Interpreting Neutron Reflectivity Profiles of Diblock Copolymer

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Interpreting Neutron Reflectivity Profiles of Diblock Copolymer Nanocomposite Thin Films Using Hybrid Particle-Field Simulations Jyoti P. Mahalik,†,‡ Jason W. Dugger,† Scott W. Sides,∥ Bobby G. Sumpter,†,‡ Valeria Lauter,*,§ and Rajeev Kumar*,†,‡ †

Center for Nanophase Materials Sciences, ‡Computational Sciences and Engineering Division, and §Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States ∥ Tech-X Corporation, Boulder, Colorado 80303, United States S Supporting Information *

ABSTRACT: Mixtures of block copolymers and nanoparticles (block copolymer nanocomposites) are known to microphase separate into a plethora of microstructures, depending on the composition, length scale, and nature of interactions among its different constituents. Confining these nanocomposites in thin films yields an even larger array of structures, which are not normally observed in the bulk. In contrast to the bulk, exploring various microstructures in thin films by the experimental route remains a challenging task. In this work, we present a modeling scheme using the hybrid particle-field simulation approach based on a coarsegrained model for representing polymer chains by continuous curves and coupling fictitious dynamics of nanoparticles to the thermodynamic forces. The simulation approach is general enough to predict microphase separation in thin films of any block copolymer nanocomposite with the specific details encoded in the interaction parameters. The approach is benchmarked by comparisons with the depth profiles obtained from the neutron reflectivity experiments for symmetric poly(deuterated styrene-b-n-butyl methacrylate) copolymers blended with spherical magnetite nanoparticles covered by hydrogenated poly(styrene) corona. We show that the hybrid particle-field approach is an accurate way to model and extract quantitative information about the physical parameters in the block copolymer nanocomposites. This work benchmarks the application of the hybrid particle-field model to derive the interaction parameters for exploring different microstructures in thin films containing block copolymer nanocomposites.

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INTRODUCTION Blending of organic polymers and inorganic nanoparticles (NPs) can bring out the best of two worlds in the form of mechanical strength as well as optical, electronic, and magnetic properties1−18, without adversely affecting the processing conditions of the polymers. Experimental1−17 and theoretical efforts19−32 have been made to understand the structural behavior of the polymer nanocomposites starting from mid-1990s when the pioneer neutron scattering experiments demonstrated the distribution of the NPs in block copolymer multilayer films.1−4,13 Since block copolymers can spontaneously form lamellar, cylindrical, continuous, or spherical microstructures,6,9,33,34 they have been used for triggering hierarchical self-assembly in the nanocomposites. Block copolymers and nanoparticles interact with each other to yield various morphologies. The presence of nanoparticles can © XXXX American Chemical Society

influence the parent morphology of the block copolymers. The polymers can undergo morphological changes such as transitions from lamellar to bicontinuous,35,36 lamellar to cylindrical,37 or spherical to cylindrical22 depending on the volume fraction of the nanoparticles and details of the block copolymers. In a well-dispersed system, the nanoparticles may prefer to form microphase-separated domains. 19−21,23−27,29,30,38−42 Both enthalpy and entropy of the polymer chains can be manipulated by anchoring short polymer chains on the surface of the nanoparticles.42,43 The theoretical phase diagrams for such systems provide general guidelines for obtaining hierarchical microstructures of diblock copolymer nanocomposites.19,34 Received: January 24, 2018 Revised: March 2, 2018

A

DOI: 10.1021/acs.macromol.8b00180 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules Confining block copolymer nanocomposites in thin films leads to an array of structures that have not been reported for bulk systems. For example, Xu and co-workers15,44,45 have reported an array of hybrid NP assemblies ranging from one-dimensional chains to two-dimensional lattices and three-dimensional networks of NPs. Lauter and co-workers1,2,4 have demonstrated using neutron reflectivity (NR) studies on thin films of an almost symmetric P(deuterated styrene-b-n-butyl methacrylate) (P(dSb-nBMA)) block copolymer that lamellae domains containing primarily deuterated poly(styrene) (dPS) expand to accommodate small magnetite nanoparticles with short hydrogenated poly(styrene) chains anchored on them. However, for a blend of asymmetric P(dS-b-BMA) diblock copolymers with the same nanoparticles, they reported coexistence of cylinder and lamellar morphologies in the thin films.13 Not only the symmetry of the morphology, but also orientation of microphase-separated domains in the confined symmetric block copolymer nanocomposites is influenced by the nanoparticles.46 For example, the lamellar domains may orient either parallel or perpendicular to the substrates depending on the concentration of the nanoparticles in the thin films.47 Furthermore, influence of the geometry of the confinement on the polymer nanocomposite structure has also been investigated.48 In particular, Pan et al.48 reported the effects of structural frustration and loss of conformational entropy of the chains on the morphologies by simulating diblock copolymer nanocomposites confined by two concentric walls. The interplay of various factors influencing morphologies in the thin films of block copolymer nanocomposites makes systematic exploration using experimental techniques quite challenging.49 In this regard, NR with its high depth resolution (∼0.5 nm) is a powerful characterization tool to detect even small structural changes in polymer thin films.4 The interpretation of neutron reflectivity profiles requires construction of scattering length density (SLD) profiles based upon an assumed physical model of the system, which is used to compute reflectivity and the structural parameters are obtained from the best fits to the reflectivity profiles. We should point out that the conventional specular neutron reflectivity data does not have phase information, and a number of models can lead to the same NR profiles. In other words, sometimes more than one model can fit the same data, and additional information is needed to distinguish between the superiority of different models. The phase information can be retrieved in experiments that require a reference layer50 but this is not always possible to perform. Nevertheless, direct comparisons of the predicted NR profiles based on the simulation techniques with the experimental data not only provide a proof of reliability of the underlying model but also allow us to explore a larger parameter space using the simulations following benchmarking and validation of the models. The physical models can be constructed51−53 either using simulation techniques such as the self-consistent field theory (SCFT), molecular dynamics (MD), etc., or in an ad hoc iterative manner respecting constraints such as mass balance while fitting the experimental data. With our intent to explore morphological changes in thin films of diblock copolymers containing spherical nanoparticles, we have developed a general model for the thin films using a hybrid particle-field approach.22,54 Motivations for using the hybrid particle-field approach include our previous work52 using the SCFT for interpreting NR profiles from thin films of polydisperse diblock copolymers and relatively fast equilibration times of the approach in comparison with particle based approaches

such as those based on classical/Newtonian molecular dynamics. Furthermore, relative ease in modeling large scale three-dimensional morphologies resulting from microphase separation in block copolymers using the hybrid particle-field approach makes it an attractive modeling scheme for interpreting and constructing physics based models of relevance to the neutron reflectivity experiments. In the hybrid approach, the polymers are treated using the standard SCFT, and the particle positions are updated based on the underlying Brownian-like dynamics (BD) with the force computed using the local field obtained from the SCFT. Polyswift++55 provides an efficient implementation of the hybrid approach and was used for all the results reported in this work. The model is used to interpret NR profiles measured for the thin films of symmetric P(dS-b-nBMA) diblock copolymers mixed with spherical nanoparticles, as reported in ref 4. Good agreement between the already reported SLD as well as NR profiles and the hybrid particle-field approach was obtained. Our approach provides confidence in the quantitative predictive power of the hybrid approach and allows us to explore morphological changes in the thin films even for asymmetric block copolymers. This paper is organized as follows: in the Methods section the general formalism of the hybrid SCFT-particle model is described, in the Results and Discussion section mapping of parameters and numerical results are presented, and we conclude with the Conclusions section.

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METHODS Hybrid Particle-Field Simulations. We considered a melt of n A−B diblock copolymer (monodisperse and flexible) chains confined between two parallel “substrates”/walls, termed as a polymer thin film. The substrates represent silicon and the air in an experimental setup, described later in this work. Two different cases of the polymer thin films were considered: (a) without nanoparticles and (b) containing np spherical nanoparticles. A local incompressibility constraint was imposed on the system so that at any location the sum of volume fractions of all the components (substrates, polymers, and nanoparticles) adds up to unity. Numerical solutions of the SCFT equations as well as the BD of NPs were simulated using Polyswift++.55 The Hamiltonian of the system, field theory, saddle point approximations, and numerical schemes to obtain the equilibrium volume fraction profiles of the different components (polymer A, polymer B, substrates, and the nanoparticles) are briefly discussed below. A schematic of the system is shown in Figure 1. Short-range pairwise interactions between different components were represented by Flory’s χ parameters. Every A−B polymer chain was represented by a continuous curve of length Nl, where N is the number of Kuhn segments, each of length l. Each chain had NA number of Kuhn segments of type A representing the A-block. We use the notation Rα(s) to represent the position vector for a particular segment, s, along the αth chain.54 The stationary substrates were modeled using the masking method54 and in particular, represented by hyperbolic tangent functions, such that each substrate had a volume fraction approaching unity at either ends of the simulation box and zero in the interior of the thin film, falling sharply over a length scale parametrized by a roughness parameter. The nanoparticles were also represented by hyperbolic tangent functions, such that they had a volume fraction approaching unity within the core volume, and falling sharply to zero over the thickness of its corona. Except for nanoparticle−nanoparticle B


Article

Macromolecules ρ0̂ ⎡⎢ ρp (r) = 1− 2 ⎢⎣ n

∑∫

ρÂ (r) =

α=1 n α=1

and nanoparticle−substrate interactions, all other pairs were modeled to interact by short-range dispersion type interaction, represented by Flory’s χ parameters. Effective forces on nanoparticles (described later) were used to generate candidate update positions. In order to avoid particle−particle and/or particle−wall overlaps, these candidate particle positions were explicitly checked and rejected for excessive overlaps. The Hamiltonian for the thin polymer film containing nanoparticles can be represented as 3 H = 2 kBT 2l + ρ0̂

−1

∑∫ α=1

0

N

⎛ ∂R (s) ⎞2 ds⎜ α ⎟ + ρ0̂ −1 ⎝ ∂s ⎠

∫ dr ∑ ∑

χkk ′ ρk ′(r)ρk̂ (r)

(2)

ρs (r) =

⎛ x − xs ⎞⎤ ρ0̂ ⎡ ⎢1 + tanh⎜ ⎟⎥ 2 ⎢⎣ ⎝ ξs ⎠⎥⎦

(3)

ds δ(r − R α(s))

NA

(6)

⎡

⎤

⎡

∫ ∏ D[R α] exp⎢⎣− kHT ⎥⎦ ∏ δ⎢⎢ρ0̂ − ∑ B

r

⎣

ρk (r)

k=p,s,a

(7)

where the delta function imposes local incompressibility so that the total number density is kept constant at ρ̂0 throughout the space. Standard field-theoretical transformations54 can be applied on eq 7 to obtain a field theory for the system leading to Z=

⎤

⎡

∫ D[ρA ]D[ρB]D[wA]D[wB]D[p] exp⎢⎣− kFT ⎥⎦

(8)

B

so that F = kBT

⎡

⎛

⎣

⎝

∫ dr⎢⎢ρ0̂ −1 ⎜⎜χAB ρA (r)ρB(r) + ∑ ∑

χkk ′ ρk (r)

k = p , s , a k ′= A , B

⎞ ρk ′(r)⎟⎟ − iwA(r)ρA (r) − iwB(r)ρB (r) − ip(r) ⎠ ⎛ ⎞⎤ ⎜ρ ̂ − ∑ ρ (r) − ρ (r) − ρ (r)⎟⎥ − n ln Q [iw , iw ] A B k A B ⎜o ⎟⎥ k=p,s,a ⎝ ⎠⎦ (9)

⎡

Q=

N

⎤ − ρÂ (r) − ρB̂ (r)⎥ ⎥⎦

where kB is the Boltzmann constant and T is the absolute temperature. The first term in eq 1 is the Wiener measure for a flexible diblock copolymer chain,56 assuming the same Kuhn segment lengths for each block (=l). The second and third terms in eq 1 represent interaction energies between different pairs within a Flory-type model, which was parametrized by dimensionless χij for species of kind i and j. ρ̂0 = nN/V is the total number density of polymer segments in a volume V. Subscripts s, a, and p represent the silica substrate, air, and nanoparticles, respectively, used to benchmark the model by direct comparisons with experiments. The substrates, nanoparticles, and the polymer segment density operators are represented as ⎛ x − xa ⎞⎤ ρ0̂ ⎡ ⎢1 − tanh⎜ ⎟⎥ 2 ⎢⎣ ⎝ ξa ⎠⎥⎦

(5)

α=1

(1)

ρa (r) =

ds δ(r − R α(s))

n

Z=

∫ dr χAB ρÂ (r)ρB̂ (r)

k = A , B k ′= s , a , p

(4)

where xk and ξk represent location and roughness, respectively, of the polymer−substrate interfaces for k = a, s. The choice of such functions for the interfaces fixes the origin of the coordinate system at the air interface, and other components are positioned in the positive x direction. The shape of the nanoparticles was set by the functional form of ρp(r), which we chose to be a hyperbolic tangent function containing a length scale Rp, which defines the radii of the particles, and another length scale, ξp, describing the thickness of the diffuse particle−fluid interface.22 The partition function for the film can be written as

Figure 1. Schematic of the system showing different components, mainly substrates, a diblock copolymer chain, and a spherical-hairy nanoparticle. All of the pairwise interactions considered in the model were represented by the Flory’s χ parameters.

n

NA

0

∑∫

ρB̂ (r) =

⎛ |r − rj| − R p ⎞⎤ ⎟⎟⎥ ∑ tanh⎜⎜ ξp ⎝ ⎠⎥⎦ j=1 np

N

∫ D[R α] exp⎢⎣− 23l 2 ∫0 ds

(

∂R α(s) ∂s

2

)

In eq 8, ρk=A,B(r) is the collective density variable and wk=A,B(r) is the conjugate field introduced through the exponential representation of the delta functional δ[ρk=A,B − ρ̂k=A,B]. p is the Lagrange’s multiplier which enforces a local incompressibility constraint. Q is the single chain partition function − i∫

0

⎡

NA

ds wA(R α(s)) − i ∫

N

∫ D[R α] exp⎢⎣− 23l 2 ∫0 ds

C

N

NA

(

∂R α(s) ∂s

⎤ ds wB(R α(s))⎥ ⎦

2⎤

) ⎥⎦

(10)


Article

Macromolecules Q can be expressed54 as Q = V−1∫ dr q(r,N) = V−1∫ dr q(r,0), ̅ where q(r,s) and q(r,s) are the so-called chain propagators that ̅ can be calculated as solutions of the modified diffusion equations subjected to the initial conditions q(r,0) = 1 and q(r,N) = 1. ̅ In particular, q and q̅ satisfy ⎧⎡ l 2 ⎤ ⎪ ⎢ ∇2 − iwA(r)⎥q(r, s), 0 < s < NA ⎦ ∂q(r, s) ⎪ ⎣ 6 =⎨ ∂s ⎤ ⎪⎡ l 2 2 ⎪ ⎢ ∇ − iwB(r)⎥q(r, s), NA < s < N ⎦ ⎩⎣ 6

FPj = −

(11)

⎧⎡ l 2 ⎤ ⎪ ⎢ − ∇2 + iwA(r)⎥q ̅ (r, s), 0 < s < NA ⎦ ∂q ̅ (r, s) ⎪ ⎣ 6 =⎨ ∂s ⎤ ⎪⎡ l 2 2 ⎪ ⎢ − ∇ + iwB(r)⎥q ̅ (r, s), NA < s < N ⎦ ⎩⎣ 6 (12)

respectively. Invoking the saddle point approximation for eq 8 with respect to ρA, ρB, wA, wB, and p yields the following expressions:

∑

χAk Nϕk (r) + η(r) (13)

k=p,s ,a

WB(r) = χAB NϕA(r) +

∑

χBk Nϕk (r) + η(r)

k=p,s ,a NA

ϕA(r) =

1 NQ

∫0

ϕB(r) =

1 NQ

∫N

ds q(r, s)q ̅ (r, s)

(14)

(15)

N

ds q(r, s)q ̅ (r, s)

A

ϕp(r) + ϕs(r) + ϕa(r) + ϕA(r) + ϕB(r) = 1

(16) (17)

where we have used the notation WA(r) = iwA(r)N, WB(r) = iwB(r)N, η(r) = ip(r)N, ϕp(r) = ρp(r)/ρ̂0, ϕs(r) = ρs(r)/ρ̂0, ϕa(r) = ρa(r)/ρ̂0, ϕA(r) = ρA(r)/ρ̂0, and ϕB(r) = ρB(r)/ρ̂0. For the numerical solution of these equations, WA and WB were updated using a steepest descent relaxation algorithm.54 The pressure field η(r) was updated by using eqs 13 and 14 so that η(r) = 0.5(WA(r) + WB(r)) − 0.5χAB N[ϕA(r) + ϕB(r)] − 0.5

∑

Nϕk (r)[χAk + χBk ]

k=p,s ,a

(18)

The pseudo-spectral method was used to solve eqs 11 and 12 in Polyswift++.55 Brownian-like dynamics (BD) was used to update the positions of the nanoparticles22 so that Δrj = DΔt FPj + λj

(20)

The volume fractions of block A, block B, air, silica substrate, and the nanoparticles were computed by eqs 15, 16, 2, 3, and 4, respectively. Experimental System. As a model we chose a well-studied system of thin films containing nearly symmetric poly(deuterated styrene-b-n-butyl methacrylate) (P(dS-b-nBMA)) diblock copolymer with a total molecular weight of 187 kDa and polydispersity of 1.18 prepared by spin-coating on a silica substrate.4 The weight fraction of polystyrene in the diblock copolymer was 0.53. Another thin film of Poly(dS-b-nBMA) was prepared by mixing 7 vol % hairy spherical nanoparticles (average core diameter of 5 nm and hydrogenated polystyrene (hPS) corona thickness of approximately 1 nm) before spin-coating. Atomic force microscopy and neutron reflectivity measurements were conducted on the as-cast thin films as well as thermally annealed (done at 165 °C for 3 h) thin films. The samples with and without the nanoparticles were measured with neutron reflectometry. The specular neutron reflectometry is a technique enabling the measurement of the profile of chemical composition perpendicular to the surface. Since neutrons are highly penetrating and sensitive to changes in the nuclear scattering length densities (SLDs) of a material at the atomic scale, the chemical evolution of buried interfaces can be detected as changes in the SLD depth profile. Considering these facts, neutron reflectometry has the capability to nondestructively probe the composition profile in thin films. The reflectivity was measured as a function of the momentum transfer or the wavevector, q = 4π sin θ/λ, perpendicular to the film plane. The reflectivity depends on the scattering length density (SLD) depth profile, the neutron wavelength (λ), and the incident angle (θ). The SLD is a quantity, which is commonly used in reflectivity measurements and depends on the atomic number density of the material involved in scattering the neutrons. It is given by SLD = ∑ni Nibi, where Ni and bi are the number density and the coherent scattering length of the ith component, respectively, in an ncomponent system. The depth dependence of the SLD profile is obtained by fitting the reflectivity data. Since the numerical calculations represent an equilibrated system, the SLD and the NR profiles of only the thermally annealed thin films were directly compared with the results of the model described below. Computation of Neutron Reflectivity. Structure of the films without and with NPs obtained from the hybrid particlefield simulations and corresponding SLD depth profiles were used for interpreting the neutron reflectivities. Reflectivity curves were calculated and compared with the experimentally obtained reflectivity data using the Motofit reflectivity analysis procedure57 within Igor Pro (Wavemetrics). We used a slab model to simulate reflectivity data from a profile described with a set of sublayers with individual thicknesses and SLDs. Continuous SLD profiles obtained using the hybrid particle-field approach were discretized into 1.37 Å thick sublayers with corresponding SLDi values for each sublayer i, so that the interfaces were described without using any particular roughness function. While the roughness parameters are meaningful for modeling experimental systems, it was not our intention at the present stage to account for specific roughness model functions. An experimental resolution function of dQ/Q = 2.5% was used for the comparisons with the neutron reflectometry experiments.

and

WA(r) = χAB NϕB(r) +

1 ∂F /kBT n ∂rj

(19)

where Δrj is the displacement of particle j located at rj in a time interval Δt, D is the diffusion constant, and λj is a Gaussian random variable. FPj is the force acting on the nanoparticle, computed using D


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Macromolecules

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molar volume for PS to be 105 cm3/mol (denoted as ρ0R−1). All other parameters were also defined with reference to the PS. The degree of polymerization of block A, block B, and the whole chain are denoted as NA′ , NB′ , and N′ = NA′ + NB′ . The effective molar volume of the diblock (denoted as ρ0−1) was evaluated using the mixing rule, N′/ρ0 = N′A/ρ0A + N′B/ρ0B, where ρ0A−1 and ρ0B−1 are the molar volumes of A and B monomers, respectively. Using N = ρ0R/ρ0N′ and NA = ρ0R/ρ0AN′A, N and NA were determined, respectively.61 Using this value of N, χABN is estimated to be between 32 and 34 at 160 °C. In the results presented here, χABN was rounded off to 30, with the intention of representing the experimental annealing temperature of 165 °C. The parameters mapping the experimental system onto a SCFT based model for block copolymers are presented in Table 1.

RESULTS AND DISCUSSION Mapping the Experimental System to the SCFT Model. In order to model the SLD and NR profiles of the experimental system, the parameters for the hybrid particle-field model were obtained from the original experimental reference,4 from other studies reported in the literature, and from fitting to the previously reported SLD profiles. Parameters fixing length scales in the model were the Kuhn segment length (l), average radius of the nanoparticle cores (Rp), thickness of the nanoparticle corona (ξp), polymer film thickness, position (xa) and roughness of the air interface (ξa), position (xs) and roughness (ξs) of the silica substrate, and the simulation box size relative to the radius of gyration of the chains. Parameters characterizing interactions between different components were χABN, χAkN, and χBkN (k = p, s, a). For comparisons with the experiments, A and B represent deuterated styrene and n-butyl methacrylate, respectively. Interactions between the nanoparticle−nanoparticle and nanoparticle−substrate were not explicitly modeled, but BD moves of any nanoparticle that led to overlap with other nanoparticles or substrates were avoided, which is equivalent to repulsive interactions between the respective components. The SLD profile of the polymer thin film in ref 4 without nanoparticles provided for an initial estimation of various parameters. As an increase in the χ parameter leads to an increase in the repulsive interactions, strong segregation of dPS (referred as block A in the hybrid particle-field model) and PnBMA (referred as block B in the model) suggests that χABN ≫ 10.5 (i.e., above the disorder−order transition for the bulk). It is well-known from the literature58 that the blocks of a symmetric diblock copolymer in bulk are miscible when χN is below 10.5, and the segregation stength increases with an increase in χN.58 The limited thickness of the film restricted by the substrate and air interfaces influences the degree of segregation between PS and PBMA, but even in the interior of the thin film, the segregation of the blocks is quite strong (indicated by sharp interface), suggesting that χABN is large. The presence of PnBMA at both the interfaces indicates that χAk > χBk (k = a, s). Narrow polymer−substrate interfacial thicknesses suggest that the roughness parameters, ξs and ξa, are small (of the order of a few Angstroms). The chemical nature as well as the SLD profile of the polymer thin film containing nanoparticles provides guidelines about χpA and χpB. Since the nanoparticles are covered with thin protonated polystyrene layer, it is expected that the nanoparticles prefer the dPS block over the PnBMA block, thus resulting in increase of the dPS domain sizes in the thin film. As NPs have lower SLDs than dPS, the SLD maxima in dPS domains were smaller in the film containing nanoparticles, strongly suggesting that the nanoparticles prefer the dPS domain. The most important parameters of the system are the bulk parameters of the block copolymers. In the absence of substrates and the nanoparticles, one would expect to obtain bulk physical properties of the block copolymers that agree with reported literature values. Bulk PS−PBMA is a well-studied system,59−63 and various groups have reported a wide range of values for the χAB interaction parameter as well as the Kuhn segment lengths. Spiro et al.63 summarized reported literature values of these parameters as well as presented their own conformationally asymmetric SCFT model for the PS−PBMA diblock copolymer. In an effort to reduce the number of parameters, we have considered a conformationally symmetric model in this work and used a numerical scheme presented in ref 63 to determine χABN. In particular, χAB reported in ref 63 is between 0.017 and 0.018 using a reference

Table 1. Parameters for the Hybrid Particle-Field Model Representing Poly(dS-b-nBMA) system

mol wt (g/mol)

N′

molar volume (cm3/mol)

N

dPS (block A) PnBMA (block B) complete block

99110 87890 187000

N′A = 884 N′B = 618 N′ = 1502

ρ0A−1 = 114.5 ρ0B−1 = 159 ρ0−1 = 132.9

NA = 964 NB = 936 N = 1900

A choice of l = 0.77 nm yielded the best fit SLD profile of the polymer film containing no nanoparticles. This value of Kuhn segment length is closer to that reported in ref 59 (l = 0.78 nm for the thin film) than ref 63 (l in the range of 0.65−0.67 nm for a bulk system). A smaller value of l yielded more layers of dPS in the thin film (three layers of dPS were reported for thin film in ref 4), whereas a larger value of l yielded a smaller number of layers of dPS in the thin film. The choice of l was crucial for fixing the length scale of the system. Once l was fitted, all other length parameters were obtained from the original paper and scaled with respect to Rgo = (N/6)0.5l (summarized in Table. 2). Values of the pairwise interaction parameters between different pairs are presented in Table 3. The interactions between the nanoparticles and the polymer blocks were parametrized as follows: since the nanoparticles were anchored with PS (assumed to be equivalent to dPS in terms of interactions), χBpN was set equal to χAB N = 30 and χApN was set to a low value of 0.2. Similarly, the substrates attract the B component; hence, χBk and χAk were chosen to be 3 and 20, respectively, to generate an order of magnitude difference in their values. We have verified that as long as χBk ≪ χAk, their absolute values did not affect the SLD profiles. This is expected based on the SCFT model for the thin films, where χBk − χAk affects the monomer volume fraction profiles near the substrates. Volume Fraction, SLD, and NR Profiles. Volume fraction profiles of the different monomers, substrates, and the nanoparticles obtained from the simulations are shown in Figures 2 and 3. The number of microphase-separated domains of block A and block B matches with that observed in ref 4, and the nanoparticles prefer the block A domain as observed in ref 64. These qualitative comparisons validated our choice of parameters in the underlying SCFT based model for the thin films of diblock copolymers. For the symmetric lamellar forming diblock copolymers, volume fractions of different components can be laterally averaged and presented in the form of one-dimensional profiles for clarity as shown in Figure 4. The obtained profiles show that the nanoparticles preferentially segregated to the middle of the dPS domains. As described in ref 4, the spatial SLD profile was obtained using SLD(x) = SLDdPSϕA(x) + SLDPnBMAϕB(x) + SLDairϕa(x) + E


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Macromolecules

Table 2. Parameters Characterizing Different Length Scales for Poly(dS-b-nBMA) Melts without NPs and Containing 7 vol % NPsa air interface

a

silica substrate

nanoparticle

system

box (Lx × Ly × Lz)

xa

ξa

xs

ξs

Rp

ξp

Np

no NP with 7% NPs

13.92 × 2.32 × 2.32 14.56 × 1.46 × 1.46

1.392 1.456

0.1 0.1

12.528 13.104

0.1 0.1

0.182

0.073

24

All the parameters are scaled with respect to Rgo = (N/6)0.5l = 13.7 nm, where N = 1900 and l = 0.77 nm.

SLDSiϕs(x) + SLDnanoϕp(x), where x is normal to the substrate, SLDdPS = 6.19 × 10−6 Å−2, SLDPnBMA = 0.55 × 10−6 Å−2, SLDair = 0, SLDSi = 2.073 × 10−6 Å−2, and SLDnano was evaluated based on the position with respect to the center of any nanoparticle. If the coordinate position lied within the radius of any nanoparticle, then SLDnano = 6.935 × 10−6 Å−2 was used, which is the SLD of Fe3O4. However, if the position lied in between Rp and RP + ξp, then SLDnano = 1.399 × 10−6 Å−2 was used, which corresponds to the SLD of hPS. For any position outside the core

Table 3. Flory’s interaction Parameters between Different Pairs for the Nanocomposite Thin Film Containing Poly(dS-b-nBMA) pairs

χijN

AB Ap Bp A k (k = s, a) B k (k = s, a)

30.0 0.2 30.0 20.0 3.0

Figure 2. Volume fraction of different components in the thin film without nanoparticles: (a) air interface and SiO2 substrate; (b) block A (dPS); (c) block B (PnBMA), showing three different domains of block A and four distinct domains of block B as observed in the experimental ref 4. The color bar for volume fraction and the coordinate axes are shown for reference.

Figure 3. Volume fraction of different components in the thin film containing nanoparticles: (a) air interface and SiO2 substrate; (b) block A (dPS); (c) 7 vol % hairy nanoparticles; (d) block B (PnBMA). The dPS block expands to accommodate the nanoparticles anchored with short PS chains without affecting the width of the PnBMA domains. The nanoparticles preferentially accumulate at the center of the PS domains to avoid a chain stretching penalty. The color bar for volume fraction and the coordinate axes are shown for reference. F


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Macromolecules

Moreover, by concentrating particles near the center of the compatible domain where the polymer ends are located, the chains can accommodate particles by moving apart rather than by stretching. Localizing particles near the center of the compatible domain thus sacrifices translational entropy of the particles but avoids an even larger chain stretching penalty incurred by distributing particles throughout the domain. NR experiment is an indirect method for determining the distribution of the nanoparticles. Depending on the quality of the experimental data the assumed SLD profile might agree with the NR profile yet miss out on this important detail as captured by the TEM images of ref 64. In contrast, the SLD profile from the model (developed here) was directly obtained from the distribution profile of different components, and the particle distribution directly observed is consistent with the TEM experiments. The model SLD profile was used to calculate the reflectivity profile and compared to the experimental reflectivity data. Very good agreement between the model and experimental reflectivity profiles was found at low wavevector transfer, q (up to 0.05 Å−1 without nanoparticles and up to 0.03 Å−1 with nanoparticles). At higher q values, deviations increased where the experimental data are very sensitive to instrumental resolution and the background and will be taken into account in future development. Our attempts to include such experimental factors in order to improve agreements between the experimental data and the computed neutron reflectivity profiles are presented in the Supporting Information. Figures S1 and S2 in the Supporting Information show that, in practice, experimental factors need to be included to get the best fits to the experimental data by starting from the physics based model obtained using the modeling scheme presented here. Furthermore, a detailed molecular description of the NPs in the simulations may lead to a better agreement at these higher q values. The overall regression coefficients (R2) were 0.9 and 0.85 for without nanoparticles and with nanoparticles, respectively. At higher q values, the model reflectivity showed a number of oscillations whereas the experimental reflectivity showed a relatively flat profile. This qualitative difference is attributed to the choice of a constant resolution function for the model, whereas it was wave-vector dependent for the results presented in ref 4 and experimental factors related to background and interfacial roughnesses were included. In particular, when we used a resolution function and included the experimental factors, the oscillations dampened resulting in a relatively flat curve as shown in the Supporting Information. Inclusion of the experimental effects while constructing model SLD improved the regression coefficients from 0.90 to 0.96 and from 0.85 to 0.90 without nanoparticles and with nanoparticles, respectively (see Figures S1 and S2). However, our intent was not to fit the experimental results but to construct a model and, at the same time, interpret experimental data so that it can be used to explore regions in the phase space. Furthermore, for the modeled films containing nanoparticles, the SLD profiles are presented with error bars from statistics resulting from the Brownian motion of the nanoparticles. In addition to the average value of the SLD, the upper and lower limit of the SLD were also used to compute the model reflectivity profiles (shown as dotted lines in Figure. 6). The reflectivity profiles of these limits agree reasonably well with the experimental reflectivity profile. These results demonstrate that the hybrid particle-field approach provides an accurate way to model and extract quantitative information on physical parameters from nanocomposites of diblock copolymers and nanoparticles. In the future, we plan to use this approach for modeling different microstructures for this particular system at different compositions.

Figure 4. Laterally averaged volume fraction profiles of various components (air interface, SiO2 substrate, dPS, PnBMA, and the nanoparticles) after averaging along y−z directions. (a) Volume fraction profiles without nanoparticles. (b) Volume fraction profiles with nanoparticles. The nanoparticles preferentially segregated to the middle of the dPS domains thus reducing the dPS volume fraction in the middle of the domains.

and cornona of the NPs, SLDnano = 0 was used. It should be noted that dPS used in the experiment4 was not fully deuterated, resulting in the SLDdPS = 6.19 × 10−6 Å−2, which is lower than 6.45 × 10−6 Å−2 for fully deuterated PS.65 In Figure 5, we show that the SLD profiles (solid lines) agree well with the experimental data, both with and without

Figure 5. SLD profiles with and without nanoparticles. The open circle and squares are results from prior experiments while the solid lines are predicted from the hybrid particle-field approach. Small deviations at the center of the PS domains in the model SLD result from preferential segregation of nanoparticles to the center of the dPS domains. Dashed and dotted SLD profiles for the thin film with the NPs result from the fluctuations of the NPs inside the dPS domain.

nanoparticles. However, the SLD profile in the presence of nanoparticles shows small deviations in the dPS domain. The model shows that the nanoparticles tend to segregate toward the center of the dPS domain causing a decrease of the SLD profile in the center of dPS domains, unlike the experimental SLD which stays flat. The segregation of nanoparticles in the center of the domain has been reported for a similar system64 using transmission electron microscopy (TEM). It has been argued64 that the particles coated with a given short homopolymer lower their enthalpy by segregating into the corresponding domain of the block copolymer. G


Macromolecules

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Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00180. Figures S1 and S2 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (V.L.). *E-mail: [email protected] (R.K.). ORCID

Jyoti P. Mahalik: 0000-0003-4448-4126 Jason W. Dugger: 0000-0002-1196-0205 Bobby G. Sumpter: 0000-0001-6341-0355 Valeria Lauter: 0000-0003-0989-6563 Rajeev Kumar: 0000-0001-9494-3488

Figure 6. NR profile of the polymer with or without nanoparticles as a function of wavevector transfer q. There is excellent agreement without nanoparticles (regression coefficient, R2 = 0.9) and good agreement with nanoparticles (R2 = 0.85). Relatively flat reflectivity profiles were observed for the experiment compared to the model reflectivity. These deviations are attributed to a choice of constant resolution function value of 2.5% for the model, whereas for the experiment, the resolution function decreased with q. Dashed and dotted lines correspond to the dashed and the dotted SLD profiles shown in Figure. 5, respectively.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was conducted at the Center for Nanophase Materials Sciences, which is a U.S. Department of Energy Office of Science User Facility. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC0500OR22725. J.P.M. acknowledges support from the Laboratory Directed Research and Development program at ORNL. B.G.S. acknowledges support from the Division of Materials Sciences and Engineering, DOE Office of Basic Energy Sciences.

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CONCLUSIONS This work provides a modeling scheme based on the hybrid particle-field approach for interpretting NR profiles for block copolymer nanocomposite thin films. A model developed with this approach is applied to study microphase separation in thin films containing symmetric poly(deuterated styrene-block-nbutyl methacrylate) diblock copolymers and spherical nanoparticles (magnetite core with hydrogenated polystyrene corona).4 The SLD and NR profiles predicted by the model are successfully tested on the experimental data. Near quantitative agreement with the experimental data not only provides confidence in the predictive power of the hybrid particle-field approach but also sheds light on the details of the microphase separation in the presence of the NPs. For example, nonuniform segregation of NPs in the interior of one of the block copolymer domains is one aspect which can be easily missed in the standard interpretation of the NR profiles. Furthermore, the model developed here for symmetric poly(deuterated styreneblock-n butyl methacrylate) diblock copolymers provides an exciting opportunity to explore the effects of compositional asymmetry on the structure in thin films and to provide a threedimensional description of the structure including the lateral correlations that result in the off-specular scattering and the grazing incidence scattering66,67 in neutron and X-ray scattering experiments. The SCFT included in the hybrid particle-field approach ignores the effects of thermal fluctuations, which have been shown to be important near the disorder−order transition temperature, and can lead to an enhanced roughness of polymeric interfaces in the thin films. As roughness of the interfaces significantly affects the neutron reflectivity, especially at higher wavevectors, a hybrid particle-field approach based on a field theoretic simulation scheme,28 which goes beyond the saddle-point approximation such as the Complex Langevin sampling scheme, should provide a better way of interpreting the experimental data at the cost of longer computational times.

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REFERENCES

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Interpreting Neutron Reflectivity Profiles of Diblock Copolymer

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