Two-Stage Model Reveals Barite Crystallization Kinetics from Solution

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 10864−10874

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Two-Stage Model Reveals Barite Crystallization Kinetics from Solution Turbidity Zhaoyi Dai,*,†,‡ Fangfu Zhang,† Amy T. Kan,† Gedeng Ruan,† Fei Yan,† Narayan Bhandari,† Zhang Zhang,† Ya Liu,† Alex Yi-Tsung Lu,† Guannan Deng,† and Mason B. Tomson†,‡ †

Department of Civil and Environmental Engineering and ‡Nanosystems Engineering Research Center for Nanotechnology-Enabled Water Treatment, Rice University, 6100 Main Street, Houston, Texas 77005, United States

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S Supporting Information *

ABSTRACT: The mechanistic modeling of mineral crystallization is essential for the understanding and control of many natural and industrial processes. In the past century, many mechanisms and models have been proposed to explain observations in different crystallization stages. However, most models only focus on a certain step or mechanism (e.g., nucleation, aggregation) and lack a comprehensive view. Incorporating nucleation, aggregation, and surface reaction together, this study developed an analytical two-stage crystallization model to simulate the particle size and number concentration versus time and correlate them with the measured solution turbidity. Through measuring solution turbidity in real time, this model can reproduce the crystallization process by predicting the key parameters: nucleation rate, particle size, number concentration, surface tension, induction time, and particle linear growth rate. Most of these values for barite crystallization match with literature data and our direct cryo-transmission electron microscopy (cryo-TEM) measurements. Moreover, the established relationships of these key parameters versus temperature and supersaturation enable this model to predict barite crystallization kinetics based only on the initial supersaturation and temperature. This study is a potential starting point to more quantitatively and comprehensively analyze and control mineral crystallization, important to various science and engineering applications.

1. INTRODUCTION

crystallization process and to guide the engineering applications in various disciplines. In the past century, many models and mechanisms, including classical nucleation theory (CNT),26−30 nonclassical nucleation theory, 31−34 surface growth, 35 aggregative growth,36−39 and Ostwald ripening,40,41 have been proposed for different crystallization stages. However, most of these models cannot be directly adapted for crystallization kinetics prediction due to the lack of a comprehensive view of the whole crystallization process or a quantitative mathematical model. For example, CNT is only focused on the nucleation stage; Ostwald ripening is only applied to describe the behavior between particle “encounters” (i.e., particles with distance less than their mean linear dimension), especially when the particle size distribution is relatively wide and the size is relatively large.41−43 In the past decade, nonclassical nucleation theories have been proposed by different research groups: Colfen, Gebauer, and their group proposed that the

Mineral crystallization is key to natural rock formation and biomineralization,1,2 which impact the chemistry of the geosphere, hydrosphere, biosphere, and atmosphere.3 It also plays an important role in various industrial disciplines, including CO2 geological sequestration, water treatment, scale formation in heating or cooling devices and in oil and gas production facilities, drug purification, and cloud seeding, to mention a few.4−13 For example, CO2 can be geologically sequestrated in the form of carbonate minerals through reacting with aquifer minerals or dissolved solids;7 mineral scale formation (e.g., calcite, barite, silicates) can cause excessive energy consumption, flow rate reduction, and economic losses in water treatment processes by forming in the heat exchange units14 or on the membrane surfaces;15−21 mineral crystallization plays an important role in the removal of heavy metals, including chromium, radium, zinc, lead, etc.;12,22−24 mineral scale formation in the oil and gas industry can cause economic loss every year of millions of dollars and raise severe environmental concerns.10,25 Therefore, a better understanding of the mineral crystallization mechanism and kinetics is important to analyze such natural and industrial © 2019 American Chemical Society

Received: Revised: Accepted: Published: 10864

March 27, 2019 May 28, 2019 June 4, 2019 June 4, 2019 DOI: 10.1021/acs.iecr.9b01707 Ind. Eng. Chem. Res. 2019, 58, 10864−10874

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Industrial & Engineering Chemistry Research

solution (CAT) contains Ba2+ as BaCl2 with 1 mol of NaCl/kg of H2O and 0.025 mol of Ca2+as CaCl2/kg of H2O as background, and the anion solution (AN) contains SO42−, as Na2SO4, and the same background electrolyte. The use of 1 m NaCl keeps ionic strength essentially constant during the crystallization process and minimizes the impacts of barite crystallization on the ion activity coefficients. Calcium ions are present to better simulate realistic conditions; for example, in nature barium and sulfate ions are generally accompanied by calcium ions. Since barite crystallization kinetics is barely impacted by pH within the range pH 3−9,59 no buffer is added in this experiment. The novel model developed in this study does not include pH impacts. Moreover, this solution composition also represents typical produced water in the oil and gas industry according to the USGS produced water database.60 In the oil and gas industry, barite is one of the most common scale minerals and can cause enormous economic loss through pipeline blockage, formation damage, or equipment malfunction.10,61 Both CAT and AN solutions were prepared and filtered with 0.25 μm cellulose filter twice to eliminate small particles that could lead to heterogeneous nucleation. In this study, the saturation indices (SI), defined as the logarithm (base 10) of the ratio of ion activity product to the solubility product, was adopted to represent the saturation status of the system. The SI values of barite were calculated using ScaleSoftPitzer, which is said to be one of the most accurate software for scale predictions.9,10,62−64 Solutions with SI values of 1.57, 1.77, 1.87, 1.98, 2.26, and 2.60 were tested at 25 °C, and the corresponding final barium and sulfate concentrations were 0.597, 0.752, 0.844, 1.32, and 1.96 mm, respectively. A 1 mL volume of CAT solution and 1 mL of AN solution were transferred by glass syringes (Hamilton GASTIGHT) and simultaneously injected into the reaction vial (polystyrene, 1.0 × 1.0 × 4.5 cm) as shown in Figure 1. Two glass syringes were clamped together to make

stable prenucleation clusters will form before the occurrence of critical nuclei;2,31,44 Myerson’s group suggest that the ions will form a cluster of solute molecules of sufficient size followed by reorganization into ordered structures, rather than directly forming a structured nucleus as suggested in CNT;4,33,45 RuizAgudo et al. found that a liquid−liquid phase separation between a denser and less dense phase happens before the formation of the primary crystalline barite nucleus.46 Unfortunately, these nonclassical nucleation theories cannot be used to quantitatively predict crystallization kinetics. Therefore, a quantitative model based on a solid theoretical basis is needed to give a comprehensive understanding of the crystallization process. Various experimental methods have been used to monitor the crystallization process and as the basis of theoretical model development. Many of their accuracies are limited by spatial resolution, temporal resolution, or statistical reliability. For example, methods that monitor solution property changes, including turbidity,28,47, refractive index,4 and conductivity,48 do not have enough spatial resolution to detect changes before the induction time (ti), which by definition is the time elapsed between the creation of supersaturation and the detection of changes.26,29,49 The quasi-time-resolved cryo-transmission electron microscopy (cryo-TEM)39,50,51 has better spatial detection limits, but is difficult to perform in real time. The advent of in situ liquid, or environmental, TEM34,52−54 improved the real-time spatial detection limits to less than 5 nm, but it is conducted only within certain limited observational regions of the solution. Colfen, Gebauer, and their group first used the novel ion selective electrode to monitor free ion concentration changes before and after the nucleation stage,2,31,55−58 and can detect the nucleation time of calcite crystallization. Unfortunately, more efforts are needed to develop a quantitative nonclassical nucleation theory to better utilize this experimental data. In order to analyze the crystallization process continuously and reliably, this study took advantage of both the high temporal resolutions and statistical reliability of turbidity measurement, and the high spatial resolutions of cryo-TEM. A novel two-stage crystallization model has been developed to predict crystallization kinetics by considering nucleation, aggregation, and surface reaction. This model also adopts the Mie light theory to correlate the changes of particle size and number concentrations with solution turbidity, which are accurately recorded using a laser apparatus in real time. Based on the real-time solution turbidity of barite crystallization, the new model can accurately predict barite crystallization kinetics, particle size, surface tension, surface reaction rate, and induction time, which match well with literature data and our cryo-TEM measurements. By establishing the correlation of nucleation rate, nucleus number concentrations versus temperature and supersaturation, this model accurately predicted barite induction times up to 90 °C. The excellent match to experimental data implies the potential application of this model to predict other crystallization processes based only on the initial solution conditions.

Figure 1. Sketch of the turbidity measurement apparatus setup.

such dual injection syringes. The stirring speed was 300 rpm to ensure the solution was completely mixed. A green light laser (wavelength is 532 nm, Edmund Optics) beam passed through the solution, and the intensity of the transmitted light detected by a Si photodiode detector was recorded using a multimeter (Radioshack) for each second. At the end of the induction period (i.e., when laser intensity starts to decrease), 3 μL of solution was collected and loaded onto the Quantifoil substrate grid (Orthogonal, 300 mesh, copper, 2 μm holes, TED PELLA, Inc.), and vitrified with liquid ethane after bolting for 2 s using Vitrobot. The samples were kept in liquid nitrogen during the whole transferring and imaging process. Cryo-TEM (JEOL 2010 with cryo) was set at 120 keV high tension, and the images were taken using a minimum dose system (MDS). The

2. EXPERIMENTAL DESIGN Barite (BaSO4) precipitation was analyzed in this study because it is a common mineral, it has only one crystalline phase, and its stoichiometry is simple. Reagent grade, or better, BaCl2·2H2O, Na2SO4, CaCl2·2H2O, and NaCl solids were dissolved in Milli-Q water to prepare the solutions. The cation 10865

DOI: 10.1021/acs.iecr.9b01707 Ind. Eng. Chem. Res. 2019, 58, 10864−10874

Article


temperature (T, K), effective interfacial energy (γ, J m−2), and the pre-exponential term (A, s−1 m−3). The slope kJ is a dimensionless function of temperature; Nav (no. mol−1) is the Avogadro constant; kB is the Boltzmann constant. In this study, the primary nuclei were assumed to be spherical which is reasonable for such small particles66,67 and consistent with the cryo-TEM observations in this study which were discussed later. 3.3. Surface Growth. The surface growth is controlled by the diffusion and surface reaction. The overall crystal growth rate is usually from first to second order.26 As with previous studies,26,68 the surface growth was assumed to be first order. For the primary nucleus, the surface growth can be described by the following equation:

electron density is always kept lower than 20 electrons/(s A2) in the imaging process.

3. MODELING 3.1. Two-Stage Crystallization Model. Focusing on simulating changes of particle size and number concentrations during the crystallization process, we developed a comprehensive two-stage crystallization model by coupling classical nucleation theory (CNT), surface growth reaction, and Smoluchowski aggregation theory. The stage of supersaturation formation is assumed to be very short when the cation and anion solutions are mixed thoroughly with a stirring bar. After the stage of supersaturation formation, the crystallization process is divided into two stages in this study. In the first stage, primary nuclei are formed through nucleation, assumed to follow CNT, which has been well-established with a solid mathematical basis.27 In the second stage the primary nuclei grow larger through surface growth and aggregation.52 In detail, the two stages can be expressed as the following reactions: first stage: c A ↔ A1c

(nucleation)

yz 4πρp n0r13 k r(4πr12)MW dt ijjj z jjC0 − − C∞zzz dV1 = 4πr1 dr1 = j zz ρp 3MW j k { 2

(5) 3

where V1 is the volume of the primary nucleus (m ) changing with time, t (s); kr is the first order surface reaction rate (m s−1); r1 is the radius of the primary nucleus increasing with time; MW is the molecular weight (e.g., 0.233 kg mol−1 for barite); ρp is the density of the particle, which is assumed to be the bulk density (e.g., 4.48 × 103 kg m−3 for barite); C0 is the initial concentration (mol m−3); C∞ is the equilibrium concentration (mol m−3). As a result, we can derive

(1)

second stage: +A

+A

+A

A1c ← → A1c + 1 ← → ··· ← → A1k

(k > c )

(surface growth)

ij yz 4πρp n0r13 dr1 jj z MW = k r jjC0 − − C∞zzz j zz ρ dt 3MW j k { p

(2)

A ia + Abj ↔ A ia++jb

(i , j > c )

(aggregation)

(3)

where “A” stands for an ion pair (e.g., BaSO4). The subscripts i, j, and k represent the number of ion pairs in the particle; the superscript represents the number of the primary nucleus in the particle. c stands for the number of ion pairs needed for a critical radius; thus A1c is the primary critical nucleus. By fitting the particle size distribution at the early stages of crystallization collected by in situ TEM, Woehl et al. find that the crystal growth is dominated by aggregation following Smoluchowski theory, rather than Ostwald ripening using the Lifshitz−Slyozov−Wagner model.52 Moreover, after a thorough review of existing theoretical and experimental research on crystallization processes, Wang et al. suggested that classical nucleation, aggregative growth, and Ostwald ripening occur in sequence, possibly with temporary overlaps between two consecutive regimes.65 Therefore, in this study Smoluchowski aggregation theory with surface growth was incorporated into our model to describe the successive crystal growth after nucleation in the second stage. 3.2. Classical Nucleation Theory. At the end of the first stage (i.e., nucleation stage, t = tn), it is assumed that there are n0 (no. m−3) primary nuclei with radius rc present in the solution, and the kinetics are assumed to follow CNT:27 16πγ 3(Vm/Nav )2 ji n zy ln(J0 ) = lnjjj 0 zzz = ln(A) − jt z 3kB3T 3[SI ln(10)]2 k n{ kJ = ln(A) − 2 SI

(6)

At the beginning of the second stage (i.e., t = tn), the radius of the primary nucleus is equal to the critical radius rc according to CNT: r1|t = tn = rc =

0.8686Vmγ RT ·SI

(7)

3.4. Aggregation. The aggregation kinetics can be described by the second order Smoluchowski theory with the following equation:69 dnk 1 = dt 2

∞

∑

βijninj − nk ∑ βik ni

i+j=k

i=1

(8)

where subscripts i, j, and k represent the number of the primary nucleus in the larger particles; n[i, j, or k] represents the number concentration of the corresponding aggregates; βij is the second order aggregation rate between particles with i and j primary particles (i.e., critical nuclei) inside. Since most of the particle sizes were in nanoscale, Brownian motion was assumed as the dominant aggregation pathway. The size dependent Brownian motion aggregation rate was calculated by βij =

(4)

2 2kBT (ri + rj) 3μ rri j

(9)

where μ is the water dynamic viscosity (0.000 891 kg m−1 s−1 at 25 °C); ri is the radius of particle i. It is commonly assumed that the aggregating particles are of equal size;52,68,70 then the rate is 8kBT/3μ. Based on this assumption, the total number concentration of all particles (np) is

where the nucleation rate (J0, s−1m−3) changes with saturation index, SI ≡ log10(aBa2+aSO42−/Ksp,barite), with aBa2+ and aSO42− being cation and anion activities, and Ksp is the solubility product. J0 depends on the mineral molar volume (Vm, m3 mol−1), 10866


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Industrial & Engineering Chemistry Research Table 1. Coefficients of the Eighth Order Polynomial Equation for Mie Theory in eq 15 j

pj

j

pj

j

pj

1 2 3

−1.524 367 186 63 × 10−3 1.952 537 067 90 × 10−2 −7.892 091 496 87 × 10−2

4 5 6

1.471 719 894 45 × 10−1 −4.768 157 832 36 × 10−2 7.622 925 016 73 × 10−3

7 8 9

−6.439 365 606 41 × 10−4 2.364 130 828 07 × 10−5 −1.204 087 695 68 × 10−7

np =

n0 1 + t /τ

(10)

and the number concentration of the particles aggregated from k primary particles (nk) is nk =

n0(t /τ )k − 1 (1 + t /τ )k + 1

(11)

where τ = 1/kan0 is the characteristic time of aggregation, and ka = 4kBT/3μka is characteristic aggregation rate. After aggregation, the primary particles can be stabilized due to particle−particle short-range interactions, especially when the particle size is small.71 Assuming that the primary particles have the same surface growth rate before and after aggregation the surface growth Vk = kV1

(12)

where Vk is the volume of the larger particles aggregated from k primary particles; V1 is the volume of primary particles changing with time as shown in eq 5. During aggregation, the structures of the aggregated particles can be very complex.72 Without an accurate quantification of the light scattering efficiency of the aggregated particle, this study assumes that the aggregated particles have the same scattering efficiency as a spherical particle with the same volume. Therefore, their effective radius (i.e., rk) for turbidity calculation can be derived as rk = k1/3r1

Figure 2. Scattering efficiency factor calculated from Mie theory through Monte Carlo simulation75 (blue), polynomial fitting (red), and Rayleigh scattering theory (green), within the range of size parameter 0 < x < 7.85. R2 = 0.999 994.

scattering deviates from the simulated values when the particle size becomes large. In addition, in order to get rid of multiple scattering, only the turbidity data less than 10 was used in the regression.76 After substituting eq 15 into eq 14, turbidity is ∞

(13)

3.5. Mie Light Theory. The solution turbidity is defined as the total scattered energy per unit volume per unit incident energy.73 Turbidity can be calculated as follows:73 turbidity = −

1 jij It yzz lnj z = l jjk I0 zz{

i

i=1

(16)

∞

turbidity =

(14)

9

=

j=1

∑

9 yz pj n0((t − tn)/τ )i − 1 ijjj z j∑ 2j − 1π jr1 j + 1i(j + 1)/3zzzz i+1 j j − 1 j z (1 + (t − tn)/τ ) j j = 1 λm k { pj n 0 2j − 1π jr1 j + 1 (1 + (t − tn)/τ )((t − tn)/τ ) λm j − 1

i

(t − tn)/τ yzz (j + 1)/3 zz i + 1 (t − tn)/τ z{ i=1 k 9 pj n0 =∑ 2j − 1π jr1 j + 1 j−1 + − (1 ( t t )/ λ n τ )((t − tn)/ τ ) j=1 m ij (t − tn)/τ yz zz Li−(j + 1)/3jjjj zz k 1 + (t − tn)/τ {

∑ jjjjj

j=1 ∞

i

(17)

where the polylogarithm function Lis(h) of order s and i s independent variable h is defined as Lis(h) = ∑∞ i=1 h /i . Unfortunately, r1(t) in eq 6 cannot readily be integrated in a form that can be explicitly substituted into eq 17; eq 6 must be solved numerically via Runge−Kutta methods. In summary, eq 17 explicitly incorporates the Mie light theory and Smoluchowski aggregation theory and implicitly, through r1(t) from eq 6, includes nucleation time and number

9

∑ pj (2πri/λm) j − 1

∑ i

where It and I0 are the transmitted and incident light energy, respectively;73 l is the light path length (0.01 m in this study); ni is the number concentration of particles (no. m−3) with radius ri; Qs,i is the scattering efficiency factor dependent on particle size and wavelength (λm). Mie light theory has been accurately simulated using Monte Carlo methods in previous studies.74,75 In this study, the particle size was less than 500 nm for most cases and thus the size parameter x (defined as 2πr/λm) was less than 7.85. Within this size range (0 < x < 7.85, 0 < rp < 500 nm), an eighth order polynomial function was used to fit the scattering efficiency factor of Mie theory simulated by Monte Carlo methods: Q s, i =

ji 9 zy 2j π n r ∑ i i jjjj∑ pj (2πri/λm) j − 1zzzzz j j=1 z i k { ∞

If we substitute eqs 11 and 13 into eq 16, we can get

∞

∑ niπri 2Q s,i

∑ niπri 2Q s,i =

turbidity =

(15)

where the parameters pj are listed in Table 1. Figure 2 shows that the fitted polynomial equation used in this study can accurately simulate the scattering regime, while Rayleigh 10867


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Figure 3. Measured and fitted turbidity changes with time are plotted in blue dotted lines and red solid lines, respectively. The simulated nucleus radius change with time is plotted in green dashed lines. The blue arrows represent the induction time according to the measured turbidity changes, and the red arrows mark the nucleation time predicted by our new model. The different SI values are listed in the title of each subfigure.

Table 2. Parameters Derived from Barite Turbidity vs Time Regression at Different SI Values at 25 °C barite SI

n0 (no. m−3)

tn (s)

rc (nm)

ln(n0/tn)

1.57 1.77 1.87 1.98 2.26 2.60

3.00 2.13 8.22 1.46 2.23 5.46

× × × × × ×

774.95 36.54 20.61 6.12 3.01 0.46

0.87 0.78 0.73 0.69 0.61 0.53

26.68 31.69 33.62 35.41 38.85 41.62

1014 1015 1015 1016 1017 1017

kr (m s−1) 4.18 4.18 4.18 4.18 4.18 4.18

× × × × × ×

C0 − C∞ (mol m−3)

measd ti (s)

0.48 0.63 0.72 0.82 1.17 1.78

1245 270 170 110 45 15

10−6 10−6 10−6 10−6 10−6 10−6

np(t=ti) (no. m−3)

r(t=ti) (nm)

pred ti (s)

C0 (Ba2+, mol m−3)

× × × × × ×

35 33 24 19 11 6

1192 230 138 94 32 16

0.575 0.725 0.815 0.915 1.265 1.875

9.40 5.32 1.03 1.55 4.98 1.00

1013 1014 1015 1015 1015 1016

4. RESULTS AND DISCUSSION Turbidity data during the induction period and the successive crystal growth period (turbidity < 10 m−1, Figure 3) was included in the regression. The blue dots in Figure 3 show the solution turbidity changes measured during barite crystallization at saturation index (SI) values ranging from 1.57 to 2.60. By fitting these six turbidity vs t curves simultaneously (red lines in Figure 3), the slope kJ in eq 4, the first order surface reaction rate (kr), and the nucleation times (tn, red arrows in Figure 3) under different SI values were generated. Note that based solely upon experimental data at least two turbidity vs t curves, at different SI values, a set of the nucleation and growth parameters can be obtained. Using the

(tn,n0) and surface reaction rate (kr). It bridges the measured solution turbidity with the changes of particle size and number concentration with time. The particle number concentration is constrained by CNT following eq 4. The particle size change due to surface growth is described by eqs 6 and 7. If the solution turbidity vs time (turbidity vs t) of mineral crystallization is measured at two SI values (at least), the slope kJ and intercept ln(A) in eq 4, the first order surface reaction rate (kr), and the nucleation times (tn, red arrows in Figure 3) for different SI values can be calculated using least squares curve fit. In this study, all six turbidity vs t curves in Figure 3 were used. 10868


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Figure 4. Particles measured by cryo-TEM at induction times when SI equals (A) 2.60, (B) 2.26, (C) 1.98, (D) 1.87, (E) 1.77, and (F) 1.57.

equations listed above, other parameters related to nucleation and growth were calculated and are listed Table 2. The good match between the fitted and measured turbidity curves demonstrates the ability of our model to reproduce mineral crystallization based only on initial conditions and solution turbidity changes with time. 4.1. Nucleation Time. In previous studies, most models fail to simulate the nucleation period distinct from the

induction period along with the successive periods of aggregation and growth. In addition, limited by the detection limits of various instruments, there are few experimental data available for the analysis of the nucleation period. In this study, a new two-stage crystallization model considers the nucleation and the crystal growth periods together in one unified model. This new model uses the complete turbidity vs t curve, instead of only the induction times, to better understand the 10869


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The fitted surface reaction rate (kr = 4.18 × 10−6 m s−1) corresponds to a linear growth rate of 0.18 nm s−1 at 25 °C and SI = 2, which matches well with the measured linear growth rate of 0.23 nm s−1 at the same conditions.28 The green dotted lines in Figure 3 show the simulated particle size increase via surface reaction. For comparison, Figure 4 shows the cryoTEM images of solution samples collected at the induction times. The measured particle radii range from 10 to 35 nm and match well with our model predictions, ranging from 6 to 35 nm. In the cryo-TEM measurements, the particles, which were frozen by liquid ethane with random orientations, show round projection shapes. This shows that the spherical particle assumption is probably valid. Figure 4E,F also shows the attachment and overlap of several particles, illustrating the possible occurrences of random aggregation. 4.4. Induction Time vs SI and Temperature. Induction time, ti, is the time elapsed between the establishment of supersaturation and the formation of a detectable crystalline phase. In this study, it is taken to be the time at which turbidity = 0.05 m−1 (see Figure 3), but other values of turbidity could have been used. This definition indicates that induction time can correspond to the time when mineral crystallization starts to cause problems that cannot be ignored, for example, pipe blocking or membrane fouling. In various engineering applications, the induction time (ti) has been used to represent the kinetics of mineral crystallization, and accurate induction time prediction is critical.4,27,85−89 The circle data points in Figure 5A show that our new model can accurately predict the induction times measured in this study. At other supersaturation values, this new two-stage crystallization model can be used to predict mineral crystallization kinetics if the interfacial energy (γ), intercept ln(A) for CNT, the first order surface reaction rate (kr), and the nucleation times (tn) are known, or can be assumed. The

crystallization process. The nucleation times predicted by the two-stage crystallization model (plotted in red arrows in Figure 3 and listed in Table 2) are much smaller than the measured induction times (blue arrows in Figure 3), when turbidity changes are first detected visually. This indicates that, after the nucleation period, a certain period of time is needed for the nuclei to grow larger enough to be detected by our laser apparatus, a turbidity meter, or visually. Such differences raise concern about the assumption made in previous studies that nucleation time and induction time are equal and related conclusions drawn based on this assumption.49,77,78 4.2. Interfacial Energy. As discussed in the last paragraph of section 3, the slope kJ, i.e. 16πγ 3(Vm/Nav)2 3kB3T 3[ln(10)]2

in eq 4, can be achieved by regression. Therefore, the effective interfacial energy between barite and 1 m NaCl solution (γ) was derived to be 75.2 mJ m−2, which agrees well with the direct measurement of barite−water interfacial energy of 76.4 mJ m−2 via adhesion tension tests.79 Some studies have reported the barite−water interfacial energy ranging from 38 to 150 mJ m−2 based on Ostwald ripening theory or CNT using various assumptions, which might be questionable. For example, based on the Ostwald equations (i.e., larger particles have less solubilities), La Mer and Enüstün and Turkevich calculated γ to be 150 mJ m−2 from the measured barite solubility of different particle sizes.80,81 The accuracy was limited by the particle size determination, which was not stable during the experiments due to Ostwald ripening. Based on CNT, Garten and Head calculated the effective interfacial energy to be 107 mJ m−2 by measuring the particle size and number concentrations microscopically and assuming a nucleation time of 10−8 s.82 Their particles observed microscopically were not critical nuclei, and the assumption about nucleation time yielded large uncertainty. In other studies, the number concentration of particles was assumed to be constant at relatively high SI, and the nucleation rate was calculated from the nucleation time based on the induction time measured photographically49 or light scattering recordings.47,83 The effective interfacial energy of barite in pure water was fitted to be 135, 38, and 93.4 mJ m−2, respectively, and that in 1 m NaCl solution was fitted to be 79.2 mJ m−2 by He et al.47 However, the reliability of these results was questioned since (1) the particle number concentration is subject to change with SI, and (2) the assumption that ti = tn was shown to be questionable by this study. Our model avoids such assumptions that may lead to errors, and thus the derived nucleation rates and interfacial energy of barite in 1 m NaCl solution based on the turbidity vs t, in this study, are probably more reliable. 4.3. Particle Number Concentration and Size. Based on eq 4 and the fitted tn, kJ, and ln(A), the critical nucleus number concentration (n0) at the nucleation time was calculated to increase with SI values from 3.00 × 1014 to 5.46 × 1017 m−3, as shown in Table 2. In previous studies, the particle number concentrations were measured at the induction time (ti) when observable changes occurred.49,84 At the induction time ti, the particle number concentrations are predicted to increase from 9.40 × 1013 to 1.00 × 1016 m−3 with SI, which are consistent with previously reported increases from 1012 to 1018 m−3.49,84

Figure 5. Barite crystallization induction time in 1 m NaCl solution predicted by our two-stage crystallization model by setting threshold turbidity to be 0.05 at (A) 25, (B) 50, (C) 70, and (D) 90 °C. The horizontal axis is the measured induction time by this study (yellow circles) and He et al.47 (blue squares). The dashed line shows the perfect agreement between predicted and measured induction time. The data in He et al.47 with the induction time less than 10 s were not included for comparison due to the large uncertainties in the measurements. 10870


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Figure 6. Linear relationship between natural logarithm of (A) fitted nucleation rate, (B) number concentration of critical nuclei, (C) nucleation time, and (D) simulated particle number concentration at induction time, versus 1/SI2.

first three values are independent of SI and are derived in this study. In addition to the linear relationship between the logarithm of nucleation rate and SI−2, which is described by CNT in eq 4, Figure 6 shows that the logarithms of nucleation time (tn) also have linear relationships with SI−2. Such a linear relationship allows us to characterize the whole crystallization process and reproduce changes of solution turbidity, particle size and number concentrations, and the induction times, based only on the initial SI value of barite. The blue squares in Figure 5A show that our new model can also accurately predict the induction times measured in other studies at 25 °C.87 In addition, Figure 6 also shows that the logarithms of the particle number concentration at the nucleation time and at the induction time have linear relationships with SI−2. This could be the fundamental theoretical reason why such linear relationships have been empirically observed in previous studies.47,49,82,83 In addition, induction times at higher temperatures, up to 90 °C, can be calculated from our results at 25 °C if it is assumed that the interfacial energy (γ) and intercept ln(A) for CNT are constant. The temperature dependencies are assumed to be represented by the slope kJ described by CNT following eq 4, and the surface reaction rate kr in eq 6. By fitting the induction times measured at 50, 70, and 90 °C as shown in Figure 5, parts B, C, and D, respectively,87 the surface reaction rates kr were calculated and plotted in Figure 7. Based on the Arrhenius equation, the activation energy of the surface growth is 30.6 kJ mol−1 (Figure 7). The activation energy value matches well with the previously published activation energy of 33.5 ± 4.0 kJ mol−1.90 These agreements further show the reliability of our two -state model to interpret barite crystallization mechanisms from the initial barite supersaturation and temperature. 4.5. Implications. Mineral crystallization is a fundamental and ubiquitous process in various applications. These applications have been limited by the lack of a comprehensive understanding of mineral crystallization mechanism and the

Figure 7. Surface reaction rate (kr) changing with T−1. The temperature is in the unit of kelvin, and the first order reaction rate is in units of m s−1.

lack of a reliable kinetics prediction model. The outcomes of this study can help fill such limitations. Mineral crystallization and deposition on flat surfaces or in the porous media can lead to tube/pipe blocking, membrane fouling, and aquifer plugging, to mention a few. On the other hand, nucleation and crystal growth are important for water and wastewater treatment to remove heavy metals, arsenic, and phosphates. Many models have been developed to analyze deposition or fouling processes. For example, Wiesner’s research group has developed models to describe nanoparticle aggregation and deposition on flat surfaces and in porous media;91−93 Han et al. have developed a network model to understand pore blockage and cake filtration by considering the properties of both membrane and individual particles.94 Such models could be more accurate by using a more fundamental model, as developed herein, for particle size and number concentrations during the nucleation and crystallization processes. 10871


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GE, Hess Italmatch, JACAM, Kemira, Kinder Morgan, Lubrizol, Nalco Champion, OASIS, OXY, RSI, Saudi Aramco, Schlumberger, Shell, SNF, Statoil, and Total. This work was supported by the NSF Nanosystems Engineering Research Center for Nanotechnology-Enabled Water Treatment (ERC1449500). The authors also would like to thank Prof. Pedro Alvarez, Prof. Joseph Hughes, and Prof. Linda Driskill for their careful comments, edits, and revisions.

It also deserves notice that mineral crystallization kinetics can be impacted by various factors, such as pH,59 cation to anion ratio,95 background electrolytes,96,97 etc. This research has not systematically studied such important impacts, but the novel model developed in this study can be potentially adopted in the future.

5. CONCLUSIONS This study has developed a novel two-stage crystallization model to predict the crystallization mechanism based on the solution turbidity changes. This model simulates the changes of particle size and number concentrations by adopting CNT in the first stage of crystallization and Smoluchowski aggregation theory and first order surface reaction kinetics in the second stage of crystallization. The Mie light theory is used to correlate the solution turbidity measured in real time with the simulated particle size and number concentrations. Based on the measured solution turbidity changes during barite crystallization under different saturation statuses at 25 °C, this model accurately predicts the nucleation rate, particle size, number concentration, surface tension, induction time, and particle linear growth rate. Most of these values match with literature data and our direct cryo-transmission electron microscopy (cryo-TEM) measurements. Moreover, this study also establishes the correlations between the saturation index and the nucleation rate, nucleation time, and particle number concentrations at the nucleation times and the induction times. Such correlations allow this model to reproduce the whole crystallization process and turbidity curve based only on the initial saturation status of barite and the temperature. As an example, the predicted induction times of barite under different supersaturation values at up to 90 °C match well with literature experimental data. Finally, this study forms a template against which to evaluate mineral crystallization kinetics from on a common theoretical basis. Using this new two-stage crystallization model, nucleation and crystal growth kinetics can be predicted either from measured turbidity vs time or from initial solution conditions. The model has implications in a wide variety of industrial and scientific disciplines.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b01707. Detailed steps of model regression using experimental data (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Zhaoyi Dai: 0000-0002-0879-1009 Ya Liu: 0000-0001-9719-4750 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge the financial support by a consortium of companies including Baker Hughes, BWA, Chevron, ConocoPhillips, Dow, EOG, ExxonMobil, FLOTEK, 10872


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Two-Stage Model Reveals Barite Crystallization Kinetics from Solution

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