Nanoparticle Composites: Extension of

Article pubs.acs.org/Macromolecules

Revisited Block Copolymer/Nanoparticle Composites: Extension of Interfacial Statistical Associating Fluid Theory Zhengzheng Feng and Walter G. Chapman* Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: The mean-field intermolecular interaction contribution in interfacial statistical associating fluid theory (iSAFT) is extended to include correlation functions, which could be evaluated from molecular simulation and the density functional theory (DFT) itself. Order−disorder transitions (ODTs) between lamellar and disordered phases for pure block copolymers and copolymer nanoparticle composites are reported, and the extended theory has been shown to give good agreement with molecular simulations and considerable improvement over the previous mean-field dispersion description. The interaction parameters at ODT are also correlated with the length of copolymers. Concentration profiles of different species are then calculated and compared quantitatively with simulation. The scaled microstructures predicted by the theory agree with simulation for all the nanoparticle size and surface chemistry investigated. Improvements over previous DFT reports have also been observed for large selective nanoparticles. iSAFT has been shown to be capable of modeling BCP from different segregation regimes using a single framework and predicting characteristic exponent for chain length dependence of lamellar spacing, in good agreement with other theories in all segregation regimes.

1. INTRODUCTION Nanocomposites of block copolymers (BCP) and nanoscale particles (NP) have drawn a significant amount of attention and interest in the past decade.1−3 This composite has several advantages over conventional polymeric materials. Nanoparticles with interesting mechanical, optical, electrical, or catalytic properties4−7 can provide unique enhancement with much lower concentrations than traditional mesoscale particle additives. Moreover, block copolymers self-assemble on the level of tens of nanometers, which enables the development of dense templates surpassing the density limitations of current optical lithography technology. Nanoparticles with tunable sizes and chemistries could distribute themselves into perfectly ordered arrays with molecular scale precision in this template. The nanolithography technique involving removal of the polymer template after the nanoparticle array has formed provides the opportunity to shrink the dimension of current electronic and storage devices.8,9 Hierarchical molecular structures formed by taking advantage of nanoparticle localizations within a block copolymer matrix could also lead to economical molecular separation, novel photonic crystals, high efficiency catalyst, and self-healing materials.10−17 The distribution state of nanoparticles is critical to the design and process of aforementioned applications. A number of experiments have been performed to study the relation between nanoparticle localizations and particle sizes and surface chemistries.11−13,18−20 It has been observed by several groups that for symmetric block copolymers with molecular weight on © 2012 American Chemical Society

the order of 100 kg/mol relatively large particles (∼20 nm) prefer the center of the chemically compatible domain while small particles (∼4 nm) tend to locate at the interfaces of two blocks, even though both types of particles are surface modified to be energetically favorable to the same copolymer block. On the other hand, carefully adjusting the composition of ligands tethered on nanoparticle surfaces as well as areal grafting densities, therefore nanoparticle chemistries, has been demonstrated to be capable of fully controlling the locations of nanoparticles in copolymer matrices. Furthermore, the presence of nanoparticles could disrupt existing polymer structure and induce interesting morphological phase transitions, without the use of external potentials.13,15,17 Phase diagrams collecting various morphological states with respect to nanoparticle fraction, nanoparticle size, and chemistry have also been created experimentally.21,22 To gain more molecular insights into nanoparticle distribution and phase transitions, extensive molecular simulation studies have been devoted to the microstructure and phase behaviors of the nanocomposite system. These studies, primarily coarse-grained molecular simulations including discontinuous molecular dynamics,23,24 dissipative particle dynamics,25,26 and Monte Carlo simulations,27,28 have shown qualitative agreement with experiments on the distribution of Received: January 25, 2012 Revised: July 6, 2012 Published: August 14, 2012 6658

dx.doi.org/10.1021/ma300197q | Macromolecules 2012, 45, 6658−6668

Macromolecules

Article

the theory predicted qualitatively different distributions compared to simulation for large selective nanoparticles. Preliminary results57 from iSAFT have shown a similar problem, which could hinder the future application of TPT1based DFT in these composite systems. To understand this problem from a phase behavior perspective, we have performed calculations to locate the order−disorder transition (ODT), which is the temperature that ordered BCP melts into the disordered state. Initial results of ODT have shown a large discrepancy between molecular simulation data and iSAFT predictions. This observation indicates that the dispersive forces or enthalpic contributions to the free energy are not accurately modeled in the theory. This inaccuracy could affect the distribution of NP through an intricate balance of enthalpic and entropic interactions. In this work, we extend iSAFT toward quantitative accuracy in the prediction of both ODT temperature and molecular structure. The extensions of the theory are tested and compared with simulation and theories23,24,58−64 first in simpler pure BCP melts and then BCP/NP composites. To our best knowledge, this work will be the first one to study the order−disorder transition with a TPT1-based theory. The remainder of this work is organized into the following sections: Section 2 describes the molecular model used and computation methods needed to ensure correct comparison with simulation; the basics of iSAFT DFT and proposed theoretical extensions are also introduced in section 2. Section 3 presents the iSAFT results for ODTs, microstructures, and spacing of nanocomposites as well as the comparison between theoretical predictions and molecular simulations. This work is then summarized in section 4.

nanoparticles in the domain but in general are not quantitatively comparable because simulations are limited to fairly short polymers. Nonetheless, various structures such as lamellas, perforated lamellas, hexagonally packed cylinders, and bicontinuous morphology are identified in phase diagrams as the concentration and chemistry of nanoparticles vary. A more economical alternative is molecular theory. Selfconsistent field theory (SCFT)29 has been the predominant describer of polymers and shows tremendous success and potential in modeling various complex polymeric systems due to its computational efficiency and accuracy in the long polymer limit. However, it still suffers from the incompressibility assumption, and it is difficult to deal with spherical particles self-consistently in the theory. Although effort30 has been made to extend a pure SCFT approach to particle polymer composite, it is limited to the dilute particle case. Researchers have managed to include in SCFT other suitable theoretical descriptions of particles, such as classical density functional theory (DFT)31,32 and Brownian dynamics simulations.33 The SCFT-DFT hybrid approach retains the efficiency and simplicity of SCFT and shows qualitative agreement with experimental observations. It has also been extended to studies of mechanical34 and optical properties35 of the composite material as well as the morphologies of confined BCP/NP films.36 However, a question has been raised27 as to whether the compressible DFT description is compatible with the SCFT framework and therefore any quantitative comparison with other theories and molecular simulations with a particular intermolecular potential. On the other hand, classical DFT naturally includes nanoparticles and compressibility effects. In particular, we consider DFT’s that account for the connectivity of spherical segments, therefore describing chain molecules and nanoparticles within the same framework. One particularly successful class of such polymer DFT is the ones developed based on Wertheim’s first-order thermodynamic perturbation theory (TPT1),37−40 which considers chain bonding as infinite strong association. These theories keep segment level information and incorporate molecular potential explicitly and thus are directly comparable to molecular simulations. Meanwhile, they share the same parameter library with an equation of state (EOS) in the uniform density limit. In this limit, most TPT1-based DFTs reduce to the statistical association fluid theory (SAFT),41,42 which is proven to be very successful in modeling bulk polymer systems and has been calibrated for many substances. With numerical efficiency retained in the long polymer limit, quantitative comparisons could also be made with experiments.43 Two types of TPT1based DFTs are widely applied in the study of polymeric systems. One is the theory of Yu and Wu44,45 which combines ideal chain formalism with the original TPT1 chain free energy evaluated at fundamental measure weighted densities. The other is interfacial SAFT (iSAFT)41,46,47 derived from the inhomogeneous extension of the original TPT1 chain free energy. Both theories have been applied with success to study structures of pure block copolymer melts,48,49 homopolymer composites with nanoparticles,50,51 and surface-induced phase transitions in (semi)confined polymer/nanoparticle thin films.52−54 Recently, the theory of Yu and Wu has been extended55,56 to the block copolymer nanocomposite system investigated by Schultz et al.24 with coarse-grained molecular dynamics simulations. However, with the exact same potential model,

2. MODEL AND METHODS One advantage of a segment based DFT is that it is directly testable by molecular simulations using the same molecular potential model. The simulations of Schultz et al.23,24 covered broad parameter space, reported phase diagrams and detailed concentration profiles of species for both pure block copolymer and nanocomposite systems. In this work, the model used therein is followed and summarized as following: (1) Diblock copolymers (two blocks labeled as “A” and “B”) are modeled as fully flexible chains formed by tangentially bonded hard spheres. Nanoparticles (labeled as “P”) are modeled as hard spheres with no structure. (2) Species interact with each other through the following potential: ⎧∞ if r < σij ⎪ ⎪ Uij(r ) = ⎨ εij if σij ≤ r ≤ σij + σchain i , j = A, B, P ⎪ ⎪ 0 if r > σij + σchain ⎩

(1)

εij is the interaction energy between species i and j. σij is the arithmetic mean of the individual diameters. Block segments in copolymers have the same diameter, denoted by σchain: σA = σB = σchain. For like species, εii = 0 and the potential reduces to a hard sphere potential. For unlike species, a repulsive shoulder exists beyond the hard core. The range of the repulsive shoulder is always σchain. Selective nanoparticles are configured to be repulsive to A block while remaining hard interactions with B block (εAP ≠ 0, εBP = 0). (3) Different nanoparticle sizes and chemistries are studied by varying reduced size and energy parameter: σ*P = σP/σchain, and ε*P = εAP/εAB. 6659


Macromolecules

Article

and hard spheres for nanoparticles. r ̅ = |r2 − r1| is molecular ref separation. gref αβ and yαβ are the pair and cavity correlation functions between the reference fluids α and β at average density ρave, respectively (kB is the Boltzmann constant; T is temperature). If gref αβ is assumed to be 1, eq 3 then reduces to the mean-field approximation. The correlations between hard chains are obtained from previous Monte Carlo simulation reports72,73 for hard chains of various lengths at various densities. The correlations between hard chains and hard spheres are generated self-consistently using iSAFT similar to the practices in previous publications.51,74 The implementation and assumption of these correlation functions are discussed in detail in the Appendix. Equations 3 and 4 are both tested in this work and are labeled iSAFT with g and y perturbation, respectively. With the Helmholtz free energy expressed, one can minimize free energy and obtain equilibrated thermodynamic properties and density distributions by solving the Euler−Lagrange equation iteratively. The excess Helmholtz free energy functionals used in iSAFT can be formulated into convolutions. The spatial integrations of these functionals in this work are carried out using fast Fourier transform (FFT),75,76 which improves the calculation efficiency substantially. Moreover, for a periodic system such as lamella formed by block copolymers, the boundary conditions are handled naturally in FFT without padding issues. The ODTs are calculated as the point where the lamellae and disordered phase could coexist.64 General phase equilibrium equations eventually reduce to the equivalence of grand potential density for these two phases:

(4) The average packing fraction of the composite (ηave = π∑iρi,aveσi3/6, ρi,ave is the average density of species i) lies mostly between 0.3 and 0.35, close to that of a polymer melt. The potential for nanoparticles could be considered as a simple core−shell model. The polymeric tethers on nanoparticles serve as the repulsive shell since they are usually chemically affinitive to one of the blocks. And the effect of mixed tethers could be easily studied by adjusting the magnitude of ε*P. Other interaction potentials have also been applied in molecular simulations.26,65 These potentials are equally accessible to the density functional theory. Throughout this work, only density variations in one dimension (lamellae) will be considered. Based on the classical SCFT phase diagram of BCP,29 even slightly asymmetric BCPs would form a complex ordered phase (cylindrical, spherical phases) as temperature is lowered below ODT. Therefore, only symmetric BCPs are studied in this work. The structures requiring two- or three-dimensional computations will be the subject of future work. The DFT in this work is based on the iSAFT of Jain et al.,46 in which the Helmholtz free energy of BCP/NP composites, a unique functional of the spatial density distribution {ργ(r)} in grand canonical ensemble (γ distinguishes different types of chemical species), is built upon the ideal gas free energy of spherical segments and includes a perturbational sum of the nonideal contributions due to hard core repulsion of these spherical species (HS), the work needed to connect them together (CH) and their dispersion interactions (DISP): A[{ργ (r)}] = Aid [{ργ (r)}] + AexHS[{ργ (r)}] + AexCH[{ργ (r)}] + AexDISP[{ργ (r)}]

Ω[{ργ (r)}]ODT /Vorder = Ω[{ργ ,bulk }]ODT /Vdisorder

(2)

where the grand potential Ω is related to Helmholtz free energy through the chemical potential μ.

We refer to previous publications and derivations for the detailed expressions of these Helmholtz free energies.41,46,66 Note that in this work the contact values of cavity correlation function (CCF) used in the chain connectivity free energy are approximated by the extended FMT.45 The mean-field description of dispersion interaction resembles that used in SCFT, and it has been successfully applied to various LennardJones (LJ) chain fluids.46,67,68 However, the neglect of correlations between species results in free energy inaccuracies. It has been shown that a quadratic density expansion treatment could improve fluid structure predictions from theory for semiconfined LJ chain fluids.69 In this work, correlations between species are incorporated through conventional perturbational method70,71 as shown in the Appendix, and the following formulas are derived for the dispersion free energy: 1 AexDISP[{ργ (r)}] = 2

1 2

(6)

The chemical (μ) and thermal equilibrium (T) are already enforced by working in the grand canonical ensemble. Fieldbased theories commonly report the Flory parameter χODT at the ODT. In this work, we report the reduced interaction parameter at the ODT εODT ** = εAB ** is the ODT/kBT because εODT energy parameter directly used in both the simulations of Schultz et al.23,24 and iSAFT calculations. The interaction energy ε** and Flory parameter χ can be compared using an equation of the form23,49,77 χαβ ≡ −

1 2kBT

** = εαβ

(3)

∫ ρnb(r)[uαα(r) + uββ(r) − 2uαβ(r)] dr

∫σ ≤|r|≤σ +σ αβ

αβ

chain

ρnb (r ) dr

(7)

In the simulations of Schultz,23,24 the integral in the rightmost expression in eq 7 is evaluated by counting and averaging the number of nonbonded neighbors (ρnb(r)) inside the interaction range for each chain segment. With the values of this integral, χNODT can then be obtained from εODT ** . Now several more methods need to be established to ensure correct comparison with simulations. First, to interpret the grand canonical DFT results in a canonical sense, the average density of each species in a calculation domain is fixed to a constant value by adjusting the bulk chemical potential. This

∑ ∑ ∫ ∫ dr1 dr2 udisp(|r2 − r1|) α

∑ ∫ dr′ ργ (r′)μγ γ

β

ref ρα (r1)ρβ (r2)gαβ ( r ̅ , ρave )

AexDISP[{ργ (r)}] =

Ω[{ργ (r)}] = A[{ργ (r)}] −

∑ ∑ ∫ ∫ dr1 dr2 udisp(|r2 − r1|) α

(5)

β

ref ρα (r1)ρβ (r2) exp( −udisp/kBT )yαβ ( r ̅ , ρave )

(4)

udisp = uαβ − uref is the potential difference between the pair potential of the fluid of interest and that of a reference fluid. The reference fluid is defined to be hard chains for copolymers 6660


Macromolecules

Article

chains, and thus the correlation hole effect, the effective repulsion is reduced and the resulting interaction parameters at ODT agree much better with the coarse-grained simulations than mean-field iSAFT. For 20mer chains, iSAFT with y perturbation gives better agreement with simulation than iSAFT with g perturbation. It is worthwhile to note that in the packing fraction range 0.35−0.45 iSAFT with y perturbation gives similar predictions in comparison (not shown) to the SCFT with fluctuation corrections by Fredrickson and Helfand (FH).58 The discrepancies between the simulation and theory results could be ascribed to the various approximations made in the last section. Another possible reason is that only a first-order expansion is adopted. A second-order expansion70,79 has been derived and shown to improve the first-order results, even though the improvement is nearly negligible. Moreover, the simulation values shown in Figure 1 are estimates between discrete states points, which might include uncertainties. The error bars of these data are displayed in Figure 1 as well. The ODT for symmetric BCP with different chain length has also been studied at ηave = 0.3. Similar to field theories which commonly report (χN)ODT, the products of chain length with interaction parameter at ODT (Nε**) ODT have been computed. The predictions from mean-field iSAFT and iSAFT with y perturbation are plotted against the variation of chain length in Figure 2 as open and closed diamonds. The

process is completed by employing a modified Powell’s hybrid algorithm,78 which solves nonlinear equations using a variation of Newton’s method. Second, the equilibrium lamellae spacing of the block copolymers corresponds to a minimum in Helmholtz free energy for our system at fixed average density. This spacing for lamellae morphology is found by employing a safeguarded quadratic interpolation minimization method78 within a searching range large enough to include the target.

3. RESULTS AND DISCUSSION To accurately describe the lamellar structure of block copolymer composites, it is crucial to first locate the phase boundaries correctly. In this section, ODT between lamellar and disordered phases for pure block copolymers will be considered first to benchmark the accuracy of the extension of iSAFT presented in the last section. Then neutral and selective nanoparticles will be added to pure BCPs and the ODTs for these composites will be computed. With these phase behavior information, microstructures and equilibrium spacing of BCP/ NP composites are then calculated and compared with molecular simulations. 3.1. Order−Disorder Transition (ODT) in Pure BCP. With the correlation functions obtained from simulations,72,73 ε*ODT * has been calculated at various average packing fractions for BCP melts with 10 and 20 segments. The comparison between aforementioned approaches and molecular simulations is shown in Figure 1. The correlation of 8mer from Figure 1 is

Figure 2. Product of chain length and interaction parameter for symmetric block copolymers at ODT as a function of chain length at average packing fraction 0.3. Symbols are predictions from iSAFT; solid lines are best fit to theoretical results.

Figure 1. Interaction parameter at the ODT for neat symmetric block copolymer chains with 10 and 20 segments at various average densities. Simulation results are from the work of Schultz et al.23,24 Error bars for simulation results are shown except the lowest packing fraction point for 20mer. Error bars for 20mer at high packing fraction lie within the boundaries of the symbols.

approach of iSAFT with y perturbation utilized the pair correlation function gHC αβ values at this density for hard chains with various lengths. More information can be found in associated content. Both sets of predictions approach asymptotic values as chain length grows, with the iSAFT with y perturbation approach constantly predicting higher (Nε**)ODT than mean-field iSAFT. (Nε**)ODT from both approaches can be well correlated to chain length by a general power law, as shown in the plot. However, since the transformation of ε** to χ (eq 5) has very weak chain length dependence, the behavior of (Nε**)ODT as a function of chain length obtained from iSAFT are quite different from the SCFT with fluctuation correction by Fredrickson and Helfand: (χN)ODT = 10.495 + 41.022N−1/3.58 iSAFT predicts stronger chain length dependence than FH theory, evidenced by the larger (absolute value) power indices. The (Nε**)ODT from

used for 10mer due to lack of simulation data for 10mers in the considered density range. In Figure 1, it is observed that the interaction energy parameters at ODT from iSAFT with meanfield dispersion are much lower than the simulation reported values. iSAFT with the mean-field term predicts that the ordered state needs to be heated to higher temperature than that in molecular simulation to melt into a disordered phase. Using a simple mean-field dispersion, iSAFT overestimates the repulsion in the theory between the two blocks in the copolymer. This overestimation becomes worse at lower average densities. By including the correlation function for 6661


Macromolecules

Article

Figure 3. Interaction energy parameter at the ODT as a function of NP concentration for different sized nonselective NPs (solid line: σ*P = 1; broken line: σ*P = 2; dotted line: σ*P = 4; dot-dashed line: pure BCP) in symmetric block copolymer matrices with 20 segments predicted by (a) mean-field iSAFT, (b) iSAFT with y perturbation evaluated at total average packing fraction, and (c) iSAFT with y perturbation evaluated at effective BCP packing fraction. The total average packing fraction is fixed at 0.35. Simulation results (symbols) and their error bars are from Schultz et al.24 The error bar for the data with highest NP fraction is not known. The error bar for the data with lowest NP fraction lies within the boundary of the symbol.

Figure 4. Interaction energy parameter at the ODT (dashed lines: mean-field iSAFT; broken lines: iSAFT with y perturbation evaluated at system total packing fraction; solid lines: iSAFT with y perturbation evaluated at effective BCP packing fraction) as a function of NP concentration in symmetric BCP matrices with 20 segments predicted by various approaches for (a) σ*P = 2; ε*P = 1 and (b) σ*P = 4; ε*P = 1. The total average packing fraction is fixed at 0.35. Simulation results (symbols) and error bars are from Schultz et al.24 The error bar for the data with highest NP fraction in (a) is not known. Error bars for the data with lowest NP fraction lies within the boundaries of the symbols.

applied to the ODT study of BCP/NP composite mixtures. The cases requiring no NP−polymer correlation (nonselective NP, εAP = εBP = 0) are considered first. The interaction parameter at the ODT ε*ODT * obtained using mean-field iSAFT for BCP/NP mixtures with total average packing fraction 0.35 and various nanoparticle sizes is plotted against NP concentration in Figure 3a. NP concentration is defined as the ratio of the NP packing fraction over the total packing fraction. The pure BCP cases shown in Figure 3 refer to the situation when the NPs are simply removed from the polymer matrix while the polymer packing fraction is kept the same as the systems containing NPs. At low NP concentration, the

iSAFT would therefore decay faster as chain length increases. It is worthwhile to point out the average packing fraction (0.3) in this study is quite low and compressibility effect might be an important factor, which is neglected in the FH theory. Note that the power indices of the chain length dependence from iSAFT predictions are relatively closer to the value of −0.9 reported by Nath et al.,61 who employed a PRISM-based DFT and dealt with a denser system with bulk packing fraction of 0.524. 3.2. Order−Disorder Transition (ODT) in BCP/NP Composite. With the successful application in pure BCPs shown in the last section, the extension of iSAFT is now 6662


Macromolecules

Article

Figure 5. Concentration profiles of BCP in mixture (solid lines) with NP (dot dashed lines) compared with pure BCP (broken lines) at the same average packing fraction of 0.28 for NP sizes (a) σ*P = 1 and (b) σ*P = 2. Dotted lines represent the total concentration in the mixture scaled by the total packing fraction.

that small NPs and highly selective NPs have strong tendency to swell their preferred domain, thus modifying the morphology of the polymer matrix. Therefore, even for symmetric BCPs, complex structures beyond the scope of this work could be formed.17,24 The ODTs of these cases are not considered in the following section. Instead, the interaction energy parameter at the ODT for relatively large NPs with medium surface selectivity is plotted against NP concentration in Figure 4. Both cases are similar to the system studied in Figure 3 in that mean-field iSAFT predicts much lower ε*ODT * than simulation. iSAFT with y perturbation evaluated either at an effective BCP packing fraction (labeled y(ηeff)) or at the total packing fraction (labeled y(ηtot)) gives better agreement with simulation. Note that for the largest NPs iSAFT with y(ηeff)correction tends to overpredict ε*ODT * at high NP concentrations. 3.3. Concentration Profile. The information on ODT for various systems enables us to locate the interaction energy parameter used in previous DFT studies,56,57 with respect to ε*ODT * in the phase diagram. Based on Figures 3 and 4, the repulsion between AB blocks in previous mean-field DFT calculations is overestimated, which could result in a more depleted interface (less solubility of A and B at interface) than simulations and drive the nanoparticles to the interface. With the extension of the dispersion free energy, the ODTs predicted by the theory are much closer to those of the simulations. The enthalpic interactions within the composite are better modeled, and therefore better predictions of the microstructures in BCP/ NP composite are expected in comparison to previous DFT calculations. First, the effect of nonselective NPs on polymer microstructure is studied. The concentration profiles of A and B blocks and different sized NPs are compared with the profiles of pure BCPs in Figure 5. The polymer packing fraction in pure BCP system is the same as that in the composite system. The concentration is given in the same form as defined by Schultz et al.:24

ODTs for BCP/NP mixtures are very close to that of pure BCP systems because NPs are just filling voids without disrupting polymer structures. As NP concentration increases, the ODTs for mixtures deviate more and more from pure BCP cases as NPs screen out unfavorable interactions of BCP molecules, so that the system could endure more chemical incompatibility before transforming into an ordered state. This finding does not agree with the report of Schultz et al.,24 in which it is claimed that the nonselective NPs would not change the ODT of the mixture compared to a pure BCP system (therefore only one set of symbols). It is highly possible this behavior is missed in simulation because, as discussed before, simulations investigated discrete state points in phase diagrams and the variations of ODT due to addition of NPs are not large in magnitude. At higher NP concentration, the smallest NPs are found to be more efficient in suppressing the system, resulting ** or lower ODT temperature. This observation in a higher εODT is in agreement with the theoretical work of Chervanyov and Balazs.63 In Figure 3b, iSAFT with y perturbation predictions are compared with simulation data. In this case, the correlation function values must be obtained with caution since they are now dependent on both the packing fraction of BCP and NP. The exact form of this dependence remains a question. The hard chain correlation functions used in Figure 3b are evaluated simply at the total packing fraction as if the system is pure BCP. With this approximation, iSAFT with y perturbation gives reasonable agreement with simulation at low NP concentration * increases at high NP concentration. but deviates more as ε*ODT This indicates the repulsion/incompatibility in the system is overestimated at high NP concentration. The reason might be that the chain−chain correlation is overcounted due to the presence of NPs. BCPs’ should really interact with each other in the volume excluded by nonselective NPs. Therefore, the effective packing fraction of BCP should be the volume taken by BCP over the total volume excluding the volume of nonselective NPs, which could be expressed as ηeff BCP = ηBCP/(1 − ηNP). iSAFT predictions with y perturbation evaluated at this effective BCP packing fraction are shown in Figure 3c, in which the agreement with simulation is much improved at high NP concentration. Now the particle−polymer correlation is included to consider selective NPs in BCP/NP composites. It is reported

φi(x) =

6 ∑i ρi σi 3

σi /2

∫−σ /2 (σi 2/4 − (Δx)2 )ρi (x + Δx) d(Δx) i

(8)

This quantity represents the local volume fraction of component i scaled by the average packing fraction. In order to compare the pure BCP results directly with BCP in 6663


Macromolecules

Article

Figure 6. Concentration profiles of polymer nanocomposite for σ*P = 1, 2, 4 and ε*P = 0, 1, 2 (εAB/kBT = 0.125). In all cases the average packing fraction of BCP and NP is 0.28 and 0.07, respectively. Solid and broken lines are BCP and NP profiles from iSAFT, respectively. Symbols are from molecular simulation of Schultz et al.24 In all plots, A blocks are on the right side and B blocks are on the left.

rich and NP-lean phases. We also checked macrophase separation for all the cases studied in Figure 6 by constructing binary phase diagrams (not shown) of nanoparticle-rich and copolymer-rich phases similar to the work of Bryk50 on athermal composites. It is confirmed that these cases would not macrophase separate within iSAFT framework. 3.4. Equilibrium Spacing of BCPs and BCP/NP Composites. For pure BCP systems, previous theoretical studies58−62,64,80,81 have reported that the equilibrium lamellar spacing follows a scaling relation L = βNα, where L is the spacing. The coefficient β is a function of the interaction strength. The exponent α differs in regimes characterized by χN, i.e., weak (W), strong (S), and intermediate (I) segregation regimes (SR). Theories60−62,64,80,81 and molecular simulations65 have consistently reported the characteristic exponent in the strong segregation regime (SSR) to be in the vicinity of 2/3, which is also observed in experiments.82 In the weak segregation regime (WSR), early SCFT and DFT calculations59,60,64 have reported the exponent to be 0.5. However, it was argued29 that this value is indeed for the regime with no segregation, and a more accurate SCFT that includes higher order cortex functions gives a characteristic exponent of 1. In the intermediate segregation regime (ISR), there have been discrepancies on the reported values60,62,64 of the characteristic exponent, varying in the range 0.72−0.95. In the following section, we intend to explore the location of different segregation regimes in the NεAB ** space and identify characteristic exponent in each regime with iSAFT. In Figure 7, the natural log of neat symmetric block copolymer equilibrium spacing scaled by the square root of chain length (ideal average end-to-end distance) is plotted

composites, the concentration for pure BCP is scaled to the average packing fraction of the composite system. In Figure 8, the z direction is perpendicular to the interface between two polymer blocks; the lamellas are moved so that the center of the interface corresponds to the coordinate z = 0. Small NPs (σ*P = 1) barely change the structure of BCP, the NPs are just filling voids and are distributed evenly inside the domain with a slight preference to the interface. For larger NPs (σ*P = 2), the BCP structure remains unaltered except that NPs have a stronger preference to the interface, and the two polymer blocks are “squeezed” to give higher concentration at the center of each block. This trend is expected to become stronger as the NPs grow larger. The total concentrations in composites, which are the sum of concentration of all species, are also plotted in Figure 5. The total concentrations are fairly uniform throughout the lamellar domain with very slight oscillations. Next, the concentration profiles of composite systems with various NP size and surface chemistry (σ*P = 1,2,4, ε*P = 0, 1, 2) are calculated and compared quantitatively with molecular simulation24 in Figure 6. The concentrations are computed in the same way as eq 8. The lamellar structures are scaled by their individual equilibrium spacing to enable direct comparison between the theory and simulation. It is shown that the scaled microstructures from iSAFT with y perturbation gives outstanding agreement with simulation for most of the cases investigated. More importantly, contrary to previous DFT reports56,57 that predicted large selective NPs prefer the interface of the two blocks, iSAFT with y perturbation captured the correct distribution preferences of NP in these cases. (e, h, and i). The largest deviation in structure is seen in case i, where the simulation reported macrophase separation between NP6664


Macromolecules

Article

conducted in ISR as well (χN = 48). However, this Nε*AB* value would fall in SSR within the framework of mean-field iSAFT. For BCP/NP composites studied in Figure 6, the equilibrium lamellar spacing as well as the variation of A, B block size found by using iSAFT with y perturbation is given in Figure 8. The change of spacing for the composite system scaled by the spacing of pure BCP is compared with the coarse-grained simulations of Schultz et al.24 in Figure 8a. The spacing predicted from theory shows similar behavior as reported in molecular simulation. However, the theory consistently overestimates the equilibrium spacing for most of the cases, which is also observed in previous study using TPT1-based DFT.56 The smallest NPs are the most effective in swelling the spacing of BCPs while strongly selective NPs tend to have a larger impact on BCP spacing as well. Note that for relatively small NP additives polymer spacing is actually reduced because NPs screen part of the unfavorable interaction between A and B blocks. The variation of A block domain size scaled by the equilibrium spacing of pure BCP is shown in Figure 8b. Since A block is repulsive to the NPs, the size of A domain is fairly unaffected by the addition of NPs and is close to the size of A in pure BCP. Therefore, the swelling in B block accounts for most of the change of total spacing. B block size scaled by the A block size is shown in Figure 8c. It is shown that the B block can be up to 45% larger than the corresponding A block in the systems studied in section 3.3.

Figure 7. Variation of equilibrium spacing scaled by √N as a function ** for neat symmetric block of the combined interaction parameter NεAB copolymers at average packing fraction of 0.3. Symbols are predictions from iSAFT. Solid lines (fixed slope given above the lines) are a guide to the eye.

against the natural log of combined interaction parameter Nε*AB*. These calculations are done by applying iSAFT with the y perturbation extension and fixing the average density. The data could be divided into three regimes, guided by the solid lines with different slopes. The boundaries between these regimes can be roughly identified as ln(Nε*AB*) = 0.307 and 1.1. The characteristic exponents in these regimes are approximately 0.5, 0.95, and 0.67, respectively, which agree well with the previously reported values in WSR, ISR, and SSR. The range of ε*AB* alone cannot be used to identify SR since data from the same ε*AB* range could penetrate different SR. Therefore, we have demonstrated the capability of iSAFT to model BCP systems from different segregation regimes consistently with a single framework. Similar as previous reports, the WSR domain is predicted to be quite narrow. The ISR domain size is ln(NεAB **)ISR−SSR − ln(NεAB **)ISR−WSR ≈ 0.8, close to the value of 0.7 found by Shull,62 who did not report a WSR. Melenkevitz and Muthukumar64 found that ISR domain size to be ln[(χN)ISR−SSR/(χN)ISR−WSR)] ≈ Ln[95/15] ≈ 2.03, which is considerably larger than our result. If mean-field iSAFT is considered in this section, all the data would be translated to the lower Nε*AB* values without qualitative change. Note that the concentration profiles in section 3.3 are obtained with NεAB ** = 20 × 0.125 = 2.5, which lies in ISR in the context of iSAFT with y perturbation. The corresponding simulations are

4. SUMMARY The mean-field intermolecular interaction contribution within iSAFT has been extended to include correlation functions of the reference fluid. In the block copolymer nanocomposite system, a first-order perturbation theory is used to calculate enthalpic interactions between unlike species. The reference fluid hard chain average segment−segment correlation functions are evaluated from molecular simulation reports. The nanoparticle−polymer correlation is calculated from the DFT itself. With these correlations, ODTs of pure BCP and BCP/NP composite systems are computed for the first time using TPT1-based DFT. These ODTs are reported in the inverse temperature−density/NP concentration space and compared with molecular simulations. It is observed that in all cases mean-field iSAFT overestimates the ODT temperature, while iSAFT with perturbation extension produces better agreement with simulation. In BCP/NP composites, the interaction parameter at ODTs are captured by iSAFT with y

Figure 8. Equilibrium spacing variations for systems studied in Figure 6. (a) Total lamellae spacing of BCP/NP composite scaled by the spacing of pure BCP with no NP added. (b) A block spacing scaled by the spacing of pure BCP. (c) B block spacing scaled by A block size. Open symbols with dotted lines are simulation results from Schultz et al.24 6665


Macromolecules

Article

1 2

perturbation at low NP concentration but underestimated at high NP concentrations. An effective packing fraction is proposed and shown to provide better predictions for BCP mixed with neutral and interacting NPs with different sizes. Using the proposed extension, microstructures/concentration profiles of the composite systems are calculated using iSAFT and compared with molecular simulation directly. Although there is a difference of the equilibrium lamellar spacing predicted by iSAFT and simulation, the scaled molecular structures agree very well between the two approaches. The composite structures for large selective NPs in this work are also improved significantly compared to previous DFT studies. iSAFT also gives very similar predictions with simulation on the variation of spacing with change of NP size and chemistry but slightly overpredicts the swelling effect of NPs. The size of the energetically unfavorable domain is found to remain unaffected by the addition of NPs and the NP incorporated block accounts for most of the spacing swelling. Furthermore, by applying iSAFT to pure BCP systems, we identified different (weak, intermediate, and strong) segregation regimes with characteristic chain length scaling on equilibrium spacing in good agreement compared to previous theoretical studies and simulations. iSAFT has been demonstrated to be capable of modeling BCP from different segregation regimes consistently with a single framework.

β

≤ AexDISP[{ργ (r)}] ≤

1 2

∑ ∑ ∫ ∫ dr1 dr2 udisp(|r2 − r1|)ρα (r1)ρβ (r2)gαβref α

β

(r1, r2)

(A3)

The task to obtain the accurate reference fluid inhomogeneous ref correlation functions gref αβ(r1,r2) and yαβ(r1,r2) is hindered by their complex nature. The form of the correlation function for an inhomogeneous hard sphere fluid is still not tractable after several decades of research. There is even less information to determine the correlation function for inhomogeneous hard chain−hard sphere mixtures, which includes additional complexities arising from the shape and size of the chains. Although the inhomogeneous correlation function of a hard chain fluid is not known, by assuming that gHC αβ (r1,r2) can be approximated as that of a homogeneous fluid at the average density in r1 and r2, the right-hand side approach in inequality A3 has been successfully applied to equation-of-state (EOS)84 and DFT79 studies of polymeric fluids with good agreement compared to experimental phase behavior data. We follow a similar approach and recognize that for BCP systems the total density is fairly uniform49 throughout the domain, as shown in section 3.3. Thus, it is further assumed gHC αβ (r1,r2) could be evaluated equivalently at the total average density

APPENDIX. EXTENSION OF THE THEORY Based on statistical thermodynamics of fluids and early perturbation theories,70,71,83 the dispersion free energy is bounded by

HC HC gαβ (r1, r2) ≈ gαβ ( r ̅ , ρave ) ≡

∑ ∑ ∫ ∫ dr1 dr2 udisp(|r2 − r1|)ρα (r1)ρβ (r2)gαβ (r1, r2) α

α

exp( −uαβ(|r2 − r1|)/kBT )

■

1 2

∑ ∑ ∫ ∫ dr1 dr2 udisp(|r2 − r1|)ρα (r1)ρβ (r2)yαβref (r1, r2)

1 mimj

mi

mj

i

j

∑ ∑ gijHC( r ̅ , ρave )

β

(A4)

≤ AexDISP[{ργ (r)}] 1 ≤ 2

where r ̅ = |r2 − r1| is molecular separation and mi is the number of segments of segment type i. gHC ij is the individual segment− segment correlation function for the hard chain fluid. Equations 3 and 4 are obtained by inserting eq A4 into inequality A3. HC Although gαβ (r,ρ ̅ ave) could be solved analytically for homogeneous hard chain fluids,85,86 the solving process itself is quite expensive computationally. Instead, Monte Carlo simulation results72,73 have been reported for the average correlation function of hard chains with different lengths at various densities, These gHC αβ (r,ρ ̅ ave) values are found to be well represented by quadratic functions in the interaction potential range of interest (extension of the repulsive shoulder). The regressed quadratic functions are used as inputs to eqs 3 and 4. To extend the simulation data over a range of density conditions, it is convenient to average the gHC αβ (r,ρ ̅ ave) values over the range of the square shoulder potential:

∑ ∑ ∫ ∫ dr1 dr2 udisp(|r2 − r1|)ρα (r1)ρβ (r2)gαβref α

β

(r1, r2)

(A1)

gαβ(r1,r2) and gref αβ(r1,r2) are the inhomogeneous pair correlation functions (PCF) for the fluids with full potential and reference potential, respectively. If the reference potential is close to the full potential, i.e., the perturbation is sufficiently small, both sides of the inequality converge to the true free energy value. This condition could be well justified in weak dispersive fluids when reference fluids with only hard repulsions are considered. For BCP/NP composites, the reference fluid is the athermal mixture of hard chains (HC) and hard spheres (HS). With this reference fluid, both sides of the inequality A1 could be reasonable approximations for the dispersion free energy. On the other hand, the pair correlation function of BCP/NP composites with full potential is related to the cavity correlation function (CCF) by

HC gave (ρave ) =


/

g HC(r , ρave ) dr

chain


gαβ (r1, r2) = yαβ (r1, r2) exp( −uαβ(|r2 − r1|)/kBT )

αβ

αβ

dr

chain

(A5)

gref ave(r,ρ ̅ ave)

The volume-averaged values can then be correlated with density and chain length. Examples of these correlations can be found in associated content. The approaches of HC incorporating the fitted gαβ (r,ρ ̅ ave) explicitly and simply applying the volume-averaged function value have been compared for pure BCP systems at various densities. It is shown (in associated content) that the relative difference

(A2)

Following convention in perturbation theory, we approximate the CCF with that of the reference fluid, yαβ = yref. It follows directly from eq A2 that for the chosen reference fluid itself yref = gref outside the hard cores of segments. Then inequality A1 is approximately 6666


Macromolecules

Article

(12) Chiu, J. J.; Kim, B. J.; Kramer, E. J.; Pine, D. J. J. Am. Chem. Soc. 2005, 127 (14), 5036−5037. (13) Kim, B. J.; Chiu, J. J.; Yi, G. R.; Pine, D. J.; Kramer, E. J. Adv. Mater. 2005, 17 (21), 2618−2622. (14) Krishnan, R. S.; Mackay, M. E.; Hawker, C. J.; Van Horn, B. Langmuir 2005, 21 (13), 5770−5776. (15) Lin, Y.; Boker, A.; He, J. B.; Sill, K.; Xiang, H. Q.; Abetz, C.; Li, X. F.; Wang, J.; Emrick, T.; Long, S.; Wang, Q.; Balazs, A.; Russell, T. P. Nature 2005, 434 (7029), 55−59. (16) Lopes, W. A.; Jaeger, H. M. Nature 2001, 414 (6865), 735−738. (17) Yeh, S. W.; Wei, K. H.; Sun, Y. S.; Jeng, U. S.; Liang, K. S. Macromolecules 2005, 38 (15), 6559−6565. (18) Kim, B. J.; Bang, J.; Hawker, C. J.; Chiu, J. J.; Pine, D. J.; Jang, S. G.; Yang, S.-M.; Kramer, E. J. Langmuir 2007, 23 (25), 12693−12703. (19) Lauter-Pasyuk, V.; Lauter, H. J.; Ausserre, D.; Gallot, Y.; Cabuil, V.; Hamdoun, B.; Kornilov, E. I. Physica B 1998, 248, 243−245. (20) Lauter-Pasyuk, V.; Lauter, H. J.; Ausserre, D.; Gallot, Y.; Cabuil, V.; Kornilov, E. I.; Hamdoun, B. Physica B 1997, 241, 1092−1094. (21) Kim, B. J.; Fredrickson, G. H.; Bang, J.; Hawker, C. J.; Kramer, E. J. Macromolecules 2009, 42 (16), 6193−6201. (22) Lo, C.-T.; Lee, B.; Winans, R. E.; Thiyagarajan, P. Macromolecules 2007, 40 (3), 641−647. (23) Schultz, A. J.; Hall, C. K.; Genzer, J. J. Chem. Phys. 2002, 117 (22), 10329−10338. (24) Schultz, A. J.; Hall, C. K.; Genzer, J. Macromolecules 2005, 38 (7), 3007−3016. (25) Chen, H. Y.; Ruckenstein, E. J. Chem. Phys. 2009, 131 (24), 244904. (26) He, L. L.; Zhang, L. X.; Liang, H. J. J. Phys. Chem. B 2008, 112 (14), 4194−4203. (27) Detcheverry, F. A.; Kang, H. M.; Daoulas, K. C.; Muller, M.; Nealey, P. F.; de Pablo, J. J. Macromolecules 2008, 41 (13), 4989− 5001. (28) Huh, J.; Ginzburg, V. V.; Balazs, A. C. Macromolecules 2000, 33 (21), 8085−8096. (29) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29 (4), 1091− 1098. (30) Matsen, M. W.; Thompson, R. B. Macromolecules 2008, 41 (5), 1853−1860. (31) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Science 2001, 292 (5526), 2469−2472. (32) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Macromolecules 2002, 35 (3), 1060−1071. (33) Sides, S. W.; Kim, B. J.; Kramer, E. J.; Fredrickson, G. H. Phys. Rev. Lett. 2006, 96 (25), 250601. (34) Shou, Z. Y.; Buxton, G. A.; Balazs, A. C. Compos. Interfaces 2003, 10 (4−5), 343−368. (35) Buxton, G. A.; Lee, J. Y.; Balazs, A. C. Macromolecules 2003, 36 (25), 9631−9637. (36) Lee, J. Y.; Thompson, R. B.; Jasnow, D.; Balazs, A. C. Faraday Discuss. 2003, 123, 121−131. (37) Wertheim, M. S. J. Stat. Phys. 1984, 35 (1−2), 19−34. (38) Wertheim, M. S. J. Stat. Phys. 1984, 35 (1−2), 35−47. (39) Wertheim, M. S. J. Stat. Phys. 1986, 42 (3−4), 459−76. (40) Wertheim, M. S. J. Stat. Phys. 1986, 42 (3−4), 477−92. (41) Chapman, W. G. PhD Thesis, Cornell University, 1988. (42) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29 (8), 1709−1721. (43) Dominik, A.; Tripathi, S.; Chapman, W. G. Ind. Eng. Chem. Res. 2006, 45 (20), 6785−6792. (44) Yu, Y. X.; Wu, J. Z. J. Chem. Phys. 2002, 117 (5), 2368−2376. (45) Yu, Y. X.; Wu, J. Z. J. Chem. Phys. 2002, 116 (16), 7094−7103. (46) Jain, S.; Dominik, A.; Chapman, W. G. J. Chem. Phys. 2007, 127 (24), 244904. (47) Tripathi, S.; Chapman, W. G. Phys. Rev. Lett. 2005, 94 (8), 087801. (48) Cao, D. P.; Wu, J. Z. Macromolecules 2005, 38 (3), 971−978. (49) Jain, S.; Chapman, W. G. Mol. Phys. 2009, 107 (1), 1−17. (50) Bryk, P. J. Chem. Phys. 2005, 122 (6), 064902.

between these two approaches is less than 1%. The volumeaveraged PCF values gref ave(ρave) are used in iSAFT calculations when no other simulation data are available. The inhomogeneous correlation between hard spheres and hard chain segments is also approximated by the averaged correlation between a particle and each chain segment in a homogeneous reference fluid: HCHS HCHS gαβ (r1, r2) ≈ gαβ ( r ̅ , ρave ) ≡

1 mi

mi

∑ gαβHCHS( r ̅ , ρave ) i

(A6)

ρave) are obtained from separate DFT calculations of polymer distributions around particles in hard chain−hard sphere mixtures with bulk packing fraction equal to the average density of interest. This procedure is documented in previous publications51,74 and will not be discussed here. gHCHS (r,̅ αβ

■

ASSOCIATED CONTENT

S Supporting Information *

Figures S1 and S2 and Table S1. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest. * Corresponding author. Phone: 713-348-4900; fax: 713-3485478; e-mail: [email protected].

■

ACKNOWLEDGMENTS The authors are grateful to Lloyd Lee, Pawel Bryk, and Roland Roth for helpful discussions on FFT. The financial support for this work was provided by the Robert A. Welch Foundation (Grant C1241) and by the National Science Foundation (CBET-0756166). This work was supported in part by the Shared University Grid at Rice funded by NSF under Grant EIA-0216467 and a partnership between Rice University, Sun Microsystems, and Sigma Solutions, Inc.

■

REFERENCES

(1) Balazs, A. C.; Emrick, T.; Russell, T. P. Science 2006, 314 (5802), 1107−1110. (2) Bockstaller, M. R.; Mickiewicz, R. A.; Thomas, E. L. Adv. Mater. 2005, 17 (11), 1331−1349. (3) Haryono, A.; Binder, W. H. Small 2006, 2 (5), 600−611. (4) Feldheim, D. L.; Grabar, K. C.; Natan, M. J.; Mallouk, T. E. J. Am. Chem. Soc. 1996, 118 (32), 7640−7641. (5) Jaramillo, T. F.; Baeck, S. H.; Cuenya, B. R.; McFarland, E. W. J. Am. Chem. Soc. 2003, 125 (24), 7148−7149. (6) Kane, R. S.; Cohen, R. E.; Silbey, R. Chem. Mater. 1999, 11 (1), 90−93. (7) Templin, M.; Franck, A.; DuChesne, A.; Leist, H.; Zhang, Y. M.; Ulrich, R.; Schadler, V.; Wiesner, U. Science 1997, 278 (5344), 1795− 1798. (8) Kang, H.; Detcheverry, F. A.; Mangham, A. N.; Stoykovich, M. P.; Daoulas, K. C.; Hamers, R. J.; Muller, M.; de Pablo, J. J.; Nealey, P. F. Phys. Rev. Lett. 2008, 100 (14), 148303. (9) Ruiz, R.; Kang, H. M.; Detcheverry, F. A.; Dobisz, E.; Kercher, D. S.; Albrecht, T. R.; de Pablo, J. J.; Nealey, P. F. Science 2008, 321 (5891), 936−939. (10) Bockstaller, M.; Kolb, R.; Thomas, E. L. Adv. Mater. 2001, 13 (23), 1783−1786. (11) Bockstaller, M. R.; Lapetnikov, Y.; Margel, S.; Thomas, E. L. J. Am. Chem. Soc. 2003, 125 (18), 5276−5277. 6667


Macromolecules

Article

(51) Bymaster, A.; Jain, S.; Chapman, W. G. J. Chem. Phys. 2008, 128 (16), 164910. (52) Cao, D.; Wu, J. J. Chem. Phys. 2007, 126 (14), 144912. (53) McGarrity, E. S.; Frischknecht, A. L.; Frink, L. J. D.; Mackay, M. E. Phys. Rev. Lett. 2007, 99 (23), 238302. (54) McGarrity, E. S.; Frischknecht, A. L.; Mackay, M. E. J. Chem. Phys. 2008, 128 (15), 159404. (55) Jin, J. Z.; Wu, J. Z. J. Chem. Phys. 2008, 128 (7), 074901. (56) Jin, J. Z.; Wu, J. Z.; Frischknecht, A. L. Macromolecules 2009, 42 (19), 7537−7544. (57) Emborsky, C. P.; Feng, Z.; Cox, K. R.; Chapman, W. G. Fluid Phase Equilib. 2011, 306 (1), 15−30. (58) Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87 (1), 697−705. (59) Leibler, L. Macromolecules 1980, 13 (6), 1602−1617. (60) Lescanec, R. L.; Muthukumar, M. Macromolecules 1993, 26 (15), 3908−3916. (61) Nath, S. K.; McCoy, J. D.; Curro, J. G.; Saunders, R. S. J. Chem. Phys. 1997, 106 (5), 1950−1960. (62) Shull, K. R. Macromolecules 1992, 25 (8), 2122−2133. (63) Chervanyov, A. I.; Balazs, A. C. J. Chem. Phys. 2003, 119 (6), 3529−3534. (64) Melenkevitz, J.; Muthukumar, M. Macromolecules 1991, 24 (14), 4199−4205. (65) Murat, M.; Grest, G. S.; Kremer, K. Macromolecules 1999, 32 (3), 595−609. (66) Segura, C. J.; Chapman, W. G.; Shukla, K. P. Mol. Phys. 1997, 90 (5), 759−771. (67) Chapman, W. G. J. Chem. Phys. 1990, 93 (6), 4299−304. (68) Emborsky, C. P.; Cox, K.R.; Chapman, W. G. J. Chem. Phys. 2011, 135 (8), 084708. (69) Li, Z. D.; Cao, D. P.; Wu, J. Z. J. Chem. Phys. 2005, 122 (17), 174708. (70) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47 (8), 2856− 2861. (71) Zwanzig, R. W. J. Chem. Phys. 1954, 22, 1420−1426. (72) Yethiraj, A.; Hall, C. K. J. Chem. Phys. 1992, 96 (1), 797−807. (73) Yethiraj, A.; Hall, C. K.; Honnell, K. G. J. Chem. Phys. 1990, 93 (6), 4453−4461. (74) Bymaster, A. PhD Thesis, Rice University, 2009. (75) Lee, L.; Bryk, P.; Roth, R. Private communication. (76) Sears, M. P.; Frink, L. J. D. J. Comput. Phys. 2003, 190 (1), 184− 200. (77) Frischknecht, A. L.; Curro, J. G.; Frink, L. J. D. J. Chem. Phys. 2002, 117 (22), 10398−10411. (78) IMSL Math Documentation. (79) Gross, J. J. Chem. Phys. 2009, 131 (20), 204705. (80) Semenov, A. N. Zh. Eksp. Teor. Fiz. 1985, 88 (4), 1242−1256. (81) Ohta, T.; Kawasaki, K. Macromolecules 1986, 19 (10), 2621− 2632. (82) Hashimoto, T.; Shibayama, M.; Kawai, H. Macromolecules 1980, 13 (5), 1237−1247. (83) Hansen, J. P.; McDonald, I. R. Theories of Simple Liquids; Academic Press Inc.: San Diego, CA, 1986. (84) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2001, 40 (4), 1244− 1260. (85) Chiew, Y. C. Mol. Phys. 1991, 73 (2), 359−373. (86) Tang, Y. P.; Lu, B. C. Y. J. Chem. Phys. 1996, 105 (18), 8262− 8265.

6668


Nanoparticle Composites: Extension of

Recommend Documents