unless necessary, we will drop the term thermal in the rest of this section. All the basic parameters such as interaction strength, masses, chain www.MaterialsViews.com Figure 5. Thermal NP and DBC mixtures in the ordered state at T ¼ 1.0 for different values of Fp (shown in percent). NPs are localized at the diblock interfaces. Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 333 L. S. Shagolsem, J.-U. Sommer www.mts-journal.de In Figure 5(a–d), we show snapshots of DBC/NP mixtures relaxed at T ¼ 1.0 for different values of Fp. Here, only minority components of the DBC and NPs are shown. In contrast to the athermal NP case, formation of NP droplets is not observed (compare Figure 2 and 5). Moreover, the bulk radial distribution functions, g3(r), show a significant difference as will be discussed later. at the diblock interfaces can be seen in Figure 5(a–d). However, ratio of the number of direct contacts in the diblock-interface region, Nc–i, and in the tail region, Nc–t, decreases as a function of Fp as shown in the inset of Figure 7(a). The decrease in the ratio Nc–i/Nc–t would suggest that NPs are increasingly delocalized from the diblock interfaces as we increase Fp. Here, Nc–t is defined as Nano-Particle Distribution The volume fraction of each species of DBC and NPs as a function of position, z, normal to the walls are shown in Figure 6. The volume fraction is calculated as follows: First, the simulation box is divided into slices parallel to the walls of thickness s. Then, we determine the total volume of each species present in a slice at position z which is then normalized by the total occupied volume, V0. The NPs volume fraction profile at Fp ¼ 0.077 (Figure 6a) shows a peak next to the walls which is a signature of NP monolayer formation. The peak grows by nearly a factor of 3 when increasing Fp up to 0.143. When we further increase Fp to 0.25, a second layer starts to form as indicated by the appearance of a new peak (Figure 6c). Higher order peaks appear for Fp ¼ 0.4. The peak corresponding to first monolayer remains constant implying saturation of NPs at both surfaces, while the second peak continues to grow when increasing Fp from 0.25 to 0.4. Around Fp ¼ 0.25, Figure 6c, a transition from 3 to 4 maxima of the majority species of DBC can be observed. This indicates a change in the period of the nano-structure. We can understand qualitatively the observed increase in the number of period. When we gradually increase Fp more NPs go to the diblockinterfaces and at the same time cylinders create more twist and bends which increases the interface area; however, more NPs are also delocalized from the diclock-interfaces upon increasing Fp. We can see this behavior in the monomer index histogram as described below. In Figure 7a, we show the monomer index histogram obtained at T ¼ 1.0 for different values of Fp. For a fixed temperature, height of the distribution in peak and tail regions increase as we increase Fp. This apparent increase of NPs 334 Figure 6. DBC and NP volume fraction as a function of distance Z from the walls at T ¼ 1.0 and different values of Fp. In the figure, walls are located at
25.0 and 25.0 on the x-axis also minority/majority species of DBC are indicated. Figure 7. Monomer index histogram of NP contacts: (a) at T ¼ 1.0 for different values of Fp and (b) at Fp ¼ 0.077 for different values of T. The region enclosed by the dotted lines belongs to AB-interface and Nc is the number of direct contacts. Inset: Ratio of the number of direct contacts in the diblock-interface region, Nc–i, and in the tail region, Nc–t, as a function of Fp and T, respectively. Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.MaterialsViews.com Order and Phase Behavior of a Cylinder Forming Diblock Copolymers . . . www.mts-journal.de Figure 8. NP radial distribution function at T ¼ 1.0 for different values of Fp in the (a) bulk region, g3(r) and (b) surface region, g2(r). the average number of contacts between NPs and a monomer index which belong to the tail region (monomer indices 25–48). The effect of temperature on the monomer index histogram is displayed in Figure 7b for Fp ¼ 0.077. Since NPs at higher temperatures are more mobile, we see a decrease in the height of the peak as we increase T. Also, the ratio Nc–i/Nc–t decreases as we increase T, see inset of Figure 7b. In Figure 8a, we display the bulk radial distribution functions, g3(r) of NPs at T ¼ 1.0 and for different values of Fp. A comparision of the peak positions of g3(r) with that of athermal NP case (Figure 3a) indicates that even up to a volume fraction of Fp ¼ 0.4 there is no signature of nanodroplets being formed. Moreover, nano-droplets cannot be observed when we visually inspect the relaxed samples shown in Figure 5. In contrast to the athermal case, g3(r) display a peak at r/ sp  1.6 which is located below the second nearest-neighbor (SNN) peak position pffiffiffiof a regular hexagonal closed packed structure r=s p ¼ 3 but, slightly above the SNN of a pffiffiffi regular square/cubic lattice r=s p ¼ 2 , see also the inset of Figure 8a. Interestingly, height of the peak at r/sp  1.6 decreases when we increase Fp to 0.25 and it disappears completely when we further increase Fp to 0.50 (see inset of Figure 8a). Since we have observed that NPs become increasingly delocalized above Fp ¼ 0.25, the peak at r/ sp  1.6 might be associated with a particular packing at the interface of the DBC phases. Radial distribution functions, g2(r) of NPs in the surface region are shown in Figure 8b. It is clear that NPs establish a long range order as we increase Fp. We observe fluid-like behavior for low overall NP volume fraction (e.g., Fp ¼ 0.143) as indicated by the presence of peaks at r/sp  1, 2, and 3. This regular peaks behavior of g2(r) were observed in the athermal NP case also, see Figure 3b. As we further increase Fp(¼0.25), g2(r) develops an additional shoulder below the peak at r/sp ¼ 2, which evolves into a pffiffiffi distinct peak at r=s p ’ 3 for Fp ¼ 0.40. Such evolution of www.MaterialsViews.com g2(r), which is related with a transition to hexagonal close packed structure, was observed earlier in simulations of 2D hard-disk fluids[30] also in an experimental study on quasi 2D granular fluids.[31] We find no significant differences in g2(r) for Fp ¼ 0.143 when compared with that of athermal NP case (see Figure 3b). Nano-Particle Segregation and Uptake Filling fraction .  of the NP layer, h, is defined as h ¼ Ns ps 2p 4A , where Ns is the number of NPs in the surface region and A is area of the surface. Surface filling fraction as a function of the overall NP volume fraction at two different temperatures is displayed in Figure 9. A surface saturation is observed at h  0.83. Linear extrapolation (dotted lines in Figure 9) of T ¼ 1.0 data points gives an estimate of Fp  0.20 at which h  0.83. At T ¼ 2.0, the filling fraction has a value of h  0.6 when increasing Fp up to 0.50. This is consistently below the results for T ¼ 1.0. The data points for T ¼ 1.0 can be fitted using a tanh-function as shown in the figure. Figure 9. Surface filling fraction, h, of NPs as a function of Fp calculated at T ¼ 1.0 and T ¼ 2.0. Dotted lines represent linear extrapolation curves. Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 335 L. S. Shagolsem, J.-U. Sommer www.mts-journal.de We define the uptake of NPs at the surface and in bulk by fs ¼ Ns/Np and fb ¼ Nb/Np, respectively. Here, Nb is the number of NPs in the bulk region. The fraction of NPs in the intermediate region is given by 1–(fs þ fb). For a fixed value of temperature both fs and fb display a non-monotonic behavior if Fp is increased, see Figure 10a and b. The surface fraction, fs, at T ¼ 1.0 and 2.0 is displayed in Figure 10a. For both the values of temperature, a maximum can be observed at Fp ’ 0:143. At low temperature (e.g., T ¼ 1.0) we observe a sharp rise in fs for Fp  0.1 followed by a nearly linear decrease as no more NP can be accommodated in the surface region. For T ¼ 2.0 a smoother behavior is displayed. Here, the saturation is not yet observed, see Figure 9, and the maximum may be related with the increasing effect of excluded volume interactions between the hard spheres (higher virial coefficient) at higher concentrations. The bulk fraction, fb, at constant temperature is shown in Figure 10b. Here, a minimum is observed at Fp  0.25. Initial drop in the bulk fraction is directly related to the segregation of NPs in the surface region. The increase of fb for Fp > 0.25 would correspond to NPs filling up the bulk region due to addition of more and more NPs. The surface fraction at constant overall NP density shows a monotonic decrease with increasing temperature, Figure 10c. Interestingly, the bulk fraction, fb, displays a non-monotonic behavior as a function of temperature as shown in Figure 10d. At higher temperatures, for T > 1.5 (Fp ¼ 0.25), the uptake of NPs in the copolymer film decreases with decreasing temperature, while at lower temperatures, for T < 1.5 (Fp ¼ 0.25), the uptake is increasing again when temperature is decreased. Using a simple mean-field model of a polymer brush with attractive NPs (see Appendix A) we can qualitatively explain this behavior. Here, we use the analogy of polymer brush and DBC in strong segregation limit as considered previously in studying density distribution of NPs and influence of NPs upon lamellar thickness and elastic constants of DBC.[32] Using this model, we obtain for the fraction of NPs inside the brush, fb, in the limit fb  1 as h  i fb  exp vp xp
v2=3 (3.4) where effective monomer-NP interaction parameter, xp  ep/T, has been introduced which is given by longer range of interaction of NP, as discussed in the Section Nano-Particle Segregation and Uptake. Furthermore, we define v ¼ (x1/2/N) with N being the chain length and np is volume of a NP in units of the monomer’s volume. It is easy to see in the limit of large N the elastic contribution is small compared to enthalpic contribution and an exponential uptake of NPs can be expected. Our model predicted a minimum of the uptake as a function of temperature as has been displayed in Figure 12 in the Figure 10. Surface fraction, fs and bulk fraction, fb as a function of Fp at T ¼ 1.0 and T ¼ 2.0, (a) and (b), and as a function of T for Fp ¼ 0.077 and Fp ¼ 0.25, (c) and (d). 336 Figure 11. Cartoon of a polymer brush with NP. D and d are respectively height of the brush and the region of constant particle concentration. Each polymer chain has length N and interfacial contact area A. Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.MaterialsViews.com Order and Phase Behavior of a Cylinder Forming Diblock Copolymers . . . www.mts-journal.de Figure 12. Numerical solution for the equilibrium volume fraction, fb, as a function of temperature, T, according to Eq. (1.6). We have chosen the interaction parameters according to the values given in the figure and R denotes the diameter of NPs. appendix. The minimum is expected at Tc ’ T ð1 þ fb Þ5=2 (3.5) which is valid for all fb and  1=2 N T  ’ 27"3p =" (3.6) being the value at zero concentration of NP. The predicted behavior is consistent with our direct simulation results: The location of the minimum of the fb(T) shifts slightly to a lower value at higher NP uptake, fb, see Figure 10d. According to our mean-field analysis, the interpretation of the non-monotonic behavior of the NP uptake as a function of temperature results from the interplay of two effects: At low temperatures, the attractive interaction between NPs and monomers dominate and decreasing temperature results in higher uptake. At higher temperatures, the osmotic effect of the brush formed by the copolymer phases becomes important. Increasing the temperature decreases the stretching of chains and a higher uptake of NP is possible. general, we find strong surface segregation of NPs in diblock-copolymer thin films for athermal to moderately attractive interactions between NPs and monomers. NPs are preferentially located at the AB-interfaces as has been shown by analyzing the number of monomer-NP direct contacts as a function of the monomer index. The observed NP localization at the interfaces is in agreement with other SCFT based studies[14] also agree with the experiments.[19] In the athermal case, the copolymer matrix can be considered as an effectively poor solvent for the NPs. This leads to the formation of NP clusters in the bulk and strong segregation effects in the polymer-wall interfaces. By fixing the value of the overall NP density at Fp ¼ 0.143, we compared the NP distribution in the symmetric/asymmetric DBC matrices. It is clear from the results of 3D radial distribution function that the symmetric/assymetric nature of the diblock influence the formation of NP clusters. However, 2D radial distribution function revels that NP distribution in the polymer–wall interfaces is not affected by the type of DBC used and NPs are packed like a 2D fluid. For the thermal case, the uptake of NPs displays a nonmonotonic behavior with respect to temperature. We give a tentative explanation for these findings based on meanfield concepts. According to this analysis, at higher temperatures the osmotic pressure of the stretched blocks dominates the temperature effect and the uptake of NPs is reduced by decreasing the temperature. At lower temperatures, attractive interactions between NPs and the monomers dominate and the uptake increases with decreasing temperature. We have derived a characteristic temperature for the crossover in between both regimes as a function of the interaction parameters and the chain length. We could understand better the distribution of NPs in the DBC matrices by calculating directly the radial distribution function. Analysis of the radial distribution functions for thermal NPs in the surface and bulk regions show a structural transition related to packing of NPs when we increase the overall NP volume fraction. In the surface region, we see a transition from fluid-like behavior at low NP density to a hexagonal close packed structure at high density. To our best knowledge, such characterization is not done in earlier simulation studies of DBC-NP thin films. Conclusion We have presented a coarse grained MD simulation to study mixtures of asymmetric A–B block-copolymers and nanoparticles confined between hard walls. Nanoparticleinteractions were non-selective with respect to the both polymer species. Considering two types of non-selectivity namely athermal and thermal NP cases, we studied the role of interfaces, segregation, and clustering behavior of NPs. In www.MaterialsViews.com Appendix A: A Simple Mean-Field Model for Mixing Nano-Particles and Strongly Segregated Copolymers In the following we consider an extremely simplified model which is aimed to illustrate the interplay of the various forces which control the uptake of small particles in a Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 337 L. S. Shagolsem, J.-U. Sommer www.mts-journal.de strongly segregated copolymer matrix. We consider a symmetric DBC for simplicity. Below ODT chains are stretched significantly, and in the strong stretching limit one can imagine pure phase of DBC form a dry polymer brush. In Figure 11, we sketch one pure phase which has an interfacial contact area, A per chain with the other species of DBC, and average height of the brush is D. As shown in the figure, d is a region above the brush where NP density is high and remains constant. This might correspond to the segregated surface state. Here, we consider this region as an reservoir of NP at high density. Let the monomer-NP effective attraction be denoted by ep. In our treatment we ignore effects of a non-homogeneous brush potential and we consider a homogeneous distribution of particles inside the copolymer phases. It is important to note that unlike a surface grafted polymer brush the grafting density in DBC is a varying quantity which depends on T and NP density. Moreover, the brush height D also depends on T and NP density. Suppose, n and x are the total number of particles and the fraction of particles inside the brush per chain unit. The volume fraction of particles adsorbed into the matrix is defined by  fb ¼ nxvp N ; (1.1) where np is the volume of an individual particles and N is length of a chain in monomer units. For simplicity we define the Kuhn segment, l as unity and the monomer’s volume as l3 ¼ 1. Incompressibility of system implies A  D ¼ N ð1 þ fb Þ (1.2) By introducing the interaction parameter for the different monomer species x ¼ e/T and between the particles and monomers xp ¼ ep/T the mean field free energy per chain can be written as
 F D2 x1=2 N fb þ ð1 þ fb Þ
xp N ’ 1 þ fb kT 2N D
 Nfb fb þ log vp 1 þ fb  1=3 D ’ x1=2 N 2 ð1 þ fb Þ1=3 (1.4) For fb ¼ 0 we reproduce (up to prefactors) the well known result for DBC in the narrow interface approximation. The second equilibrium condition, (@F/@fb) ¼ 0, yields   xp N x1=2 N N fb 1 ¼ 0: þ
þ ln D vp 1 þ fb 1 þ fb ð1 þ fb Þ2 Substitution of Eq. (1.4) leads to (1.5) v2=3   xp 1 fb 1 þ ln ¼0 þ
1=3 v 1 þ f 1 þ f ð1 þ fb Þ2 p ð1 þ fb Þ b b (1.6) where we have introduced the variable . v ¼ "1=2 NT 1=2 (1.7) Numerical solution of Eq. (1.6) provides the equilibrium volume fraction of the particles fb(T) as a function of temperature, T, and in turn the lamellar thickness D(T). An example is given in Figure 12 for two different NP sizes. A typical feature of the function fb(T) is its non-monotonic behavior. This is in qualitative accordance with the simulation results given in Figure 10d. The limit of low particle volume fraction, fb << 1, an analytic solution can be obtained: h  i fb  exp vp xp
v2=3 (1.8) In this limit the minimum of fb(T) is located at the characteristic temperature (1.3) Here, we set the chemical potential of particles outside the copolymer to zero. The first term on the rhs of Eq. (1.3) corresponds to the stretching of chains, the second term results from the interface tension, Ax1/2, where Eq. (1.2) has been applied. The third term represents the mean-field interaction of the polymer chain with the particles at the given volume fraction, fb, and the last term corresponds to the entropy of translation of the particles. The equilibrium values of 338 two variables D and fb can be obtained by minimizing the free energy. From Eq. (1.3), we get the equilibrium height of the brush as  . 1=2 N T  ’ 27"3p " (1.9) in the general case the minimum is defined by Tc ’ T ð1 þ fb Þ5=2 (1.10) Thus, the crossover temperature, Tc shifts to a lower value for larger loading of particles. This agrees to our observation (see Figure 10d) where minimum point of the fb curve shifts to a lower value of T for higher value of fb. Macromol. Theory Simul. 2011, 20, 329–339 ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.MaterialsViews.com Order and Phase Behavior of a Cylinder Forming Diblock Copolymers . . . www.mts-journal.de According to this model the non-monotonic behavior of the volume fraction of particles turns out to be the interplay between the free energy effort to accommodate the particles in the matrix which is related to an additional stretching of the chains and the (mean-field) attraction of the particles in the matrix. Acknowledgements: Support from the Deutsche Forschungsgemeinschaft (DFG) contract number SO-277/3-1 is gratefully acknowledged. MD simulation was carried out in parallel enviornment at the Center for High Performance Computing (ZIH) of the TU Dresden. Received: December 8, 2010; Revised: March 1, 2011; Published online: March 31, 2011; DOI: 10.1002/mats.201000095 Keywords: diblock copolymers; molecular dynamics; nanoparticles; phase behavior; thin films [1] M. Templin, A. Franck, A. Du Chesne, H. Leist, Y. Zhang, R. Ulrich, V. Schädler, U. Wiesner, Science 1997, 278, 1795. [2] D. Zhao, J. Feng, Q. Huo, N. Melosh, G. H. Fredrickson, B. F. Chmelka, G. D. Stucky, Science 1998, 279, 548. [3] S. Stupp, P. V. Braun, Science 1997, 277, 1242. [4] A. Sidorenko, I. Tokarev, S. Minko, M. Stamm, J. Am. Chem. Soc. 2003, 125, 12211. [5] J. Huh, V. V. Ginzburg, A. C. Balazs, Macromolecules 2000, 33, 8085. [6] J.-U. Sommer, A. Hoffmann, A. Blumen, J. Chem. Phys. 1999, 111, 3728. [7] P. Lambooy et al., Phys. Rev. Lett. 1994, 72, 2899. [8] W. Stocker, Macromolecules 1998, 31, 5536. [9] N. Koneripalli, N. Singh, R. Levicky, F. S. Bates, P. D. Gallagher, S. K. Satija, Macromolecules 1995, 28, 2897. [10] M. S. Turner, Phys. Rev. Lett. 1992, 69, 1788. [11] Q. Wang, P. F. Nealey, J. J. de Pablo, Macromolecules 2001, 34, 3458. www.MaterialsViews.com [12] G. T. Pickett, A. C. Balazs, Macromolecules 1997, 30, 3097. [13] [13a] T. Geisinger, M. Müller, K. Binder, J. Chem. Phys. 1999, 111, 5241; [13b] J. Chem. Phys. 1999, 111, 5251. [14] [14a] J. Y. Lee, Z. Shou, A. C. Balazs, Phys. Rev. Lett. 2003, 91, 136103; [14b] J. Y. Lee, Z. Shou, A. C. Balazs, Macromolecules 2003, 36, 7730. [15] [15a] V. Lauter-Pasyuk, H. J. Lauter, D. Ausserre, Y. Gallot, V. cabuil, E. I. Kornilov, B. Hamdoun, Phys. B 1998, 1092, 241; [15b] V. Lauter-Pasyuk, H. J. Lauter, D. Ausserre, Y. Gallot, V. cabuil, B. Hamdoun, E. I. Kornilov, Phys. B 1998, 248, 243; [15c] Langmuir 2003, 19, 7783. [16] B. J. Kim, J. J. Chiu, G.-R. Yi, D. J. Pine, E. J. Kramer, Adv. Mater. 2005, 17, 2618. [17] B. J. Kim, G. H. Fredrickson, C. J. Hawker, E. J. Kramer, Langmuir 2007, 23, 7804. [18] J. J. Chiu, B. J. Kim, G.-R. Yi, J. Bang, E. J. Kramer, D. J. Pine, Macromolecules 2007, 40, 3361. [19] J. J. Chiu, B. J. Kim, E. J. Kramer, D. J. Pine, J. Am. Chem. Soc. 2005, 127, 5036. [20] E. S. McGarrity, A. L. Frischknecht, L. J. D. Frink, M. E. Mackay, Phys. Rev. Lett. 2007, 99, 238302. [21] G. S. Grest, K. Kremer, Phys. Rev. A 1986, 33, 3628. [22] K. Kremer, G. S. Grest, J. Chem. Phys. 1990, 92, 5057. [23] Weak segregation theory predicted that the peak of structure factor, S(q), increases on approaching ODT from the disordered phase and diverges at ODT (spinodal point). So, we can roughly estimate TODT from the plot of S(q)
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