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J. Phys. Chem. C 2010, 114, 14876–14884

Particle Size and pH Effects on Nanoparticle Dissolution Hengzhong Zhang,* Bin Chen, and Jillian F. Banfield Department of Earth and Planetary Science, UniVersity of California Berkeley, 307 McCone Hall, Berkeley, California 94720 ReceiVed: July 1, 2010; ReVised Manuscript ReceiVed: August 5, 2010

Dissolution is an important process that alters nanoparticle abundance and properties. Here we used EDTA (ethylenediaminetetraacetic acid) to dissolve ZnS nanoparticles and develop a predictive model for the effects of particle size and pH on nanoparticle reactivity. Synchrotron in situ small-angle X-ray scattering (SAXS) was used to monitor changes in scattering intensity during the dissolution of ZnS nanoparticles over the pH range 9-10. The nanoparticle mass, size, and size distribution were derived from fitting of the SAXS data. Thermodynamic analysis showed that both particle size and pH play important roles in dissolution. Kinetic modeling of the extent of dissolution as a function of time revealed that the reaction is reversible and first order with respect to the concentration of reagents and products and the total surface area of ZnS nanoparticles. The dissolution equilibrium constant and the interfacial free energy of ZnS nanoparticles were derived from parameters obtained from the kinetic modeling. Analysis of the variation of the equilibrium constant with pH and particle size showed that the lower the pH and the smaller the particle size, the higher the solubility of ZnS nanoparticles in the experimental pH range. The method developed in this study revealed the effect of particle size on dissolution-precipitation equilibria and provides a basis for description of nanoparticle reactivity. 1. Introduction Dissolution and precipitation reactions are important steps in nanoparticle (NP) formation and can directly impact nanoparticle properties and abundance. Once formed, NPs may subsequently interact with a range of compounds, such as chemical reagents or humic acids in soil, which can cause them to dissolve. Particle size modifies NP reactivity by changing the surface site distribution and total free energies, such that small NPs dissolve faster than larger particles, or the former dissolve whereas the latter do not. In addition to its basic chemical significance, the size-dependent dissolution behavior of NPs may impact element cycling in nature and determine NP toxicity to various biological species. Thus, it is essential to study how particle size impacts NP dissolution reactions. Study of NP dissolution in a solution requires monitoring both the quantity (such as concentration or mass) and the size of the NPs simultaneously. Synchrotron small-angle X-ray scattering (SAXS) is ideal for in situ study of NP dissolution because the high X-ray flux yields strong signals suitable for rapid measurements of both NP quantity and size. For example, Rimer et al. used SAXS to study dissolution rates of silica NPs and reported that dissolution rates are nearly independent of the particle surface area at pH > 11.1 Viswanatha et al. used in situ SAXS to study the formation of 1-3 nm diameter CdS and ZnS NPs and found that higher temperatures result in better size-focusing.2 Rath et al. used SAXS to follow the growth of ZnS NPs in both conventional and microwave assisted syntheses.3 Their XRD and SAXS results showed that the synthesized ZnS NPs are 1-7 nm in diameter. In general, the particle size and size distribution derived from SAXS are in good agreement with those derived from electron microscopy observations.4 * To whom correspondence should be addressed. E-mail: heng@ eps.berkeley.edu.

The switch between dissolution and precipitation reactions for nanoparticles has been studied previously by controlling parameters including pH5 and oxygen concentration6 or by exposure to UV light.7 However, the role of the particle size in determining dissolution-precipitation equilibria for NPs has not been studied experimentally. Lack of interfacial free energy data for NPs in solutions also hinders quantitative predication of the particle size effect. In the current study, ZnS NPs in a basic EDTA (ethylenediaminetetraacetic acid) sodium salt solution was used as a model system to study NP dissolution. In situ SAXS was used to measure both the reaction extent and the NP particle size as a function of time. Thermodynamic analysis and kinetic modeling of the measured data unveiled key parameters controlling the NP dissolution-precipitation equilibria, and a predictive model for the effects of size and pH on NP reactivity was developed. 2. Experimental Section Following previous work (ref 8), we synthesized ZnS NPs via reaction of thioacetamide (CH3CSNH2; TAA; from SigmaAldrich) and zinc acetate (Sigma-Aldrich) in a basic solution. Twenty milliliters of 0.02 M TAA solution was mixed with 20 mL of 0.02 M zinc acetate solution in a 50 mL plastic tube. The pH of the solution was adjusted to 10.0 using a few drops of a pH 12.6 sodium hydroxide solution. The pH was monitored using a pH meter (Orion 290A). A pH of 10.0 was chosen to ensure the formed ZnS NPs were in a sphalerite (cubic) phase rather than a mixed cubic and hexagonal structure.8 Suspensions of the produced ZnS NPs were kept for 18 h before starting the dissolution reaction in an EDTA solution. In situ dissolution experiments were conducted in the SAXS Beamline Station 1-4, Stanford Synchrotron Radiation Laboratory. A Teflon disk (thickness ∼1 mm and diameter ∼40 mm) with a circular open window of ∼20 mm in diameter was enclosed by two thin Kapton films for use as the in situ reaction

10.1021/jp1060842  2010 American Chemical Society Published on Web 08/19/2010

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cell. A reaction solution was injected into the cell and quickly sealed and mounted onto a specially designed SAXS sample stage. The disk was slowly rotated using a direct-current driven mini motor to prevent the ZnS NPs from settling at the bottom of the cell. SAXS data collection started in less than 5 min after the solution injection. The SAXS experiments were performed at room temperature and with an X-ray wavelength of 0.1488 nm. A 2-dimensional 1024 × 1024 CCD array was used to record the scattering X-ray images, which were further converted to scattering intensity vs q (scattering vector) data using the beamline software. Each data collection lasted for ∼5 or 10 min, depending on the strength of the monitored scattering intensity (weak ones used longer times). In data reduction, dark current, solvent scattering, beam current change, and data collection time were taken into account in calibration and normalization of the scattering intensity. The usable measured q ranges from ∼0.2 to 2.5 nm-1. Two in situ dissolution experiments were conducted. In experiment 1 (expt 1), 1.0 mL of ZnS NP suspension (∼0.01 M ZnS; see above) was mixed with 0.5 mL of 0.2 M EDTA solution (pH was preconditioned to 9.2), forming a dissolution system of ∼0.007 M ZnS + 0.067 M EDTA with pH 9.0 (after minor adjustment with a dilute HCl/NaOH solution). In experiment 2 (expt 2), 1.0 mL of ZnS NP suspension (∼0.01 M ZnS) was mixed with 0.5 mL of 0.052 M EDTA solution (pH was preconditioned to 10.2), forming a dissolution system of ∼0.007 M ZnS + 0.017 M EDTA with pH 10.0 (after minor adjustment). SAXS data collection of expt 1 was continued to ∼1.5 h and that for expt 2 to ∼4 h, when no significant changes with time in the scattering intensities were observed. The pH of the expt 2 solution after SAXS data collection was remeasured to be 10.1. Three-dimensional representations (intensity-q-time) of the scattering data are shown in Figure 1. It is seen that dissolution of ZnS NPs was more extensive in expt 1 than in expt 2, indicating the effect of experimental conditions on the dissolution reaction. 3. SAXS Data Processing In SAXS, the X-ray scattering comes from the interaction between the X-ray electromagnetic wave and the electrons in a material. The scattering intensity (after background subtraction), I(q), reflects the difference in the electron densities of a targeted material and the surrounding matrices (such as a solvent). In general, I(q) is governed by the particle shape and the interparticle interaction:9

I(q) ) I0NV0 P(q) S(q) ) y0P(q) S(q) 2

Figure 1. Scattering intensity of ZnS nanoparticles (0.007 M) as functions of scattering vector and time in dissolution in (A) 0.067 M EDTA solution and pH 9.0 (expt 1) and (B) 0.017 M EDTA solution and pH 10.0 (expt 2).

with and without explicit consideration of the particle size distribution (PSD) of NPs in the fitting. 3.1. Without Explicit Consideration of PSD Function. For monodisperse spherical particles, the form factor in eq 1 takes the form9

P(q) )

(1)

where q is the scattering vector: q ) 4π sin(θ)/λ, λ being the X-ray wavelength and θ being half the angle between the scattered beam and the incident beam. P is the form factor related to the shape of the particles, S is the structure factor related to the interaction between particles, N is the total number of the particles in an X-ray irradiated space of a sample, V0 is the volume of each particle, and I0 is a proportional coefficient. The constant y0 ) I0NV02. Equation 1 shows that the scattering intensity is proportional to the number of particles and the squared particle volume. Thus, for SAXS of NPs in a solution, through fitting to the I-q data, it is possible to derive the NP size (embedded in P and S), size distribution, and quantity (N and V0) information. In the following, we consider two cases:

{

3[sin(qr) - qrcos(qr)] (qr)3

}

2

(2)

This is an oscillation function of qr (r is the radius of the particles), which smears out at large qr values. In general, most NP systems are polydisperse. If particles distribute over a wide range of particle size, a Debye approximation applies:9

(

P(q) ) 1 +

√2q2r2 3

)

-2

(3)

This is a smooth function of qr (r is the average radius of the particles), to which the upper bound of the oscillation function (eq 2) approaches as qr increases. The structure factor reflects the particle interaction (or correlation) in a system. In a very dilute solution, interaction

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of particles is minor, S is close to 1. In other cases, S is a function of q. Teixeira derived an analytical equation for the structure factor:10 S(q) ) 1 +

1 dΓ(d - 1) · × sin[(d - 1) tan-1(qξ)] (qr)d [1 + 1/(qξ)2](d-1)/2

(4)

where d is the fractal dimension, ξ is a parameter representing the size of an aggregate (or the correlation length),10 and Γ(x) is the gamma function of argument x. At high q, the unity term of eq 4 dominates (i.e., S f 1). At low q, S increases rapidly with decreasing q. Inserting eqs 3 and 4 into eq 1 generates the I vs q equation under the Debye approximation. Using nonlinear least squared fitting to the SAXS I-q data (Figure 1), the average particle radius r, the constant y0 (proportional to NV02), and the fractal dimension d can be obtained. 3.2. With Explicit Consideration of PSD Function. The reciprocal Γ distribution function is often used to describe the (number-weighted) size distribution of particles:4 PSD(r) )

( )

dN 1 1 ) Ndr Γ(1/σ2) 2σ2r0

1/σ2

( )

(2r)-1+1/σ exp 2

r r0σ2

(5)

where N is the total number of particles, dN is the infinitesimal number of particles distributed over an infinitesimal radius range dr, and r0 represents the mean radius of all the particles and σ is the variance in r0. The scattering intensity from a specific size of NPs is still described by eq 1. For all the particles, the total scattering intensity is the summation of the scattering intensities of all the particles. Referring to eq 1 and using integration for the summation, the total scattering intensity can be represented as

∫ P(q) · S(q) · V2 dN 2 ∞ 4 I0N ∫0 P(q) · S(q) · PSD(r) · ( πr3) dr 3 ∞ 16 2 π I0N ∫0 P(q) · S(q) · PSD(r) · r6 dr 9 y0′ ∫ P(q) · S(q) · PSD(r) · r6 dr

I(q) ) I0 ) ) )

(6)

∞

0

where y0′ ) 16π2I0N/9. Assuming NPs are spherical, the form factor at a specific size r takes the form of eq 2 (since in an infinitesimal dr range, the NPs can be looked as monodisperse). The structure factor still takes the form of eq 4. After eqs 2, 4, and 5 are inserted into eq 6, the total scattering equation is generated. Fitting of eq 6 to the SAXS I-q data (Figure 1), the mean size r0, the variance σ, the fractal dimension d, and the constant y0′ (proportional to N) can be obtained. The PSD function (eq 5) can also be calculated using the fitted parameters. 3.3. Fitting Results. Parts A and B of Figure 2 exemplify the fit to the SAXS data without/with explicit consideration of the PSD function as described previously. Through fitting, the form factors and the structure factors were derived as a function of the scattering vector (Figure 2C,D). From the variations of P and S with q, one sees that at low q, the variation of the scattering intensity with q is dominated by the structure factor S (as P is close to 1); at high q, it is dominated by the form factor P (as S is close to 1). Thus, the NP aggregation state

Figure 2. Fit of the SAXS data without (A) and with (B) explicit consideration of the particle size distribution (PSD) of the ZnS nanoparticles (dissolution in 0.067 M EDTA for 0.23 h in expt 1), and the corresponding form factors and structure factors (C, without consideration of PSD; D, with consideration of PSD; at mean radius 1.31 nm).

(determining the S) and the shape (determining the P) have unequal effects on the total scattering intensity at different q ranges. Figures 3 and 4 show variations with the reaction time of the fitting-derived quantities for the dissolution experiments 1 and 2, respectively. These quantities include the mean radius (Figures 3A and 4A), the relative mass (m ∼ (4π/3) × r3N; Figures 3B and 4B), and the dissolution fraction (x ) 1 - m/m0;

Nanoparticle Dissolution

Figure 3. Variations with time of number-weighted mean radius (A), mass (B), and dissolution fraction (C) of nano-ZnS in dissolution in 0.067 M EDTA at pH 9.0 (expt 1). Key: open points, without consideration of PSD; closed points, with consideration of PSD.

Figures 3C and 4C) of ZnS NPs (where m0 represents the mass at reaction time t ) 0). From Figures 3A and 4A, one sees that there is a major difference in the mean radius of the ZnS NPs obtained from fitting with and without explicit consideration of the PSD. Without explicit consideration of PSD (but with implicit wide PSD in the Debye approximation), the mean particle size decreases with increasing time, while with explicit consideration of the PSD (eq 5), the mean radius increases gradually with increasing time. The former case indicates that on average, the particles are shrinking due to the dissolution. The latter case indicates that there exists preferential dissolution of smaller particles, which leaves bigger particles in the solution and hence there is an increase in the mean radius with increasing reaction time. This is confirmed by the variation of the PSD function with the time. As demonstrated in Figure 5, with the increase in the reaction time, the normalized PSD becomes broader and moves toward larger sizes. Thus, an increase in the mean radius is produced. This effect is more obvious in expt 1 (Figure 5A)

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Figure 4. Variations with time of number-weighted mean radius (A), mass (B), and dissolution fraction (C) of nano-ZnS during dissolution in 0.017 M EDTA at pH 10.0 (expt 2). Key: open points, without consideration of PSD; closed points, with consideration of PSD.

than in expt 2 (Figure 5B), as the former had a higher EDTA concentration than the latter (0.067 M vs 0.017 M). Results also show that with the increase of the reaction time, the mass of the ZnS NPs decreases (Figures 3B and 4B) and the dissolution fraction increases (Figures 3C and 4C). In expt 1, the dissolution was close to complete after reacting for ∼1.4 h (Figure 3C). In expt 2, equilibrium was reached after reacting for ∼1 h (Figure 4C). One notes that if no explicit PSD is considered in SAXS data processing, the dissolution fraction decreases after ∼1.2 h in expt 2 (Figure 4C). This would mean that ZnS dissolves first and then precipitates again, implying oscillation that is unlikely in a closed system. This effect is more likely due to use of the Debye approximation, which assumes an unrealistically wide particle size distribution. Thus, in the following thermodynamic analysis and kinetic modeling, we use the data obtained from SAXS fitting with explicit consideration of PSD.

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[Y4-] ) CYRY4

(7)

where CY is the total concentration of EDTA in the solution and RY4 is the value of the free Y4- concentration over the total EDTA concentration:11 RY4 )

[Y4-] ) CY K1K2K3K4

[H+]4 + K1[H+]3 + K1K2[H+]2 + K1K2K3[H+] + K1K2K3K4

(8)

The R value is pH dependent: it equals 5.20 × 10-2 at pH ) 9.0 and 3.50 × 10-1 at pH ) 10.0.11 (d) Dissociation equilibria of aqueous hydrogen sulfide:

H2S(aq) ) HS- + H+ HS- ) S2- + H+

Figure 5. Normalized volume (mass)-weighted PSD function of nanoZnS in dissolution in a 0.067 M EDTA solution (A: expt 1, pH 9.0), and in a 0.017 M EDTA solution (B: expt 2, pH 10.0).

4. Thermodynamic Analysis of ZnS NP Dissolution in EDTA Solution 4.1. Dissolution of Bulk ZnS in EDTA Solution. The dissolution of bulk ZnS in an EDTA solution is described by the following reactions: (a) Dissolution of ZnS and formation of aqueous Zn2+ cations and S2- anions:

ZnS ) Zn2+ + S2-

where Ksp is the solubility product of bulk ZnS (sphalerite). (b) Coordination of Zn2+ by the EDTA anion (Y4-):

K1 ) 1.02 × 10-2

H3Y- ) H2Y2- + H+

K2 ) 2.14 × 10-3

H2Y2- ) HY3- + H+

K3 ) 6.92 × 10-7

HY3- ) Y4- + H+

where K1′ and K2′ are the dissociation constants of H2S(aq) and HS-, respectively.11 The concentration of free S2- in the solution is [S2-] ) CSRS2

(9)

where CS is the total concentration of S species in the solution and the R value of S2- is11

RS2 )

K1′K2′ + 2

[H ] + K1′[H+] + K1′K2′

(10)

Using above equation, one calculates RS2 ) 1.29 × 10-5 at pH ) 9.0 and RS2 ) 1.30 × 10-4 at pH ) 10.0. (e) The total dissolution reaction of bulk ZnS ) reaction (a) + reaction (b):

KZnY ) 3.2 × 1016

where KZnY is the formation constant of the complex ZnY2-.11 (c) Dissociation equilibria of EDTA acidic groups:

H4Y ) H3Y- + H+

K2′ ) 1.3 × 10-14

Ksp ) 2.0 × 10-25 11

Zn2+ + Y4- ) ZnY2-

K1′ ) 9.6 × 10-8

K4 ) 5.50 × 10-11

where K1, ..., K4 are the dissociation constants of various EDTA anions.11 From these dissociation equilibria, the concentration of free Y4- anion is11

ZnS + Y4- ) ZnY2- + S2Hence, the total dissolution equilibrium constant

Ktotal )

[ZnY2-] · [S2-] ) Ksp × KZnY [Y4-] ) 2.0 × 10-25 × 3.2 × 1016 ) 6.4 × 10-9

Replacing above [S2-] and [Y4-] with eqs 9 and 7, one gets

Ktotal

[ZnY2-] · RS2CS [ZnY2-] · [S2-] ) ) RY4CY [Y4-] 2[ZnY ]CS RS2 RS2 ) · ) Kcond CY RY4 RY4

(11)

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where

Kcond )

[ZnY2-]CS RY4 ) Ktotal CY RS2

(12)

is the conditional (or effective/apparent) dissolution equilibrium constant of bulk ZnS in an EDTA solution at a given condition (pH and total EDTA and S concentrations). Using values from above, we get

Kcond ) 6.4 × 10-9 × 5.20 × 10-2 /1.29 × 10-5 ) 2.58 × 10-5 at pH ) 9.0, and Kcond ) 6.4 × 10-9 × 3.50 × 10-1 /1.30 × 10-4 ) 1.72 × 10-5at pH ) 10.0 At pH 9.0 and 10.0, the conditional equilibrium constants are still much less than 1. This means that bulk ZnS is nearly insoluble in EDTA solution at these pH values. 4.2. Dissolution of ZnS Nanoparticles in EDTA Solution. The ZnS NPs have very high specific surface areas and thus possess additional surface free energy relative to the bulk counterpart. This increases the Gibbs free energy by12

∆G ) G(nano) - G(bulk) )

3Mγ Fr

(13)

where M is the molecular weight of ZnS (97.43 g/mol), γ is the interfacial surface free energy between ZnS NPs and the solution, and F is the density of the sphalerite (4.10 g/cm3; ref 13). For the dissolution of ZnS NPs in an EDTA solution

ZnS(nano) + Y4- ) ZnY2- + S2the conditional dissolution equilibrium constant is increased by a factor of exp(∆G/(RT)) relative to that of bulk ZnS (ref 12):

Kcond(nano) ) Kcond · exp

)K ( ∆G RT )

cond · exp

( Fr3Mγ · RT ) (14)

where R is the gas constant (8.314 J/mol · K) and T is the temperature (298 K for room temperature). Inserting values of M and F of ZnS into above equation, one gets

(

Kcond(nano) ) Kcond · exp 28.75

γ(J/m2) r(nm)

)

(15)

Using Kcond ) 2.58 × 10-5 and 1.72 × 10-5 for bulk ZnS at pH ) 9.0 and 10.0, respectively (above), and assuming r ) 1.5 nm and γ ) 0.5 J/m2, the conditional dissolution equilibrium constant of ZnS NPs is calculated as Kcond(nano) ) 0.37 at pH ) 9.0 and Kcond(nano) ) 0.25 at pH ) 10.0. These values are far greater than those of bulk ZnS and are close to 1. Thus, small ZnS NPs can dissolve in an EDTA solution at these pH values, as confirmed by the SAXS experiments (see Figures 3C and 4C).

If there is no EDTA in the solution, the Kcond(nano) will be that with EDTA divided by KZnY ) 3.2 × 1016 (see section 4.1(e)), i.e., 1.15 × 0-17 and 7.81 × 10-18 at pH 9 and 10, respectively, at the above given r and γ values. This means that even very small ZnS NPs cannot dissolve in a week basic solution without EDTA complexation. 5. Kinetic Modeling of ZnS NP Dissolution Reaction Figures 3C and 4C show that the dissolution of ZnS NPs in EDTA approaches equilibrium with time. This is especially obvious in expt 2 (Figure 4C) in which the dissolution fraction levels off at ∼0.6 after dissolution for ∼1 h. In the following, we analyze the dissolution kinetics and model the dissolution data based on the derived kinetic equation. The apparent dissolution reaction of ZnS NPs in EDTA (see section 4.2) and the (total) concentration (C) of each species at t ) 0 and t ) t, are

ZnS(nano) + Y4- ) ZnY2- + S2C(t ) 0): a c 0 0 C(t): a-x c-x x x where a and c are the initial concentrations of ZnS NP suspensions and EDTA in the reaction solution, respectively, x is the amount of ZnS dissolved at time t. Assuming both the forward (dissolution) reaction and the reverse (precipitation) reaction are first order with respect to the concentration of each species or the total surface area of ZnS NPs, the apparent kinetic equation can be written as

dx ) k1′AZnS(c - x) - k2x2 dt

(16)

where k1′ is the reaction constant of the forward reaction, k2 is that of the reverse reaction, and AZnS is the total surface area of ZnS NPs at time t. AZnS is approximately proportional to (a x) (coefficient being k) since the mean particle size changes only slightly in the measurements (Figures 3A and 4A). Thus,

dx ) k1′k(a - x)(c - x) - k2x2 dt ) k1(a - x)(c - x) - k2x2

(17)

where k1 ) k′k 1 is a forward reaction constant when the quantity of ZnS NPs is expressed as molar concentration. At dissolutionprecipitation equilibrium, dx/dt ) 0. Thus, from eq 17, the conditional equilibrium constant

Keq ) Kcond(nano) )

k1 xe2 ) k2 (a - xe)(c - xe)

(18)

where xe is the amount of dissolved ZnS NPs at equilibrium. Analytical solution of eq 17 at the above given initial concentrations of ZnS NPs and EDTA gives (see the Supporting Information for derivation of the following equation):

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√D √D √D x) · 2C √D √D

+B -B +B -B

e-

√D · t

-1 -

-√D · t

e

Zhang et al.

+1

B 2C

(19)

where the parameters B, C, and D are related to the kinetic constants (k1 and k2) and the initial concentrations (a and c) via

B ) -k1(a + c) C ) k1 - k2

a given pH (eqs 14 and 15). From the volume (mass)-weighted PSD obtained from the SAXS fitting (Figure 5), the average radius of ZnS NPs is 1.60 nm in expt 1 and is 1.44 nm in expt 2 (Table 1). Thus, according to eq 15, the interfacial free energy at pH ) 9.0 in expt 1 is

γ (J/m2) )

(20)

r (nm) Kcond(nano) ln 28.75 Kcond 1.60 0.94 ) ) 0.58 (21) ln 28.75 2.58 × 10-5

D ) (a - c)2k12 + 4ack1k2 In expt 1, the ZnS(nano)/EDTA ratio equals 0.007:0.067 ≈ 0.007:0.070 ) 1:10, and in expt 2, the ratio equals 0.007:0.017 ≈ 1:2.4. Thus, if we arbitrarily set the initial concentration a of ZnS as 1 (dimensionless), then the initial concentration c of EDTA would be 10 (dimensionless) and 2.4 (dimensionless) in expt 1 and expt 2, respectively. This way, x in eq 19 becomes merely the dissolution fraction (because it equals x/a), which simplifies the kinetic modeling. With above arbitrarily set dimensionless initial concentrations, one can fit the kinetic eq 19 to the SAXS derived dissolution fraction data (Figures 3C and 4C). Figure 6 shows that the kinetic equation fits the derived data very well. Table 1 lists the kinetic parameters derived from the fitting. Results show that at pH 10, both the forward and reverse reactions have higher kinetic rate constants than those at pH 9. The higher forward and reverse rate constants may be related to, respectively, higher free Y4- and free S2- concentration percentages at pH 10 than at pH 9 (see section 4.1(c),(d)). In Table 1, the equilibrium constant Keq is calculated from k1/k2 (eq 18). It is seen that although the forward and reverse reactions proceeded faster at pH 10 than at pH 9, Keq is higher at pH 9 than at pH 10. This may be related to the interfacial free energy of ZnS NPs at different pH values (eqs 14 and 15), as discussed below. 6. Variation of ZnS NP Dissolution-Precipitation Equilibrium as a Function of pH and Particle Size The conditional dissolution equilibrium constant of ZnS NPs is related to the interfacial free energy and the particle size at

Figure 6. Kinetic fitting to the dissolution fraction data for nano-ZnS in dissolution in EDTA solutions. Key: points, experimental data; solid lines, fitting.

and that at pH ) 10.0 in expt 2 is

γ (J/m2) )

1.44 0.41 ) 0.50 ln 28.75 1.72 × 10-5

(22)

The derived γ values are listed in Table 1. These values are lower than the surface energy of sphalerite in vacuum (0.86 J/m2; ref 14) due to the hydration and hydroxylation of ZnS NP surfaces in the solution. Because there is a higher degree of hydroxylation at pH 10 than at pH 9, γ (pH ) 10) is lower than γ (pH ) 9) (Table 1). In the pH range of 9-10, the variations of Kcond (bulk ZnS) and γ (ZnS NPs) with the pH can be calculated using linear interpolations based on the values at the two terminal pH values (see above). Thus, the variation of Kcond(nano) with pH and particle size can be calculated from eq 15, as shown in Figure 7. Figure 7A shows the variation of Kcond(nano) with particle size over the pH range 9-10. At Kcond(nano) ) 1, the NPs reach a dissolution-precipitation equilibrium; in the Kcond(nano) > 1 area, dissolution of ZnS NPs dominates; in the Kcond(nano) < 1 area, precipitation of ZnS NPs dominates. It is seen that at pH 9, particles greater than ∼1.6 nm tend to persist whereas those smaller than 1.6 nm tend to dissolve in the EDTA solution. At a higher pH, this size becomes smaller (e.g., ∼1.3 nm at pH 10). Figure 7B shows the variation of Kcond(nano) with pH at a given particle size (including for a “infinitely” large bulk crystal). Results show that particles ∼1 nm in radius dissolve in the EDTA solution; particles greater than 3 nm in radius do not dissolve; particles of 1.5 nm in radius dissolve at pH < ∼9.3 and do not dissolve at pH > ∼9.3. Bulk ZnS crystals do not dissolve in the pH range because of its very low Kcond. Figure 7C depicts the variation of Kcond(nano) with both the pH and the particle size. It is seen that within the pH range 9-10, the lower the pH and the smaller the particles, the higher the conditional dissolution equilibrium constant and hence the higher the solubility of ZnS NPs. This can be understood as the smaller the particles, the higher the specific surface area and the larger the additional surface free energy per mole of NPs. Thus, there is a higher driving force for NP dissolution and a higher Kcond(nano). Similarly, at a lower pH, there is a higher interfacial free energy (Table 1). Thus, there is also a higher driving force for NP dissolution and hence a higher Kcond(nano). In Figure 7C, the contour line of Kcond(nano) ) 1 is drawn on both the 3D surface and the 2D base projection. The contour

TABLE 1: Parameters Obtained from Kinetic Modeling of the SAXS Derived Data expt no.

a for ZnS NPs

c for EDTA

pH

k1 (1/h)

k2 (1/h)

Keq

average r (nm)

γ (J/m2)

1 2

1 1

10 2.4

9.0 10.0

0.43 0.78

0.46 1.90

0.94 0.41

1.60 1.44

0.58 0.50

Nanoparticle Dissolution

J. Phys. Chem. C, Vol. 114, No. 35, 2010 14883 line describes the particle size of ZnS NPs at which the dissolution-precipitation reaches equilibrium at a given pH. The higher the pH, the lower the particle size at equilibrium (e.g., ∼1.3 nm vs 1.6 nm, corresponding to pH 10 and 9). The Kcond(nano)-pH-size relationship can be expanded to a wider pH range (3-10) by combining eqs 8, 10, 12, and 15, and by assuming γ ) 0.58 J/m2 over the pH range considered. Results (Figure 7D) show that there is a maximum in Kcond(nano) at pH ∼ 4.5 at a given particle size. Below or above pH 4.5, Kcond(nano) decreases in accordance with the variation of the ratio of RY4/RS2 with the pH (ref eq 12). At pH ∼ 4.5, there is a maximum NP size of ∼1.9 nm at the dissolution-precipitation equilibrium (see the contour line at the base). Note that at very low pH, H2S(g) formation may occur. This was not considered in the calculation for Figure 7D. Thus, the contour line for Kcond(nano) ) 1 in Figure 7D at pH < 4 is shown by a dashed line. Although not fully quantitative beyond the experimental pH range, Figure 7D may be viewed as the basis for development of testable hypotheses. It predicts the complicated interplay between the pH and the particle size and its influence on the dissolution-precipitation equilibrium. 7. Conclusions Solution pH affects the dissociation equilibrium of a complexing agent, the protonation or hydroxylation of the ionic groups released by NP dissolution, and NP interfacial free energy. Thus, the influence of pH on NP dissolution is complex, depending on the structures of the complexing agents and the NPs and their interaction strengths with the solvent (such as water). For ZnS NPs in EDTA in weak basic solutions, the higher the pH, the lower the NP solubility. Particle size affects the total surface free energy of NPs, altering the driving force for NP dissolution and the equilibrium. Small NPs tend to dissolve whereas larger NPs do not. The dissolution-precipitation reaches equilibrium at a certain particle size at other fixed conditions (e.g., pH). For example, for ZnS NPs in a weak basic EDTA solution, particles with radius >3 nm do not dissolve; particles with radius