Characterization of Polymer-Silica Nanocomposite Particles with Core

ARTICLE pubs.acs.org/Langmuir

Characterization of Polymer-Silica Nanocomposite Particles with Core
Shell Morphologies using Monte Carlo Simulations and Small Angle X-ray Scattering Jennifer A. Balmer, Oleksandr O. Mykhaylyk,* Andreas Schmid, Steven P. Armes,* J. Patrick A. Fairclough, and Anthony J. Ryan Dainton Building, Department of Chemistry, The University of Sheffield, Sheffield, Brook Hill, S3 7HF, United Kingdom

bS Supporting Information ABSTRACT: A two-population model based on standard smallangle X-ray scattering (SAXS) equations is verified for the analysis of core
shell structures comprising spherical colloidal particles with particulate shells. First, Monte Carlo simulations of core
shell structures are performed to demonstrate the applicability of the model. Three possible shell packings are considered: ordered silica shells due to either charge-dependent repulsive or size-dependent Lennard-Jones interactions or randomly arranged silica particles. In most cases, the two-population model produces an excellent fit to calculated SAXS patterns for the simulated core
shell structures, together with a good correlation between the fitting parameters and structural parameters used for the simulation. The limits of application are discussed, and then, this two-population model is applied to the analysis of well-defined core
shell vinyl polymer/silica nanocomposite particles, where the shell comprises a monolayer of spherical silica nanoparticles. Comprehensive SAXS analysis of a series of poly(styrene-co-n-butyl acrylate)/silica colloidal nanocomposite particles (prepared by the in situ emulsion copolymerization of styrene and n-butyl acrylate in the presence of a glycerol-functionalized silica sol) allows the overall core
shell particle diameter, the copolymer latex core diameter and polydispersity, the mean silica shell thickness, the mean silica diameter and polydispersity, the volume fractions of the two components, the silica packing density, and the silica shell structure to be obtained. These experimental SAXS results are consistent with electron microscopy, dynamic light scattering, thermogravimetry, helium pycnometry, and BET surface area studies. The high electron density contrast between the (co)polymer and the silica components, together with the relatively low polydispersity of these core
shell nanocomposite particles, makes SAXS ideally suited for the characterization of this system. Moreover, these results can be generalized for other types of core
shell colloidal particles.

’ INTRODUCTION Colloidal polymer/silica nanocomposite particles are of considerable academic and industrial interest.1
3 Such organic/ inorganic hybrid particles have numerous potential applications, including photonic devices,4 synthetic mimics for cosmic dust,5 smart Pickering emulsifiers,6 and high performance exterior architectural coatings.7 Following initial work by Barthet et al. in 1999,8 various robust synthetic routes to vinyl polymer-silica nanocomposites have been developed over the past decade or so. In many early nanocomposite syntheses, relatively low silica incorporation efficiencies were obtained.9
11 This is problematic, since it is known that the excess (nonaggregated) silica sol can compromise the performance of nanocomposite particles when utilized as Pickering emulsifiers12 or for tough, transparent coatings.13 More recently, an improved synthesis protocol has been reported, whereby silica incorporation efficiencies can exceed 95%.13
15 Thus, both polystyrene/silica (PS/silica) and poly(styrene-stat-n-butyl acrylate)/silica {P(S-BuA)/silica} nanocomposite particles were prepared by the in situ emulsion r 2011 American Chemical Society

polymerization of monomer(s) in the presence of a glycerolmodified ultrafine silica sol using a cationic azo initiator. Welldefined core
shell nanocomposite particles of low polydispersity were obtained without the need for added surfactant, auxiliary comonomer, or nonaqueous cosolvent. It was also found that a range of nanocomposite particle diameters and silica contents could be obtained by systematically varying the styrene/n-butyl acrylate comonomer feed ratio. Many techniques have been used to characterize the particle size, morphology, and composition of these core
shell nanocomposite particles, including dynamic light scattering (DLS), disk centrifuge photosedimentometry, transmission electron microscopy (TEM), thermogravimetry analysis (TGA), aqueous electrophoresis, X-ray photoelectron spectroscopy, electron energy loss spectroscopy, and solid-state NMR spectroscopy.13
16 Received: April 11, 2011 Revised: May 25, 2011 Published: June 10, 2011 8075

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Figure 1. Schematic representation of the synthesis of nanocomposite particles by aqueous emulsion copolymerization of styrene with n-butyl acrylate using a cationic azo initiator (AIBA) in the presence of a glycerol-modified aqueous silica sol at 60
C.

In principle, small-angle X-ray scattering (SAXS) offers robust characterization of both nanocomposite particle morphology and composition. As SAXS is an in situ analysis technique, the true nanocomposite particle morphology of the aqueous dispersion can be studied, as opposed to dried nanocomposite powders or films. Since silica has a significantly higher electron density than most vinyl polymers, the overall X-ray scattering is generally dominated by the inorganic component, which means that SAXS is particularly sensitive to the spatial location of the silica within the nanocomposite particles. Moreover, SAXS is much more likely to be statistically meaningful since the scattering is averaged over millions of particles, whereas only a much smaller population (typically a few hundred particles at best) is sampled by electron microscopy-based techniques. SAXS characterization of colloidal core
shell particles has been reported previously. For example, SAXS has been utilized to interrogate core
shell particles where both the core and the shell comprise polymer and the shell is a uniform, homogeneous layer.17
19 Bolze et al.18 investigated the chemical composition of an industrially relevant copolymer latex, whereby the core predominantly consisted of a poly(styrene-co-butadiene) copolymer and the shell was found to be rich in poly(acrylic acid). Similarly, Mykhaylyk et al.19 found good agreement between SAXS and atomic force microscopy studies of latex particles comprising a hard poly(methyl methacrylate) core and a relatively soft polyurethane shell. There are also examples of the SAXS analysis of organic
inorganic core
shell particles. Yuan and co-workers20 prepared 35 nm core
shell nanocomposite particles by confining silica deposition within the cationic coronal poly(2-(dimethylamino)ethyl methacrylate) chains of diblock copolymer micelles. The more electron-dense silica overlayer could be readily observed by TEM, and silica shell thicknesses of either 5.6 or 8.0 nm were determined by SAXS for shell crosslinked or non-cross-linked micelles, respectively. Wagner21 applied a core
shell model to the analysis of a dispersion of polystyrene latex particles coated with iron oxide nanoparticles. Recently, Zhang et al.22 used ultra-small-angle X-ray scattering to investigate the distribution of zirconia nanoparticles (radius ∼2.57 nm) located near the surface of silica particles (radius ∼285 nm) in a binary aqueous dispersion at pH 1.5. It was shown that the zirconia nanoparticles self-assemble to form a “halo” around the silica particles, with a separation distance of 2.15 nm between this nanoparticle “halo” and the surface of the silica particle. A similar model to that described by Zhang et al. has

been recently used to analyze SAXS data obtained for core
shell polymer-silica nanocomposites prepared by the physical adsorption of silica nanoparticles onto sterically stabilized poly(2vinylpyridine) (P2VP) latex particles.23 In this case, the nanocomposite particle diameters, silica contents, and silica layer thicknesses obtained by SAXS analysis were in good agreement with data obtained independently using other analytical techniques (electron microscopy, dynamic light scattering, thermogravimetry, etc.). Moreover, time-resolved SAXS studies confirmed that the time scale required for the physical adsorption of the silica nanoparticles (and their subsequent exchange when challenged with excess bare latex) to form well-defined core
shell poly(2-vinylpyridine)-silica nanocomposite particles was just a few seconds at ambient temperature.24 The growing academic and industrial interest in core
shell particles comprising particulate shells clearly warrants the further development of SAXS as a characterization tool for such systems. Some refinements in the SAXS analysis of core
particulate shell nanostructures have been presented recently in the literature.22,25 A rigorous approach to this problem that combines both theoretical and experimental studies has been adopted in the present work. First, Monte Carlo simulations of various core
particulate shell structures, including calculations of their expected SAXS patterns are performed in order to examine the validity of the existing analytical equations developed for SAXS analysis of core
shell nanostructures.26 Then, these analytical equations are applied to the SAXS analysis of real core
shell particles, and the results are critically compared to data obtained using more traditional characterization techniques. Polymer/silica nanocomposite particles (comprising polymer latex cores and particulate silica shells) have been selected as the model experimental system for this study. However, it is expected that these results should also be readily applicable to other types of core
particulate shell morphologies.

’ EXPERIMENTAL DETAILS Materials. Styrene and n-butyl acrylate were purchased from Aldrich, passed in turn through basic alumina columns to remove inhibitor, and stored at
20
C prior to use. 2,20 -Azobis(isobutyramidine) dihydrochloride (AIBA; Aldrich) was used as received. The commercial glycerol-functionalized silica particles (Bindzil CC40; nominal 12 nm diameter aqueous silica sol at 40 wt % according to the manufacturer) were obtained from Eka Chemicals (Bohus, Sweden) and 8076

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Langmuir were used as received. In-house TEM study of this silica sol suggested an actual mean particle diameter of 19 nm, which was also consistent with BET surface area measurements and SAXS analyses (see below). All water used in these experiments was deionized by Elgastat Option 3A water purifier. Nanocomposite Syntheses. A typical nanocomposite synthesis was conducted as follows (Figure 1). The appropriate amount of the aqueous silica sol (8.11 g aqueous dispersion, which is equivalent to 3.0 g dry silica) was diluted with water (35.9 g) and placed in a roundbottomed flask containing a magnetic stirrer bar, followed by the addition of styrene (5.0
2.5 g) and n-butyl acrylate (0.0
2.5 g). The flask was sealed with a rubber septum, and the aqueous solution was degassed at ambient temperature using five evacuation/nitrogen purge cycles. The degassed solution was stirred at 250 rpm using a magnetic stirrer and heated to 60
C in an oil bath. The AIBA initiator (50.0 mg; 1.0 wt % based on comonomer) was dissolved in degassed water (4.0 g) and added after 20 min to give a total mass of water of 45 g. Each copolymerization was allowed to proceed for 24 h at 60
C. The resulting milky-white colloidal dispersions were purified by repeated centrifugation
redispersion cycles (5000
7000 rpm for 30 min) using a refrigerated centrifuge (cooled to 5
C). Each successive supernatant was decanted and replaced with fresh, deionized water. Redispersion of the sedimented particles was achieved with the aid of mechanical rollers (ultrasonic treatment was avoided because it is usually accompanied by a rise in temperature, which can lead to film formation instead of redispersion). This purification protocol was repeated for up to five cycles until TEM studies confirmed that all of the excess silica sol had been removed. Dynamic Light Scattering. Measurements were conducted at 25
C using a Malvern Zetasizer Nano ZS instrument equipped with a 4 mW He
Ne solid-state laser operating at 633 nm. Backscattered light was detected at 173
, and the mean particle diameter was calculated from 30 runs of 10 s duration by the quadratic fitting of the correlation function using the Stokes
Einstein equation. All measurements were performed in triplicate on highly dilute aqueous dispersions. Thermogravimetric Analysis. Analyses were conducted using a Perkin-Elmer Pyris 1 TGA instrument. Samples were predried in an oven at 70
C for 12 h and then heated in air to 800
C at a heating rate of 10
C/min. The observed mass loss was attributed to complete pyrolysis of the copolymer component, with the remaining incombustible residues assumed to be pure silica (SiO2). Helium Pycnometry. Solid-state nanocomposite densities were measured using a Micromeritics AccuPyc 1330 helium pycnometer at 20
C.

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Figure 2. SAXS results obtained for a 4.0 wt % aqueous dispersion of poly(2-vinylpyridine) latex: the experimental scattering profile (1, circles), the profile after desmearing (2, crosses) and a fitting curve (solid line). Profile 2 is shifted upward by a factor of 100 to avoid overlap. The inset represents the original 2D SAXS pattern (the side of the image is equivalent to 0.056 Å
1). The white dashed lines indicate the area used for a sector integration to obtain the experimental scattering profile. used as a sample holder. Peak positions of wet rat tail collagen were used to calibrate the q axis of the SAXS patterns. 2D SAXS patterns were reduced to one-dimensional (1D) profiles by a standard protocol available in the BSL software package.27 A sector integration of the 2D patterns was performed (Figure 2, inset) to reduce the effect of smearing caused by the rectangular shape of the beam (0.003 Å
1  0.0002 Å
1). The 2D patterns, or their corresponding 1D profiles, were subjected to incident beam intensity and background corrections. The smearing of the 1D profiles, inherited from the shape of the X-ray beam, was corrected by Lake’s method.28 The smearing function has been modeled using the Gaussian function

Field Emission Scanning Electron Microscopy (FE-SEM). Images were obtained using a FEI Inspect instrument operating at 20 kV. Samples were dried onto carbon disks adhered to an aluminum stub and sputter-coated with a thin layer of gold prior to inspection to prevent sample-charging effects. Surface Area Analysis. BET surface area measurements of silica were performed using a Quantachrome Nova 1000e instrument with dinitrogen as an adsorbate at 77 K. Freeze
dried samples were degassed under vacuum at 40
C for a minimum of 15 h prior to analysis. The particle diameter was calculated from the formula 6/(FAs), where As is the BET specific surface area and F is the particle density. Small Angle X-ray Scattering. SAXS patterns were recorded at a (now defunct) synchrotron source (STFC Daresbury Lab, UK, station 2.1, wavelength of X-ray radiation λ = 1.54 Å, camera length 6.25 m). The scattering intensity was recorded using a RAPID two-dimensional (2D) detector over a scattering vector range of 0.004 Å
1 < q < 0.06 Å
1. All measurements were conducted at room temperature on dilute aqueous dispersions (typically either 0.5 wt % or 1.0 wt % solids). A liquid cell composed of two mica windows (each of 25 μm thickness) separated by a polytetrafluoroethylene spacer of 1 mm thickness was


ðq
qc Þ2 1 f ðqÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2½bðqc Þ2 2π½bðqc Þ2

where the standard deviation, b(qc), is q-dependent and is given by
qc 2 A bðqc Þ ¼ b0 þ pffiffiffiffiffiffiffiffiffiffi e 2w2 2πw2

where qc is the mean. An aqueous colloidal dispersion of nearmonodisperse P2VP latex particles was used to obtain the following parameters for this function: b0 = 1.9  10
4 Å
1, A = 6.1  10
6 Å
2, and w = 7.3  10
3 Å
1 (Figure 2). The calibration of absolute intensity has been performed using deionized water as a standard. It was assumed that the differential scattering cross section of water is 0.0165 cm
1.29 This intensity calibration was tested using aqueous dispersions of both P2VP latex and spherical silica sols of known concentrations. P2VP mass fractions at three different concentrations were obtained by fitting SAXS patterns using the scattering form factor for spheres (0.0047 ( 0.0005, 0.0100 ( 0.0006, and 0.039 ( 0.001; see Figure 2); these data were in excellent 8077

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Langmuir agreement with the mass fractions used for the sample preparation (0.005, 0.01, and 0.04, respectively). Similarly good agreement was found for the silica mass fraction obtained from SAXS (0.009 ( 0.001) and that of the aqueous silica dispersion (0.010). Helium pycnometry densities obtained for P2VP (1.17 g cm
3) and silica (2.19 g cm
3) were used to convert SAXS volume fractions into mass fractions in these calibration studies.

’ RESULTS AND DISCUSSION Modeling of SAXS Patterns of Core
Particulate Shell Polymer-Silica Colloidal Particles. Objects of essentially arbi-

trary shape and structure can be represented by finite spherical elements. If the distance between the finite elements, d, is significantly less than the size of the objects, this approach can be used for the calculation of the expected SAXS patterns for a given object within a q range where q < 2π/d.30 Alternatively, the core
particulate shell particles can be assumed to have a simple core
shell structure, but of course this approximation neglects the true particulate nature of the shell. Moreover, this crude approach has obvious limitations, since it neglects both the form factor of the particles forming the shell (the finite elements) and the shell particle
shell particle interference cross-term. According to the literature and our own experience, a two-population model that accounts for scattering factors from both the core
shell structure and also the particles within the shell has been successfully applied to the SAXS analysis of core
particulate shell nanostructures.22,23 Starting from the methods used in the treatment of star polymers and polymer micelles,31 an attempt has been made to derive a complete analytical expression of the scattering intensity counting the shell particle
shell particle interference.25 Unfortunately, this expression cannot be used for highly polydisperse particles. Moreover, it also has limitations when assessing strong correlations within the particulate shell and, perhaps most importantly, does not appear to reproduce the experimental scattering patterns expected for a well-known core
shell structure.32 On this occasion, it is possible to make a further refinement of the two-population model22,23 such that the interparticle correlations within the shell could be represented by an appropriate structure factor. This completes the set of terms required for the two-population model to be used for an analysis of scattering patterns originating from core
particulate shell structures. The first population of the model, the core
shell structure, describes the self-correlation term of the spherical polymer core and the cross-term between the core and the particles within the shell. The second population describes the self-correlation term for the particulate shell and also the cross-term between these packed particles. This approach provides sufficient flexibility to model the size distributions of both the large core particles and the smaller particles within the shell. Moreover, it also provides an opportunity to account for the nature of the interparticle interactions within the shell by exploring different structure factors. To what extent can this simplistic approach work? Are the results obtained from such two-population model fittings actually consistent with the real nanostructure of core
particulate shell particles? A comprehensive approach to structural modeling can be undertaken to answer these important questions. The main complexity arises from the uncertainty in the arrangement of silica particles within the shell: this produces a significant number of packing variants that are distinguishable by their packing density, the number of particle layers in the shell and the extent of structural order. Given these circumstances, a Monte Carlo

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approach should be the most appropriate tool to assess the problem,19,34,35 which can be approached in the following manner. A random sampling simulation of a defined structure has to be performed first, followed by calculation of a respective scattering pattern with the subsequent fitting of this pattern using the proposed two-population model. Finally, these fittings can be compared to the structural parameters used for the Monte Carlo simulation to validate the terms used in the two-population model. It has been assumed in the simulations that the particle core comprises the organic polymer component (polystyrene) and the shell is formed by a single monolayer of silica particles. On the basis of TEM analyses of cross-sectioned polystyrene/silica nanocomposite particles, this assumption seems to be fully justified.13,15 Representative simulations comprising 1000 nanocomposite particles have been conducted at several silica loadings. The following parameters were selected for the Monte Carlo simulation of core
shell particles. (i) The polymer cores in a given sample of 1000 particles have a Gaussian (normal) distribution with a mean radius Rc of 1000 Å and a standard deviation σc of 50 Å. (ii) The silica particles comprising the shell also have a Gaussian distribution with a mean radius Rsilica of 110 Å and a standard deviation σsilica of 10 Å. Several models were considered to estimate the effect of the precise particulate structure of the shell and packing of silica within the shell on the calculated X-ray scattering patterns. The first and simplest model assumes that the silica particles are merely distributed randomly within the shell. Thus, the silica particles form a random packing pattern with no preferred order and the only parameter controlling the packing is the size of the silica particles attached to the surface of the polymer core. This model neglects interparticle forces, which could lead to rearrangements of the silica particles on the polymer core surface. Obviously, this type of random packing (disordered monolayer) will be saturated at relatively moderate concentrations of silica particles in the shell. In contrast, interparticle interactions between the silica are an inherent feature of the other two alternative models. The second model is based on the assumption that all silica particles possess anionic surface charge, causing net repulsive interactions. The third model assumes that interparticle interactions within the shell are described by a LennardJones potential which, unlike the electrostatic repulsive forces in the second model, strongly depends on the size of the silica particles. These three silica packing models will be explored in turn in increasing order of complexity. Core
Particulate Shell Particle Model with Randomly Distributed Particles in the Shell. In this model, the simulation of each ith core
shell particle is composed of two steps. The first step involves random generation of the core radius of the particle, Ri0, controlled by a size distribution function, and the second step involves random generation of silica particle positions in the shell (Rij, jij, θij), where Rij is the radius of the jth particle, controlled by a size distribution function, and jij and θij are spherical coordinates of the jth particle (0 e jij e π and 0 e θij < 2π). It is assumed that the center of the jth silica particle is located on a spherical surface of radius Ri0 þ Rij. The number of silica particles generated for the ith core
shell particle, Pi, is defined by the silica concentration within the shell. Although the random generation of spherical coordinates is used in these simulations, there is a constraint on confinement since no overlap between neighboring particles is allowed. Thus, if the generation of silica 8078

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Figure 3. Simulated core
particulate shell particles with different structural arrangements of small particles comprising the shell: (a) random distribution, packing density for particles in the shell, D = 0.45, number of particles in the shell = 190, core radius = 1020 Å, mean radius of the particles comprising the shell = 110 Å with a standard deviation = 10 Å; (b) with repulsive Coulombic interactions and respective parameters 0.45, 178, 982, 110, and 10 Å and (c) with Lennard-Jones interactions and respective parameters 0.73, 344, 1082, 110, and 10 Å. The color of the particles within the shell indicates their coordination number, as calculated using the Delaunay triangulation algorithm: pink corresponds to coordination number 4, green to 5, red to 6, blue to 7, and turquoise to 8. The following distributions of coordination numbers for the presented sets of shell particles are obtained: (a) 4 - 6, 5 - 56, 6 - 76 (red); 7 - 48, 8 - 4; (b) 5 - 30, 6 - 130, 7 - 18; and (c) 5 - 62, 6 - 232, 7 - 50. Note that the disclination charge of each set of coordination numbers is consistent with Euler’s theorem45 (which states that the total disclination charge of any triangulation of the sphere must be 12). The radii of the core and the silica particles comprising the particulate shell are scaled 2:1, respectively, to make the Delaunay triangulation net visible.

particles was conducted ad infinitum, at some point the packing density of the silica particles, randomly positioned on the core, would reach saturation, whereby no more particles could be positioned on the core. The packing density, D, is controlled by the ratio of the core radius to the silica particle radius and also by the size polydispersity of the latter component. Thus D is defined as D ¼ ðPi =2Þ½1
cosðδ=2Þ

ð1Þ

where cos(δ/2) = [(Ri0 þ Rsilica)
Rsilica ] /(Ri0 þ Rsilica).36,37 The Monte Carlo simulations indicate that, for the parameters chosen for the modeled core
shell particles, shell saturation was attained at D ∼ 0.45. Therefore, the concentration of silica particles in the shell simulated according to this model was set below this packing density limit (Figure 3a). The model bears some similarities to the well-known random sequential adsorption (RSA) model.38 The RSA maximum packing efficiency for disks placed onto a planar surface is 0.55. The limiting D value of 0.45 obtained for packing polydisperse silica particles onto the surface of a spherical core is therefore comparable to this RSA value. Model for Core
Particulate Shell Prticles with Repulsive Interactions between the Shell Particles. This model requires the arrangement of a number of spheres on a spherical surface. It is related to the Thomson problem of determining the ground state of classical electrons interacting with a repulsive Coulombic potential on the surface of a sphere.39 There is also an analogous situation in biology: the famous Tammes problem of pollen grain orifices involves packing a certain quantity of equal non-overlapping circles (spherical caps) on a sphere so as to maximize the distance between any two adjacent circles.40 Solutions for the packing problem have been established for a certain number of spherical caps36,37 among which, at small quantities, Platonic solids can be found. All these solutions belong to the family of degenerate arrangements where each circle has the same number of neighbors, at the same distance, equally spaced around itself. However, for a random number of circles arranged on a sphere, 2

2 1/2

such solutions are impractical, and a more effective algorithm, based on determining an equilibrium configuration of a certain quantity of equally charged mutually repulsive points on the surface of a sphere, should be used.41 In this more refined electrostatic repulsion model, the simulation of each ith core
shell colloidal particle comprises three steps. The first two steps are similar to those of the random distribution model. However, no confinement is required for the generation of arbitrary spherical coordinates of silica particles in the second step. Assuming the arbitrary distribution of silica particles (nodes), the final step utilizes numerical minimization algorithms in which all charged nodes repel each other according to Coulomb’s law. The superimposed Coulombic repulsive force on the jth particle is first calculated using B F j = ke ∑l=1,l6¼jPi QjQl ^dlj/djl2, where Qj and Ql are charges on the interacting silica particles, djl is the interparticle separation distance, ^dlj is the unit vector parallel to the line between the charges and ke is the Coulomb constant (which is assumed to be 1 N m2 C
2 in the calculations), and then, movement of the particle along the tangential component of the force is performed. This algorithm is run repetitively until the convergence criterion (the forces B F j are nearly normal to the surface of the core) is satisfied, and the resulting spatial distribution of the silica particles is obtained. It should be noted that the classical problem of packing of spheres on a spherical surface only deals with perfectly monodisperse particles of the same radius.36,37,41 In contrast, the current problem relates to real colloidal nanocomposite particles of finite polydispersity; this additional complexity undoubtedly affects their packing42 and hence must be incorporated into the simulations. Since electrostatic interactions have been used for equilibrating the configuration of the shell particles, it has been assumed that the anionic charge on silica particles is uniformly distributed over their surface. Thus, the charge on the jth silica particle belonging to the ith colloidal particle is proportional to its spherical surface Qj ∼ 4πRij2 in the electrostatic repulsion model simulations. 8079

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The equilibration of the silica particles confined to the spherical surface due to their mutual Coulombic repulsion produces ordered patterns resembling hexagonally close-packed structures (Figure 3b). There are apparent distortions in the packing originating from both the curvature of the spherical surface and the polydispersity of the silica particles. Using the Delaunay triangulation algorithm,43 the mean coordination number of each silica particle in the packed shell can be estimated from a corresponding number of convex polyhedral vertices of a Voronoi cell.44 Despite the polydispersity of the silica particles used in the simulations (σsilica/Rsilica ≈ 0.1) only five-, six-, and seven-coordinated particles, including pairs of tightly bound 5
7 defects with net disclination charge zero, can be found in the equilibrated shell (compare Figure 3a and b), which is consistent with the structures of two-dimensional spherical crystals.45 Model for Core
Particulate Shell Particles with LennardJones Interactions between the Shell Particles. The physical generalization of the packing problem of Pi particles constrained to the surface of a sphere has been formulated as determining the equilibrium configuration of the particles associated with the minimum potential energy, U(Pi, n), of the system UðPi , nÞ ¼

Pi 1 X k d
n ¼ minimum 2 l ¼ 1, l6¼ j jl

ð2Þ

where k is a constant.41 The situation where n = 1 corresponds to a long-range Coulombic potential. When n f ¥ the terms corresponding to the smallest distance in the sum dominate the potential energy and minimizing U(Pi, n) becomes equivalent to maximizing the smallest distance, the long-range potential (soft spheres) transforms into a short-range potential (hard spheres). Thus, increasing the exponent in the energy potential term makes equilibration much more sensitive to particle packing effects. In the limit, the problem becomes similar to that of the densest possible packing of identical small spheres on a larger sphere. Herein, the internal energy of the core
particulate shell particles is minimized with respect to potential energy of the silica particle configuration. This approach assumes that the simulated ensemble of the core
particulate shell particles is a closed system at constant temperature and volume, and the entropy of the system is constant at equilibrium. Various energy potentials can be used to describe interparticle interactions. In particular, the Lennard-Jones (LJ) potential can be used to describe particle size-controlled interactions.42 Like eq 2, the condition for the minimization of the potential energy of the system can be rewritten as 2 !12 !6 3 Pi X r þ r r þ r l j l 4 j 5 þ2 UðPi Þ ¼ ε0 d d jl jl l ¼ 1, l6¼ j ¼ minimum

ð3Þ

where ε0 is the depth of potential well (assumed to be unity in the simulations) and rj, rl are the radii of the jth and lth silica particles, respectively. In this more refined model, the simulation of each ith core
shell colloidal particle comprises four steps. The first three steps are similar to those used in the repulsive interaction model. For the additional fourth step, starting from positions of silica particles prearranged by the repulsive interactions, the Coulombic potential was replaced by the LJ potential. Thus, the positions

of the silica particles prearranged according to mutually repulsive electrostatic interactions (which are proportional to the surface of the particles) were further equilibrated using the particle sizecontrolled interactions. A Monte Carlo algorithm was used to minimize the LJ potential of the prearranged system. Spherical coordinates for each silica particle (jij, θij) were randomly changed within the neighborhood of the particle and new coordinates that further reduced the energy potential were accepted. 5000  Pi iterations were performed for ith core
shell particles. It has been estimated that the ultimate packing density of uniform spheres (or spherical caps) confined to a spherical surface depends on √ the number of spheres and lies within the √ interval (5/4)(2
2) e D e π/ 12,36,37 where the lower limit (∼0.732) corresponds to the case of five densely packed spheres and the upper limit (∼0.907) corresponds to the hexagonal close-packing of spheres on a sphere of infinite radius (i.e., a planar surface).41 Since the objective of the present work was not to identify the ultimate packing density of silica spheres on a spherical surface, the maximum silica shell concentration in these simulations was set to the lower limit packing density of 0.732 so as to avoid the undesirable overlap of silica particles within the shell. Indeed, for this lower limit no particle overlap was observed for simulated configurations of silica equilibrated with the LJ potential, as well as with repulsive interactions using the chosen values of Rc and σc (Figure 3c). Thus, this maximum concentration guarantees that the spherical morphology of the silica particles is not distorted by their packing and the spherical form factor used for the SAXS calculations remains valid. An alternative approach has been recently undertaken by Bon et al., whereby the LJ potential has been used to configure densely packed polydisperse silica particles confined by the spherical surface using metropolis Monte Carlo simulations.42 From this previous study, it can be estimated that the optimum number of polydisperse silica particles corresponding to the lowest overall energy leads to very high packing densities (D ≈ 0.924 for a standard 6
12 LJ potential and D ≈ 0.869 for a narrower 12
24 LJ potential).46 Such high densities, which would be impossible to achieve for monodisperse hard spheres,37 suggest that imaginary spherical contours of the silica particles must overlap in this configuration. Thus, such “soft” particles interacting via an LJ potential are distorted by a dense packing, which makes this approach unsuitable for the calculation of SAXS patterns of core
particulate shell colloidal particles, since a spherical form factor for the silica particles cannot be assumed with any confidence in this instance. Calculation of SAXS Patterns from the Simulated Core
Particulate Shell Structures. Following the generalized expression for spherically symmetric particles,26 the coherent differential scattering cross section of a collection of core
particulate shell particles can be expressed as M cc X  Fi ðqÞFi ðqÞ ð4Þ dσscat ðqÞ=dΩ ¼ M P Vi i ¼ 1 i¼1

where M is the number of core
shell colloidal particles generated by the Monte Carlo simulations, cc is the concentration of polymer cores of the particles expressed in a volume fraction, Vi = 4 /3πRi03 is the volume of the polymer core of the ith particle, and Pi X
i q r Fi ðqÞ ¼ vij Δξij f ðq, Rij Þ e B Bij ð5Þ j¼0

8080

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Figure 4. Calculated SAXS patterns for simulated core
particulate shell particles comprising polystyrene cores and silica particles distributed randomly within the shell (squares), distributed uniformly using repulsive interactions (circles), and distributed uniformly using LennardJones interactions (triangles). Solid lines represent fitting curves for the two-population model. The SAXS patterns are shifted upward by a factor of 5(m-1), where m is the number with which the pattern is labeled. The order of the patterns corresponds to the following core
shell structures: (1) random distribution of silica within the shell and a packing density D = 0.18; (2) repulsive interactions of silica within the shell and D = 0.18; (3) random distribution and D = 0.45 (see Figure 3a); (4) repulsive interactions and D = 0.45 (see Figure 3b); (5) repulsive interactions and D = 0.73; and (6) Lennard-Jones interactions and D = 0.73 (see Figure 3c). The volume fraction of the polystyrene particle core was fixed at 0.01 for all simulations. The lower inset plot contains patterns corresponding to the structures with D = 0.18. The upper inset plot contains patterns corresponding to the structures with D = 0.45. No arbitrary shifts have been applied to the curves in the insets.

is the form factor for ith core-particulate shell particles, where ^r ij is the radius vector describing the relative position of the center of the jth silica particle to the center of the polymer core, vij = 4/3πRij3 is the volume of the jth silica particle in the ith colloidal particle, Δξij is the difference in scattering length density between the jth silica particles or the polymer core (j = 0) and the solvent (it has been assumed that all silica particles are homogeneous and have the same scattering length density) and f ðq, Rij Þ ¼

3½sinðqRij Þ
qRij cosðqRij Þ ðqRij Þ3

ð6Þ

is the normalized scattering amplitude of a homogeneous sphere corresponding either to the jth silica particle or a polymer core in the ith colloidal particle {f(0, Rij) = 1}. The calculated scattering length density has been normalized by the volume of polymer cores, rather than by the total volume of the simulated colloidal particles (eq 4). This normalization provides an opportunity to keep one of the components of core
shell particles (the core

Figure 5. Typical fit to a scattering pattern derived for the simulated core
particulate shell particles (see Figure 4, pattern 5, particles with repulsive silica
silica interactions within the shell and D = 0.73) using a standard core
shell form factor (a, dashed curve) and the two-population model (b, solid curve) which includes scattering from both the core
shell structure (dashed curve) and a particulate structure of the shell (dotted curve). The upper patterns shown in (a), demonstrating a fit by the core
shell model, have been multiplied by a factor of 103 for the sake of clarity. The inset indicates the radial distribution of scattering length density used in the core
shell model (a). The detailed description of the structural parameters can be found in Table 1.

radius) constant in all simulations, regardless of the variable concentration of silica particles. Representative SAXS patterns have been calculated for the structural models of core
particulate shell particles obtained by Monte Carlo simulations (Figure 4). Three regions can be recognized in the calculated patterns: scattering oscillations at q < 0.01 Å
1 associated with the spherical morphology of large core
shell particles; a peak around 0.01 Å
1 < q < 0.04 Å
1 associated with the silica packing (mainly observed for structures ordered by interparticle interactions), and scattering oscillations when q exceeds 0.04 Å
1 that are associated with the spherical morphology of the small silica particles. Validation of the Two-Population Model from SAXS Patterns of Simulated Core
Particulate Shell Structures. The two-population model can be easily realized using the Irena package for small-angle scattering.47 The coherent scattering cross section of a system composed of n different (noninteracting) populations of scattering polydisperse objects can be expressed as dσscat ðqÞ=dΩ ¼

n X l¼1

Z Sl ðqÞNl 0

¥

pop

jFl ðq, rÞj2 Ψl ðrÞ dr ð7Þ

where Flpop(q, r) is the form factor, Ψl(r) is the size distribution function, Nl is the number density per unit volume, and Sl(q) is the structure factor of the lth population in the system. The terms of the two-population model proposed in this work (n = 2) are now presented in detail. The terms for the first population {l = 1 8081

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8082

1000

1000

repulsion, D = 0.73

LJ, D = 0.73

53

51

48

44

48

23

17

(P1)

σ1, Å

0.0084

0.01

0.0085

0.0087

0.011

0.014

0.011

c1 (P1)

212

219

226

218

212

222

240

(P1)

ΔR, Å

14.4

13.8

13.8 13.8

12.34

12.09

12.42

12.09

10.43

10.46

10.46 10.35

9.87

9.97

fit

111

111

112

111

111

111

113

(P2)

Rsilica, Å

7

7

8

8

9

9

9

(P2)

σ2, Å

0.004

0.004

0.0025

0.0025

0.00098

0.00098

0.00049

Simul.

0.0034

0.004

0.004 0.0034

0.0022

0.0024

0.0022

0.0023

0.00093

0.001

0.0012 0.00091

0.00048

0.00051

c2 (P2)

csilica (P1)d

silica volume fraction

106

104

130

111

214

122

289

(P2)

RPY , Å

0.41

0.35

0.35

0.2

0.392

0.052

0.35

cPY (P2)

0.4 (300)

0.4 (300)

0.25 (184)

0.25 (184)

0.098 (74)

0.098 (74)

0.049 (37)

Simul.e

0.41 (300)

0.43 (314)

0.4 (292) 0.34 (249)

0.26 (185)

0.28 (199)

0.25 (192)

0.27 (208)

0.085 (64)

0.091 (69)

0.089 (67) 0.065 (49)

0.044 (30)

0.047 (32)

c2/c1f

csilica/c1f

silica volume/polymer volume

0.75

0.79

0.73 0.62

0.47

0.51

0.47

0.50

0.16

0.17

0.16 0.12

0.08

0.08

D

Rc = mean core radius; σ1 = standard deviation of the core radius; c1 = volume fraction of the (co)polymer core; ΔR = thickness of the shell; ξshell = effective scattering length density of the shell; Rsilica = mean silica particle radius; σ2 = standard deviation of the silica particle radius; c2 = volume fraction of silica particles; RPY = Percus
Yevick hard-sphere radius of packed silica; cPY = Percus
Yevick volume fraction of packed silica; D = density of silica packing within the shell. P1 and P2 indicate to which population the fitted parameter belongs. The errors of the fitted parameters present in this table are within a unit of the last digit of the value. b Parameters used for modeling are as follows: Rc = 1000 Å; σc = 50 Å; Rsilica = 110 Å; σsilica = 10 Å; ξ0j = ξcore = 9.581  1010 cm
2 [polystyrene, (C8H8)n]; ξij(i6¼0) = ξsilica = 18.56  1010 cm
2 (silica, SiO2); ξsol = 9.42  1010 cm
2 (water, H2O). c ξshell = (ξsilicaVsilica þ ξsolVsol)/(Vshell), where Vsilica = 4/3πRsilica3 ÆPiæ, Vshell = 4/3π[(Rc þ 2Rsilica)3
Rc3], Vsol = Vshell
Vsilica and both Rc and Rsilica are the model parameters. d Volume fraction of silica in the shell, csilica, has been estimated from the fitted parameters of population 1 (P1) as the following: csilica = {(ξshell
ξsol)[(Rc þ ΔR)3
Rc]}/[(ξsilica
ξsol)Rc3c1]. e Average number of silica particles in the shell of the core
particulate shell particles, ÆPiæ, are given in brackets. f Average number of silica particles within the shell of the core
particulate shell particles, given in brackets, is obtained as ÆPiæ = (csilicaRc3)/(c1Rsilica3) or ÆPiæ = (c2Rc3)/(c1Rsilica3), where Rc, Rsilica, and c1 are the fitted parameters.

a

1000

1011

repulsion, D = 0.18

repulsion, D = 0.45

1011

random, D = 0.18

1017

990

repulsion, D = 0.09

random, D = 0.45

(P1)

Rc, Å

model typeb

Simul.c

ξshell  1010, cm-2 (P1)

Table 1. Parameters Used in the Monte-Carlo Simulation (Simul.) of SAXS Patterns for Core
Particulate Shell Particles and the Results of Fitting to Those Patterns Using the Two-Population Modela

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ARTICLE

(eq 7)}, corresponding to the core
shell particles, are the following: pop

F1 ðq, rÞ ¼ Vtotal ðrÞðξshell
ξsol Þ f ðq, r þ ΔRÞ þ V ðrÞðξcore
ξshell Þ f ðq, rÞ

ð8Þ

where ξcore, ξshell, and ξsol are the scattering length densities of the core, the shell, and the solvent, respectively, f(q, r þ ΔR) and f(q, r) can be expressed using eq 6, ΔR is the shell thickness, Vtotal(r) = 4/3π(r þ ΔR)3 and V(r) = 4/3πr3 are volume parameters of the core
shell particles, and 2 1 2 Ψ1 ðrÞ ¼ pffiffiffiffiffiffiffiffiffiffi2 e
ðr
Rc Þ =2σ1 2πσ1

ð9Þ

is the Gaussian distribution function of the polymer core radii (with a mean radius Rc and a standard deviation σ1), and c1 N1 ¼ Z ¥ ð10Þ V ðrÞ Ψ1 ðrÞ dr 0

where c1 is the volume fraction of cores. A dilute solution of the core
shell particles has been assumed in the model, so the structure factor is set to unity {S1(q) = 1}. The first population describing scattering from a core
shell structure assumes a homogeneous distribution of electron density across the shell (Figure 5a). A particulate structure for the shell, generating an additional scattering signal, is described by the second population (Figure 5b), l = 2 (eq 7), with the following terms: pop

F2 ðq, rÞ ¼ V ðrÞðξsilica
ξsol Þ f ðq, rÞ

ð11Þ

where ξsilica is the scattering length density of the silica particles comprising the shell, and 2 1 2 Ψ2 ðrÞ ¼ pffiffiffiffiffiffiffiffiffiffi e
ðr
Rsilica Þ =2σ2 2 2πσ 2

ð12Þ

is the Gaussian distribution function of silica radii in the particulate shells (with a mean Rsilica and a standard deviation σ2), and c2 N2 ¼ Z ¥ ð13Þ V ðrÞ Ψ ðrÞ dr 2

0

where c2 is the volume fraction of silica particles. A dense packing of the silica particles within the shell is described by a structure factor S2(q) using a hard-sphere Percus
Yevick model.48 Figure 5 clearly shows that accounting for this structure factor due to the silica population is essential in order to obtain a good fit to the calculated SAXS patterns obtained for the simulated core
particulate shell structures, which have a prominent broad feature at q ≈ 0.03 Å
1. Combining these two populations in the model produces good fits to the simulated scattering patterns over a wide range of silica concentrations within the shell (Figure 4). A strong correlation between the fitted parameters and those parameters used for the Monte Carlo simulations is also observed (Table 1). More specifically, the fits reproduce the mean radii of the particles, their polydispersity, and the respective volume fractions of the polymer and silica components in these core
shell nanocomposite particles. Two independent sources of information regarding the silica content are present in the fitted parameters. One source is the shell parameters of the first population such as the effective scattering length density of the shell, ξshell, and the shell

Figure 6. Representative field emission scanning electron micrographs of selected nanocomposite particles studied in this work: (a) polystyrene/silica, (b) polystyrene/silica at high magnification, (c) 70:30 poly(styrene-stat-n-butyl acrylate)/silica, and (d) partially coalesced 50:50 poly(styrene-stat-n-butyl acrylate)/silica.

thickness, ΔR, which can be converted into a silica volume fraction within the shell, csilica (Table 1). Another source is the volume fraction of the second population, which corresponds to the volume fraction of silica in the colloidal system, c2. Thus, if the chosen two-population model is correct and the only location of the silica is within the shell of the core
shell particles, then csilica and c2 must be numerically equal. The fitted parameters demonstrate a good correlation between csilica and c2. Moreover, both parameters are consistent with the parameters used for the simulations (Table 1). Some underestimation of the silica content in the colloidal system is observed from c2 at high concentrations (D = 0.73). This could be because the densely packed silica shell is described by a model developed for the three-dimensional arrangements of hard spheres in space and not for the arrangement of particles on a spherical surface. It is also noteworthy that the number of silica particles estimated from the fitted parameters closely reproduces the numbers used for the core
shell structure simulations (Table 1, penultimate column, see the numbers given in brackets). The Percus
Yevick parameters obtained from fitting the Monte Carlo simulations are perfectly reasonable, but it would be difficult to correlate these data with the structural arrangement of the silica nanoparticles within the shell. This is because this hard-sphere model does not account for the spherical geometry of the core
shell structure. Nevertheless, for the nanoparticles distributed within the shell by repulsive interactions (including Lennard-Jones interactions) the Percus
Yevick effective radius, RPY, associated with the interparticle distances correlates well with the silica packing density (Table 1). A higher silica concentration necessarily reduces the mean silica
silica separation distance within the evenly packed silica shells. The two-population model produces reasonable structural parameters for core
particulate shell particles formed by evenly distributed silica (i.e., repulsive interactions) over a wide range of 8083

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ARTICLE

Table 2. Structural Parameters for the Core
Particulate Shell (Co)Polymer-Silica Particles Obtained from SAXS Patterns Fitted by the Two-Population Modela csilicab (P1) ΔR,

ξshell  1010,

Rsilica,

σ2,

c2

RPY, Å

cPY

c1 (P1)

Å (P1)

cm
2 (P1)

Å (P2)

Å (P2)

(P2)

(P2)

(P2)

cc

120

0.0014

206

12.9

106

24

111

0.29

0.0037

88

0.0016

206

12.7

108

25

111

0.33

0.0038

styrene

Rc, Å

σ1, Å

content, wt%

(P1)

(P1)

50

755

60

834

70 80 100

971 1105 1477

76 78 108

0.0019 0.0028 0.0033

202 191 202

12.3 12.7 12.4

105 98 105

0.00057 0.00068 0.00053 0.00057 0.00047

23

0.00048 0.00064

21

0.00065 0.00052

18

0.00056

106 100 93

0.32 0.32 0.30

0.0039 0.0039 0.0041

csilica þ c1d

csilica/c1e

c2 þ c1d

c2/c1

D

0.0019

0.41 (148)

0.56

0.0021 0.0021

0.49 (177) 0.34 (157)

0.67 0.52

0.0021

0.37 (170)

0.56

0.0023

0.25 (198)

0.47

0.0024

0.26 (206)

0.49

0.0035

0.23 (330)

0.55

0.0035

0.23 (330)

0.55

0.0039

0.16 (445)

0.49

0.0038

0.17 (473)

0.52

Rc = mean core radius; σ1 = standard deviation of the core radius; c1 = volume fraction of the cores; ΔR = thickness of the silica shell; ξshell = effective electron density of this silica shell; Rsilica = mean silica particle radius; σ2 = standard deviation of the silica particle radius; c2 = volume fraction of the silica particles; RPY = Percus
Yevick hard-sphere radius of packed silica; cPY = Percus
Yevick volume fraction of packed silica; D = density of silica packing within the shell. P1 and P2 indicate to which population the fitted parameter belongs. For each fit, the scattering length densities for the cores (ξcore), the solution (ξsol) and the silica (ξsilica) were taken to be 9.581  1010 cm
2, 9.42  1010 cm
2, and 18.56  1010 cm
2, respectively. The errors of the fitted parameters presented in this table are within a unit of the last digit of the value. b Silica volume fraction, csilica, has been estimated from the fitted parameters of population 1 (P1) as the following csilica = {c1(ξshell
ξsol)[(Rc þ ΔR)3
Rc3]}/[(ξsilica
ξsol)Rc3]. c Volume fraction of core
shell particles is estimated from the mass concentration of the aqueous nanocomposite dispersion (0.005) and the core
shell particle density obtained by helium pycnometry (see Table 3). d Total volume fraction. e Average number of silica particles within the shell of the core
particulate shell particles, given in brackets, is estimated from ÆPiæ = (csilicaRc3)/(c1Rsilica3) or ÆPiæ = (c2Rc3)/(c1Rsilica3), where Rc, Rsilica, and c1 are the fitted parameters. a

silica concentrations from a dense packing, close to the theoretical limits, to packing values as low as D = 0.09 (Table 1). However, the model cannot be used to obtain a good fit to SAXS patterns corresponding to core
shell particles comprising low silica shell concentrations with no structural order (random distribution). There is a discrepancy between the fitted pattern and the calculated pattern at ultrasmall angles (Figure 4, curve 1). The random distribution of silica causes significant deviations in the local scattering length densities relative to the average (effective) coherent scattering length density of the shell, ξshell. These deviations make the scattering signal at ultrasmall angles stronger than that expected for a shell with evenly distributed particles (see the comparison of patterns in the bottom inset of Figure 4). This effect becomes unpredictable at lower concentrations of silica. Despite this discrepancy, the fitting suggests that the two-population model can still be reliable for low silica concentrations. If this ultrasmall angle regime is neglected, the fitted parameters remain reasonable (Table 1, D = 0.18, random distribution) with some inconsistency between the simulated and fitted volume fractions. At high silica concentrations, the difference in local deviations of the scattering length density between random and ordered arrangements of silica is insignificant, and the scattering signals obtained from these structures at ultrasmall angles are similar (see the top inset of Figure 4). Application of the Two-Population Model for SAXS Analysis of Real Core-Particulate Shell Structures. Model colloidal copolymer/silica particles with a well-defined core
particulate shell morphology were selected to obtain experimental SAXS patterns.13 These nanocomposite particles were readily prepared by emulsion copolymerization of styrene with n-butyl acrylate (or simply homopolymerization of styrene) using a cationic azo initiator in the presence of a commercial aqueous glycerolfunctionalized silica sol (Figure 1). Systematic variation of the

n-butyl acrylate content resulted in a concomitant reduction in the mean particle diameter from 333 nm for the polystyrene/silica nanocomposite to 191 nm for the 50:50 poly(styrene-stat-n-butyl acrylate)/silica nanocomposite, as judged by DLS. With increasing n-butyl acrylate content (and decreasing particle diameter), a progressive increase in the mean particle density from 1.22 g cm
3 to 1.34 g cm
3 was observed by helium pycnometry, which is consistent with the concomitant increase in silica content from 24 wt % to 39 wt %, as judged by thermogravimetry. FE-SEM images (Figure 6) show that these (co)polymer/silica nanocomposite particles have relatively narrow size distributions and silicarich surfaces. The high magnification image shown for the polystyrene/silica particles (Figure 6b) indicates that the silica particles on the surface of this colloidal nanocomposite are close-packed, but do not appear to exhibit any long-range order. The Irena SAS macros for IgorPro were used to fit the SAXS patterns. The maximum entropy method available for particle size distribution analysis in this software package49 suggested that a Gaussian distribution was an appropriate description of the polydispersities of both the silica and the nanocomposite particles (Figure S1, Supporting Information). Thus, such distributions (eq 9 and eq 12) were assumed in the two-population model, which was utilized to describe the core
particulate shell structure of the nanocomposite particles. Prior to any analysis of the experimental SAXS patterns obtained for the nanocomposite particles, the SAXS patterns obtained for the silica sol alone were fitted using a spherical form factor. Even though the silica nanoparticles are not perfectly spherical (see Figure 1 in ref 15), it was found from SAXS analysis of the aqueous silica dispersion (Figures S1 and S2, Supporting Information) that the assumption of a spherical morphology is a good approximation, producing a silica volume fraction that is consistent with the actual aqueous silica concentration and giving a mean particle diameter that is 8084

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Figure 7. SAXS profiles (gray circles) and corresponding fittings using the two-population model (black lines) obtained for 0.50 wt % aqueous dispersions of a series of core
shell nanocomposites plotted in increasing order of particle size (and decreasing silica content): (1) 50:50 poly(styrene-stat-n-butyl acrylate)/silica; (2) 60:40 poly(styrene-stat-nbutyl acrylate)/silica; (3) 70:30 poly(styrene-stat-n-butyl acrylate)/ silica; (4) 80:20 poly(styrene-stat-n-butyl acrylate)/silica; and (5) polystyrene/silica. For clarity, each pattern has been multiplied by a factor of 10(m-1), where m is the number with which the pattern is labeled.

comparable to that obtained by other techniques (TEM and BET). The resulting size parameters for the silica sol were subsequently used as a guide for the silica diameter and polydispersity when fitting the data for the nanocomposite particles using the two-population model. The mean silica particle diameter obtained from this fit was 22 ( 4 nm, which corresponds to a relatively high polydispersity of 18%. This is in good agreement with a number-average diameter of 19 nm estimated from TEM images, which is in turn consistent with the surface-average diameter of 19 nm calculated from BET surface area analysis.13
15 In order to fit the SAXS patterns, some of the parameters of the two-population model were fixed, while others were fitted. The scattering length densities of all components of the aqueous nanocomposite dispersions were fixed and were calculated based on the respective chemical compositions and densities of each component. The nanocomposite core consists of both polystyrene, chemical formula C8H8, (ξPS = 9.581  1010 cm
2 for polystyrene, assuming a mass density of 1.05 g cm
3 as determined by helium pycnometry) and poly(n-butyl acrylate), chemical formula C7H12O2, {ξPBuA = 9.546  1010 cm
2 for poly(n-butyl acrylate), assuming a mass density of 1.03 g cm
3}. Since the difference between ξPS and ξPBuA is insignificant, the variation in the chemical composition of the copolymer core should not significantly affect the scattering length density. Thus, it was assumed in all fittings and calculations that ξcore = ξPS (Table 2). Similarly, based on the known chemical composition of silica (i.e., SiO2) and taking the experimental silica sol density of 2.19 g cm
3, the silica scattering length density is ξsilica = 18.56  1010 cm
2. With the density of water taken to be 1.00 g cm
3,

ARTICLE

the scattering length density of this solvent is ξsol = 9.42  1010 cm
2. The two-population model produced a good fit to the experimental SAXS patterns obtained for the core
particulate shell particles (Figure 7). Multiple oscillations can be observed for each SAXS curve, suggesting that the nanocomposite particles have relatively narrow size distributions (σ1, Table 2). The mean silica diameter, or 2Rsilica, is ∼21 nm; this value was obtained from the form factor for silica (the second population) and is consistent for all five nanocomposite samples (Table 2). The silica shell thickness of approximately 20 nm obtained for each nanocomposite sample is equivalent to the diameter of a single silica particle, thus confirming that there is a monolayer of silica particles (rather than multilayers) surrounding the (co)polymer latex core (Table 2). This corroborates earlier suggestions, based on electron spectroscopy imaging studies of just a few dozen crosssectioned polystyrene/silica particles, that the nanocomposite shell comprises a well-defined monolayer of silica particles.15 The fact that a self-consistent silica particle diameter can be extracted from several independent fitting parameters confirms beyond reasonable doubt that a robust structural model for these core
shell nanocomposite particles has been established. The volume fractions of both silica in the shell, csilica, and silica in the colloidal system, c2, are similar for most of the (co)polymer/ silica samples studied, but c2 is consistently higher than csilica (Table 2). This suggests that silica is located mainly within the particulate shell of the nanocomposite particles, with apparently little or no excess silica sol present in the continuous phase. The 50:50 poly(styrene-stat-n-butyl acrylate)/silica nanocomposite is the only sample with a significant difference between c2 and csilica, suggesting some excess silica sol in this case. Unfortunately, this possible contamination cannot be confirmed from FE-SEM studies (Figure 6d). The silica packing densities (D) estimated from the fitted parameters are similar for each of the five colloidal nanocomposite dispersions (Table 2, penultimate column). This suggests that the silica shell concentration is independent of the mean nanocomposite particle diameter. The D values of ∼0.5 are comparable to the packing density observed for the random distribution of silica particles within the shell (D ∼ 0.45), which might be interpreted as evidence for no silica
silica interactions within the shell. However, a pronounced peak is observed in the experimental SAXS patterns, which indicates some degree of order within the silica shells (Figure 7). Moreover, this scenario is not consistent with the observation that a Monte Carlo simulation of the distribution of silica particles, at this silica shell concentration, does not produce any distinctive scattering features in the model based on random packing (Figure 4, pattern 3). The polystyrene/silica and the 80:20 poly(styrene-stat-n-butyl acrylate)/silica nanocomposites both exhibit a strong correlation between the total concentration of the core
shell particles obtained from the fitting (c1 þ csilica or c1 þ c2, Table 2) and the concentrations used for the sample preparation, c (Table 2). However, there is a progressive discrepancy between c1 þ csilica (or c1 þ c2) and c on decreasing the mean core radius: c1 þ csilica reduces from approximately 0.004 for polystyrene/silica (Rc = 1477 Å) to around 0.002 for the 50:50 poly(styrene-stat-n-butyl acrylate)/silica (Rc = 755 Å), while c changes only slightly (from 0.0041 to 0.0037, respectively). This comparison suggests that partial sedimentation of the nanocomposite particles occurs on the time scale of the SAXS measurements. Since the total accumulation time required for each SAXS experiment, including 8085

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Table 3. Comparison of the SAXS Data with That Obtained by Other Methods a particle diameter, nm

silica content, vol % from SAXSd

styrene

particle density,

silica content

from particle

from

form

volume

DLS

SAXSb

g cm-3

by TGA, wt %

densityc

TGAc

factor

fractions

50

191 (8)

192

1.34

39

25 (2)

23 (4)

20

24

60

206 (4)

208

1.33

37

25 (2)

22 (4)

18

19

70

246 (25)

235

1.28

31

20 (2)

18 (4)

14

15

80

255 (10)

259

1.27

27

19 (2)

15 (4)

14

14

100

333 (20)

336

1.22

24

15 (2)

13 (4)

11

12

content, wt%

a

Dynamic light scattering, DLS; helium pycnometry and thermogravimetric analysis, TGA; for the characterization of four poly(styrene-stat-n-butyl acrylate)/silica colloidal nanocomposites and a polystyrene/silica colloidal nanocomposite, with each sample possessing a well-defined core
particulate shell morphology. Errors are presented in brackets; where no bracketed numbers are shown, the errors are within a unit of the last digit of the value. b The mean SAXS particle diameter was obtained from the core radius and the shell thickness 2(Rc þ ΔR); see Table 2 for the parameters. c It was assumed in the calculations that the density of copolymer in the samples is 1.05(1) g cm
3 and the density of silica is 2.19(1) g cm
3. d The silica content derived from the core
shell form factor or from volume fractions of the populations is obtained as a ratio of the volume fraction of silica (csilica or c2, respectively) to the volume fraction of core
shell particles [c1(Rc þ ΔR)3/Rc3]; see Table 2 for the parameters.

Figure 8. Comparison of an experimental SAXS profile obtained for a 0.50 wt % aqueous dispersion of a 70:30 poly(styrene-stat-n-butyl acrylate)/ silica core
shell nanocomposite dispersion (open circles) and a SAXS profile of a corresponding core-particulate shell model with silica
silica repulsive interactions within the shell simulated using fitted structural parameters of the nanocomposite presented in Table 2 (filled squares).

preparation of the samples and collection of the data, is similar, this sedimentation can be related to the particle size and particle density. Given that the silica particles are adsorbed onto the (co)polymer cores at essentially the same packing density for each nanocomposite sample, it can be concluded that the smaller the mean core diameter, the higher the silica mass fraction in the nanocomposites. This should increase the average density of the core
shell particles facilitating the sedimentation of the particles in water. This speculation is also supported by the helium pycnometry data (Table 3).

A peak associated with a high silica packing density is observed at q ∼ 0.03 Å
1 in the SAXS patterns obtained for all five nanocomposite dispersions (Figure 7). The progressive shift in the position of this feature to higher q suggests that the correlation distance between the silica particles within the shells progressively shifts with a systematic reduction in the mean nanocomposite diameter (Table 3). However, the fitting results suggest that this trend is merely apparent, rather than a meaningful observation (Table 2). The parameters of the hard-sphere Percus
Yevick model (RPY and cPY) obtained for all samples are very similar, indicating that the silica shell structure is comparable for each nanocomposite. The correlation distance for silica packing (2RPY), which originates from the hard-sphere structure factor, also provides useful information regarding the silica shell packing. The 2RPY value is similar to the mean silica particle diameter, 2Rsilica, obtained from the form factor of the particles. This suggests that the silica particles are close-packed within the nanocomposite shell. It has to be noted that the two-population model proposed in this work is based on the assumption of a dilute solution. If the concentration of the core
shell particles in the solution is high (above 1
2 vol %), then the interference effect between the waves scattered by different core
shell particles requires a structure factor to be counted in the scattering analysis (for example, by using a hard-sphere Percus
Yevick model). In general, interparticle interference reduces the scattering intensity at small angles and simultaneously generates a structural peak associated with the average distance between neighboring particles in the solution. However, these scattering effects are negligible at qR > 2 (where R is the overall particle radius).50 Therefore, the dilute solution approximation, which neglects the interference effect, is applicable for the concentrated system in a scattering region satisfying this criterion. For example, if R ∼ 1000 Å, as in the presented work, the interference effect should be observed only at q < 0.002 Å
1, which was not accessible in the SAXS experiments (Figure 7). Thus, under such circumstances the proposed formulation (eq 7, S1(q) = 1) could be used for the analysis of more concentrated solutions (e.g., 10 vol % or even higher) of core
shell particles as well. Monte Carlo simulations of core
particulate shell particles have been developed in the modeling section of this study. This 8086

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Langmuir theoretical analysis can be extended to gain a deeper understanding of the experimental parameters obtained for the real (co)polymer/silica nanocomposite particles (Table 2). Structural data for the 70/30 poly(styrene-stat-n-butyl acrylate)/silica nanocomposite particles were chosen for the simulations assuming repulsive silica
silica interactions within the shell. It was found that the absolute intensity of the calculated SAXS pattern lies within the same range as the experimental curve and overlaps with the latter at the both ends (Figure 8) where the scattering is dominated by the particle size. A notable discrepancy is observed at intermediate q, which is associated with the X-ray scattering from a densely packed layer of silica particles. The pronounced peak in the experimental data is not reproduced in the calculated SAXS pattern despite the fact that repulsive interactions have been used for the Monte Carlo simulations in order to create ordered silica shell structures. The peak position in the experimental data indicates that the silica particles are closely packed within the particulate shell, just like the simulated structures with high packing density (Figure 2, patterns 5 and 6). However, the amount of silica available in this shell is limited (D ∼ 0.5, Table 2), and is certainly not as much as would be required for full coverage of the core associated with closely packed particles (D > 0.73, Table 1). This suggests that the silica particles in the shell of the nanocomposite particles are packed in clusters confined by a spherical surface of the polymer core. The clustering of the silica particles produces a scattering peak corresponding to a close-packed arrangement of this component (RPY is comparable to Rsilica, Table 2). Moreover, this particulate silica shell has a mean thickness comparable to the mean diameter determined for isolated silica particles (ΔR ∼ 2Rsilica, Table 2) and an overall silica content that is equivalent to the total amount of silica in each nanocomposite sample (csilica ∼ c2, Table 2). Indeed, careful examination of FE-SEM images obtained for these core
shell nanocomposite particles suggests a structural organization comprising clusters within the silica shells as indicated by the SAXS analysis (Figure 6b). The silica appears to be somewhat irregularly distributed over the surface of the nanocomposite particles. Thus, the three structural models discussed in this work, which assume a uniform distribution of silica within the shell, need some further refinement to simulate the core
particulate shell structures of these colloidal nanocomposite particles. However, this task may be problematic (and arguably overelaborate), as it requires additional parameters to describe both the size of the particle clusters and the cluster distribution within the shell. Several results extracted from fits to the experimental SAXS patterns can be compared to independent experimental data to assess whether the two-population model is both robust and selfconsistent (Table 3). The core
shell nanocomposite particle diameters determined by SAXS are in good agreement with the DLS data and also follow the same general trend: as the proportion of n-butyl acrylate comonomer is systematically increased, the nanocomposite particle dimensions are gradually reduced, from 336 nm for the polystyrene/silica particles to 192 nm for the 50:50 poly(styrene-stat-n-butyl acrylate)/silica particles. The mean silica contents for the nanocomposite particles determined from SAXS data using two different approaches (based on the core
shell form factor of the first population and the volume fraction of the second population), as well as silica contents derived from TGA data (both weight % and the corresponding volume %) and nanocomposite particle densities, are summarized in Table 3. Increasing the n-butyl acrylate content of the

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nanocomposite particles leads to a systematic reduction in the mean particle diameter and a concomitant increase in silica content as determined by thermogravimetric analyses. Such a trend is of course expected for nanocomposite particles with welldefined core
shell morphologies. The silica contents calculated from the structure factor and from the absolute X-ray intensities both increase with increasing n-butyl acrylate content. Both TGA- and density-derived silica contents lie within the range encompassed by the SAXS data. Recently, it has been reported23,24 that the robust SAXS analysis of poly(2-vinylpyridine)/silica colloidal nanocomposite particles can be achieved by using a two-population model that neglects silica
silica interactions within the shell, as described by Lewis and co-workers.22 These nanocomposite particles were prepared simply by physical adsorption of ultrafine silica nanoparticles onto a preformed sterically stabilized latex. The experimental SAXS patterns for such nanocomposite particles are similar to the patterns generated using the simplest model, in which a random distribution of silica particles within the shell is assumed (see Figure 4, patterns 1 and 3). This suggests that the silica shells in these earlier nanocomposite particles are relatively disordered since there are no SAXS features associated with ordered silica packing. In contrast, in the present work the (co)polymer/silica nanocomposite particles were prepared by in situ (co)polymerization. This alternative synthetic strategy clearly leads to a much more ordered and dense silica shell and hence a pronounced peak at q ∼ 0.03 Å
1 (Figure 7). This qualitative structural difference between these two systems is consistent with their differing propensities toward silica exchange when challenged with naked latex.51

’ CONCLUSIONS A two-population model based on standard analytical expressions for scattering is proposed for the SAXS analysis of core
shell spherical nanocomposite particles containing an inhomogeneous particulate shell formed by small particles adsorbed to the surface of the large core. The model comprises two populations: (i) the first population is the core
shell structure, which describes the self-correlation term for the spherical core and the cross-term between the core and the particles forming the shell; (ii) the second population involves the particulate shell and describes the self-correlation term of the particles comprising this shell and the cross-term between the close-packed silica particles. Verification of the two-population model using both simulated particles and real nanocomposite particles confirms that this model can be successfully applied for the comprehensive SAXS analysis of core
particulate shell structures. The structural parameters used for Monte Carlo simulations of core
particulate shell structures with randomly distributed particles comprising the shell, and ordered arrangements of particles mediated by either electrostatic interactions or Lenard-Jones interactions, have been reproduced by the fitted parameters of the two-population model. For nanocomposite particles comprising shells formed by evenly distributed (i.e., ordered) silica particles, this model is reliable for silica shell concentrations ranging from packing densities close to the theoretical limit (D ∼ 0.73) to values as low as D = 0.09. SAXS analysis of a series of poly(styrene-co-n-butyl acrylate)/ silica colloidal nanocomposite particles using the two-population model allowed the overall core
shell nanocomposite particle diameter, the core diameter and polydispersity, the mean silica shell thickness, the mean silica diameter and polydispersity, the 8087

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Langmuir volume concentrations of the two components, and the packing density and structural pattern of silica particles within the shell to be determined. The results are in very good agreement with other structural techniques (e.g., electron microscopy) and physicochemical methods (dynamic light scattering, thermogravimetry, helium pycnometry, and BET surface area analysis). Undoubtedly, the high electron density contrast between the (co)polymer and the silica components, together with the relatively low polydispersity of the core
shell nanocomposite particles, makes SAXS ideally suited for the characterization of this system. However, the results of this study should be also applicable for various other types of core
particulate shell colloidal particles. In this respect, small-angle X-ray scattering can effectively replace several analytical techniques in offering the robust in situ characterization of both morphology and composition of colloidal nanocomposite particles.

’ ASSOCIATED CONTENT

bS

Supporting Information. SAXS analysis of the glycerolfunctionalized silica aqueous sol. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail addresses: O.Mykhaylyk@sheffield.ac.uk and S.P.Armes@ sheffield.ac.uk.

’ ACKNOWLEDGMENT J.A.B. thanks AkzoNobel for an Industrial CASE EPSRC studentship. A.S. thanks The University of Sheffield for funding a Ph.D. studentship. EPSRC is thanked for a Platform grant EP/ E012949/1 to support O.O.M. Dr. P. Greenwood of Eka Chemicals is thanked for donating the aqueous silica sol. The authors are grateful to STFC for providing the beamtime at Daresbury lab and thank the former personnel of station 2.1, Dr. Kalotina Geraki and Dr. Gunter Grossman, for their helpful assistance with the SAXS experiments. ’ REFERENCES (1) Balmer, J. A.; Schmid, A.; Armes, S. P. Colloidal nanocomposite particles: quo vadis? J. Mater. Chem. 2008, 18 (47), 5722–5730. (2) Wang, T.; Keddie, J. L. Design and fabrication of colloidal polymer nanocomposites. Adv. Colloid Interface Sci. 2009, 147
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