Nucleation and Ostwald Growth of Particles in Fe-O-Al-Ca Melt

Tremendous focus has been put on the control of particle size distribution which effects the grain structure and mechanical properties of resulting metallic materials, and thus nucleation and growth of particles in solution should be clarified. This study uses classical nucleation theory and Ostwald ripening theory to probe the relationship between the compositions of Fe-O-Al-Ca melts and the behavior of particles under the condition of no external stirring. Our experimental data suggest that decreasing the initial Ca addition and Al addition is conductive to the increase of nucleation rate for calcium aluminate particles, which exhibits a same change trend with that predicted from classical nucleation theory. Based on the experimental evidence for particles size distribution in three-dimensional, we demonstrate that Ostwald ripening is the predominate mechanism on the coarsening of particles in Fe-O-Al-Ca melt at early stage of deoxidation under the condition of no external stirring but not at later stage.

compared with Al 2 O 3 , the distribution curve of CaO-Al 2 O 3 was narrower and nucleation rate was higher, indicating that CaO-Al 2 O 3 particles were fine and in large amount in Fe-10mass%Ni alloy. They also advocated that the supersaturation degree, and interfacial energy between oxide particles and liquid Fe, and the equilibrium deoxidation constant affect the nucleation and growth of particles in early stage of deoxidation under no coagulation of deoxidation particles by collision. However, they just compared the size distribution of deoxidation products of MgO, ZrO 2 , Al 2 O 3 , CaO-Al 2 O 3 and MnO-SiO 2 in an Fe-10mass%Ni alloy. In spite of many experiments performed to investigate the formation mechanism and composition control of particles in steel by thermodynamic and kinetic theories [34][35][36] , limited studies about the effect of melt composition on the nucleation and growth of particles are conducted.
In current study, the relationships between compositions of Fe-O-Al-Ca melts with not only the particle type, but also the particle size distribution were analyzed. The nucleation and growth by Ostwald ripening of particles in Fe-O-Al-Ca melt were estimated and verified by experimental data. This study will provide information to predict the nucleation and growth of particles in the melt and will be helpful for controlling behavior of particle.

Results and Discussion
Experimental results. The  CaS is no more than 5%, thus, it is ignored. Figure 1a presents the SEM images of typical particles and their average composition evolution in the melt during deoxidation process. The SEM-EDS results of typical particles in A1C1 and A2C3 can be found as Supplementary Figure S1. The samples were taken at 1600 °C after deoxidation for 360 s, 600 s, 1800 s and 3900 s and timing started at Al powder added. Experiment A1C1 ([% Al] i = 0.05 and   Figure 2 depicts the particle size distribution in three-dimensional for the experimental samples at 3900 s based on the stereological analysis 38,39 . It can be seen that the peak of the curves in Fig. 2 tends to decrease with the increasing amount of initial Ca addition, indicating that the number density of particles decreases with increasing Ca at 3900 s. Besides, the size of particles corresponding to the peaks of curves in the samples containing higher Ca content is smaller than that in the melts containing lower Ca content. It is concluded that the number density and size of particles tend to decrease with an increase of initial Ca addition after deoxidation for 3900 s. The particle size distribution for different samples after deoxidation at 1600 °C for 360 s, 600 s and 1800 s are plotted in Fig. 3. The particles with size less than 260 nm can't be detected due to the limitation of resolution of SEM. Therefore, the curves of particle size distribution in Fig. 3b,c are incomplete. However, it still can be seen that the primary particles are smaller and in significantly larger amount in the melt with high initial Ca addition ([Ca%] i = 0.78), compared with those in the melt with low initial Ca addition ([Ca%] i = 0.25). The number of particles decreases significantly and large particles form with the proceeding of deoxidation, attributing to the floatation, aggregation and growth of particles.
Calculation results. Nucleation of Calcium Aluminates. In order to study the contents of Al, Ca and O on the nucleation rates of calcium aluminates, I CxAy (cm −3 ·s −1 ), it was estimated as the following relationship based on the classical nucleation theory 32 : 2 2 x y x y x y x y x y  According to the research of Li 40 and Suito 32 , supersaturation degree of calcium aluminate S CxAy can be expressed by Eq. (2). K CaO is calculated from the relation: log K CaO = −9.08 41 (1600 °C). In this calculation, the effect of melt compositions on the activity coefficient of Ca and O is not considered.
CxAy is the critical supersaturation degree which is the value of S CxAy at I = 1 (cm −3 ·s −1 ) and it is derived from Eq. (3). A is the frequency factor (10 26 Table 1.
The interfacial energy between solid particles and metallic melt can be expressed by Young's equation: The interfacial energy between liquid particle and metallic melt can be calculated by Neumann's relation 42 : where γ CxAy and γ Melt are the surface energies of calcium aluminate and metallic melt (J/m 2 ), θ is the contact angle between solid particle and melt, ϕ is the visible contact angle of liquid particle. The relation between ϕ and θ * (the contact angle between liquid particle and melt) can be expressed by Eq. 6 42 if vertical equilibrium is considered. The surface energy of metallic melt (J/m 2 ) used in this study was calculated at 1600 °C as following [43][44][45][46] : The relevant parameters 45,47-52 used in the calculation of nucleation rates for calcium aluminates are shown in Table 3. The activities of Al 2 O 3 and CaO in particles were estimated by Factsage Software 7.0 of "Equilib" module. The calcium aluminates with mole ratio of Al 2 O 3 /CaO at the range from 1 to 3 are almost in liquid state at 1600 °C and their visible contact angle is calculated by Eq. 6. In this study, CA type particle is treated as liquid because it is often in partly or totally molten state in the experiments of measuring wettability 53 and it is observed to be spherical or semi-spherical in the samples. The surface energies of solid calcium aluminates are estimated by the relation:  Fig. 4a-d, respectively. The nucleation rate of CaO·2Al 2 O 3 for given activities of Ca and O is larger than that of CaO·6Al 2 O 3 , and the nucleation rates of various types of calcium aluminates in the region with high supersaturation degree (a o > 0.03 and a Ca > 10 −6 ) increase in the order of Figure 4e,f is obtained by adding the nucleation rates of all types of calcium aluminates together which means that it is assumed that these calcium aluminates nucleate simultaneously. In order to verify the guiding significance of the nucleation theory on the metal smelting, the theoretical nucleation rates (ln I) of calcium aluminates in experiments were calculated based on the EDS results of particle compositions as shown in Table 4 and the activities of O, Al and Ca are obtained by substituting compositions of melt and thermodynamic data in Table 5 (thermodynamic data in Table 5 are derived from ref. 37 ) to Eqs (9-10) 41 . The theoretical nucleation rates changed little as shown in Table 4 obtained by considering the effects of a o and a Al separately, which verifies that Eq. (2) is well applied in this study and the reaction between O and Al reaches equilibrium state nearly before adding Ca in this condition. The reason why the theoretical nucleation rate of calcium aluminates in the melt with lower amount of Ca addition is larger than that with higher amount of Ca addition is mainly attributed to the discrepancy of particle composition.
where a i , f i and [mass i %] are the 1 mass% activity, 1 mass% activity coefficient, and the concentration of i in mass fraction, respectively. e i j is the first-order interaction coefficient and r i j k , is the second-order interaction coefficient.
The mean values of experimental nucleation rates (I) in Exp. A1C1, Exp. A1C3 and Exp. A2C3 were obtained with Eqs. (11 and 12):  Table 3. Relevant parameters of calcium aluminates used in this study. ρ is density. V o is molar volume of oxide. γ CxAy is surface energy. θ is the contact angle between solid particle and melt. ϕ is the visible contact angle of liquid particle. θ * is the contact angle between liquid particle and melt. a is activity. C P is oxygen or calcium concentration in oxide expressed by weight per unit volume. So * is the critical supersaturation degree.
is the volume fraction of all particles in the melt after Ca adding, that is, the volume fraction of paticles at 360 s. f V(Al) is the volume fraction of Al 2 O 3 just before Ca adding). The time for equilibrium of nucleus volume is 0.2 s 18 and the critical size of nuclei r C(CxAy) is the given by x y x y x y x y The calculated results suggest that the critical size of nuclei for calcium aluminates is about 0.2-0.3 nm. The experimental nucleation rates of calcium aluminates decrease in the order of experiments A1C1< A1C3< A2C3, exhibiting a same change trend with theoretical values, which indicates that decreasing the initial Ca addition and Al addition is conductive to the increase of nucleation rate for calcium aluminate. The average nucleation rate of calcium aluminates is smaller than that predicted from classical nucleation theory, presumably attributed to the underestimation of t and overestimation of local composition of melt.

Growth of Calcium Aluminates by Ostwald Ripening.
Ostwald growth of calcium aluminates in Fe-O-Al-Ca melt controlled by oxygen diffusion can be expressed as following 18 : where r t and r 0 are the mean radius of particles at time t (m) and that at the start of Ostwald growth (m), respectively. k d is coarsening rate (μm 3 ·s −1 ). D O is the diffusion constant of oxygen (2.91 × 10 −9 m 2 ·s −1 ), C O is the dissolved oxygen concentration expressed by weight per unit volume (kg·m −3 ) and C P(CxAy) is the oxygen concentration in oxide expressed by weight per unit volume (kg·m −3 ). α is the coarsening rate coefficient. In previous study 54 , it is found that calculated coarsening rate is more accurate by using α LSW from LSW theory instead of α CN from communicating neighbour (CN model). Therefore, α values as 4/9 in this study. The coarsening rate k d of each type calcium aluminate was calculated by substituting the relevant data listed in Table 3 into Eq. (14) and are plotted with oxygen content as shown in Fig. 5. As can be seen, the coarsening rate increases with increasing dissolved oxygen content. Besides, for a given dissolved oxygen (less than 300 ppm), an increase of C/A ratio in calcium aluminates increases their values of k d (except for CA) which increases in the order of CA 6 < CA 2 < C 12 A 7 < CA.
The effect of Ca addition on the Ostwald growth of particles in Fe-O-Al-Ca melt with [Al] i of 0.04% and 0.2% was obtained by considering the oxygen diffusion and calcium diffusion as shown in Fig. 6. The coarsening rate k d(Ca) can be expressed by Eq. (15) in which the notations are similar to Eq. (14). D Ca is assumed to be equal to the D O because the solute diffusivities in liquid Fe is considered to be the same order of magnitude 18 .
x y x y x y Ca diffusion will be the rate determining step when k d(Ca) is smaller than k d(O) and vice versa. Figure 6 is obtained based on Eqs (14) and (15), in combination with FactSage modeling. Equilibrium compositions of melt with Ca addition are estimated by FACTSAGE 7.0 with the FactPS and FToxid and FTmisc databases. (based on the compositions of raw materials)."Equilib" module is used, and pure solids, and Fe-liq and A liquid slag in solution phases are selected as products. Calculated temperature and pressure are set as 1600 and 1 atm, respectively. Al diffusion will not be the rate determining step due to its high concentration in this work. It is found that with the increasing amount of added Ca (0-0.015%), the content of soluble Ca at equilibrium increases at the range from 0 to 0.6 ppm, and the coarsening rate of particles derived from Ostwald ripening decreases firstly and then increases as the liquid calcium aluminates form and increase. The Ostwald growth of Al 2 O 3 is determined by O diffusion, while Ca diffusion is the rate determining step for the coarsening of calcium aluminates at equilibrium which is marked by blue line in Fig. 6. The value of k d decreased slightly with an increase of Ca addition in the "CA6 + CA2" region ([% Ca] i = 0.0027-0.0068) due to the increasing proportion of CA2. Besides, it indicates that the optimum amount of Ca addition for inhibiting the Ostwald growth of calcium aluminate particles is 0.0027-0.0068%. In addition, when the amount of initial calcium addition is larger than 0.0027%, the coarsening e j i / r i  Table 5. First-order and second-order interaction coefficients 41 e j i and r i (j,k) of various elements in liquid steel at 1600 °C.
SCientifiC REPORtS | (2018) 8:1135 | DOI:10.1038/s41598-018-19639-w rate increases with the increasing amount of Al addition because equilibrated calcium increases and Ca diffusion is the rate determining step in this case.
The observed coarsening rates are obtained by substituting the experimental data into Eq. (13) and are plotted with the calculated values (obtained by substituting the relevant data in Table 3 and the composition of samples  in Tables 1-2 to Eqs (14 and 15)) as shown in Fig. 7. It is found that k d(cal.) tends to increase with increasing k d(obs.) during the first 600 s of deoxidation process. The data at 360 s and 600 s in Fig. 7 fall around the line k d(cal.) = k d(obs.) , although there is some deviation in those data probably caused by error of measurement for particle size (nano scale particles are excluded by Image-Proplus during the analysis of particle characteristics). In addition, it should be noted that the triangular points in Fig. 7 at later stage of melting are out of line completely. It can thus be concluded that the Ostwald ripening is the predominate mechanism of coarsening for calcium aluminate particles in Fe-O-Al-Ca melt during the first 600 s after aluminum addition under the condition of no external stirring but not at later stage of Al-Ca deoxidation. The mechanism on coarsening of calcium aluminate particles in Fe-O-Al-Ca melt at later stage of deoxidation is still going on.

Conclusion
The behavior of particles in Fe-O-Al-Ca melt under the condition of no external stirring at 1600 °C was systematically studied using experimental methods, stereological method, classical nucleation theory, as well as Ostwald ripening theory.
The nucleation rate of calcium aluminates is dependent on their type and the composition of melt. It increases with an increase of a o and a Ca , and decrease of a Al . Our experimental data suggest that decreasing the initial Ca addition and Al addition is conductive to the increase of nucleation rate for calcium aluminate, which exhibits a same change trend with that predicted from classical nucleation theory. Based on Ostwald ripening theory, for a given dissolved oxygen (less than 300 ppm), coarsening rate of particles in Fe-O-Al-Ca melt increases in the order of CA 6 < CA 2 < C 12 A 7 < CA. The optimum amount of Ca addition for inhibiting the coarsening of calcium aluminates in Fe-O-Al-Ca melt is 0.0027-0.0068%. It is experimentally confirmed that the Ostwald ripening is the predominate mechanism of coarsening for calcium aluminate particles in Fe-O-Al-Ca melt during the first 600 s after aluminum addition under the condition of no external stirring but not at later stage of Al-Ca deoxidation. The mechanism on coarsening of calcium aluminate particles in Fe-O-Al-Ca melt at later stage of deoxidation is still going on.

Methods
High temperature experiments. High-purity iron was used as raw materials in the experiment and its chemical composition (wt.%) is 99.95% Fe, 0.0016% C, 0.0033% Si, 0.01% Mn, 0.0053% P, 0.0017% S, 0.003% Al, 0.0037% Cu, 0.0038% Ni. Al powder packed in iron foil (Al >99%) was first added in the molten steel at 1600 °C and after 5 min, Si-Ca alloy (59% Si, 30% Ca) was added for deoxidation, immediately stirred by a molybdenum rod for 5 s. All the experiments were carried out in Si-Mo heating electric resistance furnace without external stirring after adding Si-Ca alloy. Samples were taken by quartz tubes (Φ6 mm) for certain holding time which were injected with Ar gas firstly to prevent molten steel from being oxidized by air, followed by rapid quenching in salt water. During the whole melting process, the argon gas was controlled at the flow rate of 5 L/min.

Characterization of particles.
The compositions and morphologies of particles were observed using scanning electron microscopy with energy-dispersive spectrometric detection (SEM-EDS). The weight percentages of Al 2 O 3 and CaO in particles were calculated based on the stoichiometric relationship and contents of Al, Ca, O which were measured by EDS. The stereological analysis (modified Schwartz-Saltykov method with the probability mass function 38,39 ) was adopted to obtain the particle size distribution in three-dimensional from that in two-dimensional. The details are described as below 54 : the back-scattered electron pictures of each steel sample were taken under 1000 times corresponding to the area of 271 µm × 271 µm. 169 successive microphotographs were obtained by designating a step of 271 µm. Besides, the planar size and number of inclusions were analyzed by Image-ProPlus software 55 . The probability mass function (PMF) is expressed as following 37 : where P is the probability of a cross section with radius r (r i − Δr < r < r i ) intersecting a sphere with radius R which is the actual radius of inclusions in three-dimensional, and Δr is interval of groups. In this study, inclusions were classified into 49 successive groups from the largest inclusions based on the measured mean radius of inclusions in two-dimensional. The diameter of largest inclusions detected in all samples is no more than 19.6 μm. Therefore, the radium of inclusions in group 1 denoted by r = 9.8 μm or d = 19.6 μm is in the range of 9.7-9.9 μm, group 2 is denoted by r = 9.6 μm and group 49 is denoted by r = 0.2 μm. For group 49, r i = 0.3 μm and Δr = 0.2 μm. According to the study of Li Tao 38 , the detected two-dimensional inclusions in group j probably belong to the three-dimensional group i (i ≤ j) as expressed by Eq. (17).
A i j i V 1 where N A and N V are the number density of inclusions in two-dimensional and three-dimensional, respectively. The transformation from two-dimensional spherical inclusion size distribution to three-dimensional inclusion size distribution can be performed based on Eqs (18), and (19) is P matrix. (P −1 is inverse matrix of P matrix) (n) Detection of sample compositions. The compositions of samples were detected by the ICP-AES method (for the detection of Al and Ca, etc.), infared absorption method after combustion in an induction furnace (for the analysis of sulfur) and Leco analyzer (for the measurement of total oxygen). The initial oxygen in all the experiments was 170 ± 20 ppm. The insoluble oxygen, alumina, and calcium contents were calculated based on Eqs (16)(17)(18)(19)(20). Data availability. The data that support the findings of this study are available from Linzhu Wang upon reasonable request.