Experimental Determination of Particle Size-Dependent Surface

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Experimental Determination of Particle Size-Dependent Surface Charge Density for Silica Nanospheres Ya-Rong Shi,†,‡ Man-Ping Ye,‡ Lu-Chao Du,† and Yu-Xiang Weng*,†,§ †

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Beijing National Laboratory for Condensed Matter Physics, CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China ‡ College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China § University of Chinese Academy of Sciences, Beijing 100049, China S Supporting Information *

ABSTRACT: Surface charge densities of spherical silica nanoparticles of varied size spanning from 4.1 to 495.7 nm in NaCl solution with a concentration from 0.003 to 1.2 mM at a pH value of 8.0 were determined by converting their corresponding measured zeta potential with Poisson− Boltzmann model. The magnitude of the derived surface charge density (negative) at a given NaCl concentration of 0.225 mM decreases monotonically with the increasing particle size and reaches almost a steady value when the size exceeds 30 nm, revealing clearly the effect of the nanoparticle curvature on the surface charge density. The experimental data provide a consensus for the validity of different models describing the relation between the surface charge density and the electric potential, that is, the surface complexation model and the Poisson−Boltzmann model for the charged nanospheres in electrolyte solution. We found that if the former includes the curvature-dependent deprotonation constant of the surface silanol groups, both models would give a consistent result.

1. INTRODUCTION Many proteins realize biological function based on their capability of size-recognition.1 With the advance in the nanotechnology, nanoparticles (NPs) have been used in various fields including DNA2,3 and protein transport,4 drug delivery,5 and DNA analysis/sequencing systems.6 NPs interacting with proteins, membranes, cells, DNA, and organelles have established a series of NP/biological interacting assemblies that depend on nanomaterial properties such as size,7 shape,8 surface chemistry, roughness, and surface coatings, 9 and this knowledge is important from the perspective of safe use of NPs.10 Among various properties, one of the key parameters is the surface charge density (SCD) of the NP, which also depends on its size.11,12 However, most existing studies assume that the SCD of a particle is a material property independent of the particle size.13−15 Despite the obvious importance of NP surface charging, relatively little information is available in the literature on the dependence of surface charging on the particle size,11 specially of wide size region. Among various NPs, the silica NPs are of particular interest owing to their good biocompatibility16 and almost wellcontrolled spherical geometry.17 Silica NPs have been used to mimic the cells to reveal qualitatively and quantitatively the structural deformation of the adsorbed protein induced by curvature5,18 which would also affect the surface charge properties of silica NPs. The surface charge of silica particles © XXXX American Chemical Society

is originated from the protonation/deprotonation of silanol functional groups, which depends on the local solution pH, ionic strength, and particle size.12 There are some experimental studies12,19−22 in which the surface charging of silica NPs less than 50 nm in diameter were investigated. However, contradictory results have been reported in the literature.11 For example, Yamanaka20 reported an enhanced SCD for 30 nm silica particles as compared with 60 nm silica particles, whereas Kobayashi et al.12 have reported nearly equal SCDs for 30, 50, and 80 nm silica particles. Sonnefeld19 reported higher SCD of silica particles of 8 nm in diameter as compared to 40 nm silica particles. Thus, the lack of the consistent and accurate data for the SCD of silica nanospheres of varied size greatly hinders the quantitative study of size effects23 in various applications, as well as the further theoretical study of size effect on the SCD which is based on the experimental support. For example, both Atalay, et al.24 and Barisik, et al.13 used the same set of experimental data to validate their theoretical predication of SCD of silica NPs at different pH or electrolyte concentration. We noted that in that experiment the silica NPs used was commercial AEROSIL OX 50 (Degussa), with a broad size distribution, that is, a small volume fraction of less than 10 nm and another large fraction with a size distribution Received: August 6, 2018 Revised: September 5, 2018 Published: September 17, 2018 A

DOI: 10.1021/acs.jpcc.8b07566 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

3. RESULTS AND DISCUSSION Figure 1 shows some typical TEM images of silica NPs. Their average sizes were also evaluated statistically from the TEM

from 40 to 70 nm (radius) from the transmission electron microscopy (TEM) measurement, giving a mean size of 58 nm in radius.25 In this work, spherical silica NPs with their size spanning from 4.1 to 495.7 nm were used for measurement of their zeta potential. Electrophoretic mobility were measured with a large span of colloidal concentration with the NaCl electrolyte concentration being varied from 0.003 to 1.2 mM at a fixed pH value of 8.0. The SCD were converted from the measured zeta potential according to the established Poisson−Boltzmann (PB) equation for spherical NPs,26 at a fixed NaCl concentration of 0.225 mM at which the effect of electrolyte concentration on the measured electrophoretic mobility is minimal for all of the silica NPs of varied size.

2. EXPERIMENTAL METHOD Silica colloidal solutions were commercially available from different sources: silica spheres with 4, 10, 14, and 20 nm in diameter were purchased from Alfa Chemicals (New York, USA); those of 6 and 40 nm were from EKA Chemicals (Bohus, Sweden).27 Monodispersed silica spheres with larger particle size (≥50 nm), that is, 60, 80, 100, 160, 200, 250, 300, 350, 400, 450, and 500 nm were obtained from Technical Institute of Physics and Chemistry, CAS (Beijing, China), which were prepared with the procedure originally described by Stöber et al.,28 that is, hydrolysis of tetraethyl orthosilicate in an ethanol solution containing water and ammonia. The resulting silica spheres were centrifugally separated from the suspension and ultrasonically washed with ethanol and further washed with water and then stocked as aqueous solution.17 From production, the dispersion was stabilized with a low concentration of Na2O (Si/Na2O ratio around 100), and it contained no surfactants, buffers, or additional stabilizers.29 The morphology of the silica NPs was examined by a JEM 2010 TEM (JEOL, Peabody, MA) and their average sizes (Table S1 in the Supporting Information) were evaluated statistically from the TEM images (Figure S1 in the Supporting Information). The zeta potential of silica NPs was measured on a Zetasizer Ver.7.01 from Malvern Instruments Ltd (Malvern, UK) and analyzed with the carried Dispersion Technology Software, by which the zeta potentials were calculated from the measured electrophoretic mobility with Henry approximation. Samples were injected into folded capillary cells, and measurements were conducted at 298 K. The equilibrium time was 600 s; deionized water was used as the dispersant for the particles to measure their zeta potential. To examine the NP (electrolyte) concentration effect on the measured electrophoretic mobility, the silica NP solutions of a given size were prepared at varied concentration ranging from 0.002 wt % to a larger value (0.8 wt %) varied on different particle size for measurement. The pH values of the initial concentrated colloidal solutions were adjusted to 8.0 by 0.01 M HCl, and then the solutions of the given concentration were obtained by dilution with deionized water with an adjusted pH of 8.0 using NaOH solution. The concentrations of NaCl for colloidal solutions of different concentration can be evaluated accordingly (Table S1). The silica NP solutions were kept stirring about 30 min before zeta potential measurement.

Figure 1. TEM images for silica NPs of typical size in diameter: (a,b) 99.7 nm; (c,d) 393.3 nm, where (b,d) are the respective enlarged images of (a,c) revealing no pores existing on the surface.

images, where we consider that all these silica NPs are spherical. All TEM images with the corresponding NP size distribution can be found in Figure S1 and the resulting average sizes in Table S1 in the Supporting Information. Zeta potential is determined from the measured electrophoretic mobility (μ) of the NPs; however, the reproducibility of electrophoretic mobility measurements should be examined because in the literature studies, it has been noted that different results were reported for the same kind particles in identical solutions even when the same measuring method was applied.30 In addition to the pH and electrolyte concentration, it has been found that the measured zeta potential depends on the NP size as well as the concentration.31 To find a proper concentration for silica of varied size, electrophoretic mobilities of a series of silica NPs of given size were examined against the concentration. Figure 2a−d displays the measured electrophoretic mobility of several selected silica NPs having typical size against the colloidal concentration and the corresponding electrolyte concentration at the same pH of 8.0. Our results show that the electrophoretic mobility against the colloidal concentration and electrolyte concentration curves can be classified into two kinds, that is, for the particles with their diameter less than 393.3 nm, the electrophoretic mobility magnitude first increases at very low concentrations of both colloid and the electrolyte then reaches almost a stable value (Figure 2a,b) as the concentrations further increase; while for those larger than 393.3 nm the electrophoretic mobility magnitude first decreases at very low concentrations and then approaches a constant value and finally decreases again as the concentrations increase (Figure 2c,d). The curves for the measured electrophoretic mobility against the colloidal concentration for other particle sizes can be found in Figure B


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Figure 2. Two typical groups of measured colloidal concentration and corresponding electrolyte concentration-dependent electrophoretic mobility (μ): (a,b) at smaller particles size (50 nm, e.g., 495.7 nm) at a pH value of 8.0. (e,f) Comparison of the measured electrophoretic mobility (393.3−453.4 nm) with the theoretical prediction of the electrolyte concentration effect on the electrophoretic mobility by the White−O’Brian theory.

converting the electrophoretic mobility to the corresponding zeta potential. Next, in this stable region, we should also consider the sole concentration effect, that is, the effect of the interparticle distance on the zeta potential. According to the literature, when the averaged interparticle distance is larger than the repulsive interaction distance of the colloid given as 2(a + λD),34 where a is the radius of the particle and λD is the corresponding Debye length, the zeta potential would not be affected by the colloidal concentration. We compared the interparticle distance at various colloidal concentration with the above repulsive interaction distance length, and it is found that the colloidal concentration would be effective only for the very small particle size (i.e., 4 nm) and at a small concentration. Therefore, the “pure” colloidal concentration effect on the measured electrophoretic mobility can be ignored in the stable region. Finally, we consider the electrolyte concentration effect on the measured electrophoretic mobility hence the zeta potential. White and O’Brian have derived a simplified analytical expression for the electrolyte concentration on electrophoretic mobilities as shown in eq 1,35 and the accuracy of this equation is to the order of 1/κa and is valid for κa > 10,36 where κ is

S2 in the Supporting Information. Nevertheless, we can always find that there is a distinctive concentration region for silica NPs of any given size at which the measured electrophoretic mobility is not or much less affected by concentration, which has been referred as the “stable region”.32 There several factors can affect the measured electrophoretic mobility when varying the colloidal concentration, for example, pH value at the extreme dilution and the concentration of the electrolytes when the colloidal solution is diluted. For the case of extreme dilution especially for the small-size particle, the light scattering signal would reach the detection limit, giving rise to a much smaller measured magnitude of the electrophoretic mobility.32 On the other hand, the pH value would easily be affected at the extremely diluted solution owing to the dissolution of the CO2 from the air, which would give rise to a lower pH value than the nominal one. Both of these effects would lead to a reduced magnitude of the measured electrophoretic mobility for smallsized NPs.33 For a reliable measurement of the electrophoretic mobility according to the literature, the silica colloidal concentration should be kept as above 0.07 wt %.32 Therefore, in the current measurement, the data points corresponding to those concentration lower than 0.07 wt % are excluded for obtaining a stable region; hence, this stable region is also used for

reciprocal of the Debye length, κ −1 = λD = C

εrε0RT /z 2F 2C0


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The Journal of Physical Chemistry C (where ε0, εr, R, T, z, F, and C0 are the permitivity of a vacuum, the medium dielectric constant, the universal gas constant, the absolute temperature, the valence number of the electrolyte, the Faraday constant, and the concentration of the electrolyte, respectively) E=

̃ − (ln 2/z)(1 − exp(−zξ )) ̃ ] 6[ξ /2 3ξ ̃ − 2 ̃ 2 2 + [κa /(1 + 3m /z )] exp( −zξ /2)

μ=

f (κa ) =

where E = 3ηeμ/(2ε0εrkBT) is the reduced dimensionless value for mobility, where η is the medium viscosity, kB is the 2ε ε RT z 2 Bolzman consntant, ξ̃ is the zeta potential, and m = 0 r ±0 Λ±

f (κa ) = 1 +

1 2{1 + 2.5/{κa[1 + 2 exp( −κa)]}}3

(6)

On the basis of Ohshima’s expression, recently, Qin, et al.42 used the least square method to fit numerically the exact function values of eq 5 calculated by Wiersema, and they gave an approximated Henry function as eq 7

(2)

i z Ψ̃ yz i ̃ y| o o kBT l zz + 4 tanhjjj z Ψ zzzo κo 2 sinhjjjj m z j 4 z} o o κ ze o 2 a k { k {o n ~

(5)

where y = eξ/kBT. Ohshima used the trial and error method to obtain an expression as eq 6 that approximates the Henry function. However, when 1 < κa < 100, the relative error is up to 3%.

where Ψ̃ is the surface potential and x is the distance at the plane of shear from the particle surface (a constant value of x = 0.3 nm was more or less arbitrarily chosen). For spherical particles, if the charge density and concentration of the electrolyte (assumed to be a 1,1-electrolyte at first) in the surrounding solution are known, eq 2 has been modified by Loeb38 to calculate the SCD (σ) on a spherical surface σ = ε0εr

E y 41

is the dimensionless ionic mobility with Λ0 being the limit equivalence conductance (for NaCl Λ0± = 126.45 cm2/Ω mol). An approximate expression for the calculation of the zeta potential from the (diffuse layer) surface potential is given in eq 2, which is derived from the Gouy−Chapman theory for flat plates37 ̃ tanh(zξ /4) = tanh(z Ψ̃/4)exp( −κx)

(4)

where f(κa) is the Henry function. Henry function f(κa) can have values between 1.0 and 1.5 depending on κa values. Wiersema40 derived eq 5 which gives a very complicated representation of f(κa) over the whole range of κa.

(1)

3η

2ε0εrξ f (κa) 3η

f (κa ) = 1 +

1 2{1 + 2.8/{κa[0.9 + exp( −κa)]}}3

(7)

As a result, this improved equation gives a better approximation than eq 6. For example, when 0.01 < κa < 1000, the relative error from this equation with respect to the exact values from eq 5 is less than 1.5%. Therefore, in the current work, we used eq 7 to calculate the zeta potential from the measured electrophoretic mobility for the silica NPs of varied sizes at a given pH of 8.0 and NaCl concentration of 0.225 mM, with a κa value ranging from 0.14 to 17.3. Figure 3 presents the as-derived zeta potential of silica NPs of varied sizes by eq 7.

(3)

and we employed eq 3 to fit our measured electrophoretic mobilities by setting the parameters σ = −0.00327 and σ = −0.00296 C/m2 at two typical particles sizes of 393.3 and 453.4 nm, respectively. Obviously, the fitting curves are in agreement with the experimental data points when κa > 10 as shown in Figure 2e,f except for a scaling factor for the particles larger than 393.3 nm, while for those of smaller particles, the measured electrophoretic mobilities are much less affected by the increasing concentration of the electrolyte. Therefore, for larger NPs at higher colloidal concentration, the decrease in the magnitude of the electrophoretic mobility is caused by the increased concentration of the electrolyte, which indicates that even in the stable region, to compare the measured electrophoretic mobility or zeta potential for silica NPs of varied size, the electrolyte concentration should be kept the same. Accordingly, to construct the SCD against particle size, we chose the measured electrophoretic mobility at a colloidal concentration of 0.15 wt % at a fixed electrolyte concentration of 0.225 mM in the stable region, an advantage of choosing this concentration is that the effect of electrolyte concentration on the electrophoretic mobility is minimal at this point for ∂μ almost all of the particle size, that is. ∂C ≈ 0.

Figure 3. Zeta potential of silica NPs as a function of the silica NP size from 4.1 to 495.7 nm at a given pH value of 8.0 and a NaCl concentration with 0.225 mM. All of the detailed data of the particle size, NaCl concentration, κa values, and calculated zeta potential with the approximated Henry function of eq 7 can be found in Table S1.

With the experimentally measured zeta potentials, the corresponding SCD thus can be determined from the different models. We first employed the surface complexation model,24 taking a spherical silica particle with a radius of a immersed in an infinite electrolyte medium. The background electrolyte contains N types of ionic and monovalence of the ith ionic species (i = 1−4 for H+, Na+, Cl−, and OH−), respectively. On the basis of the full multi-ion charge-regulation model,11 the SCD of the particle σ can be expressed as

0

At this colloidal concentration, the calculated κa ranges from 0.14 to 17.3 in the whole size region (Table S1), and neither Hü ckle approximation (κa ≪ 1)30 nor Smoluchowski approximation (κa ≫ 100)30 can be used to convert the measured electrophoretic mobility to the corresponding zeta potential ξ. Therefore, Henry approximation was used toward this purpose39

D


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Figure 4. Comparison of the measured size-dependent SCD converted from the PB model (red solid circle) at a pH value of 8.0 with that of the theoretical by Barisik et al. (blue solid line). (a) Particle sizes presented in a linear scale; (b) in a logarithmic scale.

σ = −FNtotal

KA − KB[H+]0 +

KA + [H ]0 +

the aqueous solution ψ is described by the PB equation in the spherical coordinate system

2

2 KB[H+]0

z Fψ d2ψ 2 dψ 1 ∑ FziCi0 expijjj− i yzzz + =− 2 ε0εr i = 1 r dr dr k RT {

(8)

N

where Ntotal = NSiOH + NSiO− + NSiOH2+ is the total site density of silanol functional groups on the particle surface, Ntotal = 8 sites/nm2,43 NSiOH, NSiO−, and NSiOH2+ are surface site densities of SiOH, SiO−, and SiOH2+, respectively; KA and KB are respectively the equilibrium constants for the surface reactions SiOH ↔ SiO− + H+ and SiOH + H+ ↔ SiOH2+, that is13,44 KA =

KB =

=

ΓSiOH[H+]s

σ = − ε0εr = 10−pKB(mol/L)−1 = 10−pKB− 3(mol/m 3)−1 =

where pKA = −log KA = 7.6 and pKB = −log KB = 1.9; Γ denotes the surface site density for the indicated surface groups, respectively. [H+]s is the concentration of H+ at the solid/liquid interface governed by the Boltzmann distribution. Under this approximation, [H+]s is related to the bulk concentration [H+]0 by the equation i zFψs yz zz [H+]s = [H+]0 expjjjj− z k RT {

ÄÅ É2 Fψ Ñ Å Ñ KA − KBÅÅÅÅ10−pH exp − RTs ÑÑÑÑ Å ÑÖ Ç σ = − FNtotal ÄÅ F ψ Fψ Å KA + 10−pH exp − RTs + KBÅÅÅÅ10−pH exp − RTs ÅÇ

( )

( )

ÉÑ2

( )ÑÑÑÑÑÑÖ

When σ has a unit of C/m and [H ]0 is expressed in pH unit of 10−pH mol/L = 103−pH mol/m−3,45 then +

É2 ÄÅ Fψ Ñ Ñ Å 10−pKA + 3 − 10−pKB− 3ÅÅÅÅ10−pH + 3 exp − RTs ÑÑÑÑ ÑÖ ÅÇ ÄÅ Fψs Fψ −pH + 3 −pKB− 3Å − + pH 3 Å + 10 exp − RT + 10 exp − RTs ÅÅÅ10 ÅÇ

( )

10−pKA + 3

( )

i zFψs yz 2ε0εrκRT zz sinhjjjj z F k 2RT { Ä É−2 l Å o ij zFψs yzÑÑÑÑ o 2 ÅÅÅÅ o jj zzÑÑ + 16 1 cosh + m Å j 4RT zÑÑ o o κa ÅÅÅÇ (κa)2 o k {ÑÖ n ÄÅ ÉÑÄÅ É−2 |1/2 ij zFψs yzÑÑÑÅÅÅ ij zFψs yzÑÑÑÑ o o ÅÅÅ o j z j z Å Ñ Å lnÅÅcoshjj } zzÑÑÅÅsinhjj zzÑÑÑ o ÅÅÇ Ñ Å Ñ 4 2 RT RT o k {ÑÖÅÇ k {ÑÖ o ~ r=a

(12)

where z = |zi| = 1, for H2O, ε0εr = 7.08 ×10 F/m, R = 8.31 J/(mol·K), F = 96 490 C/mol, T = 298 K, and C0 = ∑4i=1ci0, where ci0 of each species satisfies the electro-neutrality condition, that is, c10 = 10−pH+3, c40 = 10−(14−pH)+3, when pH > 7, c20 = cNaCl + c10 − c40 and c30 = cNaCl.13 Obviously, both eqs 10 and 12 lead to the SDC which requires the knowledge of the electric potential at the NP surface ψs = ψ(r)|r=a. Barisik et al. numerically solved the PB equation for some NPs of typical size ranging from 2, 4, 10, 200, 500 nm to obtain the corresponding ψs in different pH and KCl concentrations. With eq 10, a plot of SCD against the particle size is thus obtained. However, eq 12 explicitly correlates the SCD with the particle size and the surface electrical potential. Because the surface potential ψs is equivalent to zeta potential without considering the Stern layer effect,13,46 and we have measured zeta potential for silica NPs in a large size region (Figure 3), this gives us an opportunity to compare the SCD determined from a chemical complexation model and a physical PB model for electrical potential in solution generated by a charged sphere with a radius of a. With eq 12, we converted the measured zeta potential of silica particles of varied size to size-dependent SCD; the result is shown in Figure 4, which shows that the magnitude of SCD decreases with an increase in the particle size and reaches a plateau when the particle size exceeds a critical value about 30 nm. Our results are in good agreement with that of the theoretical work

(9)

σ = − FNtotal

dψ dr

−10

where z is the valence which is +1 for H+, and ψs is electrical potential at the solid/liquid interface. A general formula for the SCD in surface complexation model reads:

2

(11)

For a spherical NP with a radius of a, the approximate solution for ψ and the SCD of the particle32 with the relative error less than 1% for κa ≥ 0.524 can be derived:

[H +]0 ΓSiO− = 10−pKA mol/L = 10−pKA + 3 mol/m 3 ΓSiOH ΓSiOH2−

RTκ 2 i zFψ zy zz sinhjjj zF k RT {

ÉÑ2

( )ÑÑÑÑÑÑÖ

(10)

Though eq 10 does not contain the geometrical factors of NPs explicitly, the parameters pKA, pKB, and ψs can be sizedependent. The other one is PB model for nanospheres, where the particle size is incorporated explicitly; the electric potential in E


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Figure 5. Comparison between the size-dependent SCD derived from PB and complexation models. (a) Complexation model with a sizeindependent pKA of 7.6 (blue) compared with that from the PB model (red); (b) complexation model with size-dependent pKA values (blue), that is, pKA = 4.5 (d < 30 nm) and pKA = 8.5 (d > 60 nm), compared with that from the PB model (red).

of Barisik et al.13 except for a scaling factor as shown in Figure 4. One might expect that these two equations, that is, (10) and (12) would give rise to the same tendency of the sizedependent SCD when using the same set of the measured zeta potentials. Figure 5a compares the size-dependent SCD derived from complexation model with eq 10 using a sizeindependent pKA of 7.6 and PB model of eq 12 based on the same set of zeta potential data (Figure 3), and the result shows a significant difference between the size-dependent SCD curves calculated by the complexation model and the PB model, respectively. Such a discrepancy is possibly due to the use of size-independent value of pKA in eq 10. It has been shown that there are different types of silanol groups at the silica surface (e.g., isolated, geminal, and vicinal) that affect the surface charge. Isolated silanol groups on the surface of smaller SiO2 NPs are far apart from each group by the NPs’ high surface curvature, and thus resulting in little interaction between them.47 The surface curvature decreases with increasing radius of NPs, leading to strong hydrogen bonding (H-bonding) between silanol groups through bridging water molecules.47 The isolated and vicinal (H-bonded) silanol groups have pKA values of 4.5 and 8.5, respectively.47,48 The variation in the surface silanol structures among the differentsized SiO2 NPs leads to differences in their surface charge properties. In the case of SiO2 NPs with diameters less than 30 nm, their surfaces have a high density of isolated silanol groups,47 whereas the surfaces of relatively large SiO2 NPs (>60 nm) are fully covered with H-bonded silanol groups.49 Therefore, as shown in Figure 5b, we used two pKA values of silica NPs to substitute into eq 10, that is, pKA = 4.5 (d < 30 nm) and pKA = 8.5 (d > 60 nm), and the result is compared with that of PB model. Apparently, the curve feature matches well with that of PB model considering the missing pKA values in the size region from 30 to 60 nm. Therefore, it can be concluded that the complexion model can also well describe the size-dependent SCD of silica NPs on the condition that the pKA of the individual NPs of given size is known, and only under this circumstance, both models predict the same sizedependency of the SCD. Therefore, the PB model is more appropriate in describing the SCD without the preknowledge of the size-dependent pKA. On the other hand, the complexation model gives a molecular account for the abrupt increase of the SCD when the particle size becomes smaller than 30 nm, which is due to a much larger deprotonation constant of the isolated silanol groups at the surface. Our experimental observation is quite consistent with the theoretical calculation of the SCD for the metal oxide, that is, goethite NPs, which

reveals that as the particle diameter decreases to below 10 nm, there is considerable increase in the SCD as compared with the limiting values seen for particles larger than 20 nm. This increase in the SCD is due to the enhanced screening efficiency of the electrolyte solution around small NPs, which is most prominent for particles of diameters less than 5 nm,11 where the theory used is the corrected Debye−Hückel theory of surface complexation, which considers the change of pKA with the particle size. By using the same theory, Abbas et al. calculated the SCD of 9 and 25 nm silica particles in 0.05 M NaCl at pH = 8, and they successfully predicted that for a given pH above the pHPZC of silica (PZC stands for the potential of zero charge), the acid−base equilibrium Si−OH + OH− ⇔ Si−O− + H2O is shifted more toward the right (Si−O−) for smaller silica particles than for larger particles, resulting in a size-dependent pKA.50 Finally, we come to the absolute value of the SCD. Figure 4 predicts an SCD value of a glass plate (a → ∞) about 3.0 × 10−3 C/m2, consistent to a reported charge density for glass wall (in pure water, no electrolyte) of 2000 ± 200 e/μm2 (3.18 ± 0.318 ×10−3 C/m2),44 and 1.12 ± 0.32 × 10−3 for silica NPs with a diameter of 580 nm in deionized water.51 Also for the NPs of smaller size, it has been reported a SCD value of 0.0245 C/m2 for 9.1 nm silica at pH = 8 and 1 mM KCl,52 a condition very close to ours, and 0.025 C/m2 for silica NPs with a = 58 nm at pH = 8 and 1 mM NaCl.53 These values are quite similar to our corresponding data. Therefore, the absolute values of the SCD are reliable in our measurement as shown in Figure 4.

4. CONCLUSION We have determined the zeta potentials of spherical silica NPs of varied size with diameter spanning from 4.1 to 495.7 nm at pH = 8.0 and a background electrolyte NaCl concentration of 0.225 mM. The SCD was calculated from PB model which contains the particle size explicitly. We found that the magnitude of SCD decreases with the increase of the particle size. While an abrupt drop in SCD occurs in a size region from 4.1 to than 30 nm, and when the particle size is larger than 30 nm, the SCD is almost independent of the size. This is due to the size-dependence of deprotonation of the surface silanol functional groups, that is, surface of smaller size gives rise to a larger deprotonation constant. Furthermore, the validity of surface complexation and PB models are compared. We found that the surface complexation model with size-independent pKA would be inadequate to describe the size-dependent SCD, whereas both complexation model with size-dependent pKA and PB model are consistent in describing the size-dependent F


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SCD of silica NPs based on experimentally measured zeta potential.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b07566. All TEM images with the corresponding NP size distribution; plots of zeta potential against colloidal concentration and salt concentration for all silica NPs of different size; and all of the detailed data of the particle size, NaCl concentration, κa values, calculated zeta potential with the approximated Henry function of eq 7, and the SCDs calculated from different models (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +86-10-82648118. Fax: +86-10-82648118. ORCID

Yu-Xiang Weng: 0000-0003-0423-2266 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (NSFC grant nos. 21433014 and 11721404), the Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDPB0400), and Chinese Academy of Sciences Frontier Science Key Programs (QYZDJ-SSW-SYS017). Dr. R. Shen is kindly acknowledged for her assistance in Zeta potential measurement.

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REFERENCES

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Experimental Determination of Particle Size-Dependent Surface

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