Dissolution of Commercial Microscale Quartz Particles in Water at

Article pubs.acs.org/JPCC

Dissolution of Commercial Microscale Quartz Particles in Water at Biological-Like Conditions and Its Theoretical Description Frank Roelofs† and Wolfram Vogelsberger*,‡ †

Chemiewerk Bad Köstritz, GmbH, Heinrichshall 2, D-07586 Bad Köstritz, Germany Institute of Physical Chemistry, Chemistry and Earth Science Faculty, Friedrich-Schiller-University Jena, Helmholtzweg 4, D-07743 Jena, Germany

‡

S Supporting Information *

ABSTRACT: The solubility of two commercially available microscale quartz powders in water under biological-like conditions is investigated. The experimental results are evaluated by common phenomenological theories and by complete analysis of the Gibbs free energy of the system particles− solution (gradient method). The different theories are compared with each other, and the interrelations between the theories are discussed. The application of the gradient method always leads to the best agreement between experimental results and theory because the gradient model considers the simultaneous changes of both particle size and particle concentration during dissolution. The optimal values of saturation concentrations and dissolution rate constants are determined by statistical analyses and by calculation of the coefficient of determination. The expected influence of temperature on the dissolution behavior is confirmed. Higher values of the crystal strain of the quartz samples result in higher values of the dissolution rate constant and more rapid dissolution. The saturation concentrations and rate constants determined for the commercially available quartz samples are at least 1 order of magnitude larger than most values reported in the literature for natural quartz. In accordance with statements found in the literature, this fact should be taken into account for the assessment of the potential health risk of the investigated commercially available quartz powders.

1. INTRODUCTION Silica consists of the two most frequent elements found on the Earth’s crust which is why its crystalline or amorphous forms play a very important role for all areas of human life. An enormous number of publications exist reporting about the different properties and applications of silica. A detailed overview can be found in the book of R. Iler.1 The solubility of all forms of silica plays an essential role among all its properties, though the saturation concentration is in the range of a few millimoles per liter. Since extensive work has been published about the solubility of silica, only a small fraction can be mentioned here. It is well-known that the solubility as well as other main properties of solids depend on the particle size. This is important especially if the particle size comes in the nanometer range.2 Solubility aspects are important for the synthesis and stability of nanoparticles,3−6 and ecological aspects are of increasing importance in this context.7 For the environmental characterization of sediments8 and fly ash,9 it is of great relevance to have reliable data about the solubility of silica species. Very often silica films are used as coatings which is why it is important to know their stability in aqueous electrolyte solutions.10 Dissolution effects also have to be considered if fluorescently labeled nanoparticles are applied in biological media.11 The question of toxicity becomes more and more important the smaller the particles become.12,13 Solubility investigations © 2013 American Chemical Society

are essential for the estimation of toxicological effects of metal oxides.14−21 It should be emphasized that thermodynamically alveoli represent an open system where pulmonary fluids circulate, and an equilibrium state in terms of dissolution will not necessarily be attained (see ref 14). The role of dissolution of inorganic compounds for biopharmaceutical classification is discussed in ref 16 based on solubility and dissolution data. Some challenges for future research are formulated in ref 18 that are essential for understanding the effect of small particles in environmental processes, including their potential toxicity. A methodology from which the environmental exposure risk posed by metallic nanomaterials associated with consumer products may be assessed can be found in ref 19. The solubility behavior of man-made nanomaterials should be considered, if their ecological risk to the exposure of aquatic organisms is estimated.20 It is believed that small silica particles are quickly removed from biological systems by complete dissolution.21 The solubility of modified silica products must be considered if they are used for the recovery of toxic metal ions from toxic metal-contaminated solid waste materials.22 The health risks of metal ions emitted from welding can be minimized if they are coated by a silica film.23 The stability of these films against Received: October 11, 2012 Revised: May 13, 2013 Published: June 4, 2013 13914

dx.doi.org/10.1021/jp310072m | J. Phys. Chem. C 2013, 117, 13914−13927

The Journal of Physical Chemistry C

Article

and at 25 and 37 °C, respectively. The extracellular fluid has been modeled by a physiological electrolyte. This TRIS-buffer37 had the following composition: 6.1 g of tris(hydroxymethyl)aminomethane, 35.4 mL of 1 N HCl solution, and 6.54 g of sodium chloride per liter. The background electrolyte content (as sodium chloride) of the buffer solution was about 0.11 mol/ L. The relevant properties of the dissolution experiments are listed in Table 2. The solubility experiments of the synthetic

water is essential for avoiding toxic effects in respiratory systems. An additional aspect should be considered if crystalline quartz particles are investigated: Possible problems due to the formation of silicosis were already discussed in 1979 (see ref 1, p 769). A more recent and extensive discussion about possible toxicity issues of different silica species can be found in ref 17 where important results about the influence of particle size and crystallinity are presented and evaluated. At present, most of the toxicological research concerning silica deals with the toxicity of natural crystalline particles having a diameter in the rage of 0.5−10 μm. Differences in the solubility of small quartz particles and synthetic amorphous silicas seem to be relevant for the formation of silicosis. Quartz particles remain more or less constant during a 3-month recovery period in the lungs of rats, in contrast to the rapid dissolution of commercial silicas.15 The dissolution behavior of natural quartz at different pH values, temperatures, and mechanical conditions has been extensively studied. Here, we only refer to work that has been used for comparison with our results or for the explanation of various observations.1,24−36 The work presented in the following reports on measuring the solubility of two commercial synthetic microscale quartz samples under biological-like conditions. The dissolution process has been studied for up to 300 days. To the best of our knowledge only a few of such extensive studies can be found in the literature. The obtained results are referred to possible toxicity effects and reasons therefore. The experimental results are compared to those obtained for natural quartz. The experimental data are analyzed by different phenomenological models. Evaluation of the theoretical dissolution curves provides saturation concentrations and rate constants that are compared to each other and to the experiments.

Table 2. Summary of the Dissolution Relevant Parameters (Dissolution Temperature, T, Mass, m, Surface Area Available for Dissolution, OD, and pH Value) QFL m/g OD/m2 pH

Table 1. Main Properties of the Quartz Samples Used in the Experiments MIN-U-SIL

specificationa crystal modificationb mass density specific surface area (BET)c particle radiusd (spherical shape)

QFL

U.S. Silica Company Berkeley Springs (USA) crystalline, 99.5% content 98% < 5 μm α-quartz

Fluka Chemie GmbH Buchs (Switzerland) crystalline, p.a. ≥ 230 mesh α-quartz

2.65 g cm−3 5.406 m2 g−1

2.65 g cm−3 1.069 m2 g−1

209.4 nm

1059.0 nm

T = 310 K

T = 298 K

T = 310 K

8.4027 8.982 7.66

8.4020 8.982 7.42

6.8721 37.17 7.65

6.8784 37.21 7.44

quartz samples were carried out in 100 mL of buffer solution. The dispersions were stored in 125 mL sealable polypropylene (PP) bottles in an incubator and were continuously shaken. The reproducibility of the measurement was ensured by performing the dissolution experiments twice. The total amount of dissolved silica has been analytically determined by the molybdic acid method described by Motomizu et al.38 Thereby, each data point is the average of four single determinations. The sample was filtered through a 0.2 μm syringe filter. Additionally for selected samples the ICP-OES device SpectroFlame by Spectro, Kleve (Germany), was applied to determine the concentration of dissolved silica. An adequate amount of the undissolved solid was separated for selected times during the dissolution experiment. Therefore, a small volume of the suspension was removed from the running experiment and centrifuged at approximately 2940g for at least 10 min. The samples were dried at 110 °C after a washing procedure with water (repeated three times) to remove adsorbed molecules and ions. The remaining solid of a dissolution experiment was obtained by the same procedure at the end of the experiment. The specific surface area was determined after the BET method39 by multipoint krypton adsorption measurements at 77 K applying the sorption automated Autosorb-1 of Quantachrome Corporation. In doing so the sample had to be evacuated first at 350 °C to remove the remaining water from the surface. The structural properties of all these samples were investigated with both Xray diffraction measurements (XRD) (URD 6, Freiberger Präzisionsmechanik/D5000, Siemens-Bruker) and FT-IR spectroscopy (IFS66, Bruker).

2. MATERIALS AND EXPERIMENTS Two commercial quartz samples are used for the solubility measurements: MIN-U-SIL 5 fine ground silica offered by U.S. Silica Company, Berkeley Springs (USA), from now on referred to as MIN-U-SIL, and a quartz sample QUARTZ purum p.a. from Fluka Chemie GmbH, Buchs (Switzerland), in the following called QFL. The main properties of these two synthetic samples regarding dissolution experiments are listed in Table 1. The quartz solubility measurements were performed under physiological conditions, osmotic pressure, and near-neutral pH

manufactured by

MIN-U-SIL

T = 298 K

3. THEORETICAL DESCRIPTION OF THE DISSOLUTION PROCESS 3.1. Thermodynamic Consideration. First we give a general explanation of the symbols used throughout the work. Scheme 1 gives a schematic representation of the symbols used and the way of calculation of the Gibbs free energy of the system particles−solution. The number of molecules of a species is expressed by the capital letter, N. Three kinds of chemical species are considered, silica, N(1), water, N(2), and background electrolyte, N(3). The initial state of the system before dissolution, i.e., solid silica, water, and background electrolyte, is marked by the upper

a As manufacturer information. bXRD measurements. cMean of three krypton adsorption measurements. dAssumed radius, if all particles are spherical and of uniform size.

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calculations to assume that the whole amount of dissolved silica is present in the solution as undissociated orthosilicic acid. The amount of background electrolyte does not change during the dissolution process. Therefore, it is omitted in eq 1. The amount of water is chosen in such a way that we have N0L(2) molecules at the beginning of the dissolution and NRL (2) molecules if the substance 1 would be completely dissolved. We have solid particles and a solution of background electrolyte at the beginning of the dissolution process. After a period of time, molecules of the particles are transferred to the solution, N1L(1). The residual molecules remain in the particles, N1S(1)

Scheme 1. Schematic Representation of the Gibbs Free Energy of Different States of the System Particle−Solution That Can Be Made from N0S(1) Molecules of Solid, N0L(2) Molecules of Water, and N0L(3) Molecules of Background Electrolytea

NS0(1) = NS1(1) + NL1(1)

(2)

The actual number of water molecules, N1L(2), can be expressed by the following equation (see eq 1) NL1(2) = NL0(2) − 2NL1(1) = NLR (2) + 2NS1(1) a

Equations 2 and 3 can be of course also used to describe the corresponding amounts of substance nkj (i) = Nkj (i)/NA (NA = Avogadro constant; k = 0, 1, R, s; j = L,S). Both possibilities are used in the following. We investigate a process at constant temperature and constant pressure where the Gibbs free energy is the appropriate thermodynamic potential. Our further calculations correspond to those outlined in the references.43,44 The actual state of the system during the dissolution is compared to the reference state, where all the particles are completely dissolved: (NRS (1) = 0, NRL (1) = N0S(1), NRL (2) = N0L(2) − 2N0S(1), and NRL (3) = N0L(3)). Therefore, the change in Gibbs free energy between these two states, ΔSg, is given as follows

Δgi−j = g(final state) − g(starting state).

index 0 (N0). The state of the system during the dissolution process is marked by the upper index 1 (N1). The equilibrium state of the system, which is the saturated solution, is indicated by the upper index s (Ns), and the reference state (complete dissolution of the solid) is marked by the upper index R (NR). Species present in the solution are marked by the lower index L (NL), while solid species are marked by the lower index S (NS). Here, we consider a system consisting of N0S(1) molecules of the solid substance (1) to be dissolved, N0L(2) molecules of water (2), and N0L(3) molecules of background electrolyte (3). It is assumed that the N0S(1) molecules form uniform particles of spherical shape. The dissolution process may be described by the following overall reaction equation MeO2 + 2H 2O ⇄ Me(OH)4

(3)

ΔSg = g (NS1(1), NL1(1), NL1(2), NL0(3))

(1)

− g (NLR (1), NLR (2), NL0(3))

MeO2 represents the oxide of elements such as Si, Ti, or Zr, for example. The mass of silica dissolved is represented as Si(OH)4. It is determined by the molybdosilicate method that determines orthosilicic acid and its silicate.38 Our dissolution experiments are carried out in undersaturated solutions at pH values of about 7.5. The pKmS1 value for the deprotonation of orthosilicic acid is found to be pKmS1 = 9.8 (ref 1, p 123). This value is relatively constant in sodium chloride solutions up to ionic strength of 0.5 (ref 1, p 136). Therefore, the relation between charged and uncharged silica species in the solution is [HSiO3−]/[Si(OH)4] = 0.005. Ratios in this order of magnitude seem to be accepted generally.40 Interpolation of the ratios given in ref 40 results in [HSiO3−]/ [Si(OH)4] = 0.006. A significant contribution of HSiO3− to the total concentration of “soluble silica” determined by the molybdate reagent is observed at pH ≥ 9 (ref 1, p 47). The concentration of monomeric species in these solutions remains relatively constant in the pH range from 7 to 9, and the influence of background electrolyte concentration is not significant.41 The formation of ionic species can be observed at higher pH values (see ref 1, p 124). Furthermore, condensation reactions of orthosilicic acid in weak acidic or near neutral solutions play a significant role only in supersaturated solutions.42 This may be a justification in our

(4)

It is calculated by ΔSg = Δgs−R − Δgs−1 (see Scheme 1). Common expressions can be employed in relation to the chemical potentials of the species under investigation μL (i) = μLs (i) + kBT ln

ax(i) axs(i)

(5)

The equilibrium between the bulk phase of the particle forming species and the saturated solution is selected as a standard state indicated by the upper index, s. The ratio of activities in eq 5 is substituted by mole fractions ax(1)/axs(1) ≈ xL(i)/xLs(i) in our further calculations. This may be justified since the activity coefficient in the solution above all is determined by the concentration of background electrolyte, which remains nearly constant for each dissolution experiment. The increase in free energy of the small particles, when compared to a bulk phase, can be accounted for by the special term, h(N1S(1)). The following expression for the Gibbs free energy g(r,z) of the system particles−dissolved substance− background electrolyte−water can be obtained 13916



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⎤ ⎡ 1 − zr 3 ⎥ + (a + 2zr 3) g (r , z) = (1 − zr 3)ln⎢ ⎣ (c + zr 3)x Ls(1) ⎦

1 (1 − a′zr 3) a′ ln[1 − a′zr 3] + hzr 2 − zr 3 ln[b′] − ln[y]

g (r , z) = (1 − zr 3)ln[y(1 − zr 3)] +

⎤ ⎡ ⎡ a + 2zr 3 ⎤ b ⎥ + hzr 2 ⎥ + b ln⎢ ln⎢ 3 s 3 s ⎣ (c + zr )x L(3) ⎦ ⎣ (c + zr )x L(2) ⎦ ⎡ x R (3) ⎤ ⎡ x R (2) ⎤ ⎡ x R (1) ⎤ ⎥ ⎥ − b ln⎢ Ls ⎥ − a ln⎢ Ls − ln⎢ Ls ⎣ x L(3) ⎦ ⎣ x L(2) ⎦ ⎣ x L(1) ⎦

a′ =

NS0(1) + NL0(2) N 0(1) c0 y = Ss = Ls NL(1) cL

(6)

a=

NLR (2) , NS0(1)

b=

, z=

Z R , r= , R0 NS0(1)

NL0(3) , NS0(1)

NLR (i) , i=3 R ∑i = 1 NL (i) 4πσR 02 kBTNS0(1)

x LR (i) = h=

ΔSg kBTNS0(1)

c=

x Ls(i) =

, b′ =

NLs(1) + NL0(2) NS0(1) + NL0(2)

,

(8)

where c0L is the concentration that would be obtained if the whole solid is dissolved (c0L = N0S(1)/v), and csL is the saturation concentration.

The following quantities are used in eq 6 g (r , z ) =

NS0(1)

4. KINETIC CONSIDERATIONS The dissolution from a plane surface can be expressed by a common rate equation widely used in the literature, e.g.1,24,26,29,30,33,35,45

i=3 ∑i = 1 NLR (i) , NLR (1) NLs(i) , i=3 s ∑i = 1 NL(i)

kO dc L kO = 1 s D dt = kydt = + sD dt , k = k1OBETMv , c Ls − c L cL vc L k k+ = , respectively, c L = c Ls{1 − exp( −kyt )} OBETM

(7)

(9)

Here, R represents the radius of the particles; the number of particles, Z, is related to N0S(1); and the radius of the particles is related to the radius of a Si(OH)4 molecule, R0, if the molecular shape is assumed to be spherical. Therefore zr3 = N1S(1)/N0S(1). kB represents the Boltzmann constant and T the temperature. The Gibbs free energy value is related to N0S(1)kBT, and thus g(r,z) results in a dimensionless quantity. The quantities x(i) within eq 6 denote the corresponding molecular fractions. The surface term, h, is assumed to be proportional to the surface area of the particles. σ indicates the surface tension of the particles. All characteristic properties of the particle are accounted for by the term, h. Equation 6 enables the calculation of the Gibbs free energy of any system solid−solution that can be formed by the molecules N0S(1), N0L(2), and N0L(3). The dimension reaches from complete dissolved solid (N1S(1) = 0, N1L(1) = NRL (1), N1L(2) = NRL (2), and N0L(3)) via particle− solution systems (N1S(1), N1L(1), N1L(2) = N0L(2) − 2N1L(1), and N0L(3)) toward bulk solid pure solution background electrolyte water (N1S(1) = N0S(1), N1L(1) = 0, N1L(2) = N0L(2) and N0L(3)). This allows for the formation of a Gibbs free energy surface. For the comparison between different kinetic models of the dissolution process we used a simpler model to calculate the Gibbs free energy. This simplified model does not consider the background electrolyte (NL0 (3) = 0) and neglects the consumption of water during the dissolution process (N1L(2) = N0L(2) = constant). In terms of kinetics, we use a first-order rate equation. The dissolution from a plane surface on the one hand and of a constant number of spherical particles on the other hand is investigated. These simplifications are valid for large particles with mean sizes in the micrometer range. The following equations for the Gibbs free energy, ΔSg, are obtained considering these simplifications43

n1L(1)/v

where cL is the concentration (v is the volume of the system); csL is the saturation concentration nsL(1)/v; k1 is the rate constant of dissolution introduced in ref 1 (p 67) and ref 45; k is the rate constant suitable for irreversible thermodynamics argumentation; k+ is the rate constant in often used units to compare the results of our investigations with the results of other groups under similar conditions (T, pH, N0L(3)); OD is the surface area of the quartz sample exposed to dissolution; OBET is the specific surface area of the quartz samples; M is the molar mass of silica; y is the initial supersaturation of the system (see eq 8); and t is the dissolution time. Equation 9 can also be obtained by application of irreversible thermodynamics.46 This is demonstrated in Appendix I. Usage of the rate equation, eq 9, is referred to as model I in the following. As a next example we investigate the dissolution from a surface of a constant number of larger spherical particles (R is in the order of magnitude of some micrometers). The following rate equation can be derived, as is shown in Appendix II, eq II.3 dc L z 2/3 = 9k 0 1/3 s (c L0 − c L)4/3 (c Ls − c L) dt (c L ) c L

(10)

The dissolution from the surface of a constant number of spherical particles is from here on referred to as model II. Commonly it is assumed that the number of particles does not change during a dissolution process. This assumption is only suitable for large particles. It is well-known from nucleation experiments that a large number of particles is generated in a very short period of time, and the particles grow to a detectable size in the range of nanometers. A simultaneous change of the size and the number concentration occurs if nanoparticles are brought into contact with a solvent. Furthermore, it is known that size and number concentration change in an ensemble of small particles by Ostwald ripening. Therefore, a general description has to take into account both effects. We investigate an ensemble of spherical particles of identical size. Alternatively the results obtained may be valid for 13917



Article 0 ⎡ ⎤ ·e y0 KS,2 [H+] ⎥ k× = [P]⎢k neu + 0 y k pos + k neg ⎢⎣ ⎥⎦ [H+] KS,1·e 0

the mean size of particles with different sizes. Two variables are necessary to describe the dissolution process, and it is not possible to derive a closed formula as is the case for eq 9. It is however possible to calculate the Gibbs free energy of the system for any path on the free energy surface by eqs 6 and 8. It is assumed that the path follows the negative direction of the gradient of the Gibbs free energy, and the following rate equation is applicable dc L = −k B(c L)|∇g (r , z)| dt

kneu, kpos, and kneg are the partial rate constants describing the dissolution from the different surface sites, uncharged, positively, and negatively charged. The uncharged surface species, P, of the (polymeric) oxide is connected with the positively charged surface species by K0S,1 and the negatively 0 charged surface species KS,2 . The superscript 0 indicates intrinsic, thermodynamic constants, whereas y0 = FAψ0/RGT is a normalized, dimensionless surface potential, which is closely related to the actual surface potential, ψ0. FA means the Faraday constant. The constant k× is related to the constant k (eq 9) by the relation k× = kc0L (see Comment SI2 within the Supporting Information), and it shows that the rate constant k depends on the surface potential, ψ0, of the electric double layer near the surface of the solid and therefore on the ionic strength in the solution. The rate constant incorporates the influence of the concentration of background electrolyte on the reaction rate. Recently it has been discussed that silicate dissolution occurs through protonation and hydrolysis of bridging oxygen atoms.55

(11)

Equation 11 will be referred to as the gradient method henceforth. The gradient curves, corresponding r−z−pairs, can be calculated by solving the following differential equation.43,44

( dz = dr (

∂g (r , z) ∂z ∂g (r , z) ∂r

) )

r

z

{ = 3z{ r

h r 2h 3r

⎡ (1 − zr 3)(c + zr 3)(xL0(2))2 ⎤ − ln⎣⎢ ⎦⎥ x L0(1)(a + 2zr 3)2 ⎡ (1 − zr )(c + zr − ln⎢⎣ x L0(1)(a + 2zr ) 3

3

)(x L0(2))2 3 2

} ⎤ ⎥⎦} (12)

Equation 12 is obtained by the derivatives of eq 6 (consumption of water is considered). Equation 12 can only be solved numerically.47 The concentration, cL(r,z), is given by the first part of eqs II.1 (Appendix II) or alternatively by the solution of the complete differential of cL(r,z) c L(r , z) =

∫r

r

S

⎛ ∂ cL(r , zS) ⎞ ⎜ ⎟ dr + ⎝ ⎠z ∂r S

∫z

z S

5. COMPARISON OF EXPERIMENT AND THEORY 5.1. Representation of Experimental Results and Evaluation Procedure. The concentration in the solution is determined as a function of dissolution time for both quartz samples QFL and MIN-U-SIL and in TRIS-buffer solution. Two dissolution temperatures, 25 and 37 °C, are investigated. The concentration−time pairs and the determined errors for the concentrations (95% confidence interval) are listed in Tables 3A, 3B, and 3C. The dissolution kinetics was measured up to 310 days. The complete data listed in Tables 3A, 3B, and 3C were used for the interpretation. The three models described in the preceding section are tested by comparing the best fit of the parameters, saturation concentration, and rate constant. The criterion for the best fit is the optimal coefficient of determination, Rc2. The calculation of the best fit of the parameters and the corresponding confidence intervals (confidence level 95%) has been performed by using the software Mathematica,47 “NonlinearModelFit”. In the case of model I the differential and the integral form of the rate law (eq 9) may be used. It is only possible to apply the differential rate law, eq 10, if model II is tested. We observed an effect on the calculation results if constraints were introduced for the rate constant during the calculation/fitting procedure. The optimum of the parameters is determined by the highest coefficient of determination; e.g., the constraint 1.0 ≤ csL/(mmol L−1) ≤ 3 is used for all cases, and the constraint 0 ≤ k/h ≤ 40 led to the best fit parameters for the dissolution experiments of sample QFL at 25 °C. Model III, eq 11, calculates particle radius−number of particles pairs along the path of the steepest decent of the Gibbs free energy of the system. These pairs are not available from the experiments, where only concentrations can be determined. The same concentration determined experimentally in the system, however, can be realized by different particle radius− number of particles pairs. Therefore, the following approach is used for the determination of the optimal parameters: Theoretical dissolution curves, that is, the concentration, cL, as a function of dissolution time, t, are calculated for different

⎛ ∂c L(r , z) ⎞ ⎜ ⎟ dz ⎝ ∂z ⎠r

nS0(1) n 0(1) (zSrS3 − zr 3) = S (1 − zr 3), v v 3 Z ⎛R ⎞ zSrS3 = 0 S ⎜ S ⎟ = 1 NS (1) ⎝ R 0 ⎠

=

(13)

The last part of eq 13 follows simply from the fact that the volume of all molecules, N0S(1)4πR03, must be equal to the volume of all particles prior to dissolution, ZS4πRS3. ZS is the number, and RS is the radius of the initial particles prior to dissolution. Overall, we now obtained the concentrationdependent part of the rate equation (eq 11). The time that belongs to the r−z−pairs of the gradient curves is obtained from the time-dependent part of eq 11. t=

−1 k

∫0

ζ

dζ |∇g (r , z)|

(15)

(14)

Equation 14 is also solved numerically. Concentration and time have to be calculated for the same value of the variable ζ = N1S(1)/N0S(1) = zr3. The corresponding cL−t−pairs are obtained by this way numerically for the gradient curves. This method is called from here on gradient model or model III. A more detailed interpretation of the rate constant is possible if charged species at the surface and of the dissolved material are taken into account; see, e.g., refs 48−53. These charged species are formed by abstraction or adsorption of protons, and they are essential for dissolution and aggregation54 of silica particles. A model was developed for a plane surface, and only monovalent species are considered.50 A brief description of this model is given in the Supporting Information Comment SI2. The result is an overall rate constant, k×, of the following form 13918



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boundaries of the confidence interval of the rate constant. Therefore, it is assumed that the boundaries of the confidence interval of the saturation concentrations have the same magnitude as determined by the application of model II. Table 4 summarizes the results of the kinetic evaluation of the three models for the two quartz samples. Table 4 represents the dissolution temperatures, the saturation concentrations, csL, the rate constants, k, as well as the corresponding confidence intervals, CI, and the coefficients of determination, R2c . 5.2. Comparison of the Different Model Considerations. The three model equations, eq 9 (model I), eq 10 (model II), and eq 11, are compared to each other. Equation 11 is solved via eq 13 and eq 14 (model III). The aim of these kinetic experiments is the determination of the rate constant and the equilibrium concentration. Both these quantities act as parameters to find the optimal adaptation of the theoretical curve to the experimental results. Please note that only model III contains a term that takes into account the size of the particles, h = 4πσR20/kBTN0S(1) (eq 7). The interfacial tension, σ, of the solid−liquid interface could be used as a further parameter, therefore, if model III is applied. A variety of discussions concerning the interfacial tension of the silica−water interface can be found in the literature; see, e.g., ref 1 (p 54) and refs 33, 56, and 57. Particles in the size range of micrometers are relatively large, and the influence of the interfacial tension on thermodynamic quantities is therefore relatively small. We have examined this influence for MIN-USIL at 25 °C and a saturation concentration of 1.80 mmol L−1 between 25 and 500 mN/m. The results are shown in Table SI2 of the Supporting Information. A value of σ = 50 mN/m seems to be established in case of the silica−water interface at temperatures around room temperature (ref 1, p 54). This value has been used for our considerations since the saturation concentrations measured in our experiments are close to values reported for amorphous silica for the pH interval 6 ≤ pH ≤8 (ref 1, p 47). The saturation concentration depends on the chemical species, temperature, and ionic strength.58 The best fit between experiment and theory can be again found by calculating the optimal coefficient of determination, R2c , as described above. All experimental data listed in Tables 3A−3C are used for the fitting procedure. Figure 1 compares the results of the three model equations and the experimental values for the sample QFL (T = 298 K). The saturation concentrations and rate constants listed in Table 4 are used for the calculations. Figure 1 shows that the concentration−time curves calculated by the rate eqs 9 and 10 (models I and II) do not differ significantly. This seems to be a consequence of the fact that the curvature of a spherical particle in the micrometer size range is not too different from zero, the curvature of a plane surface. This is also evidenced by the saturation concentrations and the coefficient of determination, R2c (see the first column of Table 4). However, the rate constants, k, differ significantly when calculated by the equations of models I and II. This difference is a consequence of the fact that the reaction rate is proportional to the amount of the gradient of the Gibbs free energy and the rate constant defined by eq 11. The explanation of the situation is simple. We consider model II, eq 10, with c0L = 1.3985 mol/L, csL = 1.72 mmol/L, and z = 7.56823 × 10−12 where the approximation c0L ≫ cL is justified (cL < csL). Comparison of eq 10 with eq 9 yields the following relation between the rate

Table 3. Results of the Measurements of Dissolution Kinetics of (A) Quartz QFL, (B) Quartz MIN-U-SIL, and (C) Silica HDK T40, cL(t), in TRIS Buffer Solutions for Different Temperatures (A) QFL T = 298 K 0 2.5 6. 24. 48. 96. 168. 240. 336. 504. 672. 1368. 2232. 7320.

T = 310 K

cL/mmol L−1

t/h

0.057 0.101 0.224 0.413 0.625 0.811 0.992 1.150 1.310 1.539 1.709 1.640 1.885

0 ± ± ± ± ± ± ± ± ± ± ± ± ±

0 0.016 2.5 0.031 6. 0.016 24. 0.044 48. 0.033 96. 0.039 168. 0.074 240. 0.047 336. 0.126 504. 0.128 672. 0.083 1368. 0.045 2232. 0.043 7440. (B) MIN-U-SIL

T = 298 K 0 2.5 6. 24. 48. 96. 168. 240. 336. 504. 672. 1368. 2232. 7320.

0 0.758 ± 0.947 ± 1.166 ± 1.367 ± 1.596 ± 1.760 ± 1.747 ± 1.834 ± 1.717 ± 1.851 ± 1.780 ± 1.904 ± 2.027 ± (C) HDKT40 t/h 0 0.02 0.08333 0.25 0.5 1. 2. 4. 8. 26.1 48 75.3 168 192 336

0.162 0.252 0.405 0.669 0.849 1.076 1.139 1.265 1.431 1.543 1.697 1.529 2.300

0 ± ± ± ± ± ± ± ± ± ± ± ± ±

0.012 0.016 0.050 0.113 0.033 0.053 0.112 0.092 0.226 0.195 0.104 0.034 0.056

T = 310 K

cL/mmol L−1

t/h

cL/mmol L−1

t/h

0.058 0.038 0.168 0.077 0.065 0.204 0.163 0.063 0.054 0.074 0.060 0.109 0.049

cL/mmol L−1

t/h 0 2.5 6. 24. 48. 96. 168. 240. 336. 504. 672. 1368. 2232. 7440.

0.800 1.165 1.435 1.793 2.104 2.218 2.229 2.272 2.109 2.289 2.253 2.212 2.488 T = 298 K

0 ± ± ± ± ± ± ± ± ± ± ± ± ±

0.052 0.111 0.052 0.143 0.111 0.175 0.104 0.101 0.090 0.049 0.130 0.167 0.055

cL/mmol L−1 1.503 2.448 2.179 2.779 2.714 2.714 2.697 2.568 2.451 2.279 2.242 1.991 2.130 2.100

0 ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.190 0.148 0.137 0.135 0.138 0.138 0.138 0.143 0.147 0.154 0.156 0.167 0.161 0.162

saturation concentrations, csL. These curves are fitted to the experimental results by variation of the rate constant. The best fit has been determined by finding the optimal coefficient of determination (see Table SI1 of the Supporting Information). This procedure, however, allows only the determination of the 13919



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Table 4. Summary of Kinetic Results Obtained by the Different Model Rate Equations, For Quartz Dissolution of the Samples QFL and MIN-U-SIL (T = 298 and 310 K) QFL T = 298 K

MIN-U-SIL T = 310 K

T = 298 K

T = 310 K

1.72 1.57, 1.87 197 70.8, 323 16.8 0.98208

2.17 2.00, 2.34 178 78.8, 277 15.2 0.98663

1.72 1.57, 1.87 22.3 7.99, 36.5 197 0.98209

2.17 2.00, 2.33 20.1 8.90, 31.3 179 0.98664

1.80 1.64, 1.96 16.6 9.86, 23.4 0.99200

2.25 2.08, 2.42 19.5 12.4, 26.5 0.99461

model I csL/mmol L−1 CI/mmol L−1 (106 × k)/h−1 (106 × CI)/h−1 (1011 × k+)/mol m−2 s−1 R2c

1.72 1.60, 1.83 4.45 3.71, 5.19 1.93 0.99395

1.72 1.46, 1.97 6.83 3.79, 9.88 2.96 0.96753

1.72 1.61, 1.83 12.8 10.7, 15.0 4.46 0.99395

1.72 1.46, 1.97 19.6 10.9, 28.3 6.79 0.96754

1.75 1.64, 1.86 3.79 3.43, 4.16 0.99791

1.80 1.54, 2.06 5.50 3.63, 7.36 0.97811

model II csL/mmol L−1 CI/mmol L−1 k/h−1 CI/h−1 (106 × k*)/h−1 R2c

model III csL/mmol L−1 CI/mmol L−1 (10−2 × k)/h−1 (10−2 × CI)/h−1 R2c

simultaneously during the dissolution process. The dimensions of the rate constants k(I), k(II), and k(III) are equal. Model I is a first-order rate model. The dimension of a first-order rate constant in model II results from the fact that (c0L − cL)4/3 is compensated by (c0L)1/3csL in the denominator of eq 10. The rate constant of model III does not depend on the amount of substance used in the dissolution experiment (mass or surface area), in contrast to the rate constants of models I and II (see eq 14). k(III) is a characteristic property of the material dissolved and is therefore comparable to k+(I) which can be obtained by simple division of k(I) by the specific surface area and the molar mass of quartz exposed to dissolution in each experiment (see eq 9 and Table 4). It should be emphasized that the interpretation of the dissolution experiments with nanoparticles is only possible by model III due to the existence of a kinetic size ef fect, an unusual high solubility immediately after the start of the experiment.59 An example for the kinetic size ef fect is shown in Figure 2 where we depict dissolution results of the nanoparticulate silica HDK T40 (Wacker AG). The specific BET surface area of HDK T40 has been measured to be 376 m2 g−1 corresponding to a particle radius of 3.63 nm. The numerical values of the experimental concentration−time pairs are given in Table 3C. The red crosses with error bars in Figure 2 show the experimental results, while the black curve represents the theoretical calculations using model III. The following experimental conditions are relevant: mass of silica, 5.0223 g; dissolution volume, 0.5 L; dissolution temperature, 25 °C; pH = 7.6 (TRIS buffer); interfacial tension, 50 mN/m; saturation concentration, 2.05 mmol/L (confidence interval: 1.89−2.49

Figure 1. Concentration of dissolved silica QFL (T = 298 K), cL, as a function of dissolution time, t (black crosses with error bars = experimental results, gray curve = model I, blue dashed curve = model II, red curve = model III).

constants k(I) and k(II), k(I) ≈ k(II)·9·z2/3 (the different models are indicated by the numbers in the parentheses). The application of the numerical values given above yields k*(II) = k(II)·9·z2/3 = 4.46 × 10−6, which is close to the value k(I) = 4.45 × 10−6 (see Table 4). Thus, model I and model II give approximately the same result. This is also demonstrated in Table 4 by comparing k and k* values of the other dissolution experiments. The best agreement between the model calculations and experimental results can be obtained if model III (eqs 13 and 14) is applied, which might be explained by the fact that only model III takes the size and the concentration of the particles into consideration (see Table 4). The relation between k(I) and k(III) cannot be expressed by a simple equation because size and number of particles change 13920



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They are mainly affected by the long-term experimental results. The expected increase of the saturation concentration with temperature is only observed for MIN-U-SIL for all three models. Application of model III shows also an increase of the saturation concentration with increasing temperature for sample QFL. The difference between the two samples MINU-SIL and QFL is a consequence of the different particle size or specific surface area, respectively, as it will be explained later on in this manuscript. The rate constants and the corresponding confidence intervals are listed in rows four and five of Table 4 for each model. The curvature of the dissolution curve at the beginning of the process is responsible for the magnitude of the rate constant. The rate constant increases if the mass of the samples used in a dissolution experiment is increased since the surface area exposed to dissolution increases in the same way. The rate constants k(I) and k(II) establish this fact, if the samples QFL and MIN-U SIL are compared (see Table 4 and also Table 2). However, a difference between the rate constants still remains if the rate constants are related to the surface area exposed to dissolution of the samples. This can be seen by comparison of the rate constants k+ for the samples QFL and MIN-U-SIL. The analogous increase of k(III) is also not caused by the magnitude of the surface area of the sample exposed to solvent because k(III) does not depend on the mass of a sample used in the experiment (see eq 14). Figure 1 shows that model III gives the best approximation at the beginning of the dissolution. The same is true if the dissolution curves are calculated for MIN-U-SIL (see Figure SI2 of the Supporting Information). One reason for this is that only model III contains a term that takes into account the size of the particles, h (eq 7). The particles of MIN-U-SIL are smaller than the particles of QFL which is why the influence of h on the dissolution rate is greater in the case of MIN-U-SIL as compared to QFL. The expected increase of the rate constant of dissolution with increasing temperature is observed for both QFL and MIN-U-SIL. Model I and model II fail to show this effect for the rate constants of MIN-U-SIL. The question arises, what is the reason that QFL shows the expected increase of the rate constant with increasing temperature and MIN-U-SIL does not, if models I and II are applied. A possible explanation for this unexpected behavior could be differences in the shape of the dissolution curves. Our results indicate that the rate constant of dissolution is determined by the course of the dissolution curves at the beginning of the process. The slope of the dissolution curve of MIN-U-SIL is steeper than that for QFL. It takes about 96 h for MIN-U-SIL and about 1368 h for QFL to reach a relative constant concentration (see Table 4, Figure 1, Figure 3, and Figure SI2 of the Supporting Information). The rate constant of QFL is determined by applying eleven concentration−time pairs, whereas in the case of MIN-U-SIL a relative constant concentration is already found after five concentration−time pairs. Thus, the accuracy of the determination of the rate constant of QFL is therefore greater as compared to MIN-USIL. The influence of experimental errors on the rate constant is greater in the case of MIN-U-SIL as compared to QFL, and the accuracy of our experiments is sufficient to detect the temperature effect expected only for the case of QFL, therefore. The fifth row of Table 4 shows the dissolution rate constant, k+(I), related to the surface area exposed to dissolution in the commonly used unit, mol m−2 s−1. k+(I) can be used to compare the experiments of this work to the results of other

Figure 2. Concentration of dissolved silica HDK T40, cL, as a function of dissolution time, t.

mmol/L); rate constant, 5.36 × 103 h−1 (confidence interval, 4.71 × 103−6.01 × 103 h−1); coefficient of determination, R2c = 0.97213. The accuracy of the rate constant of HDK T40 is distinctly smaller compared to the accuracy of the rate constants determined for the quartz samples. The reason is the fast increase of the concentration immediately after the start of the dissolution experiment. Dissolution time and time necessary for the analysis are of the same order of magnitude. The dissolution time cannot be determined with sufficient accuracy, and the error of the experimentally determined concentration−time pairs is therefore large. 5.3. Dissolution of Quartz in TRIS Buffer. Dissolution experiments have been carried out with the two commercial quartz samples MIN-U-SIL and QFL in TRIS buffer at temperatures of 25 and 37 °C. The results are shown in Figure 3 up to a dissolution time of 1500 h. The complete numerical values are given in Tables 3A−3C. All of these data are used for the fitting procedure. The complete results of the fitting procedure are listed in Table 4 for the three dissolution models. The first row gives the denotation of the material. The dissolution temperatures are given in row two. The saturation concentrations and the corresponding confidence intervals are listed in Table 4 in the third and fourth row of the results obtained for each model.

Figure 3. Concentration of silica, cL, in TRIS buffer solution as a function of dissolution time, t. Magenta crosses, MIN-U-SIL (T = 310 K); green crosses, MIN-U-SIL (T = 298 K); blue crosses, QFL (T = 310 K); red crosses, QFL (T = 298 K). The results of the theoretical calculations (model III) are represented by the appropriate solid lines. 13921



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Table 6. Comparison of the Rate Constant k+ to Rate Constants Reported in the Literaturea

investigators. It is obvious that the difference of the k(I) values of the two samples persist in the corresponding k+(I) values, though they are related to the unit of the surface area. There must be a structural difference besides the difference in the specific surface area of the two samples. This structural difference is manifested by the values of the rate constant k(III), which is a constant of the material and does not depend on the magnitude of the surface area exposed to dissolution (see eq 14). The results given in Table 4 can be compared to results in the literature; e.g., values of the saturation concentration of quartz are summarized in ref 1, p 31. Some literature results obtained under similar conditions are collected and compared to the results of this work in Table 5. (The designation of the authors is used for the samples if possible.) Table 5. Comparison of the Decadal Logarithm of the Saturation Concentrations Found in the Literature to Results Determined in This Worka sample Quartzb Quartzb 5Bc 6Bc Quartzd Quartzd Quartzb Quartz

b

QFL QFL MIN-U-SIL MIN-U-SIL

T/K

I/mol L−1

pH

log10{csL/c⊖}

298 310 298 298 298 313 298

0.01 0.01 0.03 0.07 0.055 0.055 pure water pure water 0.11 0.11 0.11 0.11

− − 8.33 10.27 7.0 7.0 −

−3.75 −3.60 −4.36 −4.02 −3.94 −3.78 −3.75

24e 24e 27 27 28 28 32

−

−3.60

32

e

7.66 7.42 7.65 7.44

−2.76 −2.74 −2.74 −2.65

this this this this

310 298 310 298 310

ref

sample

T/K

I/mol L−1

pH

(1011 k+)/ (mol m−2 s−1)

5Bb 6Bb QDL1-Ic QDH8-Dc C800d C800d C800d Quartze Quartze QFL

298 298 298 313 298 298 308 298 298 298

0.03 0.07 0.10 0.10 0.01 0.01 0.01 0.1 0.5 0.11

8.33 10.27 7.0 6.75 7.61 7.64 10 6.5 6.5 7.66

0.2818 0.7943 0.3981 0.3162 37.22 21.54 280.02 0.1258 0.2512 1.93

QFL

310

0.11

7.42

2.96

MIN-U-SIL

298

0.11

7.65

16.8

MIN-U-SIL

310

0.11

7.44

15.2

reference 27 27 28 28 29 29 29 31 31 this work this work this work this work

a T = dissolution temperature; I = ionic strength. bOptically clear quartz crystals from Hot Springs, Arkansas, U.S.A., specially prepared.27 cOptically clear quartz crystals from Red Lake, Ontario, U.S.A., specially prepared.28 dCommercial ground Fontainebleau sand (British Industrial Sand Ltd.) specially prepared.29 eWater-clear Arkansas (USA) crystals specially prepared.31

29 for sample C800 at T = 298 K. This situation may be a consequence of structural features as will be discussed later on in the Discussion section. The concentration of monosilicic acid as a function of dissolution time is reported in ref 25, for crushed quartz pebbles from the bed of the Macquarie River, New South Wales. The experiments are carried out for a dissolution temperature of 25 °C and for a concentration of ammonium acetate buffer of 0.1 mol·L−1, resulting in a pH = 7.7. To apply the gradient model (model III) it is necessary to know the surface area and the mass of the quartz samples as well as the amount of water. A specific surface area of 1 m2 g−1 and a dimensionless quartz surface to water ratio (102) are given for the quartz samples A and B1 in ref 26. The dissolution conditions of these experiments (10 g of quartz in 1 kg of water) are relatively close to the conditions applied in our experiments (see Table 2). Table 7 compares the evaluation of the results of the dissolution experiments reported in ref 25 for a dissolution temperature of 25 °C to the results obtained in our investigations by model III.

work work work work

a T = dissolution temperature, I = ionic strength, csL = saturation concentration, c⊖ = standard concentration, 1 mol·L−1. bColorless crystals of Brazilian quartz, specially prepared.24 cOptically clear quartz crystals from Hot Springs, Arkansas, U.S.A., specially prepared.27 d Optically clear quartz crystals from Red Lake, Ontario, U.S.A., specially prepared.28 eNumerical values are extrapolated by suggested formulas from the authors.

The results presented in the literature are about 1 order of magnitude smaller than those measured in this work, thus the commercial available quartz samples differ strongly from natural quartz investigated in the literature. The saturation solubility of commercial quartz is close to the solubility obtained for nanodispersed amorphous silica HDK T40 or to the solubility of amorphous silica or nanoparticles reported in ref 1 p 42 and 55 or our previous results obtained for silica gel.60,61 The numerical values of the constant k+ (I) may be compared to rate constants for quartz reported in the literature summarized in Table 6. The k+ values reported in the literature are at least 1 order of magnitude smaller than the k+ values measured in this work, besides the values reported in ref 29. The rate constant can be calculated for silica and quartz as a function of temperature and pH.30 The k+ values of QFL and MIN-U-SIL determined in this work are closer to silica than to quartz (see Table SI3 of numerical values given in Comment SI3 of the Supporting Information). An approximate agreement is found between the rate constants, k+, for MIN-U-SIL with the data reported in ref

Table 7. Comparison of the Results of Our Quartz Samples to the Results Obtained for the Samples A and B1 of Ref 25, if Model III is Used for the Evaluation pH cSL/mmol L−1 CI/mmol L−1 10−2 × k/h−1 10−2 × CI/h−1 R2c a

13922

QFL

MIN-U-SIL

A

B1

7.66 1.75 1.64, 1.86 3.79 3.43, 4.16 0.99791

7.65 1.80 1.64, 1.96 16.6 9.86, 23.4 0.99200

7.7 0.40 0.36,a 0.44 374 2.44, 745 0.98370

7.7 1.90 1.71,a 2.09 896 641, 1151 0.99881

The error intervals are estimated. dx.doi.org/10.1021/jp310072m | J. Phys. Chem. C 2013, 117, 13914−13927


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Table 7 reveals that the saturation concentrations of QFL, MIN-U-SIL, and B1 are comparable and are in the same order of magnitude found for amorphous silica (ref 1, p 47). The saturation concentration of sample A (log10 csL/c⊖ = −3.4) is in the same range as for quartz (see Table 5). The samples A and B1 were ground to a particle diameter 104 h. The results of the krypton adsorption experiments establish the relative constancy of the specific surface area at least up to t = 7440 h. Longer dissolution times would be necessary to find the decrease of the specific surface area predicted theoretically, as a consequence of Ostwald ripening. The situation is analogous for the quartz QFL where the decrease in specific surface area starts only at t ≈ 106 h. The gradient curve is calculated piece by piece due to mathematical reasons. Therefore, relatively abrupt changes of the specific surface area could not be avoided when connecting the different pieces. This is demonstrated in Comment SI4 of the Supporting Information. 13923



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Quartz can be identified by its Si−O−Si vibrational band at about 690−695 cm−1.62−64 If small amounts of the undissolved solid are removed from the running dissolution process the IR spectra could be obtained, which are shown in Figure 5 and 6 together with the spectrum of the untreated material.

Figure 7. X-ray diffraction pattern of the untreated quartz samples in comparison with a rock crystal sample. The corresponding section of the 101 reflex is shown.

The band of MIN-U-SIL has been shifted to lower angle values, whereas the corresponding reflex of QFL is shifted to higher values. The cell parameters of all three types of α-quartz are listed in Table 9.

Figure 5. IR spectra of MIN-U-SIL samples before and during the solubility experiments, at dissolution temperature of 25 °C (MIN25T). The band at about 695 cm−1 (x) is characteristic for the crystalline modification of quartz. The samples were separated from the undissolved material after the time given.

Table 9. Comparison of the Results of the XRD Investigations for Rock Crystal, MIN-U-SIL, and QFLa −

rock crystal

MIN-U-SIL

QFL

a/Å c/Å c/a D/nm D(100)/nm D(001)/nm Δa/a Δa/a(100) Δa/a(001)

4.9166 5.4035 1.0990 136 115 167 0.0026 0.0026 0.0027

4.9225 5.4052 1.0981 97 80 121 0.0043 0.0042 0.0044

4.9143 5.3960 1.0980 77 68 93 0.0034 0.0033 0.0034

a

Cell parameters a and c, diameter of crystallites D, D (hkl) crystal size in 100 and 001 directions, crystal strain Δa/a and Δa/a (hkl) crystal strain in 100 and 001 directions.

The cell parameters of all three types of α-quartz differ from each other and from data given in the literature: a = 4.921, c = 5.416, and c/a = 1.101.66 Additionally, both quartz samples have a high disorder in their crystal structure compared to the reference material. The crystal strains of MIN-U-SIL are the highest compared to the other quartz samples. This may be an explanation for the high values of the rate constants k+ (model I) and k (model III) that are independent of the surface area of the quartz samples.

Figure 6. IR spectra of QFL samples before and during the solubility experiments, at dissolution temperature of 37 °C (QFL37T). The band at about 695 cm−1 (x) is characteristic for the crystalline modification of quartz. The samples were separated from the undissolved material after the time given.

From Figures 5 and 6 it can be clearly seen that no changes in the crystal modification occur during the dissolution experiments. This is in accordance with the result that no significant changes of the specific surface area can be observed up to dissolution times of 7440 h. Similar results were obtained by XRD measurements.65 Even after one year of dissolving the two quartz samples, no evidence for phase transition into both another crystalline phase or amorphous one can be observed. Characterizing the untreated sample materials by XRD as well as IR spectroscopy lead to similar phase composition. However, if we look in more detail there are slight differences observable. Figure 7 shows the corresponding section from the X-ray diffraction pattern where the 101 reflex can be recognized.

6. DISCUSSION AND CONCLUSIONS The Gibbs free energy is calculated for the dissolution of a solid in the system water−dissolved substance−background electrolyte, and the consumption of water during the dissolution is taken into account. This Gibbs free energy of dissolution considers both change of particle size and particle concentration. It includes any system solid−solution that can be formed from given numbers of molecules of solid, water, and background electrolyte. The dimension reaches from complete dissolved solid via particle−solution systems toward bulk solid pure solution background electrolyte water. This allows for the 13924



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formation of a Gibbs free energy surface in the particle concentration−particle size space.43,44 It is assumed that the path of the system from one state (determined particle size and particle concentration) to a neighboring one is only possible in the direction of the negative gradient of the Gibbs free energy. Therefore, the path from any point on the surface is supposed to follow the negative direction of the gradient of the Gibbs free energy. All states of a given ensemble of solid particles covered during the dissolution process are fixed by this way. This Gibbs free energy of dissolution is used to derive kinetic rate equations by means of common methods of statistical mechanics and a so-called gradient model. These rate equations are compared to the common first-order model of dissolution and the dissolution of a constant number of spherical particles. It is shown that the gradient model leads always to the best correlation between theory and experimentally determined concentrations of the dissolved species. The gradient model is the only model that is able to describe processes that are connected with simultaneous changes of particle size and particle concentration. These effects are the reason for an unusual maximum in the concentration−time curves, the kinetic size effect, which occurred in dissolution processes of nanoparticles, and Ostwald ripening, that may be observed after very long times of dissolution. Concentration and size of particles can be predicted by the gradient model up to the final equilibrium bulk crystal saturated solution if concentration−time pairs are measured for experimentally achievable dissolution times. The curvature of the concentration−time curves at the beginning of the dissolution determines the rate constant. Exact determination of the rate constant needs a large number of concentration−time pairs at the beginning of the process and a relatively small surface of the solid exposed to the solvent. Too large amounts of solid would increase the dissolution rate and therefore cause complications for exact measurements. To determine exact saturation concentrations the dissolution process should be continued as long as possible. Rate constant and saturation concentration are not independent from each other. This can be seen by inspection of the data listed in Table SI1 of the Supporting Information. Increasing saturation concentration causes a decreasing rate constant, if the same set of experimental results is described theoretically. The interdependence between rate constant and saturation concentration may easily be noticed by a rearrangement of eq 9 (model I). This is demonstrated in Comment SI5 of the Supporting Information. The rate of dissolution essentially depends on the magnitude of solid surface area exposed to solvent. Comparable values of the rate constant can only be obtained if the rate constant is related to the amount of solid (mass, e.g.) used in an experiment or to the surface area of this amount of solid. The rate constant becomes a measure of bulk properties of the solid considering this relation. However, this is automatically the case, if the gradient model is applied, since the rate constant determined by this model does not depend on the magnitude of solid (surface area) used in the dissolution experiment. However, the existence of different Q-species in the bulk material and at the surface and the ionization of resulting OHgroups has to be considered. A difference is observed between the rate constants determined by the gradient model for MINU-SIL and QFL that can be ascribed to different crystal strains of both materials. The amorphous nanoscale HDK T40 exhibits the largest value of the rate constant. This seems to be a confirmation that the structural features of the different quartz

samples play an important role for their solubility. The investigation of the solubility differences of different crystal surfaces, classes of surface features like steps, voids, and dislocation etch pits,34 stress conditions,36 or the role of structural units33,35 or bridging oxygen atoms55 yield important contributions to the understanding of molecular details of the dissolution process. Saturation concentration and rate constants measured for the two commercial available quartz samples are at least 1 order of magnitude larger as compared to natural quartz samples studied in the literature. These findings seem to support the interpretation that solubility aspects have to be considered if the toxicity of quartz is estimated. In contrast, to natural quartz, it seems that MIN−U−SIL and QFL dissolve more rapidly in the lung fluid.15 This result demonstrates that the assessment of the health risk, connected with the application of the two commercial quartz samples, should be done in view of their high solubility, which is comparable to the solubility of amorphous silica.

■

APPENDIX I Considerations of irreversible thermodynamics lead to the following equation for chemical reactions46 ⎡ ⎛ A ⎞⎤ w = ω⎢1 − exp⎜ − ⎟⎥ ⎢⎣ ⎝ R GT ⎠⎥⎦

(I.1)

where w is the reaction rate, in our terminology w = dcL/dt; ω is the product of the rate constant and the initial concentrations of starting products according to eq 1 (water consumption is neglected) in our terminology ω = kn0S(1)/v; and RG is the gas constant. The affinity, A, is defined by A = −d(ΔSG)/dξ; ξ is the extent of reaction; and νi is stoichiometry numbers; and ΔSG = NAΔSg is the molar Gibbs free energy. We obtain ⎡ cL ⎤ ⎥ = −A ⎣ c Ls ⎦

d(ΔSG) = dξ

∑ νiμi = R GT ln⎢

(I.2)

Inserting w, ω, y = c0L/csL, and −A/RGT into eq I.1 results in eq 9. The following rate equation is suggested in earlier work.43,44 dc L dc dζ = −kB(c L)|∇g (r , z)| = L dt dζ dt

(I.3)

Here, the concentration is a function of the variables that describe the variation of the solid during the dissolution process, B(cL). In case of dissolution from a plane surface the variable is ζ = N1S(1)/N0S(1), and according to eq 2, dcL/dζ= −n0S(1)/v = B(cL) and furthermore from eq 8 ⎡ by(1 − ζ ) ⎤ dg (r , z) = −ln⎢ ⎥ ⎣ 1 − aζ ⎦ dζ ⎡c ⎤ c ≈ −ln[y(1 − ζ )] = −ln⎢ Ls ⎥ ≈ 1 − Ls > 0 cL ⎣ cL ⎦

∇ g (r , z ) =

(I.4)

The approximations in eq I.4 a ≈ 0 and b ≈ 1 are reasonable in systems where the amount of water excides considerably the amount of solid. Furthermore, it is cL/csL < 1, and the logarithm may be expended into a power series cut after the first member. The last relation of eq I.4 shows that ∇g(r,z) = |∇g(r,z)| is valid. Comparison with eq I.3 results in 13925



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dζ = −k|∇g (r , z)| dt ⎛ c ⎞ dc = −k ⎜1 − Ls ⎟ and L cL ⎠ dt ⎝ = ky(c Ls − c L)

to the identical result because they are mathematically equivalent. Unfortunately, it is only possible to calculate the explicit expressions, t(cL) and t(r). The transformation in the form cL(t), the common representation, can be done by numerical calculation of c L −t−pairs that are used in corresponding graphical representations. The procedure is demonstrated in more detail in the Comment SI1 of the Supporting Information. The function cL(t) may also be estimated by numerical methods using Mathematica 7.047 as it is described in section 4 of this manuscript.

(I.5)

The last relation of eq I.5 is identical to eq 9. We obtained the same result. The function B(cL) is simple, B(cL) = −n0S(1)/ v, a constant. Both parts dcL/dζ and dζ/dt are independent from each other and can be integrated separately, therefore. The result is again eq 9, of course. cL =

■

nS0(1) 1 (1 − ζ ) and ζ = {y − 1 + exp(−kyt )} v y

Additional experimental details. This material is available free of charge via the Internet at http://pubs.acs.org.

■

(I.6)

We may conclude that the commonly used rate equation for dissolution processes can be obtained by consideration of irreversible thermodynamics and the suggested gradient method.43,44,46 This is reasonable since d(ΔSG)/dξ is the gradient of the Gibbs free energy in the direction of the reaction coordinate.

*E-mail: [email protected]. Tel.: +49 3641 948340. Fax: +49 3641 948302. Notes

The authors declare no competing financial interest.

■

APPENDIX II We obtain from eq 2

ACKNOWLEDGMENTS The XRD measurements have been done by Dr. B. Müller and Mrs. A. Schmidt of Friedrich-Schiller-University Jena. This is gratefully acknowledged. We thank Prof. Dr. M. Schmitt of Friedrich-Schiller-University Jena for valuable comments.

1/3 ⎧1⎛ nS0(1) c Lv ⎞⎫ 3 (1 − zr ), r = ⎨ ⎜1 − 0 ⎟⎬ , cL = v nS (1) ⎠⎭ ⎩z⎝ ⎪

⎪

⎪

3n 0(1) 2 dc L zr =− S dr v

■

(II.1)

⎧ 2h ⎡ by(1 − zr 3) ⎤⎫ ⎛ ∂g (r , z) ⎞ 2 ⎥⎬ ∇ g (r , z ) = ⎜ − ln⎢ ⎟ = 3r z ⎨ ⎝ ∂r ⎠ z ⎣ 1 − azr 3 ⎦⎭ ⎩ 3r ⎪

⎪

⎪

⎪

(II.2)

The last approximation is valid, again, in systems with large excess of water (a ≈ 0, b ≈ 1) and R≳1 μm. Comparison with eq I.3 and use of eq II.1 gives ⎤ ⎡1⎛ cs ⎞ dr = −k 3z 2r 2y⎢ ⎜1 − L0 ⎟ − r 3⎥ , ⎢⎣ z ⎝ dt cL ⎠ ⎦⎥

( a)

dc L = −3(c L0z)1/3 (c L0 − c L)2/3 , dr

(b)

REFERENCES

(1) Iler, R. K. The Chemistry of Silica; John Wiley & Sons: New York, 1979. (2) Grassian, V. H. J. Phys. Chem. C 2008, 112, 18303. (3) Andersson, J.; Areva, S.; Spliethoff, B.; Linden, M. Biomaterials 2005, 26, 6827. (4) Andersson, J.; Johannessen, E.; Areva, S.; Jarn, M.; Linden, M. J. Nanosci. Nanotechnol. 2006, 6, 2438. (5) Demadis, K. D.; Mavredaki, E.; Somara, M. Ing. Eng. Chem. Res. 2011, 50, 12587. (6) Yamada, H.; Urata, C.; Aoyama, Y.; Osada, S.; Yamauchi, Y.; Kuroda, K. Chem. Mater. 2012, 24, 1462. (7) Baccile, N.; Babonneau, F.; Thomas, B.; Coradin, T. J. Mater. Chem. 2009, 19, 8537. (8) Zhmud, B. V.; House, W. A.; Denison, F. H. J. Chem. Soc., Faraday Trans. 1997, 93, 3473. (9) Piecos, R.; Pasławska, S. Fuel Technol. 1998, 56, 201. (10) Aluri, H.; Collinson, M. M. J. Electroanal. Chem. 2011, 651, 143. (11) Mahon, E.; Hristov, D. R.; Dawson, K. A. Chem. Commun. 2012, 48, 7970. (12) Oberdörster, G.; Oberdörster, E.; Oberdörster, J. Environ. Health Perspect. 2005, 113, 823. (13) Nel, A.; Xia, T.; Mädler, L.; Li, N. Science 2006, 311, 622. (14) Borm, P.; Klaessig, F. C.; Landry, T. D.; Moudgil, B.; Pauluhn, J.; Thomas, K.; Trottier, R.; Wood, S. Toxicol. Sci. 2006, 90, 23. (15) Arts, J. H. E.; Muijser, H.; Duistermaat, E.; Junker, K.; Kuper, C. F. Food Chem. Toxicol. 2007, 45, 1856. (16) Charkoftaki, G.; Dokoumentzidis, A.; Valsami, G.; Macheras, P. Basic Clin. Pharmacol. Toxicol. 2009, 106, 168. (17) Napierska, D.; Thomassen, L. CJ.; Lison, D.; Martens, J. A.; Hoet, P. H. Part. Fibre Toxicol. 2010, 7, 39. (18) Mudunkotuwa, I. A.; Grassian, V. H. J. Environ. Monit. 2011, 132, 1135. (19) O’Brien, N. J.; Cummins, E. J. Risk Anal. 2011, 31, 706. (20) Quik, J. T. K.; Vonk, J. A.; Hansen, S. F.; Baun, A.; Van De Meent, D. Environ. Int. 2011, 37, 1068.

The gradient of the Gibbs free energy is the derivative dg(r,z)/dr of eq 8 (neglect of the water consumption in eq 1).

⎛ c ⎞ ≈ 3r 2z⎜1 − Ls ⎟ cL ⎠ ⎝

AUTHOR INFORMATION

Corresponding Author

■

⎪

ASSOCIATED CONTENT

S Supporting Information *

dc L dc dr z 2/3 = L = 9k 0 1/3 s (c L0 − c L)4/3 (c Ls − c L) (c) dt dr dt (c L ) c L (II.3)

B(cL) = dcL/dr is a more complicated function in this case. Therefore, the rate law for the dissolution can be obtained by the following two ways: integration of part (c) of eqs II.3 on the one hand and integration of part (a) of eqs II.3 on the other hand. The concentration must be substituted as a function of the radius in the latter case. This function is simply obtained by integration of part (b) of eq II.3, the result of which is already known from eq 2 and the first part of eqs II.1. Both ways lead 13926



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Dissolution of Commercial Microscale Quartz Particles in Water at

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