Silica Polymerization from Supersaturated Dilute Aqueous Solutions in

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Silica Polymerization from Supersaturated Dilute Aqueous Solutions in the Presence of Alkaline Earth Salts Marina Kley, Andreas Kempter, Volodymyr Boyko, and Klaus Huber Langmuir, Just Accepted Manuscript • Publication Date (Web): 23 May 2017 Downloaded from http://pubs.acs.org on May 25, 2017

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Silica Polymerization from Supersaturated Dilute Aqueous Solutions in the Presence of Alkaline Earth Salts M. Kley†, A. Kempter§, V. Boyko‡, K. Huber*,† †

Physical Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

§

BASF SE, Strategic Marketing and Product Development Automotive Fluids, Fuel and

Lubricant Solutions, 67056 Ludwigshafen, Germany ‡

BASF SE, Material Physics and Analytics, Colloidal Systems, 67056 Ludwigshafen, Germany

ABSTRACT The early stages of silica polymerization in aqueous solution proceeds according to a mechanism based on three steps: nucleation, particle growth and agglomeration of the particles. Application of time-resolved static and dynamic light scattering as a powerful in-situ technique in combination with spectrophotometric analysis of the monomer consumption based on the molybdenum blue method was carried out to further investigate this 3-step process. Experiments were carried out at four different initial silicic acid contents covering a range between 350 ppm and 750 ppm in the presence of either 10 mM NaCl or 5 mM of a mixture of CaCl2 and MgCl2. The process in all cases was initiated with a drop of pH to 7. Addition of the salts made possible

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an analysis of the impact of an electrolyte on the process. Independent of the presence or absence of salt, particle growth in step two proceeded as a monomer-addition process without being interfered significantly by Ostwald-ripening. The growing particles were compact with a homogeneous density. The size of the particles approached final values between 5 nm and 20 nm with the actual value increasing with decreasing initial silicic acid content. Above a certain concentration of initial silica content, which depends on the level of added salt, particle-particle interactions caused agglomeration. The presence of electrolyte shifted this level from ~ 2000 ppm to a range between 500 ppm and 750 ppm. The resulting agglomerates had a fractal dimension of 2. Independent of the conditions, particle growth could be described with a simple nucleation and growth model.

INTRODUCTION Silica polymerization in aqueous media is an important feature in many natural and geological processes including biomineralization, sinter formation and silica diagenesis. It becomes also increasingly important in material science. Inspired by biological systems additives are to be developed in order to direct silica formation into highly patterned and hierarchical structures.1,2 Last but not least, silica polymerization denoted as scaling or fouling is a challenging problem in industry, since it causes decreased water recoveries in desalination plants or a lower economic efficiency in geothermal power plants.3,4 A detailed knowledge of the mechanism and kinetics of the silica polymerization at variable parameters like the pH, salinity or temperature of the aqueous medium may indicate routes to particles with specific properties or to an inhibition of silica precipitation. Accordingly, large efforts have been undertaken in the past to meet this demand. An excellent overview on the progress accomplished by the late seventies was provided


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by Iler5 in 1979. Based on a few pioneering publications6–11 Iler5 established a mechanism for the polymerization of silica in supersaturated aqueous solution, which proceeds via three distinct steps. The first step is an induction period where nucleation takes place. In the second step particles grow. Monomers, oligomers and polymers are the species dominating throughout these two steps. The two steps are characterized by a sigmoidal increase of monomer consumption recorded by the molybdate-reactive silica. Using a particle size dependent solubility of silica, Iler5 reconciled the data9,11 with the initial formation of small particles in the range of one nm probably corresponding to nuclei, which successively grow in size up to a radius of a few nanometers. Without providing direct evidence, data on the growth step could be best matched with a monomer addition process. Monomer consumption is fastest in the range of 7 < pH < 9 with an optimum at pH = 8.3.7 At equilibrium monomeric silica approached concentrations in the regime of 120-180 ppm.7 Most likely, part of the oligomers like polycyclic octamers and decamers, which have also been observed in processes based on hydrolysis of silicic acid esters by ESI-MS12–14 turn into nuclei and the nuclei grow by adding further monomers and small oligomers, with those small oligomers acting as “monomers”. An unsettled question in this context is whether Ostwald-ripening sets in at the later stages of particle growth. Once the initial concentration of monomeric silica exceeds a certain level, the particles start to agglomerate8, thus establishing the third step in Iler’s mechanistic scheme. At high enough silica concentrations this agglomeration appears as a macroscopic gelation.6 As demonstrated by Merrill and Spencer10, the exact level of the silica content where this agglomeration sets in depends on the salinity and pH of the solution. Despite the considerable number of publications meanwhile available, knowledge on the mechanism of silica formation in aqueous media still suffers from subtle gaps or is awaiting clear


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cut confirmation. Since the present work tries to solve unsettled issues of silica formation in dilute aqueous solution close to neutral pH-values, some of the most important achievements in this field shall be briefly summarized. Aside from supporting the earlier findings mentioned above, two further studies focused on the impact of added salts15,16, and revealed an acceleration of particle growth with increasing salt concentration. Icopini et al.15 also identified a reaction rate for the monomer consumption of fourth order in terms of the monomer concentration and took this as a hint for tetramers to act as nuclei. Further insight into the origin of the induction period was provided by Noguera et al.17, who published a computational approach based on the classical nucleation theory. They successfully applied their model to experimental data from Rothbaum and Rhode18 and Tobler et al.19,20. Silica polymerization was initiated via pre-established initial supersaturation, via neutralization of a high pH solution of silica and via fast cooling. It turned out that the properties of the particle population strongly depend on the experimental conditions. Further on they postulated that the induction period has to be most likely attributed to a stage in which nucleation, growth and dissolution compete with and compensate each other. In two other recent contributions use could be made of modern scattering techniques in order to answer some of the unsettled questions. Tobler et al.19 performed time-resolved small angle xray scattering (SAXS) and dynamic light scattering at one angle together with the molybdate yellow based analysis of the depletion of monomeric silica at two different silica concentrations. From an extrapolation of the kinetic model fits to time zero, Tobler et al.19 inferred that homogeneous nucleation leads to particles of 1-2 nm. Successively, particles grow to a size not larger than 4 nm. It is the kinetic modelling of this growth (the second step) which establishes a key feature of their work19 revealing a particle growth by surface controlled monomer


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incorporation. However, no clear picture could be provided of the morphology of the particles growing in the second step. Do those particles have a spherical shape or do they have a mass fractal nature? The third stage was postulated to include Ostwald-ripening and growth via particle-particle aggregation discernible by a steep increase of particle size values to radii close to 30 nm. Kley et al.21 applied a combination of time-resolved static (SLS) and dynamic light scattering (DLS) with the molybdate blue based monitoring of monomer consumption to analyze the first stages of the particle formation mechanism covering a concentration regime of 300 – 3000 ppm of monomeric silica. Combination of these three methods provided excellent direct insitu data by yielding a time-resolved set of particle mass values in combination with the evolution of the hydrodynamic radius of particles and once particles became larger than 20 nm also the mean squared radius of gyration. In the regime of initial silica content of 350 ppm – 750 ppm, loss of monomers occurred simultaneously to the increase of size and mass of the growing particles. Together with the power laws from a correlation of particle size with particle mass, the results unambiguously proved a monomer-addition mechanism for the second step leading to homogeneous particles most likely with a spherical shape. No Ostwald-ripening could be observed in this second stage. A third step became only transparent at concentrations above 2000 ppm, with its threshold value depending on pH. In this third step, the compact particles agglomerate according to a step-growth process where any particle may stick to any other colliding particle resulting in entities with a fractal dimension close to 2. Encouraged by those results, the present work extends this preceding study21 to the impact of mixed (Mg1/5Ca4/5)Cl2 salt on the kinetics and mechanism of silica particle formation in the regime of dilute silica solutions. This salt has been selected in order to represent a typical state for hard water. In-situ SLS and DLS in combination with the molybdenum blue method is


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applied in order to get direct access to particle formation and monomer consumption and to answer the following questions. Does a change in mechanism occur upon addition of (Mg1/5Ca4/5)Cl2? If a change does occur, is it ion specific or can it also be generated by the addition of NaCl. Does the electrolyte change the tendency to form step-growth polymers in a third step and does any evidence exist for Ostwald-ripening. Experiments are performed at four different initial silica concentrations covering a regime of 350 – 750 ppm at a pH of 7 and at [(Mg1/5Ca4/5)Cl2]= 5 mM. The evolution of particle mass shall be interpreted in terms of a simple nucleation and growth model21 with its strength lying on a semi-quantitative interpretation of the evolution of particle mass in the second step.

EXPERIMENTAL DETAILS Materials. The preparation of solutions for light scattering experiments required three stock solutions: a 0.2 M sodium silicate solution (Na2H2SiO4·8 H2O, assay ≥ 98%, Sigma Aldrich); a solution with 0.2 M alkaline earth salt being composed of calcium chloride hexahydrate (CaCl2 · 6 H2O, assay ≥ 98%, Sigma Aldrich) and magnesium chloride hexahydrate (MgCl2 · 6 H2O, assay ≥ 99%, Sigma Aldrich) with a ratio of Ca : Mg = 4 : 1; a solution of sodium chloride (NaCl, assay ≥ 99%, Sigma Aldrich) with a concentration of 0.4 M. All solutions were prepared with water, which was purified by a Millipore system, leading to a conductivity of 0.055 Sm-1.

Molybdenum Blue Method. Observation of the decay of monomeric silica with time was carried out by the molybdenum blue method22. The reaction of monomeric silica as Si(OH)4 with acidified ammonium heptamolybdate to the blue silicomolybdic acid was used to determine the concentration of monomeric silica via UV-vis-spectroscopy. For this purpose a small amount of


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the silica solution which was prepared for a light scattering experiment was removed and diluted with water to a volume of 50 mL. Addition of 1 mL of 50% (vol%) hydrochloric acid and 2 mL of ammonium heptamolybdate solution with a concentration of 100 g/L while agitating led to the formation of the yellow silicomolybdic acid after the solution was allowed to stand for seven minutes. Addition of 2 mL oxalic acid with a concentration of 7.5 g/L and a second storage time of two minutes eliminated a possible influence of phosphate on the reaction. In order to obtain the blue silicomolybdic acid 2 mL of a reducing agent was added. The reducing agent was a mixture of a solution of 23.08 g Na2S2O5 in 150 mL water with a solution of 500 mg 1-amino-2naphthol-4-sulfonic acid and 1 g Na2SO3 in 50 mL water. The absorbance of the blue sample was measured at the latest 15 minutes after the addition of the reducing agent with a Lambda 19 spectrophotometer (Perkin Elmer) at a wavelength of 815 nm.

Preparation of Light Scattering Samples. The preparation of the light scattering samples was performed in three steps. In a first step a certain amount of sodium silicate stock solution depending on the desired concentration was diluted to a volume of 195 mL. Successively, the pH value was adjusted to pH 7 by addition of 2 M HCl. The pH adjustment set the starting point of the experiment. Right after the pH adjustment 5 mL of the respective stock solution of Ca2++Mg2+ or Na+ were added. The sample was transferred into a dust free scattering cell (20 mm in diameter) by filtration through a syringe filter (mixed cellulose ester) with a pore size of 0.2 m. The scattering cells had been treated with a solution of 5% chlorotrimethylsilane (98%, Janssen Chimica) in toluene, in order to avoid silica formation by the silicate solution at the inner cell walls. All experiments were performed at 37 °C, the recording time of the scattering intensity depended on the respective sample. Concentrations of silica denoted as c in


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evaluation of data from light scattering experiments are given in units of g/l, where the concentration of solid Na2H2SiO4·8 H2O is translated into a concentration of solid SiO2 with a factor of 0.215. Alternatively, concentrations of silica are expressed as ppm of solid SiO2. All solutions used for light scattering and for the molybdate blue method were 0.01 N with respect to the added (Mg1/5Ca4/5)Cl2 or NaCl in addition to the NaCl inevitably formed during the initiation of the process by neutralization. The contribution of NaCl from the neturalization amounted to 0.012 N at 350 ppm, 0.013 N at 400 ppm, 0.016 N at 500 ppm and 0.025 N at 750 ppm.

Scattering Setup. The ALV/CGS-3/MD-8 Multidetection Laser Light Scattering System from ALV-Laservertriebsgesellschaft (Langen) makes possible a time-resolved recording of angular dependent static light scattering (SLS) and dynamic light scattering (DLS) with a time resolution limited by the time required for the evolution of a correlation function in DLS. It is equipped with eight moveable detectors, which are positioned in angular increments of 8°. This set-up covers an angle of 56°, which can be moved in an angular range of 15° ≤  ≤ 136° corresponding to a q-range in water of 3.5·10-3 – 2.7·10-2 nm-1 with

4nsolv

  sin  , (1) 0 2 the momentum transfer, nsolv = 1.332 (at T = 37°) the refractive index of water,  the scattering q

angle and 0 the laser wavelength in vacuum. A He-Ne Laser with a wavelength of 632.8 nm and a power of 35 mW is used as a light source.

Static Light Scattering. The static light scattering (SLS) signal provides the Rayleigh ratios at a scattering angle , according to eq 2


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R  R , std 

r , sol  r , solv r , std

,

(2)

with R,std the tabulated Rayleigh ratio of the standard toluene, r,std the measured scattering intensity of toluene, r,sol the scattering intensity of the solution and r,solv the scattering intensity of the solvent. For the determination of the weight averaged molar mass Mw and the z-averaged radius of gyration Rg2 the Rayleigh ratios were processed according to Zimm’s approximation23 2

Rg Kc 1   q 2  2 A2 c ,  R M w 3M w

(3)

with A2 the second virial coefficient and c the mass concentration in g/L of the solid in the solution. As we were observing a reacting system an extrapolation of the scattering data to c = 0 was not possible. This forced us to neglect the influence of interactions among scattering particles on the scattering signal by setting A2 = 0. Neglect was justified as the investigated concentrations were very low (c ≤ 750 ppm). The contrast factor K introduced in eq 3 is given as 2

4 2 n 2  n  K  4 tol   ,  0 N A  c 

(4)

with NA the Avogadro constant, ntol the refractive index of the bath liquid24 and ∂n/∂c the refractive index increment of the solute in solution. The refractive index increment has been determined for an aqueous solution of Na2H2SiO4 ·8 H2O to be ∂n/∂c = 0.223 mL/g.21 This value is based on concentrations of Na2SiO3 as solid. If the monomer unit is defined as SiO2 and the value for ∂n/∂c is calculated based on concentrations of SiO2, we get ∂n/∂c = 0.454 mL/g. This value has been used in the present work for calculations of the weight-averaged molar mass. In order to directly compare the resulting data with the weight-averaged molar mass values from the previous publication21

these preceding values have to be multiplied with a factor of


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0.241 = (0.223/0.454)2. This was done accordingly with the data21 shown in Figure 6, 7, 8A, S1 and S4a. In order to directly compare the corresponding rate constants obtained from fits with the kinetic model denoted as nucleation and growth model (NG-model) in Ref. 21 with the results from model fits in the present study, the fits with the NG-model had to be re-done with the corrected weight-averaged molar mass data of Ref. 21. The new rate constants established with the NG-model reveal the same trends as discussed in Ref. 21 whereby kn is shifted by a factor of 1/0.241 or close to it, ke is shifted by a factor of 0.241 or close to it and kp remained unaffected. An overview of all rate constants and the residuals of both kinetic models obtained from fits to the light scattering data in the absence of added salt is given in Table S1 of the Supporting Information.

Dynamic Light Scattering. Measurements of dynamic light scattering (DLS) gives the fieldtime correlation function g1(). For polydisperse systems the field-time correlation function can be described by a series of exponential functions, according to eq 5 n

g1 ( )    i  ei ,

(5)

i 1

with i the time constant of species i and i the weighting factor determined by the scattering intensity of species i. According to the cumulant analysis25 the logarithm of the field-time correlation function lng1 ( ) can be described by ln[ g 1 ( )]  K 0  K 1 

1 K 2 2  ... 2!

(6)

where K0 is a constant, describing the signal to noise ratio and K1 is the z-average of the decay time constant 

z

and gives the z-averaged translational diffusion coefficient Dz according to


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K1  

z

 Dz q 2 .



The variance of  is given by the coefficient K2    



(7)

2

z

and provides information about z

the polydispersity of the sample. A linear extrapolation of Dz(q) toward q2 → 0 and to c = 0 according to

Dz  D0 (1  q 2 Rg2C  k d c)

(8) gives the z-averaged diffusion coefficient D0. In eq 8 C is a dimensionless parameter depending on the shape of the particles and kD accounts for the concentration dependence of Dz. As we observe a reacting system, an extrapolation of D for c = 0 is not possible and in analogy to the neglect of the concentration dependency in eq 3, a possible concentration dependency of Dz is neglected. The z-averaged diffusion constant D0 can be converted into the hydrodynamic radius Rh via the Stokes-Einstein equation

Rh 

kT 1  6 D0

(9)

where k is the Boltzmann constant, T the temperature and the viscosity of the solvent. The ratio  of the radius of gyration and the hydrodynamic radius is a structure-sensitive parameter and is defined as



Rg

. (10) Rh For monodisperse linear polymer chains under -conditions denoted as unperturbed chains a value of 1.504 is predicted by theory26,27, which turned out to be 17.5 % larger than values revealed by experiment.28 The fractal dimensions of such unperturbed linear chains is 2. In case of compact spheres  decreases drastically to 0.77.26 For rod-like structures a value of  ≥ 2.0 is expected.29,30 Therefore a distinction between spheres, rods and fractal structures similar to unperturbed polymer coils by means of the -parameter is possible.


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RESULTS AND DISCUSSION Effect of alkaline earth cations. The silica polymerization was investigated in the presence of a

mixture

of

divalent

calcium

and

magnesium

ions.

A

salt

concentration

of

[M2+] = [Ca2+]+[Mg2+] = 5 mmol/L at a ratio of Ca : Mg = 4 : 1 was selected in order to generate conditions typical for hard water. Experiments of four samples with silica contents of 350 ppm, 400 ppm, 500 ppm and 750 ppm were performed. Investigation of the influence of sodium ions was performed in order to enable comparison of the impact of the most simple and inert type of salt with that of divalent alkaline earth cations. The Na+ concentration amounts to 10 mmol/L introduced by the addition of NaCl. In case of divalent alkaline earth cations, Particle formation was followed by means of time-resolved SLS and DLS experiments. The consumption of monomeric silica is monitored by means of the molybdenum blue method. Figure 1A represents experiments at pH = 7 in the presence of Ca2++Mg2+-ions at variable silica contents. The growth rate as well as the finally reached molar mass strongly depends on the initial silica content. The evolution of the molar mass determined by SLS of all four samples correlates nicely with the consumption of the silica monomers determined by means of the molybdenum blue method (Figure 1B). For all samples an equilibrium concentration of 170 ppm SiO2 was reached at the end of the growth of silica particles. The three lowest concentrations (350 ppm – 500 ppm) exhibit a distinct lag-time which decreases with increasing initial silica content. During the lag-time the scattering contribution is comparable to the scattering signal of the solvent and the monomer concentration keeps close to the initial silica concentration. After this characteristic lag-time the decrease of the monomer concentration coincides with the increase of the molar mass of the silica particles and both values reach an equilibrium value.


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Only the highest silica content (750 ppm) shows a different behavior. In this case no lag-time is observed. The consumption of the monomers up to the equilibrium limit of 170 ppm is completed within t < 10 h, while the molar mass keeps increasing beyond this time without reaching a plateau value. It is highly probable that particle formation at the silica content of 750 ppm during the first hours proceeds via monomer-addition in very much the same way as in case of the lower silica contents. However, this process is completed during the first 10 hours and smoothly followed by another growth step whereby the particles resulting from monomeraddition establish the smallest units in this next growth step. As analysis of the joint SLS and

Mw / 1E4 g mol-1

DLS data will show, fractal-like aggregates are formed in this third step. 25 20

4

A

3 15 2 10 5

1

0

0

800

SiO2 / ppm

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B

700 600 500 400 300 200 100 0

10

20

30

40

50

60

70

t/h Figure 1: Formation of silica particles in aqueous solution as a function of time at pH 7 in the presence of [Ca2+]+[Mg2+] = 5 mM (Ca2+/Mg2+ = 4/1). The silica contents are 750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle) and 350 ppm (green square). (A) Weight-averaged molar mass from SLS. The experimental mass values refer to two different scales with arrows pointing to the corresponding axis; (B) consumption of the monomeric silica from the molybdenum blue method.


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Similar trends for the evolution of the particle mass and the consumption of monomers is observed for the silica polymerization in the presence of Na+-ions, which is shown by Figure 2. A good correlation between the depletion of monomers and the increase of the particle mass is observed for all four silica contents. In analogy to the measurements in the presence of Ca2++Mg2+-ions the silica contents at 350 and 400 ppm exhibit a significant lag-time during which no monomer is consumed and scattering intensity is comparable with the scattering intensity of the solvent. After this lag-time which decreases with increasing silica content a fast increase of the particle mass and simultaneously a decrease of the monomers is observed until the monomers reach the equilibrium concentration of 170 ppm SiO2. 2.0

6

Mw / 1E4 g mol-1

A

5

1.6

4 1.2

3 0.8

2 1

0.4

0

0.0

B

700 600

SiO2 / ppm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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500 400 300 200 100 0

20

40

60

80

150

300

450

t/h Figure 2: Formation of silica particles in aqueous solution as a function of time at pH 7 in the presence of additional Na+-ions at [Na+] = 10 mM. The silica contents are 750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle) and 350 ppm (green square). (A) Weight-averaged molar mass from SLS. The experimental mass values refer to two different scales with arrows pointing to the corresponding axis; (B) consumption of the monomeric silica from the molybdenum blue method.


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Figure 3 compares the temporal evolution of the hydrodynamic radius of the samples in the presence of Ca2++Mg2+-ions (Figure 3A) with the respective results collected for the samples in the presence of Na+-ions (Figure 3B). As expected from the evolution of the particle mass in the presence of Ca2++Mg2+, a distinct lag-time is followed by a short period of an increasing particle size, which approaches a constant value for low silica contents (350 ppm-500 ppm). At 750 ppm Rh increases almost linearly with t without approaching a plateau. Similar trends can be observed in the presence of Na+-ions, only the absolute lag-times at the silica contents between 350 ppm and 500 ppm are slightly longer than in the presence of Ca2++Mg2+. Final sizes of the particles formed in the presence of Ca2++Mg2+-ions and in the presence of Na+-ions are comparable. The evolution of Rh at 750 ppm SiO2 in the presence of Na+ shows the same linear increase with t as in the presence of Ca2++Mg 2+. In close analogy to the growth in water21 without added salt the final particle size decreases with increasing initial silica content in the presence of metal cations (Figure 3). This indicates that the nucleation rate increases considerably with increasing silica content. A faster nucleation generates a larger number of nuclei, which have to share the respective amount of monomers available. Hence, a lower number of nuclei, resulting from a slower nucleation, leads to larger particles. As inferred from the lag-times (Figure 4), monovalent and divalent cations accelerate the

silica

polymerization.

The

lag-time



obeys

the

following

trend:

 (salt

free) >  (Na+) >  (Ca2++Mg2+) whereby the differences between  (Na+) and  (Ca2++Mg2+) remain close to the uncertainty of establishing .. The trends discussed are nicely supported by a comparative discussion of the evolution of Mw with time done separately at each silica content (Figure S1 in the Supporting Information).


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35

A

30 25 20 15 10

Rh / nm

5 0 12

B

10 8 6 4 2 0 0

5

10

15

20

25

30

35

40

45

50

t/h Figure 3: Hydrodynamic radius from DLS of silica particles as a function of time at pH 7 at variable silica contents: (A) in the presence of [Ca2+] + [Mg2+] = 5 mM (full symbols) at 750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle), 350 ppm (green square); (B) in the presence of additional Na+ at [Na+] = 10 mM (hollow symbols) at 750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle), 350 ppm (green square).

60 50 40

/h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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30 20 10 0 300

400

500

600

700

800

SiO2 / ppm Figure 4: Lag-time versus initial silica content for the measurement series without added salt (), in the presence of additional Na+-ions () and in the presence of Ca2++Mg2+-ions (). The data recorded in the absence of added salt are taken from Ref. 21. The lag-time has been defined as the intersection of two linear approximations: One corresponding to the constant mass during the lag-time and a second one corresponding to the tangent adjusted to steepest increase of Mw versus t.


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Langmuir

As the observed particle size is very small for all measured samples no information about the particle shape can be extracted directly from the scattering curves in terms of a formfactor. However, the correlation of the hydrodynamic radius or the radius of gyration with the weight averaged molar mass gives indirect information about the particle structure. The correlation follows a power law 

Rg  Rh  M w

(11) with a shape sensitive exponent  for self-similar structures. Compact, homogeneous spheres or cubes lead to an exponent of  = 1/3 and polymer coils under ideal conditions give a value of  = 1/2.31 Such an exponent would in fact be observed if any particle reacts with any other particle in solution corresponding to a step-growth process and resulting in a monomodal size distribution. Once particles are formed via a monomer-addition process like in a chain growth, the weight-averaged molar particle mass, given by Mw = R= 0/Kc, describes an average of a bimodal ensemble of monomers and growing particles if the entire silica concentration c0 corresponding to the concentration at t = 0 is used for the calculation. Knowledge of the monomer concentration of silica cmon(t) would enable us to also calculate the concentration of growing particles cpart(t) according to eq 12 and along with it to calculate an additional weightaveraged mass for the particles only with cpart(t) used in eq 3.

cpart(t)  c0  cmon(t)

(12) However, if the mass values of an ensemble of growing particles and small monomers for a monomer-addition process are based on c0 the topology based exponent is decreased by a factor of 1/2.32,33 Accordingly, homogeneous spherical or cubic structures formed in a monomeraddition process follow an exponent of =1/6.


Langmuir

A 10

Rh / nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 12 10

B

8 6

4

2 0.01

0.1

1

10

Mw / 1E4 g mol

100

-1

Figure 5: Correlation of the hydrodynamic radius Rh with the weight averaged molar mass Mw at variable silica contents (A) in the presence of Ca2++Mg2+-ions (full symbols): 750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle), 350 ppm (green square); (B) in presence of additional Na+-ions (hollow symbols): 750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle), 350 ppm (green square). The solid black line indicates a slope of 1/2 and the dashed black line represents a slope of 1/6.

The correlation of Rh versus Mw for the measurement series in the presence of Ca2++Mg2-ions are shown in Figure 5A and in the presence of Na+-ions in Figure 5B. The mass values shown in Figure 5 have been calculated based on the initial silica concentration c0. The trends of the correlations for the two types of salts are comparable. In both types of salts two different exponents are observed. Silica contents of 350-500 ppm result in a slope of 1/6 and experiments with 750 ppm SiO2 show a slope according to an exponent =1/2. As the smallest possible exponent based on the topology is 1/3 corresponding to homogeneous spherical structures, the lower exponent of 1/6 observed below 750 ppm is only explainable by the formation of compact particles with homogeneous density (most likely of spherical shape) via a monomer-addition process. The particle growth process at 750 ppm in presence of additional metal cations exhibits


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Langmuir

no lag-time. Consumption of the monomers is fast and the weight-averaged molar mass is increasing beyond the time when the equilibrium concentration of the monomers is reached. No plateau value is reached for the particle size. The correlation of Rh with Mw gives an exponent of 1/2. The same exponent holds true for the correlation of Rg with Mw, which is only accessible for the experiment in the presence of Ca2++Mg2+ as in this case the particles have reached a sufficiently large size to detect a radius of gyration. The corresponding graph can be found in the Supporting Information (Figure S2a). For the exponent of 1/2 observed at 750 ppm two explanations are possible. (i) A particle formation via monomer-addition is assumed, where we would have to multiply the observed exponent by a factor of two to get the topology based exponent.32,33 The resulting exponent of 1 would suggest rod-like structures. However, given the state of knowledge on the formation of colloidal silica under the present conditions2,5, a rod-like structure seems to be very unlikely. Further arguments against a rod-like structure is provided by the shape-sensitive factor  = Rg/Rh (eq 10) and by the dimensionless parameter C (eq 8), which are both represented for the experiment at 750 ppm in the presence of Ca2++Mg2+ in the Supporting Information (Figure S2b and Figure S2c). For rod-like structures a value of  ≥ 2 would be expected.29,30 Values in Figure S2b are close to  ≈1.3 which is in agreement with the value observed for unperturbed polymer chains28 but significantly smaller than  ≥ 2 expected for rod-like coils. The dimensionless parameter C shown in Figure S2d approaches 0.18, which is close to 0.173 predicted for unperturbed Gaussian polymer chains26 and significantly larger than 0.033 < C < 0.044 anticipated for rod-like structures34. (ii) A growth based on a particle-particle agglomeration also denoted as step-growth is assumed, where the observed exponent can be related directly to the topology of the particles.32,33 Hence, an exponent of 1/2 indicates fractal-


Langmuir

like31,35 structures, with a fractal dimension of 2. In fact a comparison with Gaussian polymer coils, which also have a fractal dimension of 2 and  ~ 1.3 and C ~ 0.175 teaches us that explanation (ii) is appropriate. Most likely such structures are formed with the homogeneous particles generated via monomer-addition, which succeedingly react via step-growth polymerization. These results together with the perfect correlation of the evolution of the particle mass with the consumption of monomers, enable us to discriminate two concentration regimes with two mechanisms of particle growth. The particle growth at 350 ppm-500 ppm SiO2 is characterized by a distinct lag-time during which monomers are not consumed followed by particle growth via monomer addition. Above a concentration of 500 ppm the particles aggregate to fractal-like structures according to a step-growth process. These results are consistent with the findings for silica polymerization in water without added salt, which have been already presented in a previous publication21.

100

Rh / nm

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10

0.1

1

10

Mw / E4 g mol

100

1000

-1

Figure 6: Correlation of the hydrodynamic radius Rh with the weight-averaged molar mass Mw=R/Kcpart(t) at the silica contents 2000 ppm in pure water (empty, blue circle), 3000 ppm in pure water (full, pink circle), 750 ppm in the presence of Ca2++Mg2+ (full, black diamond) and 750 ppm in the presence of additional Na+ (empty, black diamond). Experimental data corresponding to silica polymerization in pure water is taken from Ref. 21. The dashed line represents a slope of 1/2.


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Langmuir

At this point, we would like to stress a comparative representation of the evolution of the hydrodynamic radius with molecular weight for the growth processes which obey a step-growth process under all three conditions. To this end values of the molecular weight of the fractals have been calculated according to Mw = R/K·cpart(t) with cpart(t) = c0 - cmon(t) and cmon(t) ≈ 170 ppm the (constant) equilibrium concentration of the non-consumed monomers. Justification for this calculation of Mw is that the scattering contribution of the non-consumed monomers is negligible which implies use of the (constant) concentration of growing fractals c0 – 170 ppm. As is demonstrated in Figure 6, the corresponding correlations do not only follow a power law with an exponent of 0.5 but also overlay perfectly. Such an overlay can be nicely reconciled with the fact that independent on the actual conditions, i.e. independent on whether additional salt is absent or present as Na+ or as Ca2++Mg2+, always the same fractals are formed and the same constituent building units are incorporated with the building units corresponding to the homogeneous particles formed respectively via the monomer-addition process. A variation of the size of the constituent particles when changing the conditions does not affect the resulting exponent of the power law in eq 11, but it would generate a parallel shift in the log-log plot as a result of a variation of the prefactor. The coincidence of all curves is particularly interesting as this implies, that the step-growth like agglomeration sets in always with constituent particles having a size in the order of 1-3 nm. Further support for the suggested mechanisms and morphologies is provided by the correlation of the finally measured hydrodynamic radii with the corresponding mass values, as it is done in Figure 7. Here, the mass values of the particles have been calculated according to eq 3 with the mass concentration of the growing particles cpart(t) from eq 12 instead of using c0. In cases where the growth process was still going on while the final measurement was done cmon(t) was


Langmuir

interpolated from the respective trend determined by the molybdenum blue method. In cases where the growth process was completed cmon(t) corresponds to the equilibrium concentration of 170 ppm SiO2. Figure 7 is a comparison of data from three measurement series: silica polymerization (i) in water without added salt from Ref. 21, (ii) in the presence of additional Ca2++Mg2+ and (iii) in the presence of additional Na+. At all conditions the final values measured for silica contents below 750 ppm follow a unique trend with a slope of 1/3. Noteworthy, these are the conditions where no aggregation of particles takes place. This result confirms the existence of homogeneous particles with a uniform characteristic density like for instance cubes or spheres.

final Rh / nm

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10

1 0.1

1

10

100

final Mw(cpart) /1E4 g mol-1

Figure 7: Correlation of Rh(t = tmax) with the final values of Mw=R/Kcpart(t = tmax) at variable silica contents: 750 ppm (black), 500 ppm (red), 400 ppm (blue), 350 ppm (green), empty symbols indicate the experiments in water without added salt, full symbols the experiments in the presence of Ca2++Mg2+ and crosses indicate the experiments in the presence of additional Na+. The straight line represents a slope of 1/3. For the sample 750 ppm in the presence of Ca2++Mg2+ two data points are shown. The filled black diamond corresponds to the final size and mass value, where the aggregation has been already started. The black star indicate one of the very early measurements, where the aggregation process of this sample has not yet started.

As expected, the situation is different at later stages of the experiments with 750 ppm silica in the presence of additional salts (black cross and black diamond in Figure 7). As no equilibrium


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Langmuir

values for Rh and Mw is reached under those conditions, we choose data at a later time, when the monomer concentration has reached its equilibrium value since long. At t = 50 h in the presence of Ca2++Mg2+ the hydrodynamic radius amounts to Rh = 26 nm and the weight-averaged molar mass is Mw = 1.40E+5 g/mol to give but an example. This data point lies far off the trend of the correlation with the exponent  = 1/3 (Figure 7). This can be nicely reconciled with the growth of fractal-like particles. However, if we consider the first data point of this TR-SLS/DLS experiment the value of the hydrodynamic radius (Rh = 3 nm) and the weight-averaged molar mass (Mw = 2.02E+3 g/mol) fits perfectly to the trend based on the exponent  = 1/3 (black star in Figure 7). The mass value Mw = 2.02E+3 g/mol has been calculated under the assumption that monomer concentration has already reached its equilibrium value. This is a hint that the early intermediates still refer to spherical particles formed by a monomer-addition process and thus nicely confirms that monomer-addition, which is observed for all lower silica concentrations, also takes place at the initial period of the process at a silica content of 750 ppm in the presence of an added salt. Only the rate of nucleation and growth is so fast that it is already completed soon after the initiation. Due to an insufficient stabilization of the spherical particles, particleparticle aggregation starts immediately and is dominating the growth process monitored by light scattering measurements. The trend in  presented in Figure S2b of the Supporting Information further confirms this aspect. Although the final particle size as well as the lag-time slightly depend on the solution condition the general trends of the evolution of the size are comparable for all conditions under consideration i.e. in the presence of additional Ca2++Mg2+-ions or Na+-ions and in the absence of any added salt. We thus can state that the presence of additional metal cations has no significant effect on the general mechanism of the silica polymerization. For all three conditions, a change


Langmuir

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Page 24 of 40

from a monomer-addition mechanism generating compact particles with a homogeneous density to a particle-particle aggregation leading to fractal-like particles with the homogeneous particles as building units is observed once a certain silica content is reached. Independent on when this change of the growth mechanism takes place, the homogeneous particles corresponding to the building units have reached a size of Rh ≈ 1-3 nm. Only the concentration threshold at which the aggregation of the particles becomes dominant is shifted to lower silica concentrations, if metal cations are present. Whereas in the presence of metal cations this threshold is smaller than 750 ppm, the threshold concentration lies between 750 ppm and 2000 ppm SiO2 in solutions without added salts.21 In none of the three steps characterizing the process of silica polymerization does Ostwald ripening play a significant role. During the second step compact homogeneous particles grow via monoer-addition as inferred from the exponent 0.5 in eq 11. Growth by Ostwald ripening during this step can be excluded as this would have led to an exponent of 1/3 indistinguishable from a step-growth process. During the third step fractals grow with a fractal dimension of 2 thus excluding Ostwald ripening also during this step. Kinetic Modelling. In a previous publication21 we introduced a kinetic model denoted as nucleation and growth model (NG-model) in order to describe the process of silica polymerization in distilled water. It is a simple process consisting of three successive reactions (i) a precursor reaction, in which the reactive monomer is formed, (ii) a nucleation step and (iii) an irreversible growth process via monomer-addition. The reactions (i) and (ii) establish the first step of the 3-step mechanism. Unexpectedly, the rate constant of the precursor step varied over several orders of magnitude as the initial silica content increased. This likely happened because the nucleation step in which one monomer acts as a nucleus is too simple to account for the complex nucleation or initiation step in reality. This deficiency may have been compensated by a


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Langmuir

variation of the rate constant of the precursor step which helped to vary the induction period with increasing initial silica content. Further on, the NG-model does not account for an equilibrium concentration of the monomers, as only growth and no degradation of the particles is permitted. An improvement of this model denoted as NGE-model shall be introduced in the present work. The new model consists of the same three basic reactions, a precursor reaction, a nucleation and a particle growth via monomer addition. However, the present model explicitly takes into account a variable size of the nuclei indicated by the degree of polymerization n and it considers explicitly depolymerization as the back reaction of monomer-addition. Precursor reaction

A

Nucleation

nB

Addition/Degradation

B+Ci

kp

kn

ke k-e

B

(13)

Ci

(14)

Ci+1

(i ≥ n =1,2,3...)

(15)

Like for the original NG-model21 it is possible to set up five differential equations which describe the first derivative with respect to time of the precursor concentration [A], the monomer concentration [B], and the zeroth [C](0), first [C](1) and second moment [C](2) of the particle ensemble. A detailed derivation of these differential equations is represented in the Supporting Information. Although the NG- and the NGE-model simulate the nucleation/initiation of the silica formation process in a very simple way, they provide a clear benefit for the interpretation of time-resolved light scattering data. Both models do not only allow to simulate the consumption of the monomers and with that the changing supersaturation with time, but also to calculate the variation of the moments of the particle ensemble with time, which makes possible for the first


Langmuir

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time a direct comparison with the weight-averaged molar mass detected via SLS. Beyond doubt the strength of the model lies on the description of particle growth, which is exactly what is recorded by TR-SLS/DLS. The weight-averaged molar mass Mw accessible via light scattering is an average over all species, including the precursor concentration, the monomer concentration and all polymeric particles signified by the degree of polymerization i. Expressed in terms of parameters and variables of the model this Mw reads

Mw (t) 





M0 A(t)  B(t)  C2 (t) A0

(16)

and is accessible by integrating numerically the differential equations for [A], [B] and [C](2) (Supporting Information eqs S3, S6 and S7). M0 is the molar mass of the monomeric unit and is set to 60.1 g/mol corresponding to a SiO2 unit and [A]0 is the initial concentration of the precursor and is given by the respective experimental silica concentration. The results of experiments under where no salt has been added21 will be used first to discuss the performance of the NGE-model in comparison to that of the simple NG-model. Figure 8 represents a comparison of the fitting results obtained by the original NG-model and the new NGE-model. A significant improvement could be achieved for the silica contents 350 and 400 ppm. The fit with the NGE-model is particularly improved in the time range of the initial increase of the particle mass. In case of the highest silica contents of 500 and 750 ppm the NGEmodel does not significantly improve the fit quality achieved with the NG-model but helps to confine kp to a range much narrower than in case of the NG-model. A more detailed presentation of the fits with the NGE-model with regard to the weight-averaged molar mass, the monomer concentration, the concentration of particles and 2 can be found in Chapter S4 of the Supporting Information.


20

0.8

A

18 16

0.7 0.6

14 12

0.5

10

0.4

8

0.3

6

0.2

4 2

0.1

0

0.0

cmon(SiO2) / ppm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Mw / E4 g mol-1

Page 27 of 40

B

700 600 500 400 300 200

(solubility limit 170 ppm)

100 0

20

40

60

80

100

120

140

160

t/h Figure 8: Comparison of the experiments without added salt21 (750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle), 350 ppm (green square)) with the fits based on the NG-model (dashed line) and the NGEmodel (solid line) in terms of the weight-averaged molar mass (A) and the monomer consumption (B). Note that the mass values at a silica content of 500 ppm and 750 ppm are rescaled by a factor of 10. The arrows point to the corresponding axis.

Classical nucleation theory36–39 predicts a decreasing nucleus size with increasing supersaturation. In the light of this theory the concentration series has an increasing initial degree of supersaturation in going from 350 ppm to 750 ppm SiO2. The NGE-model enables us to consider this feature qualitatively by varying the size n of the critical nucleus. However, one has to keep in mind that during the growth process obeying classical nucleation theory the critical nucleus size increases with decreasing monomer concentration (with decreasing degree of supersaturation), which is not accounted for by the NGE-model. In order to determine the optimal nucleus size for each initial silica concentration, the fitting has been performed for different nucleus sizes in the range of 1 ≤ n ≤ 30. The fit with the lowest χ2-value (Supporting Information, eq S9), has been used to identify the optimal nucleus size. The resulting optimal nvalues are summarized in Table 1. Unfortunately, identification of n based on 2 values could


Langmuir

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not be achieved with the same degree of significance for all four silica contents (Figure S4c of the Supporting Information). However, the data show that the most suitable nucleus size n (Table 1) is decreasing with increasing initial silica content, which is in line with the predictions of the classical nucleation theory. It has to be stressed that the formation and growth of silica particles is based on the incorporation of monomeric building units via chemical bonding corresponding to a three dimensional polymerization or polycondensation reaction, which may well differ considerable from a pure crystallization following a classical nucleation process. Table 1: Rate constants at variable silica contents in the absence of salt at pH 7 from fits with the NGE-model to data from Ref. 21. The value n describes the size of the theoretical nuclei in eq 14 which yields the smallest χ2– value, kp is the rate constant for the precursor reaction, kn for the nucleation and ke for the growth reaction in eqs 1315.

saltfree

c0, exp [ppm] 350 350 400 400 500 750

n (fixed)

n (optimal) 8

1 2 1 1 1

kp [h-1] 3.15E-2 2.05E-2 5.50E-2 4.99E-2 1.91E-1 1.94E-1

kn [L/mol)n-1h-1] 5.25E14 1.42E-5 1.04E-1 1.06E-4 1.03 5.08

ke [L(mol h)-1] 8.67E4 2.03E5 1.86E5 2.42E5 6.15E6 3.81E6

χ2 [g mol-1] 1.70E1 3.69E1 4.85E1 5.12E1 3.92E1 8.15E1

Unlike to the behavior in the presence of salt, silica particles generated in a solution of 750 ppm do not show particle-particle aggregation in the absence of added salt. In that case the concentration threshold, which marks the beginning of a particle-particle aggregation lies well above 750 ppm. However, as inferred from the poorer performance of the NG and NGE-model at the two silica contents 500 ppm and 750 ppm in salt-free solution, those two higher silica contents are much less dominated by particle growth. After a very fast increase the hydrodynamic radius remains constant within experimental uncertainty at a value of 4-5.5 nm while the weight-averaged molar mass is still slightly increasing. This behavior suggests that the increase of Mw is dominated by the formation of further particles rather than by incorporation of monomers to growing particles.


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Langmuir

The experimentally observed generation of the particles with the size around 4-5.5 nm at 500 and 750 ppm SiO2 is interpreted as follows by the NGE-model. A very large nucleation rate leads to the final amount of particles in a short time. These particles have to add the remaining monomers necessary to reach the final mass values also very fast in order not to compete with further nucleation since further nucleation would make the final particle mass too low. Accordingly, the fits lead to a constant size and weight-averaged molar mass of the particles soon after nucleation, which appears not to be fully compatible with data from 500 ppm and 750 ppm. A simple increase of the nucleation rate cannot describe gradually increasing mass at later stages of the reaction. Evaluation with the NGE-model shall be concluded with a focus on the different rate constants. Table 1 summarizes the optimized rate constant of the precursor reaction kp derived from the fits to the experimental data for silica in the absence of added salt solution at pH 7 for different nucleus sizes n. The precursor reaction was introduced in order to model the induction period, because a simple formation model consisting of a nucleation and a growth reaction with a rate constant independent of the degree of polymerization is not able to reveal an induction period. We expected a constant value for kp for all initial silica contents. In this respect the NGmodel failed, since the rate constant of the precursor reaction was increasing by several orders of magnitude with increasing silica content21. The NGE-model with its nucleation/initiation reaction of variable order n significantly improved this issue. The rate constants of the precursor reaction kp retrieved with the NGE-model all fall within one order of magnitude. A comparison of the rate constants for the precursor reaction, the nucleation and the growth reaction from fits with the NGE-model and the NG-model are given in Table S1 of the Supporting Information. The rate constant for the growth ke is in the order of 105 L(mol h-1) at 350 ppm and 400 ppm of


Langmuir

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silica and increases by one order of magnitude at the higher silica contents. This jump is a hint for deficiencies in the NGE-model as soon as the growth is not the dominating part any more. We conclude that modelling of the silica polymerization could be improved by the NGE-model with respect to the following aspects. The variation of the rate constant of the precursor reaction could be confined to one order of magnitude, which is significantly narrower than the respective variation over eleven orders of magnitude observed with the NG-model.21 If we consider only the two lowest silica concentrations, for which we have unambiguously demonstrated, that a monomer-addition mechanism is dominating, the deviations in kp are even lower. Variation of the nucleus size has shown that the silica polymerization follows the trend of a decreasing nucleus size with increasing degree of supersaturation, which is in accordance with classical nucleation theory. The equilibrium state of the silica polymerization could be successfully modelled by the implementation of a depolymerization reaction in which the polymers release monomers. A possible explanation for the deficiencies of the NG and NGE-model observed with the two higher silica contents would be an ongoing generation of nuclei as particles similar in size during the entire experimental time period, with a single type of particle being formed within a short period of time compared to the length of the overall period. Such a feature could in fact also be described by the NGE-model if n would be fixed at values large enough to correspond to 3-4 nm in size and if the rate constant of this nucleation would be very high in comparison to the rate constant of the monomer-addition. This results in a transfer of the main fraction of the active monomers into nuclei with a size large enough to fit Mw values in the order of 4·103-8·103 g mol-1 and Rh values close to 4 nm. However, such size and mass values of a nucleus which are much larger than the n values observed at low silica contents would not be compatible with the classical nucleation theory because the size of nuclei are expected to get


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smaller as the degree of supersaturation (i.e. c0) increases and not larger. A possible explanation is that the initiation reaction as part of step 1 is not fully in accordance with the nucleation in the sense of classical nucleation theory. 25 4 20 3 15 2

10

Mw / 1E4 g mol-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

5

1

0

0 1.5

5 4

1.0 3 2 0.5 1 0 0

10

20

30

40

50

60

0.0 70

t/h Figure 9: Comparison of the experimental weight-averaged molar mass Mw with the corresponding fits with the optimal nucleus size at variable silica contents (750 ppm (black diamond), 500 ppm (red triangle), 400 ppm (blue circle), 350 ppm (green square)). (A) Experiments in the presence of Ca2++Mg2+-ions; (B) experiments in the presence of additional Na+-ions. Note that the experimental data have different scales for a better overview. The arrows point to the corresponding axis. The model calculations based on the NGE-model are represented as solid lines.

After having evaluated the NGE-model with the data recorded under salt-free condition, experiments in the presence of Ca2++Mg2+ and in the presence of additional Na+ shall now be interpreted with the NGE-model. The experimental data corresponding to a silica content of 750 ppm has been excluded from the optimization process, as the correlation of the size with the mass values (shown in Figure 5) revealed that the particle formation is dominated by a stepgrowth like particle-particle aggregation. The fits with the NGE-model are represented in Figure 9 and the corresponding rate constants obtained from the optimization applied to the mass


Langmuir

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evolution of the experiments in the presence of metal cations are given in Table 2. For each silica content the results corresponding to the optimal nucleus size determined by the smallest χ2value are shown. Further details of the fits are given in the Supporting Information (Figure S5a and S5b). The variation of the rate constant of the precursor reaction in the presence of added salt extends over two orders of magnitude only and hence is similar to that of the salt-free case. The rate constants of the growth reaction lie again in the order of magnitude ~ 105 L(mol h-1) at silica contents lower than 500 ppm and can be considered as constant within the range of uncertainty. We thus can conclude that the absence or presence of added Ca2++Mg2+ or Na+ does not significantly affect the rate constant ke of the particle growth. Table 2: Rate constants at variable silica contents in the presence of Ca2++Mg2+- ions and Na+-ions at pH 7 from the optimization with the NGE-model. The value n describes the size of the theoretical nuclei which yields the smallest χ2–value, kp is the rate constant for the precursor reaction, kn for the nucleation and ke for the growth reaction. The star indicates cases when χ2 was continuously decreasing with n and where the values correspond to the fit with the maximum possible n.

c0,exp [ppm]

Ca2++Mg2+

Na

+

350 350 400 400 500 350 350 400 400 500

n (fixed)

n (optimal)

kp [h-1]

kn [(L/mol)n-1h-1]

ke [L(mol h)-1]

8

9.39E-2 5.97E-2 1.29E-1 9.89E-2 2.94E-1 1.16E-1 4.49E-2 2.16E-1 1.58E-1 2.60E-1

2.26E14 1.89E-5 5.07E51 5.83E-4 4.12 2.37E71 1.37E-4 1.12E19 8.00E-5 5.23

2.08E5 3.83E5 1.18E5 2.50E5 2.71E7 4.01E4 1.09E5 4.46E4 5.39E4 5.90E7

1 20* 1 1 30* 1 10 1 1

χ2 [g mol1 ] 7.91E1 1.43E2 6.66E1 8.98E1 3.87E1 7.03 8.12 8.82E1 1.33E2 7.29E1

The fits to data recorded in the presence of salt give an optimal nucleus size of n = 1 at the silica content of 500 ppm. At lower silica contents, the optimal size of the nuclei is larger than 1. Unfortunately in two cases a continuously decreasing χ2 has been observed with increasing n (Table 2, Figure S5d and S5f) which prohibits identification of the optimal n in those two cases. Also the significance of the optimal nucleus size is not always strong. However, the fits with the


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NGE-model suggests that the size of the nuclei is decreasing with increasing initial supersaturation, which is in line with the classical nucleation theory. A characteristic magnitude for the nucleation/initiation is the number of generated particles, which is described by the zeroth moment. Figure 10 compares the final value of the zeroth moment, depending on the silica content and the reaction conditions. The particle number is clearly increasing with increasing silica content, which confirms that the nucleation/initiation is accelerated and plays a more dominating role with increasing supersaturation. The influence of an added salt has only a minor effect on the number of formed particles and shows no clear trend.

[C](0) / 1E-6 g mol-1

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10

1

340

360

380

400

420

440

460

480

500

c0 / ppm Figure 10: Final number of particles represented as zeroth moment based on the optimal fit with the NGE-model for the corresponding silica content c0 for different conditions: Silica in water without added salt (); in the presence of additional Na+ () and in the presence of Ca2++Mg2+ ().

To summarize, the NGE-model unlike to the NG-model can reproduce the evolution of mass at variable initial silica concentration by an interplay of the rate constant of the precursor reaction and the size of the nuclei, which is connected with a change in the reaction order of the nucleation step. This works satisfactorily in case of the lowest silica concentrations 350 ppm and 400 ppm independent of the conditions, suggesting that the addition of salt does not influence the


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general mechanistic features of the reaction. In line with this the kinetic parameters do not show significant differences among the three conditions (absence of added salt, presence of Ca2++Mg2+, presence of additional Na+) except for the rate constants of the precursor reaction which are persistently lower if no salt had been added. It has to be emphasized that the precursor reaction together with the initiation/nucleation reaction of the NG- and NGE-model establish the first step of the 3-step mechanism. This first step usually denoted as nucleation step is increasingly dominating the process of particle formation, once the silica content increases beyond 400 ppm and the NGE-model like the NG-model performs progressively poorer. This becomes obvious not only in terms of the decreasing fit quality but also in terms of the resulting rate constants of the growth reaction, which at 500 ppm are deviating from those obtained at the lower silica contents.

CONCLUSIONS In a preceding work, the formation of silica particles in pure water at a pH = 7 covering a silica concentration regime, which extends from 350 ppm to 750 ppm has been analysed by timeresolved combined DLS/SLS. The work unambiguously revealed a short nucleation or initiation step, which is followed by a particle growth based on a monomer-addition mechanism. A simple kinetic scheme, the NG-model, could satisfactorily reproduce the evolution of the weightaveraged particle mass and the consumption of monomers at 350 ppm and 400 ppm, however, its performance turned out to be poorer at the higher silica contents of 500 ppm and 750 ppm. This was attributed to an increasing influence of the nucleation/initiation step as the silica content increases. The present work extends this knowledge in two different ways. First and foremost, analogous experiments have been performed in the very same concentration regime of silica,


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now in the presence of additional Ca2++Mg2+ or Na+ at a level of 0.01 N respectively in order to analyze the impact of alkaline earth cations on the formation process of silica particles. Comparison with Na+-ions does not reveal any ion specific effect for alkaline earth cations. Second, the simple NG-model was refined with respect to two aspects, the variation of the size n of the nuclei which are formed in a simple reaction of order n and the explicit consideration of a monomer release step as the back reaction of the monomer addition. The refined model is named NGE-model in the present work. Like for the NG-model a central part of the theoretical work was to develop kinetic equations, which are capable of yielding the first three moments of the resulting particle size distribution since it is this data, which is required for a comparison with light scattering results. In order to provide suitable reference data, the NGE-model was first verified with the data from the preceding work collected in the absence of additional salt.21 Several features demonstrated a better performance of the refined NGE-model. The fit improved particularly in the regime of the lower silica contents. The rate constant of the precursor reaction could be successfully confined to a range extending only over one order of magnitude which is much better than the spread of the rate constant over eleven orders of magnitude observed for the NGmodel. Further on, the NGE-model indicated a decrease of the nucleus size with increasing silica content in line with classical nucleation theory. With this in mind a more comprehensive discussion of the new results recorded in the presence of an additional salt became possible. Independent of the conditions, i.e. in the absence of added salt21 and in the presence of additional Na+ and of Ca2++Mg2+, a three step process can be identified in agreement with preceding work.5,19,21 The steps are (1) a nucleation/initiation step, (2) particle growth stage and (3) agglomeration of the particles from step 2. The present work together with Ref. 21 unravels


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further details on step 2 and 3 with combined time-resolved SLS/DLS as new in situ technique and the molybdenum blue method. Addition of salt reduced the lag-time preceding particle growth via monomer-addition. In terms of the NGE-model this is captured by a slightly increased rate constant of the precursor reaction and an increased/accelerated formation of nuclei/particles. The particles nucleated in this first step grow according to a monomer-addition mechanism where the monomers may include monomeric silicic acid and oligomers thereof. These features could unambiguously be inferred from the coincidence of disappearing molybdate-reactive “monomers” with the particle growth and from a correlation of particle size (from DLS) with particle mass (from SLS) resulting in characteristic power laws of Rh ~ Mw1/6. Ostwald-ripening can be excluded during step 2 as this would have modified the exponent of 1/6. The rate constant of the growth reaction lies in the same order of magnitude for all three conditions (without added salt, with additional Na+ at [Na+]= 0.01 N and with additional Ca2++Mg2+ at [Ca2+4/5Mg2+1/5]=0.01 N) as long as the nucleation/initiation is not dominating the particle formation, which holds for silica contents lower than 500 ppm. The decrease in lag-time together with the increase of the precursor rate constant induced by the addition of salt suggests that the added salt most significantly affects the first step of the particle formation process. In line with this, no significant variation of the growth rate constant could be discerned with the addition of salt. Although similar trends were reported in literature15,16, we hesitate to attribute this effect to a lowering of the solubility of silica by salt addition16 since we keep well below a salt content of 40 mM and vary the salt concentration only within a very narrow range of 10 mM.


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Agglomeration as the third step sets in once the concentration of silica is large enough. It proceeds via a step-growth like particle-particle aggregation. As extracted from SLS/DLS-data, the resulting agglomerates have a fractal dimension close to 2. Formation of fractals as step 3 discards Ostwald-ripening as mechanistic feature of this step. The onset of particle agglomeration according to this step-growth process, which in the absence of added salt was first observed at a silica content of 2000 ppm is shifted to lower values by adding salt. Agglomeration takes place already at 750 ppm in the presence of both types of cations. Hence the presence of cations destabilizes particles with respect to particle-particle agglomeration. This occurs most likely due to screening of electrostatic interactions among the particles. However, the presence of Ca2++Mg2+ and the presence of Na+ does not affect the fractal dimension of the growing intermediates. Even more strikingly, the presence of an additional salt does not even affect the nature of the constituting particles. Independent on whether salt is present or absent nucleation succeeded by growth via monomer-addition lead to particles with the same critical size of a few nanometers which then agglomerate to a unique type of fractal.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Comparison of the silica particle growth in salt-free solution, in the presence of Ca2++Mg2+ and in the presence of additional Na+; detailed data of structural intermediates at 750 ppm SiO2 in the presence of Ca2++Mg2+; derivation of the NGE-model and representation of the corresponding fit results for all three investigated reaction conditions.


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AUTHOR INFORMATION Corresponding Author *E-mail: (K.H.) [email protected]. Telephone: (+49) 5251602125. Fax: (+49) 5251 604208.

Notes The author declares no competing financial interest.

ACKNOWLEDGMENT The BASF SE is acknowledged for funding this work within the project “Polymer Assisted Silica Polymerization”.

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