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Nonlinear Concentration Dependence of the Collective Diffusion Coefficient of TiO2 Nanoparticle Dispersions J. Perez Holmberg, Z. Abbas, E. Ahlberg, M. Hassell€ov, and J. Bergenholtz* Department of Chemistry, University of Gothenburg, SE-41296 G€oteborg, Sweden ABSTRACT: Aqueous dispersions of titania nanoparticles are shown to yield collective diffusion coefficients in dynamic light-scattering measurements that depend nonlinearly on particle concentration under dilute conditions. From theory, one expects a linear dependence for monodisperse systems except for strongly interacting charged particles in low ionic strength media. Angularly resolved dynamic light-scattering measurements reveal that aggregates are present, which explains the collective diffusion coefficient tending to lower values in the dilute limit than the StokesEinstein diffusion coefficient of the nanoparticles. A simple theoretical model based on mixtures of charged nanoparticle spheres and small amounts of larger-sized neutral or weakly charged spheres, modeling the presence of aggregates, is applied and shown to yield predictions in qualitative accord with the experimental trends. In particular, the downward curvature of the collective diffusion coefficient on diluting the system arises in the model from nanoparticles being driven into close proximity to the larger particles by electrostatic interactions. Similar experimental trends observed in silica dispersions suggest that the behavior is not an isolated finding. This study clearly shows that a small number of larger aggregates dramatically change the measured value of the collective diffusion coefficient; thus, care must be exercised when characterizing nanoparticles with dynamic light scattering.
’ INTRODUCTION Dynamic light scattering on dilute samples is frequently the method of choice for particle sizing and for detection of aggregates. It is one of a few techniques that are noninvasive, and it generally yields excellent statistical averages. The drawbacks are well-known; because it relies on bulk time averages of intensities rather than particle counting, particle size distributions and their averages become intensity-weighted. For large particles, this method works splendidly as samples can be brought to high dilution without overly compromising the scattering power of samples and thereby the statistical quality of the data. This can be done because the scattered intensity is proportional to the square of the particle volume. However, for the same reason, sizing of small, nanoscopic particles with dynamic light scattering can be more troublesome. As samples are diluted, the scattering power decreases. One has to compensate for the loss of statistics by increasing the number or duration of runs, and many times one has to resort to measuring on nondilute samples. In this case, to obtain a reliable hydrodynamic radius, aH, one should extrapolate to vanishing particle concentration to obtain the StokesEinstein value of the diffusion coefficient, D0 = kBT/6πμaH, where kBT is thermal energy and μ is the viscosity of the solvent. Because of the intensityweighting, the hydrodynamic radius is actually a high-order, so-called z-average radius, given by aH = Æa6æ/Æa5æ,1 where Æ 3 æ denotes an average weighted by the size distribution. In this context, it is common to restrict the measurements and analysis to a single scattering angle. This is almost invariably done when r 2011 American Chemical Society
dealing with nanosized particles because it is only for particles of sizes approaching the wavelength of light that one expects angular variations in the intensity. However, in so doing, it becomes more difficult to assess whether any aggregates are present. This should be especially important for inorganic particles because they are typically characterized by a chargeregulating surface chemistry, thereby being more sensitive to changes in the physicochemical conditions, and high values of the Hamaker constant.2 For dispersions of small-sized particles, measurements under nondilute conditions generally do not present any problems. The diffusion coefficient that one extracts from the intensity correlation function is the collective diffusion coefficient, and one expects a linear dependence on particle concentration3,4 with a slope that depends on the particleparticle interaction.5 However, in some cases, a nonlinear dependence on concentration is obtained, even under dilute conditions. The reasons for this could be that the ionic strength is sufficiently low so that the extent of the diffuse part of the electrical double layer is governed mainly by counterions. This leads to a concentration-dependent screening length, and it is known that, for such cases, short-time transport coefficients assume a nonlinear dependence on concentration.68 However, for salt concentrations in the millimolar range, the screening is mostly due to ions from added salt, and in such instances, the reason for the Received: March 18, 2011 Revised: May 30, 2011 Published: May 31, 2011 13609
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The Journal of Physical Chemistry C deviation from a dilute-limiting linear concentration dependence remains unclear. In the present work, we follow up on a previous study,9 henceforth referred to as “I”, in which we presented a possible route toward well-defined, nanoscopic titanium dioxide particles of controlled morphology and size. These particles are highly interesting in and of themselves. TiO2 nanoparticles find uses in a wide range of important application areas, including photocatalysis,10 water purification,11 solar energy conversion,12 sensors,13 and optoelectronics.14 In I, hydrodynamic radii were determined by extrapolating collective diffusion coefficient data recorded under nondilute conditions to vanishing particle concentration. In general, rather good agreement was observed with the results of an electrospray technique that sorts particles based on their aerodynamic mobility in an electric field. However, a more careful analysis of the collective diffusion coefficient, undertaken here, shows that it, in fact, depends nonlinearly on particle concentration at millimolar concentrations of added electrolyte. In short, this work is an account of a more careful assessment of the collective diffusion coefficient of these dilute nanoparticle dispersions and it also contains an attempt to rationalize the nonlinear trend seen in the concentration-dependent data. To this end, a simplified model based on some larger nanoparticle aggregates interspersed among the primary nanoparticles is presented and is shown to reproduce qualitatively the same trend as followed by the experimental data. The nonlinear concentration dependence within this model derives from the complex interactions in the system; in particular, as the charge of the nanoparticles is increased, depletion-like interactions turn into a particle haloing effect, whereby the small charged particles accumulate close to the aggregates. In what follows, we begin by presenting the experimental protocol and continue with a brief review of the relevant theory of dynamic light scattering from polydisperse sphere dispersions. Experimental data for the collective diffusion coefficient as a function of TiO2-particle concentration are then reported, demonstrating the presence of aggregates and a nonlinear dependence on concentration. Results from the model, qualitatively reproducing the experimental data, are presented and analyzed, which leads us to propose nanoparticle haloing of larger aggregates as the cause of the nonlinear concentration dependence of the collective diffusion coefficient in reasonably well-screened dispersions.
’ EXPERIMENTAL SECTION Titanium dioxide nanoparticles with number-average sizes of 810 and ∼20 nm were synthesized through a controlled hydrolysis route described in detail in I. The particles were also extensively characterized in I, and they were found to consist mainly of anatase. Concentration series of dispersions of TiO2 particles were prepared by diluting a stock dispersion with a concentration of 18.6 g 3 L1 with a pH 2.5 (∼3 mM) HCl solution. Diffusion coefficients were measured by dynamic light scattering as a function of angle on an ALV CGS-8F DLS/SLS-5022F instrument (ALV, Langen, Germany), equipped with an ALV6010/160 correlator and dual APD detectors, at a wavelength of 633 nm. The detected signals were cross-correlated to extend the measurements down to delay times of approximately 0.01 μs. Further dynamic light-scattering measurements were made at a fixed angle of 173 with a Zetasizer Nano ZS (Malvern, Worcestershire, U.K.) at a wavelength of 633 nm. Measurements were
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conducted on both centrifuged and uncentrifuged samples. Centrifugation of diluted samples was done in 2 mL polypropylene tubes for 2 h at 6000 rpm and a maximum RCF of 2000g (Galaxy mini 37000700, VWR). The top 1 mL fraction of the centrifuged sample was used in the analysis. Number-average particle size distribtions were determined by ES-SMPS, which is a technique that combines electrospray (ES model 3480, TSI Inc.) and a scanning mobility particle sizer (SMPS model 3936, TSI Inc.). It is described in greater detail elsewhere.15 Zeta potentials were also determined by converting electrophoretic mobilities measured with the Zetasizer Nano ZS using the Smoluchowski equation. For ∼810 nm TiO2 particles, this typically yielded zeta potentials of ≈þ50 mV, and for ∼20 nm particles, zeta potentials of ≈þ40 mV were obtained.
’ SCATTERING THEORY In dynamic light-scattering experiments, one measures the intensity correlation function from which, via the Siegert relation, the normalized field correlation function g(1)(q,t) is obtained. The angle of detection, θ, enters via the modulus of the wavevector q = 4π(n/λ0)sin(θ/2), where n is the refractive index of the solvent and λ0 is the laser wavelength in vacuo. The normalized field correlation function is related to the normalized, so-called measured or effective structure factor16 g ð1Þ ðq;tÞ ¼
SM ðq;tÞ SM ðqÞ
ð1Þ
where SM(q) = SM(q,t f 0) is the measured static structure factor.17 Labeling quantities as “measured” or “effective” refers to the fact that, in a polydisperse system, it is not possible to separate cleanly interference effects that derive from the shape, size, and size distribution of particles, that is, form factor effects, from those caused by interactions, that is, structure factor effects. Rather, the intensity scattered at a certain time and angle from an m-component mixture of colloidal particles is a complex superposition of partial dynamic structure factors SRβ(q,t) weighted by scattering amplitudes of, for example, type-R particles, fR(q), according to16 Iðq;tÞ ¼ nf 2 ðqÞSM ðq;tÞ ¼n
m X
ðxR xβ Þ1=2 fR ðqÞfβ ðqÞSRβ ðq;tÞ
ð2Þ
R;β ¼ 1
where n is the total number density and xR is the mole fraction of type-R particles. The so-called measured form factor of the system 2 is given by f2(q) = ∑m R=1xRfR(q) and is obtained from a static scattering measurement at high dilution where SM(q) ≈ 1. The initial decay of the normalized field correlation function in eq 1 is usually characterized by a cumulant expansion,18 g(1)(q,t) = (2) (3) 2 3 exp[Γ(1) M (q)t þ μM (q)t þ μM (q)t þ ...], where the first cumulant can be identified as ð1Þ
ΓM ðqÞ ¼ lim
tf0
∂ ln SM ðq;tÞ ∂t
ð3Þ
and μ(n) M (q) are higher-order cumulants in the expansion. Proceeding as for monodisperse systems, one can define a q-dependent short-time diffusion coefficient ð1Þ
DM ðqÞ ¼ 13610
ΓM ðqÞ HM ðqÞ ¼ 2 q SM ðqÞ
ð4Þ
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Figure 1. Collective diffusion coefficient as a function of particle mass concentration, corresponding to a range of volume fractions up to 0.0109, for aqueous dispersions of Ludox HS silica particles. The error bars represent (3%, as quoted by Finsy et al.20 In addition, the StokesEinstein diffusion coefficient using the TEM value of 16.5 nm for the diameter with a 0.21 normalized standard deviation21 is 2.42 1011 m2/s assuming that the size distribution follows a Schulz distribution;1,22 it is shown as a filled square at c = 0.
where the last equality comes from carrying through the differentiation in eq 3 assuming that the system is governed by diffusive Smoluchowski dynamics.19 The measured diffusion coefficient is given by the ratio of an apparent hydrodynamic function and the measured static structure factor. It embodies the fact that the motion of colloidal particles is coupled through rather long-range hydrodynamic interactions. Neglecting hydrodynamic interactions decouples the motion of particles, and the hydrodynamic function simplifies considerably such that DM ðqÞ
m X 1 xR fR2 ðqÞD0R SM ðqÞf 2 ðqÞ R ¼ 1
ð5Þ
where D0R = kBT/6πμaR is the StokesEinstein diffusion coefficient of a type-R sphere with a radius of aR embedded in a solvent with a viscosity of μ. In the dilute limit, SM(q) f 1, when, in addition, qa , 1, with a being the mean radius, the form amplitude fR(q) becomes independent of q and proportional to a3R so that1 m P
kB T R ¼ 1 DM ðqÞ m 6πμ P R¼1
xR a5R xR a6R
¼
kB T 6πμaH
ð6Þ
where aH = Æa æ/Æa æ is the hydrodynamic or z-average radius. 6
5
’ RESULTS AND DISCUSSION In Figure 1, we show data for the collective diffusion coefficient of Ludox HS silica particles as a function of particle concentration. These data were recorded and tabulated by Finsy et al.,20 and they serve to demonstrate the nonlinear dependence on concentration of the collective diffusion coefficient that we address in this work. As seen, on diluting the system from the highest concentration, the collective diffusion coefficient decreases as expected for mutually repulsive, charge-stabilized particles,35 seemingly toward the StokesEinstein value, 2.42 1011 m2/s, inferred from electron microscopy.21 However,
Figure 2. Dynamic light-scattering measurements of the collective diffusion coefficient as a function of the mass concentration of TiO2 particles, corresponding to a range of volume fractions up to 1.3 103, in a 3 mM HCl solution at a scattering angle of 173. Data are shown for particles produced by hydrolysis of TiCl4 and dialysis and aging close to 0 C (filled circles) and room temperature (open circles), resulting in smaller- and larger-sized particles, respectively. The filled square shows the StokesEinstein diffusion coefficient calculated from the number size distribution obtained from ES-SMPS measurements of the dispersions of the smaller of the two particle sizes shown.
as the particle concentration is decreased further, the collective diffusion coefficient begins to deviate from a linear concentration dependence, dipping below the StokesEinstein value at the lowest concentrations. Finsy et al.20 suggest that, although the particles are small compared to the length scales, q1, probed in the experiment, an angular dependence of the collective diffusion coefficient may be responsible for the nonlinear dependence on concentration, but they present no analysis in support of their suggestion.
’ EXPERIMENTAL RESULTS In I, TiO2 nanoparticles free of organic additives were synthesized by hydrolysis of TiCl4. The reaction/dialysis and storage time and temperature determine the morphology and size of the particles. Particle sizes and size distributions were determined in I by both an electrospray technique, ES-SMPS, based on sorting particles according to their aerodynamic mobility, and dynamic light scattering conducted at a single scattering angle. Whereas good agreement was found between the two measurements in general, a more careful examination of dynamic light scattering data for the smaller-sized particles reveals that the measured diffusion coefficient depends nonlinearly on particle concentration. This behavior is shown in Figure 2, where the collective diffusion coefficient has been more finely resolved over a more extended concentration interval than in I. Figure 2 shows data for small- and larger-sized particles, with diameters of about 8 and 20 nm, respectively. Whereas the collective diffusion coefficient of the large-sized particles exhibits a linear dependence on concentration, the small-sized particles do not. From Figure 2, it is easy to see how this nonlinear concentration dependence can be missed if a few measurements are conducted in a limited concentration interval. Moreover, the collective diffusion coefficient of the smaller particles at the higher concentrations is seen to roughly tend toward the value corresponding to the ES-SMPS size, shown in 13611
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Figure 3. Angular dependence of the normalized intensity correlation function, g(2)(t) 1, shown as a function of q2t, for TiO2 nanoparticles in a 3 mM HCl solution at a particle concentration of 1.86 g L1. The left panel shows correlation functions at four different scattering angles, and the right panel shows results for the same samples obtained after mild centrifugation.
Figure 2 at the low concentration of the ES-SMPS measurement. However, on diluting the system further, just as for the measurements in Figure 1, the diffusion coefficient dips below the apparent dilute-limiting value. Routine dynamic light-scattering experiments are usually conducted at a measurement angle of 90. For polydisperse samples, examining the angular dependence can be very revealing1 because, at smaller angles, the light scattering will be dominated by the larger-particle fraction and the opposite will hold at higher angles. It follows that angularly resolved measurements can be equally revealing for assessing whether aggregates are present.23 The current investigation of the angular dependence served a two-fold purpose. First, it served as a check for the presence of larger aggregates in the TiO2 dispersions. Second, it was used to examine whether the aggregates influenced the size data obtained at larger angles, where the routine size measurements for the TiO2 dilution series were conducted. The angular dependence of light scattering by TiO2 nanoparticles in a 1.86 g L1 dispersion, corresponding to a volume fraction of ϕ = 4.8 104, is presented in Figure 3. The intensity correlation function g(2)(t) is shown as a function of time for four different measurement angles. For clarity, the correlation functions have been normalized to a common intercept of 2; in addition, they are shown as a function of q2t rather than t, where q is the modulus of the scattering vector. Because dynamic light scattering probes correlations in density fluctuations extending over length scales given by q1, this presentation removes the trivial effect of altering the length scale over which the collective diffusion process is probed. In other words, any differences observed in this graph as a function of angle are caused by an angular dependence of the diffusion coefficient DM(q). As seen in Figure 3, the correlation functions exhibit a dependence on the scattering angle. At the largest angle, the decay is nearly exponential, resulting in a sigmoidal curve on the semilogarithmic graphs of Figure 3. As the angle is decreased, a second, slower component in the decay of the correlation function appears. It becomes more pronounced as the angle of detection is further decreased, yielding a well-developed, two-step decay at
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Figure 4. Normalized intensity correlation function as a function of time and TiO2 nanoparticle concentration in terms of dilution factor from a stock sample with a concentration of 18.6 g L1 at a constant scattering angle of 173. The dispersion medium is a 3 mM HCl solution, and all samples were centrifuged for 2 h to remove larger aggregates.
the lowest angle. Moreover, the initial, short-time decay slows down as the angle is decreased. Because the abscissa in Figure 3 is q2t, the decay constant of the initial decay is the measured short-time diffusion coefficient DM(q). It follows that the diffusion coefficients possess a dependence on angle. Such an angular dependence can come from polydispersity,1 but it can, in principle, also be caused by interactions among particles.68 Titanium dioxide has a high refractive index, and as a consequence, even for these nanosized particles, the systems still scatter sufficiently at low concentrations. However, long-range electrostatic interactions can be important even in very dilute systems, but only if the ionic strength is low such that the Debye length is dominated by counterions.68 Because the number of counterions in solution is directly coupled to the particle concentration, the Debye length becomes dependent on particle concentration such that it increases with decreasing particle concentration. In this way, very dilute particle suspensions can still be influenced by interactions. Moreover, in such cases, it has been shown that a number of short-time transport coefficients acquire a nonlinear dependence on particle concentration,24,25 which is what we observe for the collective diffusion coefficient in Figure 2. For instance, the short-time sedimentation coefficient has been found to be well described by H(0) ≈ 1 1.8ϕ1/3 for ϕ f 0 in strongly deionized systems26 However, at a pH of 2.5 the ionic strength is not low enough to bring the samples into this counterion-dominated regime. Indeed, as is also shown in Figure 3, mild centrifugation of samples removes much of the slow component in the correlation functions at small angles. It follows that the angular dependence of the correlation functions is caused by aggregates so that the samples, in effect, behave as a polydisperse system. These aggregates scatter mainly at small angles and cannot easily be observed in measurements at higher angles. Figure 4 shows the intensity correlation function as a function of concentration at a constant scattering angle of 173. The stock sample was centrifuged to remove the larger aggregates, and as seen, a separate long-time decay is not observed. Nevertheless, it is clearly seen that there is an increased slowing down of the 13612
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Figure 5. Binary sphere model consisting of small colloidal particles (left) and irreversible aggregates of particles (right), both with associated counterions shown as points. For simplicity, the former are treated as small charged particles, whereas the latter are modeled as large net neutral (hard) spheres, as schematically shown by the dashed outline to the right.
correlation function as the particle concentration is decreased; the decay of the correlation function is pushed to gradually longer times as the concentration is lowered. Moreover, the degree of slowing down is not linearly proportional to the concentration; rather, it becomes more pronounced at the lowest concentrations. This results in a collective diffusion coefficient with a nonlinear dependence on concentration, as seen in Figure 2.
’ SIMPLIFIED MODEL To rationalize the experimental results, we propose a simplified model consisting of two colloidal components differing in size, interactions, and concentration. For the sake of simplicity, we take both components as spherical in shape and the hydrodynamic interaction between all particles is neglected because a porous aggregate is expected to interact more weakly hydrodynamically than a solid sphere of comparable size.27 As illustrated in Figure 5, we investigate the measured diffusion coefficient, DM(q), for a hypothetical system comprising two differently sized spheres. For simplicity, we treat them both as homogeneous spheres even though the larger-sized species is meant to capture finite-sized, irreversible aggregates of the smaller-sized component. Furthermore, we shall set the surface charge of the large-sized spheres to zero, again to keep matters simple, and also because we expect the surface charges in the aggregate in Figure 5 to be strongly screened due to the entrained counterions.28 The interaction between particles is taken, in addition, to a hard core, as the repulsive, screened Coulomb part of the DLVO potential29 uRβ ðrÞ=kB T ¼
8 > > L Z Z > : B R β
ekaR 1 þ kaR
! ekaβ ekr 1 þ kaβ r
r < aR þ aβ aR þ aβ < r
ð7Þ where LB is the Bjerrum length, ZR is the number of surface charges on particles of radius aR belonging to type R, and k1 is
Figure 6. Normalized measured diffusion coefficient as a function of the volume fraction of the majority small-sphere component, ϕsmall, for three different scattering angles as labeled. The volume ratio of small- to largesized particles is constant throughout so that ϕlarge = 103ϕsmall. The inset shows, in addition, the apparent linear concentration dependence (dashed line) that results when the minority large-sphere component is removed.
the Debye screening length given by k2 ¼ 4πLB
m X R¼1
nR jZR j þ k2salt
ð8Þ
where nR is the number concentration of type-R colloidal particles. The screening length above includes a contribution from the counterions that results from the dissociation of surface charges on the colloidal particles as well as the usual contribution from added monovalent salt, k2salt = 8πLBnsalt, in terms of the bulk number density nsalt of added salt. The first term in eq 8 introduces a dependence of the screening length on particle concentration, which can be important when the ionic strength from the added salt is small.68 Given the electrostatic interactions in eq 7 together with the composition of the system, the diffusion coefficient in eq 5 can be calculated. However, in practice, as seen from eq 2, this requires determining the partial structure factors that enter SM(q). This has been done using hypernetted chain (HNC) theory,30 which produces accurate structure predictions for systems with longrange interactions. For the calculations at hand, the interest is in dilute systems for which the HNC theory becomes exact. The form amplitudes fR(q) are needed in eq 5 and were taken as those of homogeneous spheres, that is fR ðqÞ ¼
4πΔnR ðsin qaR qaR cos qaR Þ q3
ð9Þ
where ΔnR is the refractive index difference between type-R particles and the surrounding solvent. Given the rather large parameter space formed by particle charges, concentrations, and sizes, we refrain from attempts at a quantitative reproduction of the experimental data because it would likely not be meaningful; instead, we aim for obtaining qualitative agreement between model and experiment. On exploring a variety of compositions and size ratios with the larger particles being the minority component, it was found that some produced a measured diffusion coefficient that qualitatively follows the observed experimental trend of (i) DM(q) values 13613
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Figure 7. Normalized measured diffusion coefficient as a function of wavevector and the volume fraction of the majority small-sphere component, ϕsmall, as labeled. The volume ratio of small- to large-sized particles is constant throughout so that ϕlarge = 103ϕsmall. Note that q = 6.84 103, 1.87 102, and 2.64 102 nm1 correspond to scattering angles of θ = 30, 90, and 173.
that do not differ too greatly at higher concentrations from the small-sphere StokesEinstein value and (ii) DM(q) values that curve downward and drop below the small-sphere Stokes Einstein value on dilution of the system. Figure 6 shows results for the concentration dependence of the measured collective diffusion coefficient for three separate angles of detection, θ = 30, 90, and 173. As seen, regardless of the scattering angle, the same qualitative behavior is obtained, with the diffusion coefficient at higher concentrations exhibiting what appears to be close to a linear concentration dependence on concentration. However, on lowering the concentration, the diffusion coefficient begins to curve downward such that the value becomes smaller than the StokesEinstein value of the small-sized particles. These results were obtained using small particles with a diameter of 10 nm, carrying 25 charges, and 100 nm neutral hard spheres representing aggregates. The concentration ratio between the two was kept constant at 1 large particle per 1 million small particles. In other words, in Figure 6, the volume fractions of the minority, large-particle component can be found from ϕsmall = 103ϕlarge. This behavior is in stark contrast to what is obtained using the same model in the absence of the large-sized particles, which is shown in the inset to Figure 6. With only small-sized, charged particles present, the collective diffusion coefficient exhibits what appears to be a linear concentration dependence with a positive slope with a large magnitude. This is precisiely what one expects for monodipserse colloidal dispersions,35 at least when the ionic strength is not too low.31 To understand this behavior, in particular, the reason for the measured collective coefficient dropping below the value of the majority, small-sphere component, Dsmall, it proves useful to examine the angular dependence. This is done in Figure 7, where the normalized collective diffusion coefficient is shown as a function of the wavevector for a range of volume fractions in both the presence and the absence of a large-sized particle component. As seen, in the absence of a minority, large-sized component, the collective diffusion coefficient becomes independent of q for sufficiently small q, where it becomes equal to the gradient diffusion coefficient.32 The scattering angles of
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Figure 8. Radial distribution functions as a function of dimensionless separation distance and volume fraction of the majority, small-sphere component, ϕsmall. Going from left to right along the abscissa, the smallsmall, smalllarge, and largelarge sphere radial distribution functions are shown. The inset shows the correponding measured static structure factors as a function of wavevector for the same volume fractions. The volume ratio of small- to large-sized particles is constant throughout so that ϕlarge = 103ϕsmall.
θ = 30, 90, and 173 all roughly fall within this low-q regime for a small-sphere diameter of 10 nm. Moreover, the decrease of the low-q-limiting DM(q) occurs toward Dsmall nearly linearly with decreasing concentration, as shown in the inset to Figure 6. At large q, the angular dependence disappears, which occurs when the collective diffusion coefficient approaches the selfdiffusion coefficient. The scattering from the large-sphere component is confined to small q, and it does not contribute at such large q. Hence, it is the small-sphere self-diffusion coefficient that is measured at large q, and it shows no concentration dependence because hydrodynamic interactions have been neglected. The form factor has zeroes, beginning at qσsmall ≈ 9, which are seen in Figure 7 at large q. These would disappear if polydispersity in the small-sphere size were introduced. This behavior changes dramatically on introducing neutral hard spheres with a diameter of 100 nm at a proportion of one to a million small-sized spheres. In this case, the low-q behavior of the measured collective diffusion coefficient is entirely different, again because the scattering from the larger particles occurs at small angles correponding to low q. Even though the concentration difference is large, in favor of the small spheres, the large-particle component contributes because the scattered intensity is proportional to the square of the particle volume. In TEM or ES-SMPS measurements, only the majority-component smaller-sized spheres are sampled with reasonable statistics, and this is the reason the measured collective diffusion coefficient drops below the majority-component value, Dsmall. However, this does not explain the nonlinear concentration dependence obtained in Figure 6. The only concentration-dependent term in eq 5 is the measured static structure factor because the hydrodynamic interactions have been neglected. It follows that any nonlinearities in concentration dependence must derive from SM(q). Indeed, as shown in the inset to Figure 8, the measured static structure factors exhibit a complex dependence not only on wavevector but also on particle concentration. At not too small q, 13614
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The Journal of Physical Chemistry C the measured structure factors show the usual progression with concentration. The main correlation peak migrates to lower q as the particle concentration is decreased, and in the absence of the large-sized component, the q f 0 intercept, which, in this case, is the isothermal osmotic compressibility, increases toward unity. However, SM(q) at low q approaches unity in a slightly nonlinear fashion, something that is masked to some extent by the inversion to DM(q), which is the reason it is not seen in Figure 6. As for DM(q) in Figure 7, the introduction of the large-sized component changes the low-q behavior dramatically. The measured static structure factor now acquires an upturn at low q, which produces the suppression relative to the monodisperse system at low q in Figure 7. Moreover, as can be seen from the intercept values in the inset to Figure 8, SM(q) clearly varies nonlinearly with particle concentration at low q, which causes the nonlinear concentration dependence of DM(q) in Figure 6. These effects are brought on by interactions despite the low overall particle concentration. The HNC theory produces the partial static structure factors that combine to form the measured static structure factor in eq 2. These are related to pair correlation functions, which are shown in Figure 8 as functions of separation distance and particle concentration. The pair correlation function for the small-sized particles shows that they are excluded from one another at short distances, leaving a correlation hole, as expected for the screened-Coulomb interaction. When the particle concentration is decreased, the extent of the correlation hole increases because the screening of the repulsion is more efficient at the higher concentrations due to contributions to the ionic strength from the dissociated counterions. However, the extent of the correlation hole does not scale with the Wigner Seitz cell radius,33, that is, n3 small, with nsmall being the smallparticle number density, because the acidic pH yields a constant electrolyte background that is completely dominant at the lowest particle concentrations. The peak of this correlation function is not much above unity for the conditions investigated here. This does not immediately imply that we are dealing with a relatively weakly correlated system.33 Indeed, examination of the crosscorrelation function, which is related to the probability of finding a small particle at a particular distance from a large particle, establishes that the system is strongly correlated; at the higher concentrations, there is a pronounced correlation peak near particleparticle contact, which shows that small particles accumulate around the large ones. The effect is gradually decreased when the concentration is lowered. The largelarge correlation function shows that the large particles are excluded from one another at distances less than about a small-sphere diameter. This is quite the opposite of the depletion effect often at play in mixtures of differently sized components.34 Here, the small spheres can decrease the number of close-lying charged neighbors by assembling adjacent to large particles. Theoretical calculations similar to the ones done here show this effect in systems with large size asymmetry and a high concentration of small, compared to large, spheres,35 where it can, even under some circumstances, lead to phase separation. In addition, the behavior observed in Figure 8 is reminiscent of the halos formed by small charged spheres around large, very weakly charged spheres,36 though the composition of the system here is quite different. Experimental investigations carried out on colloidal suspensions containing silica microspheres and hydrous zirconia or polystyrene nanoparticles3638 describe the haloing effect as a consequence of long-range Coulombic repulsion between the highly charged nanoparticles, as is the case with the present
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model study. However, this effect is also observed to be enhanced by a weak attraction between the microsphere and nanosphere species,39,40 which is absent in the model studied here because the large-sphere component is not intended to capture the behavior of solid spheres, but porous aggregates instead. This is also a reason for neglecting hydrodynamic interactions, which are weaker for permeable particles.27 If the charge on the small particles in Figure 8 is decreased, this haloing effect gradually turns into a depletion effect in the sense that the largelarge correlation function exhibits a peak at contact. However, for the parameters investigated here, the collective diffusion coefficient never obtains a negative slope with increasing particle concentration, which can occur when sufficiently strong attractions act between the large spheres5 and when the scattering of the small-sphere component is negligible in mixtures.41 To describe the effects of various parameters on model predictions further, we define the particle charge ratio, RZ = Zlarge/Zsmall, the particle concentration ratio, Rn = nlarge/ nsmall, and the particle size ratio, Ra = alarge/asmall. According to the simple binary sphere model, the downward curvature observed in Figure 6 with increased dilution is a robust finding in that it persists for a generous range of concentration and size ratios. In addition, it is also observed when the large spheres are made weakly charged, slightly increasing RZ, but it disappears when the size ratio Ra and/or the concentration ratio Rn is decreased sufficiently. Moreover, it disappears entirely, leaving a linear concentration dependence for the collective diffusion coefficient, when the charge on the small-sized spheres is removed, which illustrates the electrostatic origin of the effect. Whereas the qualitative form of the nonlinear dependence on concentration of the diffusion coefficient is insensitive to changes in system parameters, the dilute-limiting intercept is not. In general, one recovers Dsmall only for small Ra and Rn. It should also be noted that modeling the system as composed of a single small-sized species, but with a short-range attraction, in addition to the screened Coulomb repulsion, does not reproduce the experimental trends. In particular, the collective diffusion coefficient dipping below the small-sphere StokesEinstein value at very low concentrations requires essentially irreversible aggregates, that is, a separate large-sized component.
’ CONCLUSIONS Dilute dispersions of TiO2 nanoparticles have been investigated by dynamic light scattering. Because the particles are small relative to the wavelength of light, the measurements probe collective diffusion. Collective diffusion coefficients as functions of nanoparticle concentration exhibit a nonlinear concentration dependence that also extrapolates to dilute-limiting values well below the StokesEinstein diffusion coefficient of the particles. The effect is observed only for dispersions of small particles, with diameters j 10 nm. Dynamic light-scattering measurements conducted at small angles together with mild centrifugation establish that aggregates are present in the dispersions, which explains why the collective diffusion coefficient does not approach the StokesEinstein diffusion coefficient in the dilute limit. Calculations based on a simple binary sphere system, composed of small charged particles and larger neutral particles, are shown to reproduce qualitatively the observed experimental trends. Analysis of the model predictions suggest that the nonlinear concentration dependence derives from complex electrostatic interactions in the system, whereby nanoparticles accumulate near the surfaces of larger particles. The 13615
dx.doi.org/10.1021/jp202585e |J. Phys. Chem. C 2011, 115, 13609–13616
The Journal of Physical Chemistry C results clearly show that the presence of small amounts of larger aggregates in nanoparticle dispersions might lead to collective diffusion coefficients that do not reflect the nanoparticles.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT Financial support from the Swedish Research Council (20053070, 2007-4931) and the Swedish Environmental Research Council FORMAS is gratefully acknowledged. J.P.H. was supported, in part, by the University of Gothenburg Strategic Research Platform “Nanoparticles in interactive environments”.
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