Article pubs.acs.org/JPCB
Decoration of Microparticles by Highly Charged Nanoparticles Haohao Huang*,† and Eli Ruckenstein*,‡ †
School of Materials Science and Engineering, South China University of Technology, Guangzhou, 510640, China Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260, United States
‡
ABSTRACT: A negatively charged microparticle is immersed into an aqueous solution containing highly negatively charged nanoparticles and an electrolyte. Using a modified Poisson−Boltzmann equation which takes into account the van der Waals interactions between microparticle and nanoparticles and in which the latter are treated as macro-ions, it was found that there are conditions under which the concentration of nanoparticles close to the microparticle becomes larger (even much larger) than the bulk one and halos are formed around the latter particles. Halos are formed when the van der Waals attractive interactions between the microparticle and nanoparticles are stronger than the repulsive interactions between the same particles plus the repulsive interactions between the nanoparticles.
equation theories.12−16 Two mechanisms have been proposed to explain the experimental results of Toher et al. In the first, the repulsion between nanoparticles was considered responsible for their deposition upon the microparticle and for the increased repulsion between microparticles.1,2 In the second, an attraction was considered to be responsible for the formation of a halo of highly charged nanoparticles around the microparticle and for the increased stability of microparticle dispersions.12 The latter authors noted that the average distance among nanoparticles in the dispersion is too large for the electrostatic repulsion among them to be responsible for the formation of halos. The goal of this paper is to calculate the concentration distribution of nanoparticles and to identify the conditions under which halos of nanoparticles are formed around microparticles. The distribution of nanoparticles around microparticles is determined using a modified Poisson− Boltzmann equation which includes the van der Waals interactions between nano- and microparticles and consider the nanoparticles as macro-ions.
1. INTRODUCTION In the past decade, a technique for colloidal stabilization has been proposed by Tohver et al.1,2 Their experiments involved aqueous dispersions of silica spherical microparticles possessing a relatively low negative charge and of highly negatively charged zirconia nanoparticles. In the absence of nanoparticles, as well as in the presence of very low concentrations of nanoparticles, the microspheres aggregated because of van der Waals attraction between them. By increasing the nanoparticle concentration, the aggregation could be prevented and the suspension stabilized because halos of highly charged nanoparticles were formed around the microparticles. At high nanoparticle concentrations, attraction between microparticles was observed and aggregation reappeared. Following the Tohver et al. experiments, the halo formation was extensively examined.3−11 The amounts of nanoparticles present in the halo were determined by Zhang et al. who empolyed the ultra-small-angle X-ray scattering procedure.3 Chan et al. examined the effect of size ratio between nanoparticle and microparticle on the halo formation.4 Chang et al. conducted a series of experiments by adding various volume fractions of zirconia nanoparticles in negatively charged microlatex dispersions in water.5 They confirmed that the zirconia nanoparticles do stabilize the microcolloidal suspension. McKee and Walz6 used atomic force microscopy to measure the interaction force between a microsize glass sphere and a planar glass substrate (both negatively charged) in aqueous solutions containing highly negatively charged polystyrene or zirconia nanoparticles. They found that the repulsion between microparticle and the planar surface became stronger as the nanoparticle concentration in the dispersion increased. The haloing mechanism was examined theoretically by Monte Carlo simulations and statistical mechanisms integral© 2013 American Chemical Society
2. THEORETICAL FRAMEWORK A microparticle of radius RL is immersed into an aqueous solution containing nanoparticles of radius RS and an electrolyte. Each of the nanoparticles has a charge ze, (where e is the protonic charge and z the number of charges assumed negative) and the microparticle has a surface potential Ψ0, also assumed negative. The nanoparticles cannot be located in a layer of thickness RS around the microparticle due to hardReceived: February 22, 2013 Revised: April 9, 2013 Published: April 26, 2013 6318
dx.doi.org/10.1021/jp401889m | J. Phys. Chem. B 2013, 117, 6318−6322
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sphere repulsion. Outside this region, the distribution of nanoparticles is assumed to be Boltzmannian. Because the microparticle is assumed spherical, the Poisson equation for the electrical potential Ψ has the form
As boundary conditions, we use Ψ(r = RL) = Ψ0
where Ψ0 is constant, and
2
Ψ=0
ρ dΨ 2 dΨ + =− 2 r dr εε0 dr
(1)
RLR S AH 6 (RL + R S)(r − RL − R S)
(for r → ∞)
(9)
Equation 1 and the boundary conditions (8) and (9) can be solved numerically to obtain the electrical potential distribution for selected bulk concentrations of electrolyte and nanoparticles. Once the distribution of electrical potential is obtained, the distribution of nanoparticles is obtained by assuming Boltzmannian distribution.
where ρ is the charge density in the solution, ε is the dielectric constant, and ε0 is the vacuum permittivity. The charge density ρ involves the ions of the electrolyte, the charged nanoparticles, and the counterions of the nanoparticles (which are considered the same as the cations of the electrolyte). The concentration of the counterions of the microparticles is neglected. The nanoparticles have van der Waals interactions with the microparticle, with a van der Waals interaction energy given by17 Uvdw = −
(8)
3. RESULTS AND DISCUSSION The microparticles employed in experiments have a zeta potential in the range of −20 to −70 mV. The effect of this surface potential on the concentration distribution of nanoparticles is examined in Figure 1, which shows that there are
(2)
where AH is the Hamaker constant. The van der Waals interaction was calculated assuming a cutoff distance of 3.5 Å from the microparticle surface. Because at thermodynamic equilibrium the chemical potential of each species i (cations, anions, and nanoparticles) in the system is the same as that in the bulk, one can write that for dilute solutions μi = αie Ψ + kT ln ci = kT ln ci,0 (i = + for the cations and − for the anions)
(3) Figure 1. Effect of the surface potential of the microparticle on the concentration distribution of nanoparticles around the microparticle: (1) −10 mV; (2) −20 mV; (3) −40 mV; (4) −50 mV; (5) −80 mV. For the other parameters, the following values were selected: RS = 5 nm, RL = 500 nm, z = −10, c = 0.01 M, AH = 5 × 10−21 J, the volume fraction of nanoparticles ϕ = 10−2.
for the ions of the electrolyte and μs = ze Ψ + kT ln cs −
RLR S AH 6 (RL + R S)(r − RL − R S)
= kT ln cs0
(4)
for the nanoparticles, where ci is the concentration (c+ of the cations and c− of the anions), ci,0 is the value of ci in the bulk, cs is the concentration of nanoparticles, cs0 is their concentration in the bulk, αie is the charge of species i, k is the Boltzmann constant, and T is the absolute temperature in K. The concentration of cations in the bulk includes the counterion dissociated from the negatively charged nanoparticles. Consequently, c+ = c + |z|cs0, where c is the bulk concentration of the electrolyte. The concentrations of cations and anions can be expressed in the form ⎛ eΨ ⎞ c+ = c+0 exp⎜ − ⎟ ⎝ kT ⎠
(5)
⎛ eΨ ⎞ c − = c −0 exp⎜ ⎟ ⎝ kT ⎠
(6)
concentration peaks (hence halos) close to the surface of the microparticle for surface potentials of the latter less negative than −40 mV. These peaks occur because the van der Waals attraction between the microparticle and nanoparticles surpasses the repulsion between the nanoparticles combined with the repulsion between microparticle and nanoparticles. The effect of nanoparticle radius on the distribution of electrical potential is presented in Figure 2 for a constant
and that of the nanoparticles as ⎧ ⎛ −ze Ψ − Uvdw ⎞ ⎟ (for RL + R S ⎪ cs0 exp⎜ ⎠ ⎝ kT ⎪ < r < ∞) ⎪ cs = ⎨ (for RL ⎪0 ⎪ ϕ. The dotted line which separates halos from not halos is marked in the figure. The electrolyte concentration affects the distributions of the electrical potential and nanoparticle concentration. As the electrolyte concentration increases, the concentration of nanoparticles at the surface of the microparticle also increases because of their lower repulsion by the surface potential of the microparticles (Figure 5). However, when the electrolyte concentration becomes sufficiently large (0.1 M in the system under consideration), the concentration distribution of nanoparticles does not change because of too strong screening.
Figure 3. Effect of the charge of nanoparticles on the concentration distribution of nanoparticles around microparticles. The charge of nanoparticles is as follows: (a) AH = 5 × 10−21 J; (1) −5e; (2) −10e; (3) −15e; and (4) −20e; (b) AH = 1 × 10−20 J; (1) −5; (2) −10; (3) −15; (4) −20; (5) −40; (6) −45; and (7) −60. The following values of the other parameters were selected: RS = 5 nm, RL = 500 nm, c = 0.01 M, Ψ0 = −20 mV, and ϕ = 10−2.
distribution through the electrostatic interactions between themselves and between the microparticles and the nano ones. When the charges of micro- and nanoparticles are sufficiently small, the electrostatic interactions are weak, and only the van der Waals attraction has a significant effect on the concentration of nanoparticles close to the microparticle. When the charge of nanoparticles becomes more negative, their concentration close to the microparticle decreases. There are critical charges of nanoparticles for halo formation (about −20e for AH = 5 × 10−21 J (Figure 3a) and about −45e for AH = 1 × 10−20 J (Figure 3b)) at which the nanoparticle concentration near the microparticle surface becomes close to its value in the bulk; for less negative values of the charge halos with higher nanoparticle concentrations are formed. Comparing parts a and b of Figure 3 (in which AH = 5 × 10−21 J for Figure 3a and AH = 1 × 10−20 J for Figure 3b) at various charges, one can observe that when the Hamaker constant increases from 5 × 10−21 to 1 × 10−20 J, the 6320
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Figure 5. Effect of the concentration of electrolyte on the concentration of nanoparticles. The electrolyte concentrations are (1) c = 1 M; (2) c = 0.1 M; (3) c = 0.01 M; (4) c = 0.005 M; and (5) c = 0.001 M. For the other parameters, the following values were used: RS = 5 nm, RL = 500 nm, z = −15, Ψ0 = −20 mV, AH = 5 × 10−21 J, the volume fraction of nanoparticles ϕ = 10−2.
Figure 6. Effect of the size ratio between nanoparticle and mciroparticle on the distribution of nanoparticles for (1) RS = 12.2 nm (effective diameter = 26 nm); (2) RS = 15.4 nm (effective diameter = 34 nm); (3) RS = 17.5 nm (effective diameter = 40 nm); (4) RS = 23.7 nm (effective diameter = 57 nm). The values of experimental parameters were RL = 590 nm, c = 0.001 M, Ψ0 = −1 mV, the volume fraction of nanoparticles ϕ = 10−3, and Hamaker constant AH = 2 × 10−21 J, and the cutoff distance is 0.15 nm. The charges of nanoparticles are calculated from their measured zeta potentials.
Using ultra-small-angle X-ray scattering (USAXS), Zhang et al.3 found that the nanoparticle concentration in the halo is significantly higher than its bulk value in solution and that the lateral separation distance between nanoparticles within the halo is larger than their characteristic size. They determined the number of nanoparticles in the halo of each microparticle by three different methods, USAXS, adsorption, and zeta potential. Their experimental data are listed in Table 1 together with the values obtained on the basis of the present approach.
them. The number of nanoparticles adsorbed on the surface per unit area is approximated by the number of nanoparticles present in a layer RS/2 thick around the microparticle. The results are listed in Table 2. There are differences between Table 2. Number of Nanoparticles near the Microparticles’ Surfacea
Table 1. Comparison of the Number of Nanoparticles in the Halo of Each Microparticlea
no. of nanoparticles in halo nanoparticle vol fraction lateral separation dist (nm)
USAXS3
adsorption3
ζpotential3
calculation
1935
3280
7569
1834
0.039 22.9
0.068 17.7
0.154 11.6
0.038 23.6
pH
expt, μm‑2
calcd, μm‑2
halo
2.75 3.0 4.6 5.6
160 50 4 ∼0
156 37 ∼0 ∼0
exists exists does not exist does not exist
a
The values of some parameters used in the calculation were taken from ref 11. They are RS = 10 nm, z = −86, Ψ0 = −20, −28, −61, −71 mV for pH 2.75, 3.0, 4.6, and 5.6, respectively. c = 2 mM for pH 2.75, c = 1 mM for pH 3.0, 4.6, and 5.6. ϕ = 10−3. The value of Hamaker constant AH = 1.75 × 10−20 J and the cutoff distance 0.15 nm were selected for fitting the data.
a
The following values of the experimental parameters were used in the calculation: RL = 285 nm, RS = 2.6 nm, Ψ0= −10 mV, c = 0.0315 M. z = −44 (calculated from the zeta potential of nanoparticles). The value of Hamaker constant AH = 4.4 × 10−20 J and the cutoff distance 0.15 nm were selected for fitting the data. The number of the nanoparticles was counted from the microparticle surface to the distance at which the nanoparticle concentration reaches its bulk concentration. The volume fraction and lateral separation distance are calculated with equations used as in ref 3: ϕ ∼ [Nnano/(4Nmicro)](RS/RL)2 and L = (πRS2/ϕ)1/2.
experiment and calculations. However, the number of nanoparticles per unit area is nonuniform (and the calculations considered that it is uniform) and the plates were washed with aqueous solutions of various pHs. The washing will make the number counted smaller than the real one and less uniform. In the calculations, we consider that halo forms when the concentration of nanoparticles close to the microparticles becomes larger than the bulk one.
The size ratio effect on the halo formation was investigated experimentally by Chan et al.4 They found that the number of nanoparticles in the halo of each microparticle increases with increasing nanoparticle size. Figure 6 shows that the calculations based on our approach qualitatively provide the same behavior. The experimental results obtained by Ji et al.11 are now compared with the results of the calculations. In their experiments, a silica plate was immersed in a nanoparticle solution in water of polystyrene for 30 min, rinsed gently with nanoparticle-free solutions of various pHs, and then dried in air. The number density of nanoparticles was determined from scanning electron microscopy images. The charge of a nanoparticle was calculated using the Gouy−Chapman theory for the zeta potential of −60 mV determined experimentally by
4. CONCLUSIONS One can conclude that the formation of halos of highly charged nanoparticles around microparticles involves both van der Waals and electrostatic interactions. The attractive van der Waals interactions play the main role in the generation of halos because they stimulate adsorption on the microparticle, whereas the electrostatic repulsive ones (between nano- and microparticle as well as those between the nanoparticles) oppose their formation. Critical values are determined for the surface potential of microparticles below which halos are formed, for the charge of the nanoparticles below which halos are generated, and for the electrolyte concentration above which halos occur. 6321
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (H.H.); feaeliru@buffalo.edu (E.R.). Notes
The authors declare no competing financial interest.
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REFERENCES
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