Depletion Stabilization in Nanoparticle–Polymer Suspensions: Multi

Article pubs.acs.org/Langmuir

Depletion Stabilization in Nanoparticle−Polymer Suspensions: MultiLength-Scale Analysis of Microstructure Sunhyung Kim,† Kyu Hyun,‡ Joo Yong Moon,§ Christian Clasen,† and Kyung Hyun Ahn*,§ †

Department of Chemical Engineering, KU Leuven, University of Leuven, W. de Croylaan 46, B-3001 Heverlee, Belgium School of Chemical and Biomolecular Engineering, Pusan National University, Jangjeon-Dong 30, Busan 609-735, Republic of Korea § School of Chemical and Biological Engineering, Institute of Chemical Process, Seoul National University, Seoul 151-744, Republic of Korea ‡

ABSTRACT: We study the mechanism of depletion stabilization and the resultant microstructure of aqueous suspensions of nanosized silica and poly(vinyl alcohol) (PVA). Rheology, small-angle light scattering (SALS), and small-angle X-ray scattering (SAXS) techniques enable us to analyze the microstructure at broad length scale from single particle size to the size of a cluster of aggregated particles. As PVA concentration increases, the microstructure evolves from bridging flocculation, steric stabilization, depletion flocculation to depletion stabilization. To our surprise, when depletion stabilization occurs, the suspension shows the stabilization at the cluster length scale, while maintaining fractal aggregates at the particle length scale. This sharply contrasts previously reported studies on the depletion stabilization of microsized particle and polymer suspensions, which exhibits the stabilization at the particle length scale. On the basis of the evaluation of depletion interaction, we propose that the depletion energy barrier exists between clusters rather than particles due to the comparable size of silica particle and the radius gyration of PVA. stabilization arises. Semenov24 showed that, in semidilute polymer solution, depletion energy barrier exists at h ≈ ξ, where h is the particle separation and ξ is the correlation length of polymer concentration fluctuation. Very recently, depletion stabilization has found its promising application in the field of nanobioscience by Zhang et al.,32,33 who demonstrated depletion stabilization as a novel way to stabilize the nanoparticles in a wide range of pH and ionic strength, which charge stabilization cannot achieve. Furthermore, in contrast to steric stabilization, depletion stabilization does not require a specific binding with polymers; thus, it enables the particle surface to be accessible to reactions with various kinds of molecules in the medium. To utilize such advantages of depletion stabilization in advanced applications, the variables that affect the suspension microstructure should be systematically understood. One important variable in depletion interaction is the size ratio of the particle and polymer, ρ = 2Rg/d, where Rg is the radius of gyration of polymer and d is the particle diameter. For ρ ≪ 1 (i.e., the size of polymer is much smaller than the size of the particle), it is assumed that the polymer chain, a sphere with a radius of Rg, is located between two parallel walls, and the range of depletion interaction is of the order of polymer size (i.e., depletion layer thickness is approximately Rg). Theories26,27,34 based on the

1. INTRODUCTION Dispersion stability of nanosized colloidal particles in polymer solution is an important scientific and technological subject in various disciplines including nanoparticle printing,1,2 chemical and biological sensing,3 and polymer nanocomposite.4−6 In polymer solution, the dispersion stability of nanoparticles is affected by both adsorbed and nonadsorbed polymers. At a relatively low polymer concentration (ϕ), when surface coverage is much below the saturation, suspensions may be destabilized due to polymer bridging. As ϕ increases and the surface of particle is sufficiently covered by adsorbed polymer, steric stabilization dominates. With a further increase in ϕ after polymer saturates particle surface, nonadsorbed polymers play a role by introducing depletion interaction. The magnitude of depletion interaction is proportional to the osmotic pressure in polymeric medium. This results in strong depletion attraction, which gives rise to a depletion flocculation.7−17 Further increase in ϕ above depletion flocculation, the particles are stabilized again, which is termed as depletion stabilization.18 Depletion stabilization has been observed in many experimental studies,19−23 but its mechanism still remains a subject of debate.24−29 Recent theoretical calculations2829,30 and experimental works31 reveal that the depletion interaction has both short-range attractive minimum and long-range repulsive barrier, and both the magnitudes increase with the increase in polymer concentration. Above a certain polymer concentration, the repulsive energy barrier becomes high enough to allow particles kinetically stabilized; thus depletion © 2015 American Chemical Society

Received: November 24, 2014 Revised: January 22, 2015 Published: January 22, 2015 1892

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where σ is the width parameter of the Schulz distribution function, from SAXS intensity with 0.3 wt % charge-screened silica suspension.39 Poly(vinyl alcohol) (PVA, Sigma-Aldrich, molecular weight (31−50) × 103 g/mol, density 1.2 × 103 kg/m3, and degree of hydrolysis (DH) 87% according to the supplier) was used to prepare 15 wt % of PVA stock solution by dissolving PVA in distilled (DI) water at 80 °C for 5 h. A proper amount of colloidal silica and PVA solution were mixed to prepare silica/PVA suspensions of ϕPVA from 0.25 to 8 wt % at the fixed silica concentration of 10 wt %. After preparing the suspension, we reduced pH of the suspensions to pH 3 (near isoelectric point) to minimize the charge contribution on the particle interaction.38 Once reducing pH to 3, the silica particles lost the charge stabilization and started to form a flocculated microstructure which only depends on the contribution of adsorbed and nonadsorbed polymers. The zeta potential of the silica in the suspension was measured to −3 ± 2 mV at pH 3 for the whole suspensions, which allows us to ignore the effect of charge on the particle interaction. All the measurements were carried out after 24 h of stirring at room temperature. The size ratio of PVA and silica, ρ = 2Rg/d, needs to be specified, where Rg is the radius of gyration of PVA and d is the particle diameter. It is hard to determine accurate Rg of PVA in the suspension, but literatures show that 2Rg of PVA in dilute solution with similar molecular weight and degree of hydrolysis as used in this study is within the range of 14 and 20 nm, depending on the measurements.40 On the basis of the literature, we assume 2Rg = 17 nm at ϕPVA = 0.25 wt % which depends on ϕPVA according to the following scaling dependence, Rg ∝ ϕ−1/8.41 Calculated 2Rg is varied from 13.5 nm at ϕPVA = 2 wt % to 11.3 nm at ϕPVA = 8 wt %, resulting in ρ varied from ρ = 0.89 at ϕPVA = 2 wt % to ρ = 0.75 at ϕPVA = 8 wt %. This allows us to approximately, but reasonably, assume that ρ ≈ 0.8 for the silica/ PVA suspensions used in this study. Rheological Measurements. All the rheological measurements were conducted with an ARES strain-controlled rheometer (TA Instruments) using a cone and plate fixture. The geometry of the cone was 50 mm diameter and 0.0398 rad angle. Temperature was maintained at 25 °C during measurements. Small-Angle Light Scattering (SALS). The SALS experiments were performed to characterize the structure of silica/PVA suspension. The setup consisted of a 632 nm He−Ne laser, a semitransparent screen, and a 10-bit high-resolution digital camera (TM-1300 from PULNIX), connected to a digital frame grabber (TCi-Digital SE from Coreco). The sample was placed between two slide glasses with 100 μm gap. The sample-to-detector distance was 27 cm, covering a q range of 0.12−2.35 μm−1. The images were analyzed with the software developed in KU Leuven.42 After removing the background intensity and relavant uncertainties around the beamstop, q range under the investigation was determined to 0.26−1.7 μm−1, corresponding length scale of 3.69−24.2 μm. Small-Angle X-ray Scattering (SAXS). X-ray scattering experiments were performed at an undulated PLS-II 9A U-SAXS beamline of PAL (Pohang accelerator laboratory, Korea). The two-dimensional scattering images from the sample of 1 mm thickness were recorded on a CCD detector (Rayonix 2D SX165). The wavelength of X-ray beam was 1.12 Å, and the sample-to-detector distances was 4511 mm, covering a q range of 0.06−1 nm−1. After removing the background intensity and relavant uncertainties around the beamstop, the q range under investigation was determined 0.12−0.7 nm−1. The corresponding length scale was 8.97−52.3 nm, which represents approximately the length scale up to four particles. Adsorption Measurement. The amount of adsorbed polymer on silica surface was measured from the polymer concentration in the medium before and after polymer adsorption.22 Particles in the suspension were sedimented by centrifuging at 23140g force for 2 h, and the supernatant solutions were separated. The supernatant solution was dried at 70 °C for 24 h. The concentration of polymer in the supernatant was measured by weighing the dried polymer.

Asakura−Oosawa model35 can adequately predict the particle interaction and phase behavior of particle−polymer mixtures for ρ ≪ 1. However, when the particle size becomes nanoscale, which is similar to the size of polymer chain (i.e., ρ increases to unity), as far as its interaction with the particles is concerned, the overall shape of the polymer becomes nonspherical; thus, the depletion layer thickness deviates significantly from Rg. In addition, polymer−polymer interaction may not be nonadditive anymore (i.e., limited overlap of polymer chains), which is different from the Asakura−Oosawa model assuming purely nonadditive polymer interaction (complete overlap of polymer chains).36 Therefore, the prediction of phase behavior by these models26,27,34,35 is expected to be invalid anymore. A study by Ramakrishnan et al.,37,38 using the polymer reference interaction site model (PRISM), showed that, when ρ approaches unity, the polymer−polymer interaction should be taken into account on depletion interaction and resulting phase behavior. Although there have been considerable efforts to understand microstructure and phase behavior by depletion interaction for comparable particle to polymer size ratio (ρ ≈ 1),9,29,30 the majority of the studies were focused only on the depletion flocculation, which occurs at a relatively low polymer concentration much below the concentration that depletion stabilization occurs. Furthermore, most experimental19−22,31 and theoretical studies24−30 on the depletion stabilization have treated micron-sized particles in polymer solution; thus, they mainly focus on the condition of ρ ≪ 1. Therefore, they may not be adequate to explain the mechanism of depletion stabilization when the particles are nanosized, which would be comparable to the radius of gyration of polymers in the medium. In this study, the mechanism of depletion stabilization and the resulting microstructure of nanoparticle/polymer suspensions are investigated where the polymer to particle size ratio is close to unity (i.e., ρ ≈ 0.8). Colloidal silica and poly(vinyl alcohol) (PVA) suspensions with varying PVA concentration (ϕPVA) are employed as a model nanoparticle/polymer suspension. Together with rheology, small-angle light scattering (SALS), and small-angle X-ray scattering (SAXS) techniques, we characterize the suspension microstructure in a broad length scale from particle to cluster size. As ϕPVA increases, we observe consecutive stages of dispersion stability from bridging flocculation, steric stabilization, depletion flocculation to depletion stabilization. When depletion stabilization occurs, the suspension exhibits stabilization at the cluster length scale, while maintaining the fractal aggregates at the particle length scale. This shows a marked difference from the depletion stabilization of microsized particles in polymer solution (i.e., at ρ ≪ 1), which displays the stabilized microstructure at the particle length scale.22−30 The evaluation of depletion interaction suggests that the depletion energy barrier exists between clusters, rather than between particles, because of the similar size of silica particle and the radius of gyration of PVA chains.

2. EXPERIMENTAL METHODS Sample Preparation. Aqueous suspensions of silica and PVA were prepared by mixing colloidal silica and PVA solution with an appropriate ratio. Electrically stabilized 30 wt % colloidal silica (Ludox HS-30, Sigma-Aldrich, specific surface area 220 m2/g, density 2.37 × 103 kg/m3 according to the supplier) is used as received. The diameter of silica particle was measured to 15 nm with σ/d = 0.145, 1893

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Figure 1. (a, c) Steady shear viscosity of silica/PVA suspensions at different ϕPVA as indicated. (b, d) Frequency-dependent storage modulus (G′, closed symbol) and loss modulus (G″, open symbol) of silica/PVA suspensions at different ϕPVA.

3. RESULTS Rheology. Rheological properties of silica/PVA suspensions at different ϕPVA are displayed in Figure 1. Steady shear viscosity at ϕPVA from 1.5 wt % to 4 wt % (Figure 1a) shows the increase in zero shear viscosity (η0) with strong shear thinning behavior as ϕPVA increases. The frequency-dependent storage (G″) and loss moduli (G″) at the same range of ϕPVA (Figure 1b) show the behavior of a viscous liquid at ϕPVA = 1.5 wt % (i.e., G′ ∼ ω2, G″ ∼ ω1) but a gradual increase in both G′ and G″ as ϕPVA increases. The observed rheological behavior indicates the silica particles form a flocculated structure as ϕPVA increases. Notably, G′ and G″ show the scaling behavior at ϕPVA = 3 and 4 wt % (i.e., G′ ∼ G″ ∼ ω0.5 at ϕPVA = 4 wt %). The power law behavior is a characteristic feature of a critical gel, which is observed at the gel point of sol−gel transition.43 Microstructure of the critical gel is known to consist of selfsimilar fractal clusters,43−45 which should be distinguished from sample-spanning percolated 3D networks characterized as frequency-independent G′ at low ω and yield behavior.46 Therefore, the existence of η0 and the observed power law behavior at the range of ϕPVA from 3 to 4 wt % may suggest that the particles form a cluster of fractal aggregates in this range. Further increase in ϕPVA above ϕPVA = 4 wt % results in a reduction of η0 with a weakening of shear thinning behavior (Figure 1c) and a reduction of both G′ and G″ with a loss of the power law behavior (Figure 1d), indicating that the suspension exhibits the stabilization. It is notable that the suspension at ϕPVA = 7 wt % shows a Newtonian behavior (Figure 1c) with a slight elasticity (G′ ∼ ω1.6 in Figure 1d). The viscoelastic property of PVA solution at the same ϕPVA was measured as purely viscous (i.e., unmeasureble G′ and G″ ∼ ω1, result not shown here). This implies the Newtonian behavior is a contribution of PVA medium, and the slight elasticity reflects the presence of some microstructure in the suspension. To understand the evolution of rheological behavior with increasing ϕPVA, we characterize the microstructure of silica/

PVA suspension by means of SAXS and SALS techniques in the following section. Small-Angle X-ray Scattering (SAXS). In this section, we characterize spatial arrangement of silica particles in the suspensions with SAXS measurement as we increase ϕPVA from 0 to 8 wt %. In silica/PVA suspension at pH 3, PVA is known to be strongly adsorbed onto silica surface by hydrogen bonding, which significantly influences particle interaction as well as microstructure and rheolgy.47 To determine the contribution of polymers on the particle interaction and the resultant microstructure, we analyze the SAXS intensity together with the PVA adsorption. Figure 2 shows the amount

Figure 2. Admount of PVA adsorption Γ. Solid line is the Langmuir isotherm eq 1.

of adsorbed PVA on the silica surface (Γ) as a function of ϕPVA. Γ exhibits a gradual increase to approach a plateau, which is expressed as a form of Langmuir isotherm48 Γ = Γ0

KϕPVA 1 + KϕPVA

(1)

where Γ0 is the adsorption at saturation and K is the Langmuir equilibrium constant (in Figure 2, K = 200 and Γ0 = 0.44 mg/ m2). It is known that bridging attraction dominates dispersion stability at low Γ, but steric repulsion increasingly dominates as 1894

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Figure 3. (a, b) SAXS spectra and (c, d) structure factor of silica/PVA suspensions with respect to ϕPVA from 0 to 1.25 wt % as indicated. The solid line in (a) is the form factor.

increasing upturn at the lowest q as ϕPVA increases (Figure 5b), indicating again a flocculating process. In particular, I(q) (Figure 5a) shows a power law slope at ϕPVA = 4 wt % without a shoulder at high q, suggesting that the particles form fractal clusters, which is also supported by a power law behavior of G′ and G″ in rheological measurements (Figure 1b). Formation of fractal cluster has been understood as a consequence of depletion flocculation in particle/polymer suspensions.12,16,51 In silica/PVA suspension, Γ is nearly saturated and approaches a plateau in this range of ϕPVA; thus, the increased amount of PVA mainly remains in the medium rather than in the adsorption layer. The increase in PVA in the medium would increase the magnitude of depletion interaction; thus, the microstructure at 1.5 wt % < ϕPVA < 4 wt % can be explained based on depletion flocculation.11,12,14,15 When ϕPVA increases more than 4 wt %, I(q) still shows a power law behavior (Figure 5b), indicating silica particles form fractal clusters. It should be emphasized that in this range of ϕPVA a reduction of both η0 and G′ (Figures 1c and 1d, respectively) was observed, implying stabilization of particles in the suspension. Current SAXS data at this range of ϕPVA do not explain the reduction of both η0 and G′ from the standpoint of microstructure; thus, we analyze the microstructure of the suspension at the cluster length scale by means of SALS. Small-Angle Light Scattering (SALS). In this section, the microstructure of silica/PVA aqueous suspension is characterized at a length scale of clusters by means of SALS. Figure 6 shows a SALS intensity of the silica/PVA suspensions at 2 wt % < ϕPVA < 8 wt %. A plateau in I(q) at low q for ϕPVA = 2 wt % changes to a steep slope as ϕPVA increases up to 4 wt % (Figure 6a), indicating a formation of larger clusters with increasing ϕPVA. Together with the SAXS intensity (Figure 5a) and rheology (Figure 1a,b), SALS intensity implies that, as ϕPVA increases in the range of ϕPVA from 2 to 4 wt %, silica particles surrounded by adsorbed PVA are flocculated into fractal

Γ increases. The transition from bridging flocculation to steric stabilization occurs at around Γ = Γ0/2.49,50 Figure 2 shows Γ0/ 2 = 0.22 mg/m2 at ϕPVA = 0.5 wt %; thus, the transition may occur at this range. Figure 3 shows SAXS intensity, I(q), and structure factor, S(q), in the range of ϕPVA from 0 to 1.25 wt %. S(q) is calculated from I(q) = P(q)S(q), where P(q) is a form factor measured with charge-free silica particles at 0.3 wt % (solid line in Figure 3a). At ϕPVA below Γ0/2, S(q) shows a significant rise in peak intensity at q = 0.42 nm−1 and a deeper minimum at q = 0.2 nm−1 (Figure 3c), indicating a formation of highly ordered short-range particle aggregates. On the other hand, as ϕPVA increases above Γ0/2, S(q) displays a reduction of both the peak intensity at high q and the upturn at the lowest q (Fig 3d), indicating a disappearance of particle aggregates. This implies that the silica/PVA suspension exhibits a transition from bridging flocculation to steric stabilization as ϕPVA increases. Figure 4 shows a maximum settling behavior at ϕPVA = 0.5 wt %, and it becomes less severe as ϕPVA increases, which also supports the transition from bridging flocculation to steric stabilization.49,50 When ϕPVA increases more than 1.5 wt %, SAXS intensity in Figure 5 shows another systematic trend. S(q) shows an

Figure 4. Images of silica/PVA suspensions with ϕPVA from 0.1 to 1.5 wt % after sedimentation for 1 h. 1895

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Figure 5. (a, b) SAXS spectra and (c, d) structure factor of silica/PVA suspensions with respect to ϕPVA from 1.5 to 8 wt % as indicated.

arise from the presence of remaining aggregates which are not fully dispersed at high temperature. Their arguments have some consistency with our SAXS results that support the existence of fractal aggregates at ϕPVA > 4 wt %, where the structural transition from fractal cluster to dispersed liquid is suggested by rheological measurements (Figure 1c,d). Therefore, the transition of the shape in SALS intensity at ϕPVA > 4 wt % supports the stabilization at cluster length scale, with particle aggregates remained wherein. Strong low-q dependency in SALS spectra has also been observed in suspensions with attractive interactions, where individual clusters are aggregated again into a larger cluster, which can be called a “secondary cluster”.54−56 To further understand the SALS result, we compare them with a theoretical model developed by Morbidelli and co-workers54 which considers the mass fractal clusters bounded by surface fractal secondary clusters.57 In the model, S(q) is expressed by S(q) = Sm(q) + Ss(q)

(2a)

where Sm(q) = Figure 6. SALS intensity (symbol) and S(q) obtained from eq 2 (solid line) at different ϕPVA as indicated.

Ss(q) =

clusters at broad length scale, owing to the increasingly stronger depletion attraction. With a further increase in ϕPVA above 4 wt % (Figure 6b), we observe a gradual transition in the shape of S(q) that shows a strong q dependency at low q range. Rheological measurement in this range shows a drastic reduction in η0 and G′; thus, the observed transition in SALS intensity may reflect the structural evolution associated with this rheological measurement. In fact, the anomalous evolution of SALS intensity to exhibit strong low-q dependency has been observed in thermoreversible colloidal suspensions, where its phase transition from gel to fluid occurs as temperature increases.52,53 This was suggested to

sin[(Dm − 1) arctan(qξ1)] (Dm − 1)(qξ1)[1 + (qξ1)2 ](Dm − 1)/2

(2b)

K sin[(Dm − Ds + 2) arctan(qξ2)] (qξ2)[1 + (qξ2)2 ](Dm − Ds + 2)/2

(2c)

where Sm(q) is the structure factor of mass fractal clusters,42,53 which expresses a mass fractal scaling, S(q) ∼ q−Dm at qξ1 ≫ 1. Dm and ξ1 in eq 2b are respectively the mass fractal dimension and the cutoff length of the cluster indicating the upper length limit of the fractal scaling. Ss(q) in eq 2c corresponds to the structure factor of the surface fractal secondary cluster which expresses the surface fractal scaling, S(q) ∼ q−(Dm−Ds+3) at qξ2 ≫ 1.54,57 Ds and ξ2 in eq 2c are the surface fractal dimension and the cutoff length of secondary clusters, indicating the upper length limit of surface fractal scaling, respectively. From SALS 1896

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Langmuir intensity in Figure 6, it is presumed that the form factor P(q) = 1 in this range of q due to the small size of silica particles, which allows us to consider SALS intensity as a structure factor, S(q), as it is. Using Dm obtained from SAXS spectra in Figure 5 (values in Table 1), we obtain theoretical S(q) which shows an

previously reported particle/polymer suspensions exhibiting stabilization at particle length scale.22−30 We speculate the marked contrast is attributed to the polymer to particle size ratio (i.e., ρ ≈ 0.8 for our system, whereas ρ ≪ 1 for reported ones). In the next section, we discuss how ρ affects the depletion stabilization and propose a possible mechanism of the observed depletion stabilization in silica/PVA suspensions.

Table 1. Parameters Used for the Calculation of Eq 2 ϕPVA (wt %) ξ1 (nm) ξ2 (nm) Dm Ds K

2

3

4

6

7

8

600

750 2700 2.1 1.6 1.71

960 4500 2.0 1.6 4.33

410 3500 1.9 1.45 16.0

300 4000 1.9 1.38 21.7

200 4000 1.85 1.45 13

2.1

4. DISCUSSION When the particle is significantly larger than polymer (ρ ≪ 1), depletion stabilization is reported to arise owing to the repulsive energy barrier between the particles, which kinetically stabilizes the suspension.26−31,34,35 At ρ ≪ 1, it can be assumed that the polymer chains are located between two parallel walls (part of big particles), and the range of depletion interaction is of the order of polymer size. However, as ρ increases to unity, as in this study (ρ ≈ 0.8), it is not possible to assume that polymers are located between two walls, and the depletion layer thickness is expected to be deviated significantly from the depletion thickness predicted for ρ ≪ 1. Therefore, the theories developed for ρ ≪ 1 may not clearly explain the depletion stabilization when ρ is close to unity. Our main finding in the depletion stabilization of silica/PVA suspension with ρ ≈ 0.8 (similar size between particle and polymer chain) is that the microstructure is stabilized at the cluster length scale with the microstructure at particle length scale unchanged. A few studies have reported the stabilization at the cluster length scale in particle/polymer suspensions, which should be compared to our observation. Ogden and Lewis21 showed that the clusters formed by van der Waals attraction can be dispersed into smaller clusters with the addition of free polymers. They proposed that the depletion energy barrier exists between the clusters, thus resulting in kinetic stabilization of the clusters. The argument on the energy barrier between the clusters has also been reported in particle/polymer mixtures in polar medium,58,59 where the depletion-flocculated clusters are stabilized by long-range electrostatic repulsive energy barrier. These studies allow us to speculate that the depletion energy barrier could be located between clusters, which may be a driving force for the observed depletion stabilization of silica/ PVA suspensions. To check the existence of depletion energy barrier between the clusters, we evaluate the depletion interaction not only between the particles (Vd,p) but also between the clusters (Vd,ξ,). We employ the model developed by Mao et al.,29 which can predict the depletion energy barrier, unlike the other models which consider only the attractive part of the depletion interaction.7,25−27 The Mao model assumes a bimodal suspension with a large size ratio and calculates the

excellent agreement with the measured SALS intensity. This implies that SALS intensity can be properly described in terms of the presence of cluster and secondary cluster. ξ1, the characteristic length scale of the cluster, is summarized in Table 1. When ϕPVA increases up to ϕPVA = 4 wt %, ξ1 increases to 960 nm, corresponding length scale of 63d, where d is the diameter of silica particle. However, for 4 wt % < ϕPVA < 8 wt %, ξ1 reduces down to 200 nm, corresponding length scale of 13d, as ϕPVA increases. The dependency of ξ1 on ϕPVA shows a fairly good consistency with the rheological behavior with increasing ϕPVA, implying that a rheological behavior reflects a microstructure at a cluster length scale. ξ2 is also obtained in the range of ϕPVA > 3 wt %, which shows a maximum at ϕPVA = 4 with a slight reduction at higher ϕPVA. Unfortunately, the measured S(q) from SALS does not fully cover the low-q range which enables us to obtain valid scaling behavior related to ξ2 and Ds in eq 2c. The details of Ss(q) need to be further investigated by obtaining SALS intensity at lower q range by means of other techniques.54 Therefore, in this study, we do not further discuss on the structure of the secondary clusters. Finally, the rheology, SALS, and SAXS results allow us to propose the evolution of the microstructure with increasing ϕPVA as schematically displayed in Figure 7. As ϕPVA increases up to ϕPVA = 4 wt %, the suspensions follow the steps of dispersion stability from bridging flocculation (Figure 7a), steric stabilization (Figure 7b) to depletion flocculation (Figure 7c). When ϕPVA increases above 4 wt %, rheolgy and SALS support a stabilization of microstructure which is believed to be depletion stabilization at the cluster length scale, whereas SAXS shows fractal aggregates at the particle length scale (Figure 7d). The inconsistency of structural evolution between the cluster length scale (from rheology and SALS) and particle length scale (from SAXS) of depletion stabilization sharply contrasts the

Figure 7. Schematics of the dispersion stability of silica/PVA suspension as ϕPVA increases: (a) bridging flocculation, (b) steric stabilization, (c) depletion flocculation, and (d) depletion stabilization. 1897

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Langmuir interactions between large spheres where small spheres act as a depletant, using the following equations29 Vd(λ) =

−3dϕkTλ 2 dϕ2kT + (12 − 45λ − 60λ 2) 2σ 10σ (3a)

for h < σ Vd(λ) =

dϕ2kT (12 − 45λ + 60λ 2 − 30λ 3 + 3λ 5) 10σ

for σ ≤ h ≤ 2σ Vd(λ) = 0

for h > 2σ

(3b) (3c)

where d and σ are the diameters of large particle and small particles, respectively. h is the surface to surface separation between large particles, and ϕ is the volume fraction of small particles. λ is the scaled length, λ = (h − σ)/σ. In this study, we consider d = dsilica + 2δ, where δ is the thickness of adsorption layer and σ is analogous to 2Rg of PVA. To determine δ, we assume that the averaged volume fraction of adsorption layer at saturation is 0.2 based on the results given by Yang et al.,60 and it is independent of the distance from the particle surface. For simplicity, we assume δ independent of ϕPVA within the range of varied ϕPVA. From Γ0 = 0.24 mg/m2, we obtain δ = 2.6 nm; thus, a resulting d is determined to be 20.2 ± 2 nm from dsilica = 15 ± 2 nm. σ is 13.45 nm at ϕPVA = 2 wt % and reduces to 2Rg = 11.3 nm at ϕPVA = 8 wt % with a scaling law, Rg ∝ ϕ−1/8 41 (σ = 12.8, 12.3, 11.7, and 11.5 nm for ϕPVA = 3, 4, 6, and 7 wt %, respectively). It should be noted that the Mao model29 assumes the situation which is different from our current condition to some extent. First, the Mao model assumes monodisperse size distribution of small and large particles. Second, it assumes large size ratio between the particle and polymer, ρ < 0.1. The silica and PVA used in this study have a considerable degree of polydispersity. Furthermore, the clusters are not spherical, and the size in Table 1 has a fairly large polydispersity. Thus, it would be reasonable to think that calculated Vd,p and Vd,ξ from MAO model in this study needs to be discussed only in a qualitative manner. Calculated depletion interaction between the particles, Vd,p is displayed in Figure 8a in a range of ϕPVA from 2 to 8 wt %. Vd,p at contact (h = 0 nm) becomes more negative with higher ϕPVA, suggesting that the particles experience stronger depletion attraction. The stronger attraction is fairly consistent with the SAXS intensity at ϕPVA < 4 wt %, but there exists a slight mismatch at ϕPVA > 4 wt %, which is nearly independent of ϕPVA. Nevertheless, the calculation of Vd,p and SAXS intensity can allow us to conclude that the particles are dominated by attractive interactions at the particle length scale, and there exists no structural evolution relevant to depletion stabilization. Now we evaluate depletion interaction between the clusters, Vd,ξ. For the evaluation of Vd,ξ, we treat each cluster as a large sphere, and the diameter is analogous to ξ1 obtained from SALS measurement (values in Table 1). The resulting Vd,ξ between the clusters are displayed in Figure 8b in the range of ϕPVA from 2 to 8 wt %. As ϕPVA increases up to 4 wt %, Vd,ξ shows a stronger attractive interaction, implying that larger clusters are likely to form. However, as ϕPVA increases above ϕPVA > 4 wt %, Vd,ξ exhibits again a weak attractive interaction, implying the small clusters are likely to form. The behavior of Vd,ξ as a function of ϕPVA shows a qualitatively consistency with SALS intensity in Figure 6, suggesting that the underlying mechanism of cluster formation could be explained in terms of

Figure 8. Depletion interaction calculated from eq 3 for silica/PVA suspensions with varying ϕPVA from 2 to 8 wt % as indicated: (a) particle−particle interaction, Vd,p; (b) cluster−cluster interaction, Vd,ξ.

Vd,ξ. One important observation in Figure 8 is that the level of the energy barrier of Vd,ξ (= 0.25kT) is 10 times larger than Vd,p (= 0.025kT) at ϕPVA > 4 wt %. The Mao model (eq 3) provides the expression of energy barrier, Vd,max, as follows.29 Vd,max =

6dkTϕ2 5σ

(4)

Equation 4 indicates that, at fixed ϕPVA, Vd,max depends on d/σ = 1/ρ; thus Vd,max ∼ 1/ρ, implying that larger particles result in a higher level of energy barrier. This implies that the energy barrier between the clusters is larger than that between the particles. When we consider the kinetic aspect of cluster formation in silica/PVA suspensions, this also suggests that the energy barrier of Vd,ξ between the clusters becomes larger as the they grow. Interestingly, the magnitude of energy barrier in Vd,ξ at ϕPVA > 4 wt % seems to be nearly constant, independent of ϕPVA. The nearly constant energy barrier in Vd,ξ implies that the energy barrier increases up to a critical value, which prevents clusters from further aggregation. This allows us to explain the mechanism of depletion stabilization at ϕPVA > 4 wt % in terms of the energy barrier between the clusters21,58,59 as follows. The pH of the silica/PVA suspension is initially around 10, and the silica particles are charge-stabilized in the PVA solution. When pH is reduced to 3, the particles lose the electrostatic repulsion and start to aggregate to form clusters due to depletion attraction. During the formation of clusters, the distance between the clusters increases and the level of energy barrier increases in proportion to the size of the clusters.29 When the distance between the clusters and the level of energy barrier becomes large enough to prevent clusters from approaching and aggregating with each other, the clusters do not grow anymore, resulting in a formation of stabilized clusters. This suggests that the energy barrier between the clusters would prevent further aggregation of the clusters, providing a kinetic effect on the stabilization in silica/PVA suspensions at the cluster length scale. 1898

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5. CONCLUSIONS The mechanism of depletion stabilization and resultant microstructure was investigated for nanosilica and poly(vinyl alcohol) suspensions where the radius of gyration of PVA chain (Rg) is comparable to the diameter of the particle (d) (e.g., ρ = 2Rg/d ≈ 0.8). The equilibrium structure of the suspension was characterized in a broad length scale covering from the particle size to cluster size by means of small-angle X-ray scattering (SAXS), small-angle light scattering (SALS), and rheological measurements. As PVA concentration (ϕPVA) increases up to 4 wt %, the particles experience a consecutive evolution from bridging flocculation, steric stabilization, and depletion flocculation, followed by depletion stabilization at higher concentration. Surprisingly, depletion stabilization shows stabilization only at the cluster length scale, while maintaining fractal aggregates at the particle length scale. It is different from the depletion stabilization in previous reports, in which the stabilization occurs at the length scale of individual particles. On the basis of the evaluation of potential energy of depletion interaction, we propose that the depletion energy barrier exists between the clusters, not between the particles because of the similar size of silica and PVA chain.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected], Ph +82-2-880-8322 (K.H.A.). Present Address

S.K.: Institute for Mechanical Process Engineering and Mechanics, Karlsruhe Institute Technology, Gotthard-FranzStraße 3, Building 50.31, 76131 Karlsruhe, Germany. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S. Kim and C. Clasen acknowledge financial support from the ERC Starting Grant No. 203043-NANOFIB. K. H. Ahn acknowledges the NRF of Korea for funding through No. 2013R1A2A2A07067387 and K. Hyun through No. 20100024466. U-SAXS measurements at beamline 9A, PLS-II, were supported in part by MEST and POSTECH. We thank J. Vermant and K. Naveen for helpful comments on the analysis of scattering intensity.

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REFERENCES

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