On Precipitation of Sparingly Soluble Fluoride Salts - Crystal Growth

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On precipitation of sparingly soluble fluoride salts Ricco T. Kügler, and Matthias Kind Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01115 • Publication Date (Web): 13 Dec 2017 Downloaded from http://pubs.acs.org on January 4, 2018

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Crystal Growth & Design

On precipitation of sparingly soluble fluoride salts Ricco T. Kügler*, Matthias Kind** Institute of Thermal Process Engineering, Karlsruhe Institute of Technology, Kaiserstrasse 12, 76131 Karlsruhe, Germany KEYWORDS: Particle formation, Calcium fluoride, Strontium fluoride, Nucleation, Growth, Population balance modeling

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ABSTRACT

The classical approaches for nucleation rate and diffusion-limited crystal growth are checked for their validity for sparingly soluble salts, i.e. calcium fluoride and strontium fluoride. Particle size distributions are calculated by population balance modeling as a function of the initial supersaturation and free lattice ion ratio. Theoretical results are compared with experimentally determined particle sizes which are measured by dynamic light scattering technique. Before measurement, suspensions were stabilized by dilution with deionized water or surfactant solution. The dilution of suspensions was done after solid liquid equilibrium had been reached. Consequently, particle formation mechanisms, such as nucleation and crystal growth which depend on supersaturation, are assumed to be unaffected. If supersaturation is calculated correctly by considering activities and ion complex formation, we found that experimental results are predictable by modeling with classical approaches. Hereby, interfacial energy is the only fitting parameter. Furthermore, the diffusion-limited crystal growth rate decreases with deviation from stoichiometric precipitation condition due to the lower transport flux of the shortfall ion.


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1. INTRODUCTION Precipitation of sparingly soluble, inorganic salts is used frequently to investigate the fundamentals of the particle formation process. It is currently difficult or impossible to predict the properties of the final particles, for example, the size distribution, morphology, tendency to agglomeration and aggregation, with sufficient accuracy. The main reason is that all process steps of particle formation involved proceed simultaneously: Buildup of the supersaturation by mixing of reactants, primary processes, such as nucleation and crystal growth, and secondary processes, such as agglomeration, aggregation, ripening and aging. Consequently, quantitative knowledge about precipitation kinetics is often hard to get. Solid formation paths of 2:2 electrolytes, such as barium sulfate, calcium carbonate and calcium oxalate, have especially been extensively described in the past.1 Söhnel and Garside,1 Nielsen,2 Nielsen and Söhnel,3 and Kind and Mersmann4 were among those who particularly introduced classical nucleation theory (CNT) kinetic approaches to determine nucleation rates theoretically. Thenceforward, approaches modified from CNT have been used for the estimation of nucleation rates. Thereby, the main parameter in these equations is the interfacial energy between solution and crystal.3,5-9 This parameter is commonly determined by matching theoretical models to experimental data. A more direct method for measuring this material property has not be discovered to date. Nevertheless, it can be estimated by theoretical concepts.3,10 However, it is generally known that the nucleation rate shows a strong nonlinear dependence on the prevailing supersaturation. The dependency of formation mechanisms on the supersaturation is responsible mainly for the final properties of the particles precipitated. The particle size is a particularly important property for the characterization.


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We have already shown in an earlier publication that a CNT approach for nucleation and a mass transfer approach for diffusion-limited growth are appropriate to describe the particle formation of barium sulfate.11 There, calculations used activity-based supersaturation and experimental parameters were chosen to suppress aggregation. Because generic kinetics was adopted, the approach used should be valid for other salts in addition to barium sulfate, for example, for 2:1 electrolytes, such as calcium fluoride (CaF2) and strontium fluoride (SrF2). In contrast to 2:2 electrolytes, investigations about the particle buildup fundamentals of 2:1 electrolytes at high supersaturation are rare in literature. In the case of calcium fluoride and strontium fluoride precipitation, only a few discussions exist about their formation mechanisms assuming classical nucleation.12-19 Nearly all publications about calcium fluoride precipitation comprise the determination of integration-limited growth rates,13,15,17 or the measurement of the induction time at very low supersaturation.12 A measurement of the primary crystal size is difficult, especially at high supersaturation, due to agglomeration.20-22 Publications about the precipitation of strontium fluoride from aqueous solution are even rarer. Bochner14 and Hamza16 carried out seeded precipitations at low supersaturation. They studied both the integration-limited crystal growth and its inhibition by other ions or additives. In addition to supersaturation, the ratio between cation and anion concentration, the so-called free lattice ion ratio (Eq. (3)), has a significant impact on primary process kinetics, for example, in the case of barium sulfate precipitation.23-26 Furthermore, the free lattice ion ratio is not often regarded in literature since experiments are carried out mainly in a stoichiometric condition. However, if a distribution of free lattice ion ratio appears in a solution, which mainly occurs at the precipitation in stirred reactors, it is important to know the influence of the free


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lattice ion ratio on the growth rate. Hereby, it can be considered in predictions, for example, of the particle size distribution. The aim of this work was to review the theoretical kinetic approaches for nucleation and growth known for barium sulfate precipitation and to extend them to other sparingly soluble salts, especially to fluoride salts. In the theoretical section, the same primary mechanisms are assumed for the formation of all three inorganic precipitates. In the experimental section, data about the agglomeration-free particle size are collected at high supersaturation and at varied free lattice ion ratio. These data are then compared to simulation results, whereby, the agglomeration term is not considered within population balance modeling. Well-known mixing conditions are realized and ensured by using a well characterized confined impinging jet mixer (CIJM). 2. MATERIAL AND METHODS 2.1 Experimental Methods All experiments were carried out at a temperature of 25 °C. The stock reaction solutions were prepared by dilution of potassium fluoride (Carl Roth, no. 2617.4), calcium chloride di-hydrate (Carl Roth, no. 5239.3) and strontium chloride hexa-hydrate (Carl Roth, no. 4473.3) in deionized water. They were mixed in CIJMs (Y- or T-mixer) with equal flow rates of 300 ml/min per jet.27 A Reynolds number of 7,132 was reached in the mixing zone of the CIJM with these settings. The same experimental setup was used to accomplish these flow rates, as described in Kügler and Kind.27 Stabilization of the suspension was achieved either by ensuring a high electrostatic potential between the crystals or by steric hindering with a surfactant adsorbed on the crystal surface. Electrostatic potential could be raised by potential determining ions, precipitating at free lattice ion excess or dilution of the suspension with deionized water.11,28,29 The precipitated


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suspension caught in a beaker may also be diluted by adding an aqueous additive (BASF SE, Melpers0045) solution.11 Diluted suspensions were stirred for some hours. Dynamic light scattering was used to quantify the particle size distributions (Malvern Instruments, Zetasizer Nano ZS). Water or surfactant solution is always added after solid liquid equilibrium is reached. Hence, an influence of dilution on the primary mechanisms of the particle formation could be excluded. Conductivity of the mixtures was monitored to estimate the time span between mixing reactants and obtaining solid liquid equilibrium (WTW, LF 197). The mother liquor of an untreated suspension was separated after centrifugation and the slurry was washed with deionized water. After a further centrifugation, the slurry was dried at 80 °C overnight. A fine powder was made by grinding with pestle and mortar for the SEM images. 2.2 Theoretical Calculations The precipitation reactions of electrolytes mostly contain one cation species , here calcium Ca2+ or strontium Sr 2+ , and one anion species , here fluoride F - .

+ ⇌ ↓

(1),

where is the stoichiometric coefficient of the cation and of the anion; and are the charge coefficient of the cation and the anion, respectively. The supersaturation is calculated activity-based by $

± ̃ ∙ ̃ % = ⊖ ∙ # ̃ SP,

where

&'

(2)

represents the solubility product of the salt, ± the mean activity coefficient, which is

determined by Bromley’s approach,30 ̃ ⊖ is a standard normalization constant for the definition


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of activity by molar concentration, which is 1 mol/l, and ̃ and ̃ are the molar concentrations of the free ions, which form no ion pairs with other ions in solution (no complexation). Important ion pair associations are considered and are subtracted from the total ion concentration of a species,

IP,CaF +

= 0.94 according to the PHREEQC data base31 and

solubility products

SP

of the two salts under investigation are

1004.33.1 The free lattice ion ratio is defined by

=

SP,CaF2

̃ ̃

IP,SrF +

= 5.75.14 The

= 100$1.23 and

SP,SrF2

=

(3),

according to Kucher.29 Modeling of the particle size distribution is carried out by a onedimensional population balance equation (Eq. (4)) coupled with component balances (not shown, see Kügler11 for further details).

5DE7 , 89 ∙ 6789F 56789 = ;hom 7 9 ∙ ?78crit 9 − 5: 58

(4),

where 6789 is the particle number density in size class 8, and ;hom and E represent the nucleation

rate and total growth rate, respectively. The function ?78crit 9 is the particle number density

distribution of the nuclei formed. A Gaussian function with a standard deviation of 5 % is used to describe the size distribution of the nuclei.32 The critical nucleus diameter 8crit is calculated with the CNT approach by Volmer and Weber.33 Equation (4) is a simplified population balance, since all agglomeration and deagglomeration rates are set to zero. This simplification is justified because it reflects our experiments, where coagulation was suppressed by particle stabilization.


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Homogeneous nucleation is assumed to be the only nucleation mechanism, because of the high level of supersaturation applied.34 Nucleation rate is calculated by the CNT approach of Kind and Mersmann4

;hom

\ 3 cl 16 cl L Im J/L = H Im M ∙ exp U− W X Y X Y ] 7 + Z 9 ∙ ln 2 OB Q 3 OB Q

(5),

where H is the diffusion coefficient of the salt calculated by Eq. (6), proposed by Nielsen,35

with the cation and anion diffusion coefficients H and H given in Table 1.

H =

7| | + | |9 ∙ H H | | ∙ H + | | ∙ H

(6)

Table 1: Diffusion coefficients of ions in water at 25 °C.36-38 Ion

Ca2+

H ∙ 10_ in m²/s

Sr 2+ F-

0.79 1.22 1.46

In Eq. (5), Im is the molar volume, OB is Boltzmann’s number and Q is the temperature. Schwarzer and Peukert32 give the concentration of monomers which collide during

nucleation by

= ∙ `A ∙ ̃ ⊖ ∙ 7

$ b %b SP 9

(7),


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where ` is Avogadro’s number. The interfacial energy cd of a salt can be estimated by the equation of Mersmann10 which is constituted only of material properties. Diffusion-limited growth is assumed due to the high supersaturation adjusted. The total growth rate E is mathematically derived from Fick’s mass transfer approach.32,29,40 Hence, it is determined by the activity gradient’s minimum of lattice ion species.41

E=

f ∙ ̃ ⊖ 2 ∙ Sh ∙ e ∙ minhH ∙ 7i − i ∗ 9, H ∙ 7i − i∗ 9k g ∙ 8

(8),

f is the molar mass and g is the density of the salt. where Sh is the Sherwood number, e Small, spherical particles will be present (with diameters < 2 µm) and, thus, Sh = Shmin = 2.1

The numerical solution of Eq. (4) and coupled component balances was carried out with the solver PARSIVAL (CiT GmbH).


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3. RESULTS AND DISCUSSION 3.1 Degree of Supersaturation Calcium Fluoride Nanometer-sized particles of calcium fluoride can be precipitated at stoichiometric reaction conditions, depending on the initial degree of supersaturation, see Figure 1. The mean particle size measured of 831,L for ,1 ≤ 100 agreed well with the simulated size, and a rapid reduction of both experimentally determined and simulated particle sizes occurred with increasing supersaturation. For ,1 > 100, the particle size increased up to more than a decade without dilution, because the suspensions were no longer stable and agglomeration occurred. Nucleation is the solid forming step, but secondary processes like agglomeration disturb the quantitative analysis of the primary particle size after reaching solid liquid equilibrium of the solution. Agglomeration rate mainly depends on particle number and rises with increasing supersaturation due to nucleation rate.32 It was possible to reach a significant stabilization effect in this range by diluting with pure deionized water or surfactant solution. The CNT, see Eq. (4), states that the nucleation rate depends strongly on the degree of supersaturation. This strong dependency in population balance modeling leads to the non-linear course of the mean particle size versus initial supersaturation observed. However, this is only the case if agglomeration of the primary particles is neglected in the simulation calculations. Experimental data match the simulation results well. This is strong evidence that primary particles are also obtained in the experiments, and that the classical kinetic approaches are


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applicable in the case of calcium fluoride precipitation (for critical free energy and critical diameter see Supporting Information). It should be mentioned that this good accordance is achieved only if the supersaturation is calculated by activities and under provision of the formation of the CaF+ complex. If the initial supersaturation was computed with concentrations rather than with activities and without ion complex formation, experimental data points would be shifted to much higher values of supersaturation. However, considerable differences between experimental and simulated particle size are obtained if the theoretical approach of Mersmann10 is used for the calculation of the interfacial energy cl . Nevertheless, a change of cl has no large influence on the course of particle size versus supersaturation, but it is very important in terms of its absolute value. Hence, interfacial energy cl is still the most uncertain factor in CNT and must be seen as a parameter to be fitted by regression analysis. The SEM images affirmed the trend of the particle sizes measured; see Figure 2. By qualitatively comparing the sizes of particles in the SEM image and measured at ,1 = 30, 50 and 125, it seems that the particle sizes were somewhat larger than determined by dynamic light scattering technique. A main reason could be that the refractive indices of water (1.33) and calcium fluoride (1.43) do not differ fundamentally. This is why calcium fluoride suspensions appear transparent. At ,1 = 200, it is recognizable that crystal sizes are definitely lower than 100 nm by comparing data by DLS (approx. 500 nm) to Figure 2d. Here, the measured particle size is mainly influenced by agglomeration. The particles showed the typical morphologies of single crystals, such as smooth surface and sharp edges from a macroscopic view. Therefore, they seemed to be primary particles. Cubic


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crystal morphology, especially at low supersaturation of ,1 = 30, is reminiscent of the crystal growth model proposed by Kossel42 and Stranski43, which deals with the integration-determined growth mechanisms. Strontium Fluoride A good accordance between experiments and simulation was also obtained in the case of strontium fluoride if suspensions were stabilized and the fitting of the interfacial energy cl (mentioned previously) was applied, see Figure 3. Strontium fluoride particles agglomerated without dilution of the suspensions, as has already been seen for calcium fluoride. Agglomerate sizes, even in the micrometer scale, were observed. Nanometer-sized particles were obtained in diluted suspensions, confirmed by the SEM images of the powder, see Figure 4. The size of the primary particles decreased strongly with an increasing initial degree of supersaturation until a minimal particle size of approximately 30 nm was reached. Simulation results were in good agreement with the experimental data. In contrast to results expected from simulation, a particle size of less than one micrometer was obtained at a supersaturation of ,1 = 10. At ,1 = 10, the particles were found to have a rose-like morphology. Such morphology is difficult to explain. Crystal structures of particles are the same, which was verified by powder X-ray diffraction (see Supporting Information). Integration-limited crystal growth mechanisms dominate the growth rate at low supersaturation. The growth rates of these mechanisms could be theoretically described whereby semi-empirical equations are often used with parameters which can only be estimated, for example, the diffusion coefficient of a growth unit on the crystal surface. This is why integration-limited crystal growth kinetics is not


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considered to be population balance modelling, see Eq. (4). At ,1 ≥ 30, nanometer-sized particles with ordinary crystal morphology were obtained. Here, the growth rate was mainly limited by the velocity of ions diffusing from bulk to particle surface. It should be mentioned that the modeling approaches used are purely theoretical. Therefore, they cannot be expected to predict experimental results perfectly. However, a quite good accordance was reached between experiment and simulation, which considers the activity-based calculation of supersaturation and kinetic approaches used. By customizing the interfacial energy, the equation for nucleation rate is applicable to the precipitation of calcium fluoride and strontium fluoride.

3.2 Free Lattice Ion Ratio A further test of the applicability of CNT was about its ability to predict the influence of varying free lattice ion ratio at constant supersaturation. The free lattice ion ratio as well as the

supersaturation change during a precipitation experiment. Therefore, their respective initial

values 1 and 1 are given in the following. Suspensions were stabilized by dilution with surfactant solution or deionized water. Diffusion-limited growth is influenced by activity gradients, see Eq. (8). Therefore, it was expected that various free lattice ion ratios lead to lower growth rates and, hence, to lower particle sizes when deviating from the stoichiometric condition (1 = 0.5). Calcium Fluoride The expected dependency of the particle size on the free lattice ion ratio is approved by experimental data, as shown in Figure 5. The SEM images also proved this finding, see Figure 6.


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Primary particles are observable at ,1 = 50 which have ordinary crystal morphology characterized by clear edges and smooth planes. Whereby crystal morphology looks more cubic when there is calcium ion excess than when there are stoichiometric or fluoride excess conditions. Strontium Fluoride It is shown in Figure 7, that the expected relation of particle size versus free lattice ion ratio for strontium fluoride is not as clearly observable as for calcium fluoride. The main reason for this observation may be that the determination of primary particle size was difficult at fluoride ion excess due to agglomeration. 4. CONCLUSION The scope of this study scrutinized the applicability of the CNT for the simulation of the precipitation of different sparingly soluble precipitates. Initial experimental conditions, such as supersaturation and free lattice ion ratio, can be well adjusted by using a mixing nozzle process which leads to highly reproducible results. A high accordance between experiment and simulation is reached by population balance modeling with classical kinetic approaches for nucleation and growth. It is possible to estimate experimental results in good accordance by simulation calculations. The degree of supersaturation has to be calculated with activities rather than with concentrations for a correct description of precipitation processes of sparingly soluble substances. Associations or complex building of ion pairs must also be considered. Precipitations with ion excess lead to lower diffusion-limited growth rates compared with stoichiometric reaction conditions, because a lower activity gradient of the shortfall ion limits its diffusion to


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the surface. Consequently, the initial supersaturation depletes more slowly, which results in more nuclei. Therefore, crystal size decreases at precipitations with ion excess. We conclude that – at least for barium sulfate11, calcium and strontium fluoride – classical kinetic approaches are suitable for the description of the particle formation by precipitation. Supporting Information Gibbs free energy barrier ∆Epqrs as well as critical cluster size 8pqrs for CaF2 and SrF2, Powder X-

ray diffraction patterns of SrF2 particles at 1 = 0.5 and various ,1 . This material is available

free of charge via the Internet at http://pubs.acs.org.


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AUTHOR INFORMATION Corresponding Author *[email protected] Principal corresponding Author **[email protected] Funding Sources Deutsche Forschungsgemeinschaft (DFG) Project no. KI709/27-1 Notes The authors declare no competing financial interest. ACKNOWLEDGMENT Julyan Hanna and Maximilian Ailinger are gratefully acknowledged for their excellent work in the lab. Financial support by the Deutsche Forschungsgemeinschaft (Project no. KI709/27-1) is gratefully acknowledged. NOTATION

– anion species i – activity

– cation species

– concentration (#/m³)

̃ – molar concentration (mol/l)

̃ ⊖ – normalization constant (1 mol/l) H – diffusion coefficient (m²/s)

? – Gaussian distribution function (1/m) ACS Paragon Plus Environment

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E – growth rate (m/s)

; – nucleation rate (#/m³s) IP

SP

– ion pair product

– solubility product

OB – Boltzmann’s number (1.381·10-²³ J/K)

`A – Avogadro’s number (6.022·10²³ #/mol) f – molecular mass (kg/mol) e 6 – number density (#/m4)

– free lattice ion ratio

– degree of supersaturation Sh – Sherwood’s number : – time (s)

Im – molar volume (m³/mol)

Z – stoichiometric coefficient 8 – diameter (m)

831,L – mean volume weighted diameter (m)

– ion charge Greek Letters

± – mean activity coefficient

cl – interfacial energy between crystal and liquid (J/m²)

W – mathematical constant (3.1416) g – density (kg/m³)

Sub- and superscripts 0 – initial (: = 0) – anion species


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i – activity-based

– cation species – salt

crit – critical hom – homogeneous

Abbreviations CIJM – confined impinging jet mixer CNT – classical nucleation theory

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FOR TABLE OF CONTENTS ONLY

On precipitation of sparingly soluble fluoride salts Ricco T. Kügler and Matthias Kind

The precipitation of sparingly soluble, inorganic fluoride salts was demonstrated to provide crystalline nanoparticles. The agglomeration of the nanometer-sized crystals generated was suppressed after solid liquid equilibrium had been reached. In addition, the simulation of the particle sizes was established by modeling with activity-based supersaturation, considering ion complex building, population balance equation and classical kinetic approaches for nucleation and growth. A good accordance between particle sizes of experimental data and simulation was achieved by solely adjusting interfacial energy.


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Calcium fluoride: experimental and simulated mean particle size versus initial supersaturation. Green squares: undiluted suspension; red filled circles: diluted after precipitation with deionized water; blue diamonds: diluted with surfactant solution; dashed line: calculated results with γcl = γcl,0 = 0.169 J/m2 according to Mersmann10; solid line: γcl = γcl,0 = 0.187 J/m2 obtained by data fitting. 233x224mm (96 x 96 DPI)


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Calcium fluoride particles precipitated at R0 = 0.5 and various initial degrees of supersaturation Sa,0 = 30 (a), 50 (b), 125 (c) and 200 (d). 289x200mm (96 x 96 DPI)



Strontium Fluoride: experimental and simulated mean particle size versus initial supersaturation. Green squares: undiluted suspension; blue diamonds: diluted with surfactant solution; dashed line: calculated results with γcl = γcl,0 = 0.128 J/m2 according to Mersmann10; solid line: γcl = γcl,0 = 0.161 J/m2 obtained by data fitting. 74x70mm (300 x 300 DPI)


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Strontium fluoride particles precipitated at R0 = 0.5 and various initial degrees of supersaturation Sa,0 = 10 (a,b), 30 (c) and 75 (d). 280x193mm (96 x 96 DPI)



Mean size of calcium fluoride nanoparticles precipitated at Sa,0 = 50 and at various initial free lattice ion ratios. Red filled circles: suspension diluted with deionized water; blue diamonds: suspension diluted with surfactant solution; solid line: calculated results with γcl = 0.187 J/m2 . 234x218mm (96 x 96 DPI)


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Calcium fluoride nanoparticles precipitated at Sa,0 = 50 and at various free lattice ion ratios R0 = 0.005, 0.5 and 50 (from top to bottom). 142x300mm (96 x 96 DPI)



Mean size of strontium fluoride nanoparticles precipitated at Sa,0 = 40 and various initial free lattice ion ratios. Blue diamonds: suspension diluted with surfactant solution; solid line: calculated results with γcl = 0.161 J/m2 . 74x70mm (300 x 300 DPI)


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On Precipitation of Sparingly Soluble Fluoride Salts - Crystal Growth

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