Article pubs.acs.org/Langmuir

Mechanistic Studies of Silica Polymerization from Supersaturated Aqueous Solutions by Means of Time-Resolved Light Scattering M. Kley,† A. Kempter,‡ V. Boyko,‡ and K. Huber*,† †

Physical Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany BASF SE, Material Physics, Properties of Colloidal Systems, 67056 Ludwigshafen, Germany

‡

S Supporting Information *

ABSTRACT: Silica polymerization in a supersaturated aqueous solution of sodium silicate is a fundamental mineralization process with broad relevance for technical applications as well as for biological processes. To contribute to a better understanding of the mechanism underlying the polymerization of sodium silicate under ambient conditions, a combined multiangle static and dynamic light scattering study on the evolution of particle mass and size is applied for the first time in a timeresolving manner. The light scattering experiments are complemented by a timeresolved analysis of the decay of the concentration of monomeric silicate by means of the silicomolybdate method. Particle formation was investigated at a variable concentration of silicate at pH 7 and 8. The joint experiments revealed a loss of monomers, which is parallel to the formation of compact, spherical particles growing by a monomer-addition process. An increase in the silicate content of up to 750 ppm increased the extent of nucleation and at the same time decreased the lag time observed between the start of the reaction and the actual onset of the growth of particles. Once the silica content is considerably larger than 1000 ppm, the formation of particles is succeeded by particle−particle agglomeration leading to larger fractal-like particles. By the time agglomeration becomes noticeable with light scattering, the monomer concentration has already reached its equilibrium value. An increase in the pH to 8 again revealed particle formation via a monomer-addition process. However, the extent of nucleation was increased and particle−particle agglomeration was inhibited even at an initial silica content of 2000 ppm.

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temperatures (1.90 °C) using the freezing-point method to determine the degree of polymerization of the silica particles and got a third-order reaction at low pH 4.36, with a rate constant that depends on the number of functional groups. Goto4 found in general the tendency of an increasing size and reaction rate with increasing pH in the regime of pH 7−10. Baumann5 was the first who identified an induction period of silica polymerization under certain conditions, in which no change in the molybdate-reactive silica took place. The length of this induction period strongly depended on the initial silica concentration. This behavior has been confirmed by Rothbaum and Rhode7 and by Makrides et al.8 Makrides et al. focused for the first time on the nucleation of the silica polymerization. On the basis of a linear trend of the induction period with the inverse of supersaturation, Makrides et al. postulated that the predominant number of nuclei are formed during the induction period. They explained this behavior in terms of a simplified version of the classical nucleation theory that differed from exact classical nucleation theory by the following feature. The free energy of the activation for nucleation was assumed to be constant and independent of the critical nuclei size, thereby

INTRODUCTION The present work will focus on the silica polymerization based on the polycondensation of monomers with four reactive groups, which leads to 3D polymeric networks. Polycondensation occurs in supersaturated aqueous solutions of sodium silicate and is highly relevant for water-based industrial systems as well as for natural processes such as biomineralization. In industrial facilities, silica polymerization is usually undesirable because it leads to silica fouling and scaling on equipment and membranes used in desalination, water treatment systems, and cooling or heating cycles. A knowledge of the reaction mechanisms at variable environmental conditions is expected to promote the development of specific additives to inhibit the polymerization.1,2 The first investigations in this field focused on the mechanism and kinetics of the polymerization of monosilicic acid in aqueous solution as a function of the pH value, the temperature, the salinity, and the initial silica concentration.3−8 It was found that the highest rates for silica polymerization via condensation occur between pH 6 and 9 and for the hydrolysis of Si−O−Si bonds at a pH close to 2. The polymerization proceeds until the solubility limit of monosilicic acid is reached. Rothbaum and Rhode7 found a second-order reaction for the reaction among monomers and a higher reaction order for the reaction of monomers with polymers in the pH regime from 7 to 10 at a temperature of between 5 and 90 °C. Alexander3 investigated the pH regime between 1 and 6 at very low © XXXX American Chemical Society

Received: July 10, 2014 Revised: September 9, 2014

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increasing silica content. Later on, a nucleation and growth model is applied to the mass evolution of the silica process, which substantiates the picture of a significant induction period followed by a monomer-addition process. The investigated concentrations also extend into a regime where the agglomeration of the resulting small particles sets in. Experiments in the latter regime offer insight into the impact of the initial silica content and the pH value on the transition from monomer-addition to particle−particle agglomeration as well as the morphology of the resulting agglomerates in the absence of any salt other than that formed during polymerization.

leading to a constant nucleation rate that is independent of the degree of supersaturation. On the basis of these pioneering works and his own extensive studies, Iler6 postulated a mechanism for the silica polymerization process, which has become generally accepted. The condensation rate from supersaturated solution is proportional to the H+ concentration for pH 2. The process sets in with the formation of small cyclic oligomers because these structures enable more siloxane bonds to be formed than for other structures with the same number of Si atoms. These small cyclic oligomers grow by the addition of monomers to form condensed particles with an outer shell of SiOH groups. The resulting small particles act as nuclei for further monomeraddition growth. With increasing salinity and pH, the particles can also gel or precipitate via particle−particle agglomeration. In regimes where agglomeration is suppressed, Ostwald ripening may occur and is accelerated by the ease of hydrolysis or bond formation. In recent publications, which focused on silica polymerization in the negative pH regime at variable salinity9 and under alkaline conditions,10,11 the characterization of growing silica colloids by means of inductively coupled plasma mass spectroscopy (ICP-MS), dynamic light scattering and viscosity measurements in combination with the recordings of the monomeric silica by means of the silicomolybdate method had been applied. In the neutral pH regime, Icopini et al.12 and Tobler et al.13 investigated the impact of the ionic strength on the kinetics of silica formation and observed a fourth-order reaction for the decrease in monosilicic acid. On the basis of SAXS and DLS measurements, Tobler et al.13 described the silica polymerization as a three-step process similar to that described by Iler.6 They postulated homogeneous and instantaneous nucleation with a critical nucleus diameter of 1.4 to 2 nm followed by first-order particle growth and finally Ostwald ripening or the agglomeration of particles. Recent developments in light scattering instrumentation make available time-resolved multiangle light scattering in static and dynamic mode,14−16 which for the first time offers the chance to record simultaneously the weight-averaged molar mass, the radius of gyration, the hydrodynamic radius, and, once particles get large enough, also the form factor from the static scattering curves as they evolve with time. This direct access to size, mass, and shape data on growing particles in combination with recently developed approaches to analyze time-resolved scattering data17−19 promises new insight into the highly relevant process of silica particle formation under ambient conditions and hence suggests a revision and refinement of the pieces of information already available on the process. Therefore, the present work is designed to address the mechanism of silica particle formation and its connection to agglomeration and Ostwald ripening. To this end, we describe the first time-resolved, multiangle, combined static and dynamic light scattering study of silica polymerization from supersaturated solutions in combination with the detection of the accompanying monomer consumption via the silicomolybdate method, which yields the most complete set of data on changes referring to monomers and particles. The study is performed close to neutral pH (pH 7 and 8) at variable initial silica concentrations covering the regime of 350 to 3000 ppm. Under conditions where silica polymerization forms condensed, small particles, growth by monomer addition could be unambiguously unraveled, where nucleation increases in extent with

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EXPERIMENTAL DETAILS

Materials. Stock solutions with a concentration of 0.2 M Na2SiO3 were prepared by dissolving sodium metasilicate (Na2SiO3·9H2O, assay ≥98%, Sigma-Aldrich) in water. The water was purified by a Millipore system. This treatment led to a water conductivity of 0.055 μS m−1. Silicomolybdate Method. The consumption of the silicic acid during the growth process is determined by means of the molybdenum blue method.20 This method makes use of the reaction of monomeric Si(OH)4 with acidified ammonium heptamolybdate to silicomolybdic acid, which enables us to determine the concentration of monomeric silica. The analysis is also possible in the presence of polysilicic acid because the formation of silicomolybdic acid is much faster than the depolymerization of polysilicic acid. Two different species are suitable for the UV−vis analysis: The yellow and the reduced blue silicomolybdic acid. In the present work, the blue species was used, which is recommended for very low concentration samples because the absorbance of the blue species is much larger than that of the yellow species. The sample preparation proceeds as follows.6,20 Depending on the silica content, a sample with a volume of between 150 and 230 μL was removed from the reaction mixture for the analysis of monomeric silica. This removed volume was diluted with water to a volume of 50 mL. After the addition of 1 mL of 50 vol % hydrochloric acid in water and 2 mL of an ammonium heptamolybdate solution with a concentration of 100 g/L, the sample was agitated and was allowed to stand for 7 min. Then, 2 mL of oxalic acid with a concentration of 75 g/L was added, followed by a second storage time of 2 min. Finally, 2 mL of the reducing agent was added. The UV spectrum of the sample could be recorded 5 min after the addition of the reducing agent. Because silicomolybdic acid is not stable, the spectrum had to be taken at the latest 15 min after addition of the reducing agent. The reducing agent was prepared by mixing a solution of 23.08 g of Na2S2O5 in 150 mL of water with a solution of 500 mg of 1-amino-2-naphthol-4-sulfonic acid and 1 g of Na2SO3 in 50 mL of water. The sample absorbance was measured with a Lambda 19 spectrophotometer (PerkinElmer) at a wavelength of 815 nm. Sample Preparation for Light Scattering. Samples for light scattering experiments were prepared from the silicate stock solution by dilution with water to a volume of 200 mL. The concentrations investigated covered a concentration regime of between 350 and 3000 ppm. The pH value of the solutions was adjusted to 7 or 8 by adding 2 M HCl. The volume of added 2 M HCl was appropriately taken into account for the calculation of the initial silica concentration. The pH adjustment of the solution set time zero for recording the time. After the initiation of the reaction, the sample was filtered directly through a syringe filter with a pore size of 0.2 μm into a dust-free scattering cell, which has been treated before use with a solution of 5% chlorotrimethylsilane (98%, Janssen Chimica) in toluene. This treatment was expected to avoid silica formation from silicate solution on the inner wall of the scattering cell. To accelerate the growth process, all measurements with sodium silicate solutions were performed at 37 °C. The time regime analyzed by light scattering measurements depended on the sample and varied from 2 to 160 h. Static Light Scattering. The scattering intensities from static light scattering (SLS) obtained as Rayleigh ratios ΔRθ at an angle θ were processed in a Zimm plot21 in order to obtain the weight-averaged B

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The ρ ratio adopts a value of 0.77 for compact spheres and a value of 1.5 for monodisperse linear chains.24 Scattering Setup. The ALV/CGS-3/MD-8 multidetection laser light scattering system provides eight detectors that are positioned in angular increments of 8°. A He−Ne laser with a wavelength of 632.8 nm and a power of 35 mW was used as a light source. The multiangle device enabled a time-resolved recording of angular-dependent SLS and DLS, whereby the time resolution is limited by the measuring time for the evolution of a correlation function in DLS. Differential Refractometer. The differential refractometer from SLS-Systemtechnik detects the beam displacement of a laser beam with a wavelength of 635 nm as a function of the analyte concentration in solution after passing a two-chamber cuvette (solvent/solution). The beam displacement is proportional to the difference in the refractive index Δn between the solvent and the solution. The detection of the dependence of Δn on the concentration of the solution gives the refractive index increment. For the determination of the ∂n/∂c value of the silica solution, a concentration series of between 12 and 6.1 × 10−3 g/L SiO2 was measured. For the silica solution, a ∂n/∂c value of 0.223 mL/g was obtained.

molar mass Mw and the z-averaged radius of gyration Rg2 according to eq 1:

Rg2 2 Kc 1 = + q + 2A 2 c ΔR θ Mw 3M w

(1)

A2 is the second virial coefficient, and c is the mass concentration of the solid in the solution. The quantity q is the momentum transfer given by

q=

⎛θ⎞ 4πnsolv sin⎜ ⎟ ⎝2⎠ λ0

(2)

with the wavelength λ0 in vacuum and the refractive index of the solvent nsolv. Because an extrapolation of the scattering data to c = 0 is not possible in an aggregating or reacting system, we were forced to neglect a possible impact of interactions among scattering particles on the scattering signal. In the range of c ≤ 3000 ppm, this is justified. In the case of a distinct bending of the scattering curves, a quadratic Guinier22 approximation has been applied:

⎛ Kc ⎞ ⎛ 1 ⎞ Rg2 2 q ln⎜ ⎟ = ln⎜ ⎟+ 3 ⎝ ΔR θ ⎠ ⎝ Mw ⎠

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KINETIC MODEL: NG MODEL Static light scattering provides the temporal evolution of the weight-averaged molar mass Mw, which is proportional to the total intensity of scattered light. It is this evolution that is suitable to be interpreted with kinetic models. If a growth process via monomer addition is considered, the weightaveraged mass values include the monomers as well as the growing species. We will compare the experimental results of Mw with a nucleation−growth model (NG model) based on the considerations of Tsapatsis et al.25 The model consists of three fundamental steps. In the first step, precursor A forms reactive monomer B.

(3)

In eqs 1 and 3, K is the contrast factor and is defined as K = (4π nsolv2/ λ04)(∂n/∂c)2, with Avogadro’s constant NA and the refractive index increment ∂n/∂c of silica in the solution. The refractive index increment for sodium metasilicate in aqueous solution was determined to be 0.223 mL/g by means of a differential refractometer. This value was used for the evaluation of all weight-averaged molar mass Mw. Normalization of the scattering curve with forward scattering ΔRθ→0 leads to the form factor 2

P(q) =

ΔR θ ΔR θ→ 0

(4)

which provides direct information about the particle structure. Form factors are established from SLS for particles with radii larger than 100 nm because they start to discriminate significantly among different shapes only beyond this size. Dynamic Light Scattering. Dynamic light scattering (DLS) provides the normalized field-time correlation function g1(τ). The field-time correlation function was evaluated by means of the cumulant analysis:23 ln[g1(τ )] = K 0 − K1τ +

1 1 K 2τ 2 − K3τ 3 + ... 2! 3!

nucleation

(5)

addition

(6)

(7)

(i = 1, 2, ...)

(11)

(12)

where kn, ke, [B], [Ci], and δi1 are the rate constant of nucleation, the rate constant of monomer addition, the monomer concentration in mol/L, the concentration of the growing particles with the degree of polymerization i in mol/L, and the Kronecker delta, respectively. We now present a route to establish expressions for the moments of species Ci as a function of time. This route is

Rg Rh

ke

B + C1 → Ci + 1

d[Ci] = ke[B][Ci − 1](1 − δi1) − ke[B][Ci] + kn[B]δi1 dt

with k being the Boltzmann constant, T being the temperature, η being the viscosity of the solvent, and Dz,q→0 being the diffusion coefficient, which has to be extrapolated to q = 0 for a proper evaluation of Rh. A possible slight concentration dependence of Dz,q→0 has been neglected for the same reason as given for the neglect of the concentrationdependent term in eqs 1 and 3. Additionally, we can extract more information on the structure of the particles from structure-sensitive ratio ρ.

ρ=

(10)

The rate constant of the addition of monomers is independent of the degree of polymerization of the particle. The assumption of a single growth constant introduces a considerable simplification because growing particles offer an increasing number of sites for further monomer addition. This qualifies an exact physical interpretation of the resulting rate constant but does not inhibit the successful application of the simplified model. The change in the concentration of growing particles can be expressed by a simple master equation

Coefficient K2 = ⟨(Γ − ⟨Γ⟩) ⟩ is the variance of Γ and serves as a measure of the polydispersity of the sample. Hydrodynamic radius Rh is given by eq 7

kT 1 6πη Dz , q → 0

kn

B → C1

Particles Ci with a degree of polymerization i grow by the irreversible addition of monomers B.

2

Rh =

(9)

In the second step, reactive monomers B nucleate by forming species C1.

Coefficient K0 is a measure of the signal-to-noise-ratio, and K1 is the z average of the time constant ⟨Γ⟩ and is related to the z-averaged diffusion coefficient Dz according to

K1 = ⟨Γ⟩ = Dzq2

kp

A→B

monomer formation

(8) C

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similar to that published by Lomakin et al.26 In general, the kth moment of an ensemble of a species Ci can be defined as [C](k) =

∑ ik[Ci]

with M0 being the molar mass of the monomeric unit. The ordinary differential equation (eq 20) was solved numerically to obtain the evolution of the first moment with time. The numerical calculations were performed with Matlab software. For this purpose, ordinary differential equation solver ode45 was applied. The calculation of the second moment according to eq 21 was carried out by numerical integration via the trapezoidal method. For the fitting procedure, the derivative-free “Nelder−Mead simplex direct search” algorithm was used.

(13)

i

Inserting the rate of change of the particle concentration [Ci] introduced by eq 12 into the time derivative of eq 13 gives a set of recursive equations describing the first derivative of the kth moment with time. k−1 ⎛ ⎞ k d[C](k) = k n[B] + ke[B] ∑ ⎜ ⎟[C]j dt ⎝ j⎠ j=0

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RESULTS AND DISCUSSION Variation of the Silica Content at pH 7. The concentration dependence of the silica polymerization was investigated by means of a concentration series of six samples having SiO2 contents of 350, 400, 500, 750, 2000, and 3000 ppm at pH 7. Figure 1A summarizes the evolution of the

(14)

On the basis of eq 14, the first derivative with time of the first three moments (k = 0, 1, 2) can be established as eqs 15−17. d[C](0) = k n[B] dt

(15)

d[C](1) = k n[B] + ke[B][C](0) dt

(16)

d[C](2) = k n[B] + ke[B][C](0) + 2ke[B][C](1) dt

(17)

The mass conservation is expressed by the initial concentration of the precursor [A]0, which is at any time equal to the sum of the concentrations of precursor A, monomer B, and the concentration of monomers B that are already incorporated into growing particles C i. The latter concentration is represented by the first moment [C](1) of the ensemble. [A]0 = [A] + [B] + [C](1)

(18)

The change in the precursor concentration is given by d[A] = −k p[A] dt

(19)

Equations 15, 16, 18, and 19 can be used to get a general equation for the derivative of the zeroth moment. Figure 1. Formation of silica particles as a function of time in water at pH 7 at variable silica concentrations: 750 (◊), 500 (△), 400 (○), and 350 ppm (□). (A) Apparent weight-averaged molar mass from SLS. (B) Consumption of the monomeric silica from the silicomolybdate method.

d2[C](0) d[C](0) d[C](0) −k pt = − − [A] k k e k k [C](0) 0 p n n e dt dt dt 2 (20)

A numerical solution of eq 20 provides the zeroth moment and its time derivative as a function of time. These values enable us to calculate the second moment of the particle ensemble via the integration of eq 17. ⎡ ⎡1 [B]⎢k n + 2ke⎢ [C](0) (t ) ⎣2 ⎣ ⎤⎤ + ([A]0 − [A](t ) − [B](t ))⎥⎥ ⎦⎦

[C](2) (t ) = [C](2) t=0 +

∫0

apparent weight-averaged molar mass for the concentration series up to a silica content of 750 ppm. The corresponding monomer consumption is depicted in Figure 1B. The silica polymerization shows a distinct lag time depending on the initial silica content. During this lag time, the scattering of the sample remains close to the scattering of the pure solvent. The onset of monomer consumption nicely correlates with the end of this lag time. As can be extracted from Table 1, the lag time is reduced with increasing initial silica concentration, which is in line with the results of Rothbaum and Rhode.7 The sample with the highest silica content exhibits no lag time and shows a fast decay of the monomer concentration. All samples approach a plateau value in mass and size (Figures 1 and 2). It is noteworthy that the monomer content shown in Figure 1B decreases whereas the weight-averaged particle mass is increasing toward the plateau value and the time where this plateau value is reached correlates with the time at which the monomer concentration reaches the solubility limit. This

t

(21)

It has to be mentioned that the initial concentration of growing particles is set to zero. This fact is accounted for by assuming (1) (2) [C](0) t=0 = [C]t=0 = [C]t=0 = 0. The weight-averaged molar mass Mw, which is accessible via light scattering, is an average value over all particle species in the solution comprising precursors, monomers, and growing particles, M w (t ) =

M0 ([A](t ) + [B](t ) + [C](2) (t )) [A]0

(22) D

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A special situation is given if the growth of the particles followed by light scattering obeys a monomer-addition mechanism. In such a case, the weight-averaged molar mass obtained via light scattering refers to all solute species including nonconsumed monomers and particles, which leads to an exponent smaller by a factor of 1/2 than the value of the exponent given by the respective topology.18,19 The molar mass values corresponding to the time-resolved data in Figure 3 were

Table 1. Lag Time, Final Hydrodynamic Radius, and Final Weight-Averaged Molar Mass Observed for the Respective SiO2 Contents Measured at pH 7

pH 7

SiO2 content (ppm)

lag time (h)

final Rh (nm)

final Mw [1 × 105]

350 400 500 750

57.4 19.4 2.12 0

17 14 5.7 4.6

7.02 3.96 0.312 0.183

Figure 3. Correlation of the hydrodynamic radius with the weightaveraged molar mass at variable silica contents: 750 (◊), 500 (△), 400 (○), and 350 ppm (□). The dashed line indicates a slope of 1/6. Full symbols show the corresponding correlation of Rh(t = tmax) with the final values of Mw = ΔRθ=0/Kcpart(t = ∞). The straight line indicates a slope of 1/3.

Figure 2. Hydrodynamic radius of silica particles as a function of time in water at pH 7 at variable silica contents from DLS: 750 (◊), 500 (△), 400 (○), 350 ppm (□).

strongly suggests that particle growth occurs by an addition of monomers. As we will demonstrate in the following paragraphs, a detailed analysis of our data confirms this mechanism. However, the exact nature of the monomer has to remain unknown. The final plateau values in Table 1 indicate an increase in the size and weight-averaged particle mass with decreasing silica concentration, whereas monomers are consumed faster with increasing silica content. These trends can be explained by assuming a nucleation rate that increases with increasing silica content. The more nuclei that are formed, the smaller the averaged size and mass of particles because the monomers have to be shared by an increasing number of growing particles. Along with this, the monomers are consumed faster, the larger the number of growing particles becomes. As it can be inferred from the very small hydrodynamic radii of the silica particles, a form factor of the particles is not accessible via angle-dependent SLS. To obtain structure information on the particles, we use the correlation of the particle size with the weight-averaged molar mass of the ensemble. The correlation of the radius of gyration or the hydrodynamic radius with the weight-averaged molar mass, all available by light scattering, provides important information about the form of the growing particles. For self-similar structures, the correlation results in power laws with a shapesensitive exponent α, which in general exhibits the same value for Rg and Rh.

R g ∝ Rh ∝ Mw α

calculated on the basis of the constant initial silica concentration that at any time represents the weight average of monomers and particles. Samples with silica contents of 350 and 400 ppm show good agreement with α = 1/6. In light of the just-mentioned division of the exponent by 2, a value of α = 1/6 corresponds to compact structures such as spheres or cubes that are formed via a monomer-addition mechanism. Unlike the two low silica contents, samples with silica contents of 500 and 750 ppm do not show a distinct power law because no significant regime of an increasing hydrodynamic radius could be detected anymore. Therefore, we also consider the correlation of the final hydrodynamic radius and mass for all four silica contents. As has been outlined by eqs 1 and 3, weight-averaged mass values of the solute are given by Mw = ΔRθ=0/(Kc). If the entire weight c0 of the solute corresponding to the monomer concentration at t = 0 is used, then the weightaveraged mass refers to an ensemble in which all intermediate states include the monomers. If the concentration of particles cpart(t) c part(t ) = c0 − cmon(t )

(24)

is known, then an alternative value can be given for the ensemble referring to particles only. In eq 24, it is c0, the initial silica content, and cmon(t), the monomer concentration. In cases where particle formation is still going on, an estimation of cmon(t) can be given from the monomer consumption recorded by the silicomolybdate method. In cases where final plateau values for the averaged particle mass Mw are reached, the extent of consumption of monomers can be assumed to correspond to c0 − cmon(t = ∞), with cmon(t = ∞) ≈ 170 ppm of SiO2 corresponding to the equilibrium concentration. Because plateau values are reached for all four

(23)

In the case of spherical particles, α assumes a value of /3, and for polymer coils under ideal conditions, the power law is α = 1 /2. 1

E

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silica contents, final Mw values were calculated by using the concentration cpart(t = ∞) according to eq 24, which does not include the nonconsumed (soluble) monomers anymore. The resulting correlation is plotted in Figure 3 and reveals an exponent of α = 1/3. This is the value expected for compact structures based on topological considerations. The behavior of the exponent based on mass values including and excluding the monomer concentration is a clear hint for a monomer-addition mechanism of silica polymerization under these conditions. The silica polymerization at considerably higher silica contents of 2000 and 3000 ppm obeys a mechanism different from that observed in the regime of low silica contents at pH 7. Noticeably, the particle growth predominantly occurs in a time regime in which no further change in the monomer concentration is observed. As is nicely demonstrated in Figure 4, monomers have been used up to the solubility limit at t < 0.2

Figure 5. Evolution of the radius of gyration Rg, the hydrodynamic radius Rh, and the corresponding ρ ratios according to eq 8 at pH 7 for silica contents of 2000 (Rg, ■; Rh, □) and 3000 ppm (Rg, ▼; Rh, ▽).

Figure 4. Evolution of the silica particle growth at pH 7 for the two highest silica concentrations investigated: 2000 (■) and 3000 ppm (▼). (A) Time dependence of the apparent weight-averaged molar mass. (B) Consumption of the monomeric silica vs time.

h, whereas the main increase in the molar mass occurs after this solubility limit has been reached. This indicates that the particles grow via agglomeration of constituent particles formed at t < 0.2 h, thereby exhibiting much higher values in mass and in particle size than those observed at the lower silica contents of 350−750 ppm. The large values of the particle size enable the determination of the radius of gyration in addition to Rh (Figure 5). Therefore, a correlation of Rg and Rh with the weight-averaged molar mass is possible. Because growth occurs via agglomeration of particles, a constant cpart could be used for the determination of Mw = ΔRθ=0/Kc. Figure 6 summarizes the correlation of the radius with mass. The radius of gyration and the hydrodynamic radius exhibit a power law of 1/2 over a wide regime of mass Mw, suggesting a fractal dimension similar to that of polymer coils. Only for very high molar masses does the correlation of the radius of gyration decrease slightly. The onset of this slight deviation may be due to a gradual shift of the Guinier regime toward lower q as the particles are growing. Thereby the Guinier regime may gradually leave the q-range accessible by the instrument once Rg gets close to 200 nm.

Figure 6. Size of the silica polymer at pH 7 for the two silica contents: (□) 2000 and (▽) 3000 ppm. (A) Correlation of Rg with Mw. (B) Correlation of Rh with Mw. The lines indicate a slope of 1/2.

However, a form factor (Figure 7) of the particles, which was taken at t = 19 h after the initiation of the reaction, shows good agreement with the form factor of a Gaussian coil,27 even in the latest stages of the agglomeration process. The agreement of the experimentally determined form factor with that of a polymer coil clearly suggests that the coalescing silica particles form fractals with a fractal dimension close to that of polymer coils. Whereas at low silica concentrations (350−750 ppm) compact structures are formed via monomer addition, the morphology of the particles changes as agglomeration sets in. It is this switch in morphology that excludes Ostwald ripening as the main process driving the growth at silica contents higher than 1000 ppm. However, we suggest that the trend of an increase in nucleation rate accompanied by a decrease in the F

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nucleation is accelerated by an increase in the pH value from 7 to 8, which is confirmed by the decreased mass and size of the particles at pH 8 in comparison to those at pH 7 because a faster nucleation leads to a larger number of nuclei and therefore to smaller particles for a given amount of monomer. The observed hydrodynamic radii are at the limit of detection and exhibit higher uncertainties than at pH 7; therefore, only the final values of Rh are accessible. The general trends in the final Mw and Rh values as a function of the initial silica content are summarized for both pH values in Figure 8.

Figure 7. Form factor of the sample with a silica concentration of 2000 ppm (□) at t ≈ 19 h compared to a form factor of a hard sphere28 (green −−), a coil27 (), and a stiff rod29 (red −−).

size of the spherical particles, formed during monomer addition growth, extends into the regime of higher silica contents (2000 and 3000 ppm). Hence, a short time after initiation an even greater number of small spherical particles is formed at silica contents of 2000 and 3000 ppm. This is nicely supported by the first data point recorded for the experiment with a silica content of 2000 ppm at t = 4.8 min (Mw = 3.5 × 103 g/mol, Rh = 2.8 nm), which perfectly fits into Figure 3. Obviously, this particle number concentration at contents of 2000 ppm or larger is high enough to overcome the stabilization effect caused by surface charges, and particle−particle agglomeration sets in. From the literature, it is known that silica sols begin to gel above pH 6 if the salt concentration amounts to 0.1−0.2 mol/L and the silica concentration is at least 1−2%.6 The presented measurements show that an agglomeration of spherical sol particles starts if only the counterions of sodium metasilicate, which serve as the silica source, are present. In the case of a SiO2 content of 2000 ppm, which is equal to a silica concentration of 0.2%, the corresponding sodium ion concentration amounts to 0.07 mol/L. The agglomeration tendency and the agglomeration rate depend on the initial silica content. The light scattering results clearly show the existence of fractal-like structures. These structures are probably formed by the collision of spherical particles and fractal-like particles, which are then connected via siloxane bonds from surface to surface. Influence of the pH Value. A second sample series at variable silica content has been studied at pH 8. The initial silica concentration has been varied between 350 and 2000 ppm. The results for the evolution of the weight-averaged molar mass and the monomer consumption depending on the time and the correlation of the finally reached mass and size at pH 8 are shown in the Supporting Information as a direct comparison to the results at pH 7. Here, we want to discuss only the main aspects of silica polymerization at pH 8. In general, a decreasing final weight-averaged molar mass with increasing silica content can be observed for both pH values between 350 and 750 ppm. Only the absolute values are lower for pH 8. In the case of the four lowest silica concentrations (350−750 ppm), faster monomer consumption is observed. This trend is accompanied by a reduced lag time (Figure SI-1A of the Supporting Information). Hence, the

Figure 8. Final weight-averaged molar mass corresponding to Mw = ΔRθ=0/Kcpart(t = ∞) at pH 7 (■) and pH 8 (□) and final hydrodynamic radius Rh(t = tmax) at pH 7 (▲) and pH 8 (Δ) as a function of the initial silica content.

For higher silica contents (c0 > 750 ppm), the results recorded at pH 8 deviate from those at pH 7. Whereas at pH 7 for very high silica contents (2000 and 3000 ppm) an agglomeration process sets in, at pH 8 the particles maintain a small size (∼4−5 nm) with a similar weight-averaged molar mass as observed with 750 ppm of SiO2, hence excluding an agglomeration of particles. From this, we can conclude that the concentration threshold for the onset of an agglomeration process is shifted to higher silica contents as the pH increases. A reason could be the better stabilization of the particles in a solution with higher pH as a result of a larger number of charged groups on the particle surface. A correlation according to eq 23 also for the data from pH 8 shown in Figure 8, which is given in the Supporting Information as an extension of Figure 3, indicates the same power law of α = 1/3 and hence suggests the formation of compact structures also at pH 8. Kinetic Modeling. The applied model is based on three independent parameters that have been varied within the optimization process: The rate constant of the precursor reaction kp, of the nucleation kn, and of the growth reaction ke. The molar mass of the monomeric unit M0, which corresponds to the initial monomer concentration [A]0, has been fixed to the molar mass of one SiO2 unit with M0 = 60.1 g/mol. One disadvantage of the model is that it does not result in an equilibrium concentration of the monomeric units and therefore cannot reproduce the equilibrium concentration cmon(t = ∞) of 170 ppm SiO2, which was detected by means of the silicomolybdate method. To still adjust model calculations to final equilibrium concentrations of the monomers, we used [A]0 values corresponding to G

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Langmuir [A]0 =

Article

c0 c (t = ∞ ) − mon M0 M0

Table 2. Model Parameters Obtained from the Optimization Process with the NG Model in Terms of Equation 20a

(25)

c0,exp/ ppm

with the experimentally used initial concentration c0 and the equilibrium concentration cmon(t = ∞). Another disadvantage is that the consideration of the nucleation step according to eq 10 may be oversimplified.30 Figure 9 compares the evolution of the weight-averaged molar mass and the consumption of the monomeric silica at pH

350 400 500 750

[A]0/ mol L−1 3.0 × 10−3 3.8 × 10−3 5.5 × 10−3 9.7 × 10−3

kp/h−1

kn/h−1 −2

2.1 × 10

4.8 × 10−2 2.6 × 108 6.7 × 108

2.2 × 10−6 8.7 × 10−6 3.6 × 10−4 2.4 × 10−3

ke/L (mol h)−1 1.5 × 10

5

3.3 × 105 1.6 × 104 1.4 × 104

χ2

J/cm−3 s−1

1.3 × 10−6 9.5 × 10−7 6.7 × 10−6 1.9 × 10−4

2.1 × 109 9.7 × 109 5.0 × 1011 5.0 × 1012

Parameters kp, kn, ke, and χ2 correspond to the rate constant of the precursor reaction, the nucleation, the growth step, and the residuals, weighted by the experimental values. Column 7 provides initial nucleation rates determined as J = kn((c0NA)/M0).

a

interplay of a precursor reaction more complicated than that described by eq 9 with nucleation being highly complex on its own if appropriately described in a kinetic model.30 For the even higher silica contents (2000 and 3000 ppm), the number density of particles gets large enough to make particle−particle agglomeration the main process observed by our light scattering experiments. Agglomeration rules out further application of our NG model. As it can be extracted from the Supporting Information, a fit in which species A acts as a reactive monomer or species A and B act as reactive monomers without any precursor reaction leads to higher residuals. This failure indicates the need for a suitable reaction such as precursor formation according to eq 9 in order to account for the appearance of a lag time. The consideration of an equilibrium in the precursor reaction also does not lead to decreasing residuals of the fit. Thus, the introduced NG model among all of the applied models seems to describe the real chemical reactions of the silica polymerization in the best way. On the basis of findings by Belyakov et al.,31 the precursor reaction may be related to an increased reactivity with an increasing degree of polymerization. In light of the model presented by Iler,6 the formation of small oligomeric rings may be responsible for the lag time because they induce nucleation and provide the building units necessary for the monomer addition step. At this point, we direct the reader’s attention to an observation made by Fouilloux et al.32 Although it derives from an experiment based on tetraethylorthosilicate as a precursor and thus on a reaction pathway that slightly differs from the conditions applied in the present work, the observation offers an interesting view on the nucleation step. At the end of the lag time corresponding to the maximum degree of supersaturation, Fouilloux et al.32 have noticed an increase in the volume fraction of particulate mass at constant particle size. If the density of the particles remains constant, such a trend can be reconciled only with an increase in the number of particles having the same size. However, Fouilloux et al.32 favored an alternative reasoning that adopts a densification of particles at a constant number of particles during that initial stage of particle formation. If such an interpretation could be transferred to the present system, then the formation of nuclei would be accomplished by a densification of loose prenuclei clusters and would thus provide one reason for an even more complex nucleation than classical nucleation theory33,34 would already suggest. Currently, we can speculate only on the nature of the precursor reaction where even the exact nature of the monomer is not known to us.

Figure 9. Comparison of the experimental results in terms of the evolution of the weight-averaged molar mass (A) and in terms of the decay of the monomeric silica concentration (B) at variable initial silica contents with the corresponding fits with the NG model represented by the continuous lines: 350 ppm SiO2 (□), 400 ppm SiO2 (○), 500 ppm SiO2 (△), and 750 ppm SiO2 (◊). The inset of panel A shows the enlargement of the experimental and fitted curves with silica contents of 500 and 750 ppm in order to visualize deviations of the model.

7 with model simulations based on the NG model. The χ2 value, which is a measure of the deviation of the simulated trends from the experimentally obtained results, is calculated according to eq 26 χ2 =

1 N

N

∑ i=1

(M w,exp − M w,fit)2 M w,exp2

(26)

with the number of fitted data points N, the experimental mass values Mw,exp, and the calculated mass values Mw,fit. As can be extracted from Table 2, χ2 is of the same order of magnitude for the two lowest silica contents. The fits show good agreement between theory and experiment over the entire range of time only for the two lowest silica contents of c0 = 350 and 400 ppm. In the case of silica contents of 500 and 750 ppm, the NG model describes the behavior less satisfactorily, and this becomes obvious in the inset of Figure 9A and by the higher χ2 values. Rate constants kp and kn had to be drastically increased yet without being able to reach the proper plateau value. As has been inferred from the data of our kinetic experiments, nucleation plays an increasingly dominating role. Very likely, it is this dominating role of nucleation that makes our NG model less appropriate because it cannot describe in adequate detail the induction period, which turns out to be an H

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by the trends in monomer consumption, recorded by means of the silicomolybdate method, monomers are consumed during monomer addition but not during agglomeration. It is this increase in the extent of nucleation with the silica content that eventually generates a number density of primary particles large enough to undergo particle−particle agglomeration. The shape of the silica particles was characterized via a correlation of the molar mass with the size, indicating compact spherical particles in the case of growth via monomer addition. Once particle− particle agglomeration sets in, this correlation indicates the formation of fractal-like structures. The formation of fractals could be further verified by a comparison of the radius of gyration with the hydrodynamic radius and by the form factor of large aggregates. From a detailed consideration of the monomer addition growth, it turned out that the process can be characterized by two features, the lag time and the plateau value of particle size and mass. During the lag time, the sample generates no significant net scattering. The lag time decreases with increasing initial silica content. Along with this, final particles decrease in size and mass with increasing initial silica content simply because the nucleation rate increases with increasing silica content, resulting in a larger number of growing particles. The increase in the nucleation rate with increasing silica content could be confirmed by fitting the experimental results with a simple nucleation−growth model. The model well describes the nucleation and growth of the silica samples with initial silica contents of 350 and 400 ppm. For silica contents of 500 and 750 ppm, the model was less satisfactory, probably because of the preponderance of precursor formation and nucleation followed by only minor growth. Classical nucleation can hardly be the sole origin of an elongated induction period if full strength supersaturation is existent from the onset of the process. However, an induction period could easily be reconciled with an appropriate precursor reaction, which provides the building unit capable of nucleating and being added to growing particles as monomers. If silica polymerization is induced at pH 8, then the lag time is reduced and along with it the nucleation rate is increased, which is also confirmed by the decreased size and mass of the final silica particles. Unlike the case at pH 7, an agglomeration process has not been observed at pH 8. An increase in charge density on the particles probably increases the concentration threshold for the onset of agglomeration with increasing pH drastically. This result is in line with a decrease in the gelling time6 of silica and an increasing pH in the regime of pH >6, as the rate of particle collisions decreases as a result of increasing charges on the particle surface. The achievements established in the present work are summarized in Figure 11 and are related to the mechanistic features given by Iler.6 Silica polymerization starts with nucleation, followed by a monomer-addition process under all conditions applied in the present work. As a striking feature, the increased rate of nucleation observed with increasing silica content is reflected in the decrease in particle size. Combined time-resolved light scattering enabled us to discriminate this monomer-addition process unambiguously from a second process, which corresponds to particle−particle agglomeration and sets in only once the silica content (and with it the number density of particles) is large enough and the pH is low enough. The latter aspect could be inferred from the fact that no agglomeration was detectable at pH 8.

Finally, the nucleation rates determined with the NG model at pH 7 shall be compared to classical nucleation theory (CNT). In CNT, the nucleation rate J correlates with the supersaturation according to eq 27.35,36 J = kn

⎛ −K ⎞ c N c0NA = K a exp⎜ 2 b ⎟ 0 A ⎝ σ ⎠ M0 M0

(27)

In eq 27, the supersaturation σ is expressed as ⎛ ⎞ c0 σ = ln⎜ ⎟ ⎝ cmon(t = ∞) ⎠

(28)

with c0 being the initial silica concentration and cmon(t = ∞) being the equilibrium concentration, respectively. To compare nucleation rate constants obtained from the fits of the NG model with classical nucleation theory, we use a plot of the nucleation rate constants established with the NG model versus the supersaturation in Figure 10. The exponential increase in

Figure 10. Variation of the nucleation rate constant with the supersaturation of the solution ln(c0/cmon(t = ∞)), where c0 indicates the initial silica concentration and cmon(t = ∞) indicates the silica concentration at equilibrium. The solid line describes the best fit according to eq 27.

the nucleation rate constant with supersaturation is fitted according to eq 27 and provides parameters Ka = (5.7 × 10−6) ± (3 × 10−7) s−1 and Kb = (4.7 ± 1) × 10−1. Nucleation rate J = kn((c0NA)/M0), as shown in Table 2, adopts values with orders of magnitude between 1 × 109 and 1 × 1012 and with that is much closer to predictions of classical nucleation theory than the values published by Vekilov et al.35,36

■

CONCLUSIONS The silica polymerization from supersaturated solutions of sodium metasilicate was investigated with time-resolved static and dynamic light scattering by varying the initial silica content. The reaction was initiated by adjusting the pH value to 7 and 8. At pH 7, lower initial silica contents (350−750 ppm) suggest a growth mechanism via monomer addition. Monomer consumption gets accelerated with increasing silica content, predominantly because of an increased rate of nucleation. At high initial silica contents of 2000 and 3000 ppm SiO2, only a short period of particle formation based on monomer addition precedes particle−particle agglomeration. As is demonstrated I

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(7) Rothbaum, H.; Rohde, A. Kinetics of Silica Polymerization and Deposition from Dilute Solutions between 5 and 180°C. J. Colloid Interface Sci. 1979, 71, 533−559. (8) Makrides, A. C.; Turner, M. J.; Slaughter, J. Condensation of Silica from Supersaturated Silicic Acid Solutions. J. Colloid Interface Sci. 1980, 73, 345−367. (9) Gorrepati, E. A.; Wongthahan, P.; Raha, S.; Fogler, H. S. Silica Precipitation in Acidic Solutions: Mechanism, pH- Effect, and Salt Effect. Langmuir 2010, 26, 10467−10474. (10) Nordström, J.; Nilsson, E.; Jarvol, P.; Nayeri, M.; Palmqvist, A.; Bergenholtz, J.; Matic, A. Concentration- and pH-Dependence of Highly Alkaline Sodium Silicate Solutions. J. Colloid Interface Sci. 2011, 356, 37−45. (11) Böschel, D.; Janich, M.; Roggendorf, H. Size Distribution of Colloidal Silica in Sodium Silicate Solutions Investigated by Dynamic Light Scattering and Viscosity Measurements. J. Colloid Interface Sci. 2003, 267, 360−368. (12) Icopini, G. A.; Brantley, S. L.; Heaney, P. J. Kinetics of Silica Oligomerization and Nanocolloid Formation as a Function of pH and Ionic Strength at 25°C. Geochim. Cosmochim. Acta 2005, 69, 293−303. (13) Tobler, D. J.; Shaw, S.; Benning, L. G. Quantification of Initial Steps of Nucleation and Growth of Silica Nanoparticles: An in-situ SAXS and DLS Study. Geochim. Cosmochim. Acta 2009, 73, 5377− 5393. (14) Egelhaaf, S. U.; Schurtenberger, P. Micelle-to-Vesicle Transition: A Time-Resolved Structural Study. Phys. Rev. Lett. 1999, 82, 2804−2807. (15) Meng, Z.; Hashmi, S. M.; Elimelech, M. Aggregation Rate and Fractal Dimension of Fullerene Nanoparticles via Simultaneous Multiangle Static and Dynamic Light Scattering Measurement. J. Colloid Interface Sci. 2013, 392, 27−33. (16) Michels, R.; Sinemus, T.; Hoffmann, J.; Brutschy, B.; Huber, K. Co-Aggregation of Two Anionic Azo Dyestuffs at a Well-Defined Stoichiometry. J. Phys. Chem. B 2013, 117, 8611−8619. (17) Abécassis, B.; Testard, F.; Spalla, O.; Barboux, P. Probing In Situ the Nucleation and Growth of Gold Nanoparticles by Small-Angle Xray Scattering. Nano Lett. 2007, 7, 1723−1727. (18) Liu, J.; Rieger, J.; Huber, K. Analysis of Nucleation and Growth of Amorphous CaCO3 by Means of Time-Resolved Static Light Scattering. Langmuir 2008, 24, 8262−8271. (19) Liu, J.; Pancera, S.; Boyko, V.; Shukla, A.; Narayanan, T.; Huber, K. Evaluation of the Particle Growth of Amorphous Calcium Carbonate in Water by Means of the Porod Invariant from SAXS. Langmuir 2010, 26, 17405−17412. (20) Standard Methods for the Examination of Water and Wastewater; American Public Health Association: Washington, DC, 1998. (21) Zimm, B. H. Apparatus and Methods for Measurement and Interpretation of the Angular Variation of Light Scattering; Preliminary Results on Polystyrene Solutions. J. Chem. Phys. 1948, 16, 1099−1116. (22) Guinier, A.; Fournet, G. Small-Angle Scattering of X-rays; Wiley: New York, 1955. (23) Koppel, D. E. Analysis of Macromolecular Polydispersity in Intensity Correlation Spectroscopy: The Method of Cumulants. J. Chem. Phys. 1972, 57, 4814−4820. (24) Burchard, W.; Schmidt, M.; Stockmayer, W. H. Information on Polydispersity and Branching from Combined Quasi-Elastic and Integrated Scattering. Macromolecules 1980, 13, 1265−1272. (25) Drews, T. O.; Katsoulakis, M. A.; Tsapatsis, M. A Mathematical Model for Crystal Growth by Aggregation of Precursor Metastable Nanoparticles. J. Phys. Chem. B 2005, 109, 23879−23887. (26) Lomakin, A.; Teplow, D. B.; Kirschner, D. A.; Benedek, G. B. Kinetic Theory of Fibrillogenesis of Amyloid β-Protein. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 7942−7947. (27) Debye, P. Molecular-weight Determination by Light Scattering. J. Phys. Chem. 1947, 51, 18−32. (28) Rayleigh, L. On the Diffraction of Light by Spheres of Small Relative Index. Proc. R. Soc. London, Ser. A 1914, 90, 219−225. (29) Neugebauer, T. Berechnung der Lichtzerstreuung von Fadenkettenlösungen. Ann. Phys. 1943, 434, 509−533.

Figure 11. Schematic overview of the mechanistic features of silica polymerization at variable silica contents at pH 7 and 8 established in the present work. Blue arrows denote the growth of primary particles by monomer addition, and green arrows indicate particle−particle agglomeration. As reflected by the final particle size reached via growth by monomer addition, nucleation increases with the silica content at both pH values but is smaller in general at pH 7. Particle−particle aggregation starting from primary particles can be observed only at pH 7 and only once the number density of primary particles gets large enough.

■

ASSOCIATED CONTENT

S Supporting Information *

Detailed comparison of the light scattering results at pH 7 and 8. Fitting of kinetic data with various nucleation and growth models. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: (+49) 5251602125. Fax: (+49) 5251 604208. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was funded by the BASF SE within the project “Polymer Assisted Silica Polymerization”. REFERENCES

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K

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