Theoretical Insight into Dispersion of Silica Nanoparticles in Polymer

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Theoretical Insight into Dispersion of Silica Nanoparticles in Polymer Melts Zhaoyang Wei, Yaqi Hou, Nanying Ning, Liqun Zhang, Ming Tian, and Jianguo Mi J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b01399 • Publication Date (Web): 02 Jul 2015 Downloaded from http://pubs.acs.org on July 7, 2015

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The Journal of Physical Chemistry

Theoretical Insight into Dispersion of Silica Nanoparticles in Polymer Melts Zhaoyang Wei,a,b Yaqi Hou,a Nanying Ning,a,b Liqun Zhang,a,b Ming Tian,a,b,* and Jianguo Mia,* a

State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing, China b

Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer Materials, Beijing University of Chemical Technology, Beijing, China

Abstract: Silica nanoparticles dispersed in polystyrene (PS), poly(methyl methacrylate) (PMMA), and poly(ethylene oxide) (PEO) melts have been investigated using a density functional approach. The polymers are regarded as coarse-grained semiflexible chains and the segment sizes are represented by their Kuhn lengths. The particle−particle and particle−polymer interactions are calculated with the Hamaker theory to reflect the relationship between particles and polymer melts. The effects of particle volume fraction and size on the particle dispersion have been quantitatively determined to evaluate their dispersion/aggregation behavior in these polymer melts. It is shown that theoretical predictions are generally in good agreement with the corresponding experimental results, providing the reasonable verification of particle dispersion/agglomeration and polymer depletion.

*

Corresponding author. E-mail address: [email protected]. [email protected]. ACS Paragon Plus Environment


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I. Introduction Incorporating nanoparticles into polymer matrixes can dramatically improve the mechanical, optical or electrical properties of these polymer nanocomposites.1−5 The availability of particles of various sizes and diverse chemical properties opens up the possibility of imparting anticipated functionality to composite materials.5−10 Clarifying the mechanisms which govern the properties of polymer nanocomposites is of great interest in order to design new materials with dedicated and controllable excellent properties. In polymer nanocomposites, the size ratio of nanoparticle to polymer monomer is exceedingly large, and polymer chains can contact and entangle with neighbor chains. Therefore, highly asymmetric size and complicated configurations are the main characteristic physical features of the systems. Their structures and properties are generally determined by an intricate balance of entropic and enthalpic forces.11, 12 The enthalpic force comes from surface energy and dispersion force. The entropic contribution, on the other hand, arises mainly from size anisotropy between the polymer segments and particles and from polymer conformation. Previous studies focused mainly on the effect of packing entropy on the interface structure variation, whereas the effect of conformational entropy is also important for particle/polymer systems.13 For example, chain stiffness affects both the range and strength of the attractive depletion force arising from the polymer chains. The combined effects of these forces, which occur at a molecular scale, on behavior at mesoscopic and macroscopic scales are an active area of research with many unknowns. Separating the different contributions from these components in a polymer matrix is difficult through experiment due to their intricate attributes. Molecular simulation has proven to be a useful tool in this regard due to its ability to link molecular level features of the polymer and particle additives to the resulting morphology within the nanocomposite. In order to understand the variations in structure and properties caused by particles at the microscopic level, many advances have been made with molecular simulations to predict the interactions and the structural details in the context of polymer composites.14−18 Compared with experiment, molecular simulation offers a unique approach to identify and separate individual contributions to the phenomenon or process of interest. However, dilute limit of particle volume fraction and the highly asymmetric size are not easily accessible by molecular simulation

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since it is difficult to achieve the small fraction of the large sphere component while maintaining good statistics and computational efficiency. It is also difficult to equilibrate the system with such vastly different length scales, especially for the systems with high-density. As a natural consequence of the computational effort required, particles are rarely more than ten times the diameter of monomers due to equilibration difficulties and computational cost. Another important way is to develop suitable theoretical approaches for the systems. In recent years, the polymer reference interaction site model (PRISM) integral equation theory,19−22 self-consistent field theory (SCFT),23−26 and density functional theory (DFT)27−30 have been widely applied to study the structure and phase behavior in polymer nanocomposites. PRISM theory is a microscopic statistical mechanical approach originally developed to describe polymer solutions, melts, blends and block copolymers. When extended to polymer nanocomposites, the theory can provide a broad exploration of the effective interactions and equilibrium structure for varying particle shapes, polymer architectures and intermolecular interactions. Results from the PRISM theory show qualitative agreement with experiments and simulation results. However, the theory suffers from several major defects. The most notable defect lies in its intrinsic thermodynamic inconsistency, which could leads to inaccuracy and poor predictability. In addition, the theory is lack of freeenergy expression, and the computational performance for structure description depends on the selection of closure equation. SCFT is based on a critical element that the evaluation of the configurational partition function for any segment can be expressed using formally methods in field theory context. DFT is similar to SCFT in spirit in that the starting point for such theories is a model for the free-energy of the system as a functional of the inhomogeneous density field. SCFT can be more computationally efficient than DFT and can provide overall description of the interface structure of particle/polymer blends. However, the theory, which is a coarse−grained representation, cannot connect the model parameters directly to the chemical and morphologic details of polymer nanocomposites. Without consideration of the detailed asymmetrical conformation, such field theory is insufficient to capture the segment-level characteristics of polymer nanocomposites. In contrast, the DFT approach can retain chemical and structural details of polymer matrix, which are important for capturing the depletion effects induced by the liquid-like polymers. ACS Paragon Plus Environment


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In this study, we present a DFT approach which incorporates the contributions of highly asymmetric particle−polymer interactions and polymer configuration, focusing on their microstructures. Unlike previous theoretical investigations,29, 30 we concern real silica nanoparticles dissolved in polystyrene (PS), poly(methyl methacrylate) (PMMA), and poly(ethylene oxide) (PEO) melts, respectively. For these polymer melts, we utilize coarse-graining chains for modeling the conformational behavior of polymer with bending stiffness. Chain stiffness (or flexibility) reflects the dependence of bending energy on the angle between two nearby bond or tangent vectors, which is represented by the Kuhn length of the chain. Meanwhile, we keep the particle size and the effective particle−particle and particle−polymer interactions in theoretical calculations consistent with the experimental ones. As such, we take the parameters from the TraPPE-UA force field31,32 and compare the theoretical predictions directly with the reported experimental results to address the above issues. Accordingly, the effects of particle volume fraction and size on the interfacial structure and properties in polymer nanocomposite melts have been quantitatively evaluated.

II. Theoretical Model We apply the weighted DFT integrated with the interfacial statistical associating fluid theory33 to study the structure and phase behavior in polymer nanocomposites. To simplify the complicated polymer conformation, a coarse-grained method is employed to describe polymer chains. Each polymer chain is represented by N tangentially bonded spheres and obey the semiflexible chain conformation. The sphere diameter is given by Kuhn segment. The theorem of DFT states that there exists a density distribution ρα (r ) of the constituent species α that minimizes the functional Ω [ ρα (r ) ] . It is formulated in the open ensemble, specified by the temperature T , the total volume V , and the chemical potentials of all sites in the system, µα . Considering a uniform polymer−particle mixture around a fix particle β , the grand free-energy is a functional of the site densities ρα (r ) Ω [ ρα (r ) ] = F [ ρα (r ) ] − ∑ ∫ dr  ρα (r ) ( µα − Vα (r ) )  α


(1)

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where F [ ρα (r ) ] is the intrinsic Helmholtz free-energy, and the external field Vα (r ) = U αβ (r ) is the pair potential between species α and the fixed particle β . U αβ (r ) is given by the Hamaker theory34

 r 2 − (α1 + α 2 ) 2   2α1α 2 2α1α 2 A Uαβ (r ) = −  2 + + ln  2   r − (α − α ) 2   6  r − (α1 + α 2 ) 2 r 2 − (α1 − α 2 )2 1 2     2 2 2 2  r − 7r (α1 + α 2 ) + 6 (α1 + 7α1α 2 + α 2 ) r + 7r (α1 + α 2 ) + 6 (α12 + 7α1α 2 + α 22 )    + 7 7 6 r − − r + + α α α α   ( ) ( ) Aσ LJ 1 2 1 2 +  2  37800r  r + 7r (α1 − α 2 ) + 6 (α12 − 7α1α 2 + α 22 ) r 2 − 7r (α1 − α 2 ) + 6 (α12 − 7α1α 2 + α 22 )  − −  7 7 + − r α α ( ) ( r − α1 + α 2 ) 1 2  

(2)

6 in which A = 4π 2ε LJ ρ1 ρ 2σ LJ is the Hamaker constant, α1 and α 2 are the radii of polymeric segment and

particle, r is the distance between their centers, ε LJ and σ LJ are the Lennard−Jones (LJ) energy and distance cross-interaction parameters for the pair of interaction sites, ρ is the density of interaction sites in the macroscopic body. The detailed calculation method for Hamaker constant can be seen elsewhere.16 Silica particle contributes only the interaction of its oxygen atoms35 with the polymeric matrix. The LJ parameters for C, CH, CH2, CH3, and O sites in PS, PMMA and PEO are taken from the TraPPE-UA force field,31, 32 and listed in Table 1. As a result, the Hamaker constants for silica particle, PS, PMMA, and PEO are 6.43×10-20, 5.84×10-20, 4.34×10-20, and 1.31×10-19 J, respectively. The equilibrium Helmholtz free-energy functional for particle-polymer mixture can be generalized as

F [ ρα (r )] = kBT ∑ ∫ drρα (r ) ln ( ρα (r )Λα 3 ) − 1 + F hs [ ρα (r )] + F att [ ρα (r )] +F chain [ ρα (r )] +F stiff [ ρα (r )] (3) α

where the terms on the right-hand side represent the intrinsic Helmholtz free-energies of ideal gas, hard-sphere repulsion, dispersive attraction, chain constraint, and chain stiffness, respectively. Here kB is the Boltzmann constant, and Λα is the thermal wavelength. The contribution of hard-sphere repulsion is given by the fundamental measure theory36

F hs [ ρα (r )] = k BT ∫ dr Φ hs [nγ (r )]

(4)



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and Φ hs  nγ ( r )  is the Helmholtz free-energy density which stems from the modified fundamental theory37 including both the scalar and vector contributions 3  n1n2 − nV1 ⋅ nV2 1  n32  n2 − 3n2nV2 ⋅ nV2  Φ  nγ ( r )  = −n0 ln (1 − n3 ) + +  n3 ln (1 − n3 ) +   1 − n3 36π  n33 (1 − n3 )2    hs

(5)

where nγ (r ) and ϖ α(γ ) ( r ) with γ = 0,1, 2, 3, V1 , V2 are the weighted densities and the weight functions, respectively. The details have been given elsewhere.37 The attractive contribution to the free-energy functional can be simplified as F att [ ρ α ( r ) ] =

1 ∑ ∑ d r ′ ∫ d r ′′ρ α ( r ′) ρ α ′ (r ′′)uααatt ′ ( r ′-r ′′ ) 2 α α′ ∫

(6)

att where uαα ′ (r ) is the attractive interaction potential between any two spheres of species α and α ′ . The

dispersion interactions of polymer−polymer are weak, which can be approximately treated with purely hardcore repulsion. To compute the free-energy contribution due to the chain connectivity, we use the Tripathi−Chapman functional,31 which is based on the Wertheim’s thermodynamic perturbation theory.38, 39 The chains are treated as a sequence of N tangentially bonded segments enforced by giving each segment a label and allowing segments to exclusively bond to their specific matching segments. It is indicated as F chain [ ρ α ( r ) ] =

k BT 2

{α }  N   δ ( r ′ − r ′′ − σ αα ′ ) ′ ′  1 − ln  ∫ d r ′′ d r ( r ) yαα ′ ( r ′, r ′′) ({ξ } ) ρ α ′ ( r ′′)   ρ ∑ α 2 ∫ ∑  4πσ αα ′ α =1 α′     

(7)

where σ αα ' = (σ α + σ α ' ) / 2 is the interaction diameter of two species, the notation {α } on the innermost summation of this functional indicates this sum is performed over all segments α ′ , which are connected to the segment α (e.g., segments α − 1 and α + 1 for a middle segment in a linear chain). yαα ' (r ', r ") is the cavity correlation function between the two segments α and α ' in consideration of the total particle and polymer packing.33 The functional F stiff [ ρ ] is constructed to represent the contribution of configurational entropy given by the polymer stiffness, which is implemented to improve the accuracy for description of polymer chains ACS Paragon Plus Environment

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F stiff [ ρ α ( r ) ] = −


1 ∑ ∑ d r ′ ∫ d r ′′ρ α (r ′) ρ γ ( r ′′) cαγstiff ( r ′-r ′′ ) 2 α γ ∫

(8)

with the approximation c stiff ( r ) = c semiflexible ( r ) − c flexible ( r ) . Here the subscripts α and γ represent merely the species of polymer segments, and csemiflexible ( r ) and c flexible ( r ) denote the direct correlation functions of semiflexible and flexible polymer chains. They are calculated from the polymer reference interaction site model integral equation40

r r r r r r h(r) = ∫ dr ' ∫ dr ''ω(| r − r '|)c(| r '− r ''|)[ω(r '') + ρh(r '')]

(9)

where h(r ) is the total correlation function, and ω (r ) is the intramolecular correlation function. Here we use the Koyama model41 to represent the linear chains with flexible or semiflexible structure. the polymer physics

parameters for ω (r ) calculation are summarized in Table 2. To solve the equation, the Percus−Yevick approximation46 is adopted. The details have been given elsewhere.47 Minimization of the grand potential from eq 1 with respect to the density profiles of each component produces a set of Euler−Lagrange equations

  δ ( F hs [ ρα (r ) ] + F att [ ρα (r ) ] + F chain [ ρα (r ) ] + F stiff [ ρα (r ) ]) − Vα (r )  ρα (r ) = exp  µα −   δρα (r )  

(10)

In order to account for the asymmetric sizes and interactions between particles and polymer chains, we employ δ F [ ρα (r ) ] / δρα ( r ) instead of ρα ( r ) for the iteration convergence. Therefore, the above equation can be rewritten as

δ ( F [ ρα (r )]) = µα − Vα (r ) δρα (r )

(11)

An ordinary Picard iteration scheme is used in the process for solving the equations. The procedure is repeated until the average fractional difference over any grid point between the old and the new

δ F [ ρα (r ) ] / δρα ( r ) is less than 1.0 × 10 −4 . The structure factor S n (q ) provides an experimental link to the particle microstructure as it is the Fourier transform of the pair correlation function48 ACS Paragon Plus Environment

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S n (q ) = 1 + 4πρ n ∫ ( ρ n (r ) ρ n − 1)r 2 0

sin(qr ) dr qr

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(12)

Here ρ n is the particle number density with ρ n = 3φn 4π R 3 , in which R is the particle radius, φn the volume fraction, and q the scattering vector. ρ n (r ) varies as a function of distance between particles, making it a critical link to particle microstructure. The pair correlation function asymptotes to ρ n (r ) ρ n = 0 when r < 2 R as particles cannot interpenetrate and ρ n (r ) ρ n ≈ 1 as r → ∞ as the likelihood of finding a particle becomes proportional to the average particle density.

III. Results and Discussions Before application, the present theoretic model needs to be verified. An effective way to test theory is to compare its calculation results with simulation data, which can in principle represent the behavior of a system. Here we focus on the examination of the chain stiffness contribution, since other terms in the free-energy functional has been well addressed. For simplicity, we use molecular dynamics simulation method to provide the intuitive density distributions of nanoparticle/polymer blends confined in two parallel hard walls. Particles and polymer beads have the same mass m . A polymer chain contains 100 beads with diameter σ p . The diameter of spherical particles is set to σ n = 10σ p , and the total packing fraction is η = π ( ρ nσ n3 + ρ pσ p3 ) 6 = 0.3 ,

ρ n and ρ p are the number densities of particles and polymer beads, respectively. The particle volume fraction φn = ρ nσ n3 ( ρ nσ n3 + ρ pσ p3 ) = 10% . The simulation contains 5150 chains and 56 particles. The polymers are modeled as bead-spring chains using the finite extensible nonlinear elastic (FENE) potential49

EFENE

  r 2   σ p 12  σ p 6  = −0.5 K F R ln 1 −    + 4ε   −    + ε R r  r       0   2 0

(13)

where K F is the spring energy, R0 is the maximum extension between two consecutive monomers, ε is the interaction strength for the monomer interactions, respectively. We set K F = 30ε / σ p2 , R0 = 1.5σ p . The


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semiflexible chain model41 is employed to implement the stiffness of the chains with a bending potential E = ε stiff (1 + cos θ ) , where ε stiff is the bending energy, and θ is the angel between two consecutive bonds. We

focus on the comparison of flexible chain with ε stiff = 0 and semiflexible chain with a moderately strength of

ε stiff = 5.0 . The pairwise interactions between all the particles are described by a shifted LJ potential  σ 12  σ 6 1  ELJ,ij ( r ) = 4ε ij  p  −  p  +  , r < rc + ∆ij  r − ∆ij   r − ∆ij  4   

(14)

where the well depth for the site interaction is ε ij = 1.0ε , and the shift parameters are ∆ pp = 0 , ∆ nn = σ n − σ p , and ∆ np = (σ n − σ p ) / 2 for polymer−polymer, particle−particle and particle−polymer interactions, respectively. 1

All the interactions between the particles are repulsive, thus the cutoff distance is rc = 2 6 σ p . All simulations are performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) in a three-dimensional cubic box with side lengths Lx = Ly = Lz = 100σ and periodic boundary conditions along the x and y directions. The reduced units of all quantities are adopted. σ and ε are the basic length and energy scales, respectively. The reduced temperature T * is defined as T * = kBT / ε . Two smooth walls are placed perpendicular to the z-axis on either side of the box. To generate the initial configurations, we place the polymer chains in a large box, and obtain a system of low density adopting canonical (NVT) ensemble. Then, the isothermal–isobaric (NPT) simulation is used to compress the system of low density to the target density. The simulations are continued in the NVT ensemble. The structure is equilibrated over a long time to make sure that each chain has moved at least 2 Rg . The blends are firstly equilibrated for 3×109 MD steps with a time step of ∆t = 0.001τ where the dimensionless time unit is τ = m / ε . The data for post-processing are collected from after equilibrium runs of an additional 107 MD steps. Figure 1 presents the density profiles of particles and polymer chains at the end of equilibration runs for two different systems. In Figure 1a, one sees that the aggregations of particles mediated by the flexible and semiflexible polymers are different. As the chain stiffness increases, the particle concentration drops in the ACS Paragon Plus Environment


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vicinity of the surface as evidenced by the height of the first peak. Compared with the flexible chain, the semiflexible chain provides enlarged configurational entropy loss and more significantly elevated packing entropy. The reduced particle aggregation in semilflexible polymer matrix can be explained in terms of a competition between the two entropies. On the other hand, the polymer packing displays a reversal tendency, which is shown in Figure 1b. As the configurational entropy loss of the polymer chains increases, the beads are more likely to be away from the wall, whereas the packing effect plays a dominate role in the chain accumulation, leading to the enhanced polymer packing. For comparison, the corresponding theoretical predictions are also presented in Figure 1. It is show that the theoretical predictions are generally in accordance with the simulation data for polymer nanocomposites with different chain stiffness. The results indicate that the current theory is quantitatively reliable for describing the dispersion/aggregation behavior of particles in polymer melts. Validated by molecular dynamics simulation, the theoretical model is then extended to investigate different polymer nanocomposites. At the beginning, we consider the effect of volume fraction of silica nanoparticles on the dispersion/aggregation behavior in PS melt. Three different particle volume fractions are considered with

φn = 5% , 10%, and 20% for different three systems. The calculated results are shown in Figure 2. The total packing factor is 0.5, and the particle size is 28.8 nm. In Figure 2a, particles display multilayer adsorption around the central particle because of the self-aggregation tendency arising from the depletion effect. At different volume fractions, the three systems share a common shape but have quantitative differences. Such density fluctuations has also been predicted by the integral equation theory.19 There are significant contact peaks at r = σ n , which quickly decay to below the random value of unity when the surface-to-surface separation is only ~ 0.05σ p . The contact values are extremely large: ρ n (r = σ n ) ρ n ~ 501 , 601, and 637 at

φn = 5% , 10%, and 20%, respectively. As the volume fraction increases, the density fluctuation increases. Meanwhile, the distributions of the polymer layers are coherently related to the particle distributions. In Figure 2b, the adsorbed polymer chains accumulate on the particle surface to form the multilayer adsorption, which is attributed to the strong segment−particle attractions. As the particle volume fraction increases, the absorbed ACS Paragon Plus Environment

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polymer chains can be pushed away due to the particle occupation. Nevertheless, the reduction effect is unobvious in each adsorption layer. Figure 3 displays the calculated structure factors of particle aggregation with various particle volume fractions. These results are compared with the corresponding experiment data.50 It is shown that, the different and complicated structure factors vary with increasing wavevector. The first layer of peaks corresponds to the particle cluster. The system with 5% loading shows cluster effect. As the loading increases, the effect increases and moves to the low- qσ n region, indicating that the population in this phase is enhanced. The peaks for particle direct contact occur at q ~ 0.0288 Å-1 ( qσ n = 8.27 ). These peaks quantify the local ordering of the “cage” of particles surrounding a tagged particle. The sharpening of the cage peak and its shift to lower wavevector with increasing φn indicate that particles are becoming more ordered and closer together. As the wavevector increases, the intermediate scaling regime emerges. The intermediate scaling regime is of particular importance, as it can characterize the local packing features which can be represented partially by the particle−particle and particle−polymer interactions, as well as the contribution of chain conformation. One can see from the figure that although some discrepancies in the intensity of the peaks and valleys are perceivable, the overall shape of theoretical curves and the position of the main peaks can match the experimental results. The deviations are probably due to the approximation of the model, in which the tacticity or torsional angle effects are neglected. Figure 3 validates that the current theoretical model is suitable to quantitatively evaluate particle dispersion in polymer melts. According to the distribution function of particles, we compute the second virial coefficient B2 through the relation ∞ 2 B2 = πσ n3 + 2π ∫ [1 − ρ n (r ) ρ n ]r 2 dr σn 3

(15)

where the first term is the particle contribution, and the second one is the polymer-mediated contribution. Positive value of B2 indicates stable particle dispersion (effective particle−particle repulsion), while negative value signifies unstable dispersion. Figure 4 shows that, as particle volume fraction increases, the second virial ACS Paragon Plus Environment


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coefficient decreases monotonically over the range studied. This declining tendency corresponds to the improving attractive effect, indicating that the short-range attraction controls the overall destabilization. In order to make clear the effect of highly asymmetric structure on the particle dispersion/aggregation behavior, we consider silica particles with lager diameters dissolved in the same PS melt. The density profiles in four polymer nanocomposites are shown in Figure 5 with different particle sizes (10.8 nm , 18.0 nm , 27.0 nm , and 36.0 nm ) at φn = 5% . The total packing factor is fixed at 0.5. Similarly to Figure 1a, the contact values of particles at r = σ n are extremely large, which can be seen in Figure 5a. As the particle size increases, the oscillation behavior becomes more unobvious, and particles are more disordered. At constant particle volume fraction, the decrease of particle size can improve the specific surface, resulting in more polymer segments adsorbed on the surface and then squeeze out other particles. In Figure 5b, the polymer chains form a monolayer adsorption around the central particle, and the polymer packing decreases with increasing particle size. Figure 6 summarizes the structure factors of particles with various sizes. At given volume fraction φn = 5%, although there are few particles in contact with each another, the overall distribution becomes more homogeneous as the particle size increases. The peak in the low- q scattering corresponds to cluster size. We conclude that the cluster diameter is 3024 Å (roughly twenty-eight particle diameters) at small particle size ( σ n = 10.8nm ). If the particle size increases to 18 nm, the cluster diameter declines to 1685 Å (about nine particle diameters). Further increasing the particle size to 27 nm leads to a more smaller cluster (1210 Å, about four particle diameters). Once the particle size increases up to 36 nm, the cluster could vanishes. This variation tendency agrees with the experiment observation.51 As the particle size increases, the cage peak shifts to higher wavevector, suggesting that the particles distribution tends to homogeneous state. Figure 7 shows the second virial coefficients for different particle sizes with constant particle volume fraction

φn = 5% . The results are in agreement with our previous conclusion: the tendency to dispersion increases as the particle size increases.15 One can see that B2 is independent of the particle size when it is larger than a certain value. The effect of particle size on the pair correlation function becomes insignificant once the size reaches a threshold value. Before the threshold value, B2 increases but the increasing amplitude declines as the ACS Paragon Plus Environment

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particle size increases. It confirms that large particles are more accessible to achieve stable dispersion than small particles. Figure 8 illustrates the effect of the material-specific variables of polymer on particle distribution. The total packing factor and particle volume fraction are fixed at 0.5 and 10%, respectively. In Figure 8a, silica particles dissolved in PS and PMMA display significant contact peaks at only one location of r = σ n , but two evident peaks emerge at r = σ n + σ PEO and r = σ n + 2σ PEO in PEO melt. The three locations represent three different stable states of particle organization: (a) direct contact, (b) one polymer layer between particles, and (c) two polymer layers between particles. Due to the strong depletion effects of PS and PMMA, the contact values in the two nanocomposites are extremely large, correlating to particle aggregation. In contrast, PEO can produce good particle dispersion because it is a highly active surface molecule, which is able to strongly adhere on both hydrophilic and hydrophobic surfaces. In the particle−PEO system, the particles form multilayer peaks, and polymer segments tend to be adsorbed on the particle surface and then squeeze out some other particles. Accordingly, it is unfavorable to particle aggregation. In Figure 8b, PEO exhibits density enhancement at the particle surface, which is mainly attributed to the strong particle−PEO attractions. In contrast, PS and PMMA produce relatively weak attraction. Figure 9 illustrates these different interaction strengths to account for the different dispersions of particles in PS, PMMA and PEO. When the distance between polymer and silica particle is smaller than 0.6 σ n , the attraction becomes effective. In contrast, PEO exhibits relatively higher effective attraction and longer ranges compared to PMMA and PS. The structure factors of silica particles dissolved in PS , PMMA, and PEO are displayed in Figure 10. Similarly to the density distribution curves, there are clear peaks in the low- q scattering in PS and PMMA melts, corresponding to the close clusters. Particles in PEO, however, have only the hard-sphere peak ( qσ n = 7.55 ) , which shifts towards larger wavevector. This implies that PEO can give stable particle dispersion. The second virial coefficients of particles in PS, PMMA and PEO are -3.14, -1.85, and -0.48, respectively. Obviously, the dispersion tendency of silica particles in these three polymer melts is PEO > PMMA > PS. The reason is that PEO has the strongest attraction to particles, nevertheless it has the most flexible configuration. ACS Paragon Plus Environment


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On the other hand, the conformation of PMMA and PS chains are similar, whereas the enthalpic contribution given by the attractive force of PS is slightly weaker than that of PMMA. In fact, the enthalpic contribution plays a main role in determining nanoparticle dispersion/aggregation.

IV. Conclusion In summary, a DFT approach has been presented and validated to analyze the microstructure of nanopaticles dissolved in different polymer melts. The dispersion/aggregation behavior of particles has been largely influenced by the particle size and volume fraction, as well as the polymer conformation at the equilibrium states. It is shown that particle size plays the key role in the dispersion/aggregation behavior. If the size deceases, their aggregation could be enhanced exponentially. Meanwhile, the particle volume fraction also affects the aggregation extent. Larger volume fraction is conductive to higher packing, and the variation tendency is approximately linear. Finally, the chemical characteristics and chain morphology affect the particle dispersion through the enthalpic contribution and depletion effect. Apart from structure description, the current theory can also provide the reliable interfacial energy, which is particularly important for the ongoing activities in analyzing the thermodynamic equilibrium of entropy and enthalpy, and the dynamic nonequilibrium during the flow of particles and polymer chains. As a result, this work provides a reliable method for studying the dispersion/aggregation behavior of nanoparticles in polymer melts that may find wide applications in the field of material technology.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 21276010 and 21476007), and by Chemcloudcomputing of Beijing University of Chemical Technology.


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Table 1 United-atom force field parameters31,32 Site

σ (Å)

ε / kB (K)

C(carbonyl)

3.82

40

C

3.85

22

CH(aliphatic)

3.73

47

CH(aromatic)

3.695

50.5

CH2

3.95

46

CH3

3.75

98

O(ether)

2.80

55

O(carbonyl)

3.05

79

O(in silica )

3.0

230


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Table 2 Kuhn lengths and bond angles for different polymer melts Polymer

Kuhn length (nm)

Bond angle (deg)

PS

1.8

140a

PMMA

1.7

122b

PEO

1.1

130c

Data for Kuhn length are taken from Ref 42. a Reference 43. b Reference 44. c Reference 45.



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Figure Captions Figure 1 Density profiles of nanoparticles (a) and polymer chains (b) at solid surface with N = 100 , η = 0.3 , and σ n = 10σ p . The theoretical predictions are compared with simulation data.

Figure 2 Effect of particle volume fraction on the density distributions of silica particles (a) and PS chains (b) around a central silica particle with N = 100 , η = 0.5 , σ n = 28.8nm , and σ p = 1.8nm . To enhance visual clarity, the profiles at φn = 10% and 20% are shifted rightward by 0.25 and 0.5, respectively.

Figure 3 Effect of silica particle volume fraction on the structure factor. The specific parameters are the same as Figure 2. Full lines are the theoretical calculations, and dotted lines are the experiment data.

Figure 4 Second virial coefficients for silica nanoparticles dissolved in PS melt with different silica particle volume fractions. The specific parameters are the same as Figure 2.

Figure 5 Effect silica particle size on the density distributions of particles (a) and PS chains (b) around a central particle with N = 100 , η = 0.5 , and σ p = 1.8nm , and φn = 5% . To enhance visual clarity, the profiles of 18 nm, 27 nm, and 36 nm are shifted rightward by 0.25, 0.5, and 0.75, respectively.

Figure 6 Effect of silica particle size on the particle structure factor. The specific parameters are the same as Figure 5.

Figure 7 Second virial coefficients for different silica particle size. The specific parameters are the same as Figure 5.

Figure 8 The density distributions of silica particles (a) and polymer chains (b) around a central silica particle corresponding to different polymer species with N = 100 , η = 0.5 , σ n = 28.8nm , and φn = 10% . To enhance visual clarity, the profiles of PMMA and PEO are shifted rightward by 0.25 and 0.5, respectively.

Figure 9 The interaction potential between silica and PS, PMMA and PEO.


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Figure 10 Effect of polymer species on the silica nanoparticle structure factor. The specific parameters are the same as Figure 8.



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Figure 1


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Figure 2



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Figure 3


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Figure 4



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Figure 5


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Figure 6



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Figure 7


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Figure 8



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Figure 9


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Figure 10



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For Table of Contents Use Only

Theoretical Insight into Dispersion of Silica Nanoparticles in Polymer Melts Zhaoyang Wei,a,b Yaqi Hou,a Nanying Ning,a,b Liqun Zhang,a,b Ming Tian,a,b,† and Jianguo Mia,*


Theoretical Insight into Dispersion of Silica Nanoparticles in Polymer

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