Adsorption of Small Cationic Nanoparticles onto Large Anionic Particles from Aqueous Solution: A Model System for Understanding Pigment Dispersion and the Problem of Effective Particle Density

The present study focuses on the use of copolymer nanoparticles as a dispersant for a model pigment (silica). Reversible addition–fragmentation chain transfer (RAFT) alcoholic dispersion polymerization was used to synthesize sterically stabilized diblock copolymer nanoparticles. The steric stabilizer block was poly(2-(dimethylamino)ethyl methacrylate) (PDMA) and the core-forming block was poly(benzyl methacrylate) (PBzMA). The mean degrees of polymerization for the PDMA and PBzMA blocks were 71 and 100, respectively. Transmission electron microscopy (TEM) studies confirmed a near-monodisperse spherical morphology, while dynamic light scattering (DLS) studies indicated an intensity-average diameter of 30 nm. Small-angle X-ray scattering (SAXS) reported a volume-average diameter of 29 ± 0.5 nm and a mean aggregation number of 154. Aqueous electrophoresis measurements confirmed that these PDMA71–PBzMA100 nanoparticles acquired cationic character when transferred from ethanol to water as a result of protonation of the weakly basic PDMA chains. Electrostatic adsorption of these nanoparticles from aqueous solution onto 470 nm silica particles led to either flocculation at submonolayer coverage or steric stabilization at or above monolayer coverage, as judged by DLS. This technique indicated that saturation coverage was achieved on addition of approximately 465 copolymer nanoparticles per silica particle, which corresponds to a fractional surface coverage of around 0.42. These adsorption data were corroborated using thermogravimetry, UV spectroscopy and X-ray photoelectron spectroscopy. TEM studies indicated that the cationic nanoparticles remained intact on the silica surface after electrostatic adsorption, while aqueous electrophoresis confirmed that surface charge reversal occurred below pH 7. The relatively thick layer of adsorbed nanoparticles led to a significant reduction in the effective particle density of the silica particles from 1.99 g cm–3 to approximately 1.74 g cm–3, as judged by disk centrifuge photosedimentometry (DCP). Combining the DCP and SAXS data suggests that essentially no deformation of the PBzMA cores occurs during nanoparticle adsorption onto the silica particles.


■ INTRODUCTION
It has been known for more than five decades that amphiphilic AB diblock copolymers can undergo spontaneous self-assembly in solution. 1−4 Various copolymer morphologies can be obtained depending on the diblock composition, including spherical micelles, worm-like particles, or vesicles. 5−20 It is also well-known that a range of diblock copolymers can be used as dispersants for numerous inorganic and organic pigments. 21−27 For example, diblock copolymer micelles have been adsorbed onto polystyrene (PS) latex, silica, or carbon black. 28−30 Similarly, we reported that a commercial hydrogenated polyisoprene−polystyrene diblock copolymer can form starlike spherical micelles in n-dodecane and hence act as a dispersant for carbon black 31 or diesel soot. 32 Many research groups have reported that polymerizationinduced self-assembly (PISA) offers an efficient and versatile route to a wide range of well-defined diblock copolymer nanoparticles. 33−35 One important advantage of this approach is that the copolymer is prepared directly in the form of nanoparticles at high solids, which eliminates any requirement for post-polymerization processing. Depending on the choice of monomers, colloidally stable nanoparticle dispersions can be obtained in water, ethanol, or n-alkanes. 20,35−39 Of particular relevance to the present work, poly(2-(dimethylamino)ethyl methacrylate)−poly(benzyl methacrylate) (PDMA−PBzMA) copolymer nanoparticles can be prepared directly in either ethanol or ethanol/water mixtures using reversible addition− fragmentation chain transfer (RAFT) dispersion polymerization. 40−43 There is a considerable volume of literature describing the controlled adsorption of small particles onto larger particles. 30,44,45 Depending on the conditions used, this is sometimes known as heteroflocculation or heterocoagulation. 46−49 Well-known examples include the electrostatic adsorption of a submicrometer-sized anionic PS latex onto a micrometer-sized cationic PS latex, 50 the heterocoagulation of a small cationic poly(butyl methacrylate) latex onto a large anionic PS latex followed by thermal annealing, 51 and the adsorption of globular proteins onto polypyrrole particles. 52 Previously, we reported the adsorption of 20 nm-diameter silica nanoparticles onto 463 or 616 nm diameter poly(2vinylpyridine) latexes. 53 Such heteroflocculation offers an interesting and potentially rather generic route to colloidal nanocomposite particles. 54,55 In the present study, we examine a complementary system comprising small cationic diblock copolymer nanoparticles adsorbed onto large anionic silica particles. The former component can now be readily accessed using PISA formulations 40−43 while the latter colloidal substrate is commercially available with narrow particle size distributions over a range of mean diameters. In principle, such nanoparticlecoated silica particles are a useful model system for understanding the behavior of certain paint and coating formulations. For example, copolymer nanoparticles are sometimes adsorbed onto titania particles to maximize their gloss potential. 56 They may also offer new opportunities for the preparation of colloidally stable inkjet inks 57 or carbon nanotube dispersions. 58 ■ EXPERIMENTAL SECTION Synthesis of PDMA 71 Macromolecular Chain Transfer Agent (Macro-CTA) via RAFT Solution Polymerization. A roundbottomed flask was charged with 2-(dimethylamino)ethyl methacrylate (DMA; 60.0 g), 4-cyano-4-(2-phenylethanesulfanylthiocarbonyl)sulfanylpentanoic acid (PETTC; 1.30 g; synthesized as reported by Jones et al. 41 ), 4,4-azobis(4-cyanovaleric acid) (ACVA) initiator (1.1 mg), and tetrahydrofuran (THF; 60 g). This formulation corresponds to a target degree of polymerization (DP) of 75 (assuming perfect RAFT agent efficiency) and a macro-CTA/ACVA molar ratio of 10. The sealed reaction vessel was purged with nitrogen and placed in a preheated oil bath at 66°C for 8.5 h. The crude PDMA homopolymer (DMA conversion = 75%; M n = 11 500 g mol −1 , and M w /M n = 1.24) was purified by precipitate into excess petroleum ether. The mean DP of this PDMA macro-CTA was calculated to be 71, using proton nuclear magnetic resonance ( 1 H NMR) spectroscopy by comparing integrated signals corresponding to the aromatic protons at 7.2−7.4 ppm with those assigned to the methacrylic backbone at 0.4−2.5 ppm.
Synthesis of PDMA 71 −PBzMA 100 Diblock Copolymer Nanoparticles via RAFT Dispersion Polymerization. A typical protocol for the synthesis of PDMA 71 −PBzMA 100 diblock copolymer nanoparticles at 15% w/w solids is as follows: benzyl methacrylate (BzMA, 5.0 g), PDMA 71 macro-CTA (3.26 g), and azobisisobutyronitrile (AIBN) (9.9 mg; macro-CTA/AIBN molar ratio = 5.0) were dissolved in a 85:15 w/w ethanol/water mixture. This reaction mixture was sealed in a round-bottomed flask, purged with nitrogen gas for 20 min, and then placed in a preheated oil bath at 70°C for 8 h. The final BzMA conversion was determined using 1 H NMR analysis in CDCl 3 by comparing the integral of the two benzylic protons at 4.9 ppm assigned to PBzMA to that of the two vinyl protons corresponding to the BzMA monomer at 5.2 and 5.4 ppm.
Purification of Silica Particles. Commercial silica particles (AngstromSphere microspheres; nominal 500 nm diameter) were purified to remove any surface contamination via ultrasonication of a silica dispersion in isopropanol for 30 min, followed by centrifugation at 20 000 rpm for 30 min. 59 The alcoholic supernatant was carefully decanted and discarded. This clean-up protocol was repeated twice, and the purified silica particles were dried under vacuum at 30°C overnight.
Heteroflocculation Protocol. Silica particles were dispersed in deionized water (pH 6.5) at 5.0 % w/w by stirring at 20°C for 24 h. PDMA 71 −PBzMA 100 diblock copolymer nanoparticles dispersed in water (pH ≈ 6.5) at 1.0 % w/w were added to this aqueous silica dispersion, with nanoparticle adsorption occurring at 20°C during roller mill mixing for 24 h. The number of nanoparticles per silica particle, N, was systematically varied to identify the critical value corresponding to the saturation coverage.
Gel Permeation Chromatography (GPC). The molecular weight distribution of the PDMA 71 −PBzMA 100 diblock copolymer was assessed using GPC. The GPC set-up comprised two 5 μm (30 cm) "Mixed" columns and a WellChrom K-2301 refractive index detector operating at 950 ± 30 nm. THF eluent containing 2.0 w/w % triethylamine and 0.05 w/v % butylhydroxytoluene (BHT) was used at a flow rate of 1.0 mL min −1 . A series of ten near-monodisperse linear poly(methyl methacrylate) standards (M p values ranging from 1280 to 330 000 g mol −1 ) were purchased from Polymer Laboratories (Church Stretton, UK) and used for calibration, with the above refractive index detector.
Ultraviolet (UV) Absorption Spectroscopy. Copolymer adsorption onto silica particles was verified using UV spectroscopy. Aqueous dispersions of nanoparticle-coated silica particles were centrifuged at 5000 rpm for 25 min; then, the supernatant was carefully decanted in each case. An accurate volume of supernatant solution (2.00 mL) was dried under vacuum, and then the solid residues were dissolved in 1,4-dioxane. Absorption spectra were recorded using a dark-sided quartz cuvette of 1.0 cm path length and a Varian Cary 50 UV−visible spectrophotometer operating at a scan speed of 600 nm min −1 . For the Beer−Lambert calibration plot, aqueous PDMA 71 −PBzMA 100 dispersions were dried at 30°C using a vacuum oven and then dissolved in 1,4-dioxane to produce known copolymer concentrations. These solutions were serially diluted, and each absorbance was monitored at a λ max of 307 nm. The molar absorption coefficient (ε) was determined from the linear absorbance versus concentration plot (R 2 > 0.99).
Diluted supernatants were analyzed for their absorption at λ max = 307 nm, and the corresponding concentration of non-adsorbed PDMA 71 −PBzMA 100 copolymer nanoparticles was calculated. Hence, the adsorbed amount in each case was determined by the difference.
Helium Pycnometry. The solid-state density of the purified silica particles was determined using a Micromeritics AccuPyc 1330 helium pycnometer operating at 20°C.
Transmission Electron Microscopy (TEM). Copper/palladium TEM grids (Agar Scientific, UK) were surface-coated in-house with a thin film of amorphous carbon. The grids were subjected to a glow discharge for 30 s to create a hydrophilic surface. Each aqueous copolymer-coated silica dispersion (10 μL) was adsorbed onto a treated grid for 60 s and then blotted with filter paper to remove excess liquid. Uranyl formate (10 μL, 0.75 w/w % aqueous solution) was placed onto each loaded grid for 20 s and then blotted to remove excess liquid. The grids were dried using a vacuum hose. Imaging was performed using a Philips CM100 instrument operating at 100 kV and equipped with a Gatan 1k CCD camera.
X-ray Photoelectron Spectroscopy (XPS). Dispersions of copolymer-coated silica particles were centrifuged for 25 min at 5000 rpm. Each supernatant was carefully removed, and the sediment was dried under vacuum overnight. The dried copolymer-coated silica particles were pressed onto an indium foil before measurement. Control samples comprising bare silica particles and PDMA 71 − PBzMA 100 copolymer nanoparticles were also prepared on the indium foil. XPS studies were conducted using a Kratos Axis Ultra "DLD" instrument equipped with a monochromatic Al Kα X-ray source (hν = 1486.6 eV) operating at a base pressure in the range of 10 −8 to 10 −10 mbar.
Dynamic Light Scattering (DLS). Hydrodynamic particle diameters were determined using a Malvern Zetasizer NanoZS model ZEN 3600 instrument, equipped with a 4 mW He−Ne solidstate laser, operating at 633 nm. Back-scattered light was detected at Langmuir Article DOI: 10.1021/acs.langmuir.6b04541 Langmuir 2017, 33, 1275−1284 173°, and the mean particle diameter was calculated by the quadratic fitting of the correlation function using the Stokes−Einstein equation over thirty runs of 10 s duration. Highly dilute aqueous dispersions were analyzed using disposable plastic cuvettes at 25°C; all measurements were performed three times, and mean particle diameters were reported. Aqueous Electrophoresis. Zeta potentials and mobilities were determined for silica sols, copolymer nanoparticles, and various nanoparticle-coated silica particles dispersed in the presence of 1 mM KCl background salt using the same Malvern Zetasizer NanoZS instrument described above. The solution pH was adjusted manually using either 0.1 M KOH or 0.01 M HCl.
Small-Angle X-ray Scattering (SAXS). SAXS patterns were collected at a synchrotron source (Diamond Light Source, station I22, Didcot, UK) using monochromatic X-ray radiation (wavelength λ = 0.0998 nm, with q ranging from 0.01 to 0.2 Å −1 , where q = 4π sin θ/λ is the length of the scattering vector and θ is half of the scattering angle) and a 2D Pilatus 2 M pixel detector (Dectris, Switzerland). Polycarbonate capillaries of 2.0 mm diameter were used as a sample holder. SAXS data were reduced (normalization and integration) using the Dawn software supplied by Diamond Light Source. Scattering data were further analyzed (background subtraction and data modeling) using Irena SAS macros for Igor Pro. 60 The structural model used for the SAXS data analysis is given in the Supporting Information. This model is based on the analytical expression for a spherical micelle form factor. 61 Disk Centrifuge Photosedimentometry (DCP). A DC24000 instrument (CPS Instruments, USA) was used to obtain the weightaverage particle size distributions. This disk centrifuge employs a 405 nm diode sensor for particle detection at the disk boundary. The rate of centrifugation used in the present study was 5124 rpm. After reaching this speed, a density gradient was generated in situ by filling the empty disk with an aqueous spin fluid (total volume = 14.4 mL). An 8−24 w/w % aqueous sucrose gradient was prepared by injecting 1.50 mL of nine aqueous sucrose solutions in turn (beginning with the most concentrated solution), followed by n-dodecane (0.50 mL) to extend the gradient lifetime. After equilibration for 20 min, the instrument was calibrated by injecting 100 μL of 476 nm nearmonodisperse poly(vinyl chloride) latex particles (Analytik Ltd., UK) before each measurement. Particle size distributions for both bare silica and nanoparticle-coated silica particles were recorded after injection of a 0.25 w/w % aqueous dispersion in each case.
Thermogravimetric Analysis (TGA). Analyses were conducted using a Q500 TGA instrument (TA Instruments) on dried sediments of aqueous dispersions of nanoparticle-coated silica particles obtained after centrifugation for 25 min at 5000 rpm, followed by drying under vacuum. Each sample was heated under a nitrogen atmosphere and initially held at 100°C for 30 min to ensure complete removal of water before heating up to 800°C at 10°C min −1 . The observed mass loss was attributed to the complete pyrolysis of the copolymer component, with the remaining incombustible residues attributed to silica. The bare silica particles also exhibited a mass loss of 4.5% owing to surface moisture. This value was used to correct the mass losses obtained for the nanoparticle-coated silica particles.

■ RESULTS AND DISCUSSION
PDMA macro-CTA was synthesized by RAFT solution polymerization in THF using a trithiocarbonate-based RAFT agent (PETTC; prepared as reported previously). 41 Following purification, 1 H NMR studies indicated a mean DP of 71 for this PDMA macro-CTA, which was then chain-extended with BzMA via RAFT dispersion polymerization in an 85:15 ethanol/water mixture (see Scheme 1a). 1 H NMR confirmed that almost complete BzMA polymerization occurred after 8 h at 70°C; THF GPC analysis showed an increase in the molecular weight relative to that of the PDMA macro-CTA, and the relatively low final copolymer polydispersity (M w /M n = 1.28) indicated a well-controlled polymerization (see Figure   S1). PDMA−PBzMA diblock copolymer nanoparticles were synthesized at 15 w/w % solids and then diluted to 1.0 w/w % in water before mixing with silica particles, which serve as a model pigment.
An AngstromSphere silica sol was purified before use to remove any organic surface impurities. 59 Helium pycnometry indicated the density of the dried, purified silica sol to be 1.99 g cm −3 . This silica sol was then redispersed in deionized water at 5.0 w/w % solids before analysis. Figure 1a shows the TEM image of these purified silica particles. The particle size distribution was also assessed using DCP, which reported a D w of 470 ± 17 nm for the major population, with an additional minor population at 580 nm corresponding to doublets ( Figure  1b).
The PDMA 71 −PBzMA 100 diblock copolymer nanoparticles were characterized using TEM, DLS, and SAXS (see Figure 2). DLS gave an intensity-average hydrodynamic diameter (D H ) of 30 ± 4 nm; TEM studies confirmed a spherical morphology with a number-average diameter (D n ) of 25 ± 3 nm (counting  Figure 2c. This curve was fitted using a micelle model 61,62 assuming a Gaussian size distribution, yielding an overall weight-average diameter (D w ) of 29 ± 0.5 nm and a weight-average PBzMA-only core diameter of 19.6 ± 0.9 nm (see Supporting Information for further details). From this analysis, the radius of gyration of the PDMA stabilizer was calculated to be 2.4 nm and the mean aggregation number (N agg ) for the nanoparticles was determined to be 154 (with negligible solvation of the coreforming PBzMA block). The difference in size simply reflects the differing moments reported by each technique, as previously reported. 43 In addition, TEM is largely insensitive to the PDMA stabilizer layer, whereas SAXS and DLS account for this structural feature.
Aqueous dispersions of silica (5.0 w/w %) and PDMA 71 − PBzMA 100 nanoparticles (1.0 w/w %) were mixed overnight to obtain nanoparticle-coated silica particles. The copolymer/silica mass ratio was systematically varied to identify the conditions corresponding to the monolayer coverage. The packing efficiency was calculated using the method reported by Balmer et al. 53 Hence the precise copolymer/silica mass ratio required for saturation coverage was determined. The number of silica spheres (n s ) and copolymer spheres (n p ) were calculated using eqs 1 and 2, respectively. Here, n s and n p depend on the mass of the components (m s for the silica sol and m p for copolymer nanoparticles), the mass density of silica (ρ s ), N agg , and the number-average molecular mass of the copolymer chains (M n ). The mean silica radius (r s ) is taken from the weight-average diameter reported using DCP. N is the mean number of added copolymer nanoparticles per silica particle, as shown in eq 3. The mean copolymer nanoparticle radius (r p ) is taken from the intensity-average diameter determined using DLS. The final packing efficiency (P) can then be calculated using the N was systematically varied by fixing the silica concentration and varying the copolymer concentration to enable the construction of a plot of DLS hydrodynamic diameter against N (see Figure 3a). This plot was then used to estimate the critical value of N corresponding to the monolayer coverage, as described by Balmer et al. 53 For N values greater than that of the inflection point (indicated by a vertical dashed line), the apparent particle diameters are relatively small and approximately constant, which indicates a high degree of dispersion owing to steric stabilization at or above saturation coverage of the silica particles by the cationic copolymer nanoparticles. In contrast, much larger apparent diameters were observed at lower N values, which correspond to lower coverage and hence bridging flocculation of the silica particles. This interpretation was supported by TEM observations: Figure 3b−d shows images recorded at selected N values. When N < 465 (i.e., below monolayer coverage), partially coated flocculated silica particles are observed (Figure 3b). At N = 465, the silica particles are well-coated with cationic nanoparticles (Figure 3c). Higher N values lead to excess copolymer nanoparticles being observed in the background, in addition to nanoparticle-coated silica particles (Figure 3d). For this system, saturation coverage was obtained at N ≈ 465, which equates to P = 0.42 (or 42%). This fractional surface coverage is somewhat lower than that observed by Balmer et al., who reported P = 0.69 (or 69%) for 463 nm-diameter P2VP latex particles coated with 20 nmdiameter silica particles. 53 The lower surface coverages observed in the present study reflect both the sterically stabilized nature of the cationic nanoparticles and their strong electrostatic interactions with the anionic silica particles, which reduces the possibility of nanoparticle rearrangement after adsorption.
Aqueous electrophoresis was used to assess the electrostatic adsorption of the PDMA 71 −PBzMA 100 diblock copolymer nanoparticles onto the silica particles. The silica particles are anionic at high pH as a result of ionization of their surface silanol groups. On the other hand, the diblock copolymer nanoparticles are cationic over a wide pH range because the PDMA stabilizer chains are protonated below their pK a of approximately 7.5. 63 Thus adsorption of these cationic nanoparticles onto the anionic silica particles at around pH 7 leads to surface charge reversal and therefore a change in the overall zeta potential from negative to positive.
Electrophoretic mobilities (μ) were determined using phase analysis light scattering. As the nanoparticle radius is relatively small compared to the Debye length (κ −1 ), the Helmholtz− Smoluchowski equation is not applicable. 64 However, as the magnitude of the zeta potential is relatively low (< 50 mV), 64 the Henry equation can be used instead. 65 This equation includes a prefactor f(κa) term that has been calculated using Ohshima's equation for any value of κa, where κ is the inverse Debye length and a is the particle radius. 66 Figure 4 shows the zeta potential versus pH curves obtained for PDMA 71 −PBzMA 100 nanoparticles, bare silica particles, and the resultant nanoparticle-coated silica particles at saturation coverage (N = 465). The diblock copolymer nanoparticles are cationic at or below pH 9, the bare silica sol is anionic above pH 4, and the nanoparticle-coated silica particles exhibit an intermediate isoelectric point at around pH 7. Thus these data suggest that the cationic nanoparticles indeed adsorb onto the anionic silica particles. (PDMA 71 −PBzMA 100 nanoparticles become anionic above pH 9 as a result of ionization of the terminal carboxylic acid groups derived from the RAFT CTA. Similar observations have been reported previously for this system 40,41 ).
UV spectroscopy was used to assess the efficiency of copolymer nanoparticle adsorption. Figure 5a shows the absorption spectra recorded at 307 nm for copolymer solutions of known concentration in 1,4-dioxane. This is a good solvent for both blocks, which leads to nanoparticle dissolution and hence eliminates the problem of particle scattering at shorter wavelengths. The corresponding calibration plot (see Figure  5b) yielded the molar absorption coefficient for the trithiocarbonate end-group at this wavelength, which was used to determine the concentration of non-adsorbed nanoparticles present in the supernatant. After mixing, the dispersions were centrifuged at 5000 rpm for 25 min, and the supernatant was carefully decanted. This was then dried, and the non-adsorbed copolymer was dissolved in 1,4-dioxane for analysis using UV spectroscopy. Figure 5c shows the amount of excess copolymer present for various N values. At N ≤ 465, there is almost no residual copolymer, indicating that all nanoparticles have adsorbed onto the silica particles. Above this critical value, the amount of copolymer present in the supernatant increases as more copolymer is added, as expected.
To assess the mass of the adsorbed nanoparticles directly, TGA studies were conducted on the sedimented nanoparticlecoated silica particles obtained after centrifugation. Samples were heated at 100°C for 30 min to correct for any mass loss due to the remaining solvent before heating up to 800°C. The observed mass loss between 100 and 800°C is attributed to the pyrolysis of the adsorbed copolymer nanoparticles. Below saturation coverage, the nanoparticle-coated silica particles exhibited a monotonic increase in mass loss up to N = 465, for which the mass loss was approximately 3.3%. This is consistent with the UV spectroscopy analysis, which indicated that essentially all copolymer nanoparticles were adsorbed onto the silica particles up to N = 465. It was anticipated that all samples prepared at or above monolayer coverage would exhibit the same mass loss because all excess copolymer nanoparticles remaining in the supernatant should be removed before TGA. However, in practice, the experimental mass loss varied by ±0.5%, which is an indication of the experimental error associated with this protocol. Nanoparticle-coated silica particles prepared at three different copolymer/silica mass ratios were analyzed using XPS. Theoretical surface coverages of the silica by the nanoparticles were 31, 42, and 52%, respectively, as calculated using eq 4. A summary of the XPS data is given in Table 1. X-ray photoelectron survey spectra confirmed the presence of the N1s signal (which serves as a unique elemental marker for the PDMA 71 −PBzMA 100 copolymer nanoparticles) and the Si2p signal (which serves as a unique elemental marker for the silica component 67 ). In all cases, underlying silica particles are detected in addition to the adsorbed layer of copolymer nanoparticles. Because the mean nanoparticle diameter exceeds the typical XPS sampling depth of 2−5 nm, 68 this indicates gaps between the adsorbed copolymer nanoparticles. This was expected given the relatively low theoretical fractional surface coverages calculated using eq 4 and reported in Table 1. Experimental values are calculated by comparing the surface nitrogen content of a given sample with that obtained for the pristine copolymer nanoparticles. The data in Table 1 indicate that the experimental fractional surface coverages determined using XPS follow the same trend as the theoretical values calculated using eq 4. However, the former values are somewhat higher in all cases, which may indicate perturbation of the original copolymer morphology after nanoparticle adsorption onto the silica particles. The experimental fractional surface coverage of 0.51 (or 51%) is comparable to that determined by Balmer et al. for the physical adsorption of silica nanoparticles on P2VP latexes. 53 Aqueous electrophoresis measurements were performed for both bare and nanoparticle-coated silica particles (at pH 6.4). Electrophoretic mobilities (μ) were converted into zeta potentials using the Henry equation, as described earlier. The bare silica particles were highly anionic, as expected. On addition of cationic copolymer nanoparticles, this negative surface charge was first neutralized and then reversed (see Figure 6). Zeta potentials recorded for the nanoparticle-coated silica particles increased approximately linearly up to N = 290, and as N was further increased, the zeta potential reached a plateau value of around +40 mV. If the variation in the zeta potential at low N is subject to linear interpolation, then neutralization of the particle surface charge occurs at N ≈ 144. According to Figure 3a, this critical value lies well within the bridging flocculation regime. The zeta potential continues to increase monotonically as more cationic nanoparticles are adsorbed, and the "knee" of this plot corresponds to approximately N = 465, or monolayer coverage.  Adsorption of low-density copolymer nanoparticles onto silica leads to a lower effective density for the resulting nanoparticle-coated silica particles compared with that of bare silica particles. This is potentially of interest when formulating paints and coatings because such composite particles should be more buoyant within the aqueous phase and hence be less susceptible to gravitational settling over extended time periods. As stated earlier, DCP studies reported a mean weight-average diameter of 470 ± 17 nm for the bare silica sol (Figure 1b). However, using the solid-state silica density determined using helium pycnometry (1.99 g cm −3 ) for the nanoparticle-coated silica particles leads to an erroneous underestimated weightaverage diameter of 437 ± 20 nm. This artifact highlights the importance of correcting for the effective particle density in such centrifugal sizing measurements. Figure 7 shows two possible scenarios for the PDMA 71 − PBzMA 100 nanoparticles after their electrostatic adsorption onto the 470 nm silica particles: (1) no deformation of the PBzMA cores and (2) partial deformation of the PBzMA cores to form hemispheres (and hence a somewhat thinner adsorbed layer). The effective density of the nanoparticle-coated silica particles can be calculated theoretically from the sum of the volume fraction (φ)-weighted mass densities (ρ) using eq 5.
Here the core is silica and the shell is a mixture of copolymer nanoparticles and the entrained solvent. To a good first approximation, the density of this shell can be taken to be 1.00 g cm −3 because of the relatively high volume fraction occupied by the solvent (water). ρ φ ρ φ ρ = + particle core core shell shell (5) If no deformation of the PBzMA cores is assumed, then an upper limit of the shell thickness can be calculated. Based on SAXS measurements, the overall volume-average diameter of the sterically stabilized nanoparticles in solution is 29 nm. The R g of the PDMA stabilizer chains is approximately 2.4 nm, suggesting a mean stabilizer layer thickness of 4.8 nm.
Assuming that the stabilizer chains in direct contact with the silica surface are fully collapsed after nanoparticle adsorption, the shell thickness is calculated to be 24.2 nm. This gives a theoretical effective particle density of 1.74 g cm −3 . Alternatively, the PBzMA cores may become partially deformed after their adsorption onto the silica particles. In this case, the shell thickness can be estimated by assuming that the PBzMA cores form hemispheres on the silica surface, giving a mean adsorbed layer thickness of 14.5 nm and a corresponding theoretical effective particle density of 1.83 g cm −3 .
DCP was used to assess which of the two nanoparticle adsorption scenarios depicted in Figure 7 is likely to be physically realistic. Inputting a theoretical effective density of 1.74 g cm −3 produced a weight-average particle diameter of 524.9 nm (see Figure 8). This value lies quite close to the mean diameter of 470 nm + (2 × 24.2 nm) = 518.4 nm, calculated for the nanoparticle-coated silica particles; this corresponds to scenario 1 (i.e., retention of the original spherical nanoparticle morphology after adsorption). However, inputting the theoretical effective density of 1.83 g cm −3 suggested by scenario 2 produced a weight-average particle diameter of only 482.8 nm. This value lies further from the overall weightaverage diameter of 470 nm + (2 × 14.5 nm) = 499 nm calculated for this alternative scenario, which assumes partially deformed adsorbed nanoparticles. In summary, the combined DCP and SAXS data suggest that the PBzMA cores remain essentially unperturbed after nanoparticle adsorption onto the silica particles, as represented by scenario 1 in Figure 7. The modest discrepancy between the theoretical and DCP-derived weight-average diameters is at least partly the result of assuming that the adsorbed PDMA chains in direct contact with the silica surface occupy negligible volume. We note that the lack of deformation of the nanoparticle cores during adsorption is consistent with the minimal solvation of the PBzMA chains indicated by SAXS analysis. Moreover, TEM studies of Figure 6. Change in the zeta potential recorded at pH 6.4 for nanoparticle-coated silica particles as a function of the number of copolymer nanoparticles added per silica particle, N. The zeta potential becomes approximately zero when N = 144 and reaches a plateau value of +40 mV for higher N values. Figure 7. Schematic cartoon showing PDMA 71 −PBzMA 100 diblock copolymer nanoparticles adsorbed onto 470 nm silica particles for two differing adsorption scenarios. Scenario 1 corresponds to no deformation of hydrophobic PBzMA cores during adsorption, whereas scenario 2 depicts a partial deformation of these PBzMA cores during adsorption to form hemispheres (and hence a somewhat thinner adsorbed layer of nanoparticles). In both cases, the red cationic stabilizer chains in direct contact with the anionic silica surface are collapsed and occupy negligible volume.

■ CONCLUSIONS
In summary, we have examined the electrostatic adsorption of small sterically stabilized cationic diblock copolymer nanoparticles onto large anionic silica particles, which serve as a model pigment for aqueous formulations. This study demonstrates that copolymers can act as effective dispersants when present in nanoparticle form, rather than as molecularly dissolved chains. Relatively high degrees of dispersion can be achieved either at or above saturation coverage of the silica particles, whereas extensive bridging flocculation is observed at lower coverages. The combination of SAXS and disk centrifuge data suggests that the copolymer nanoparticles adsorb onto silica without significant perturbation of their original morphology. Such systems enhance our understanding of the problem of effective particle density and are also likely to be of interest for certain advanced aqueous pigment and coating formulations.
■ ASSOCIATED CONTENT

Notes
The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS
We thank EPSRC for the postdoctoral support for E.R.J. (EP/ J018589/1). Lubrizol is thanked for partial support of this project in the form of project consumables. S.P.A. is the recipient of a five-year ERC Advanced Investigator grant (PISA 320372). The authors are grateful to Diamond Light Source (Didcot, U.K.) for providing SAXS beam-time. The personnel of I22 station are thanked for their help with the SAXS experiments. Figure 8. Normalized weight-average particle size distributions obtained using DCP for bare silica particles (density = 1.99 g cm −3 ) and nanoparticle-coated silica particles before and after correction for their effective particle densities. The combined DCP and SAXS data suggest that the PBzMA cores remain essentially unperturbed after nanoparticle adsorption onto the silica particles (see schematic cartoon depicting scenario 1 in Figure 7).

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