When dealing with small particles, the curvatures of small surfaces play a key role in the thermodynamics. The surface energy is not negligible, and its contribution enters the energy balance of a system. One of the properties affected by the change of size and curvature is the melting point. Small particles may have a melting point depression of several hundred degrees.
Stølen and Grande [80] analyze the surface effect on the melting point, one should start by analyzing the equilibrium between a solid and a liquid droplet, having the same mass. The solid and the liquid are made of the same substance, to simplify the calculation. The chemical potentials µ of the liquid and the solid droplet are equal:
µ𝒔(𝑻, 𝒑𝒔) = µ𝒍(𝑻, 𝒑𝒍) Equation 2
85 The equilibrium conditions require equal temperatures, but different pressures. The superscript * refers to the reference phase, “s” for solid and “l” for liquid. The previous equation can be rearranged in the form below, after the chemical potential as a power series, and by deriving partially with respect to temperature and pressure.
µ𝒍∗(𝑻∗, 𝒑∗) − µ𝒔∗(𝑻∗, 𝒑∗) + (𝑺𝒍− 𝑺𝒔)(𝑻 − 𝑻∗) +𝑴
𝝆𝟏(𝒑𝒍− 𝒑∗) −𝑴
𝝆𝒔(𝒑𝒔− 𝒑∗) = 𝟎 Equation 3
In this equation, the first two terms are the chemical potential of the chosen reference states at the temperature and pressure T* and p*. They are both assumed to be zero. The third term contains the difference in the entropy in the solid and liquid. It corresponds, by definition, to the enthalpy of fusion ΔfusHm, with changed sign, and divided by the reference temperature T*. The fourth and the fifth term are the partial derivative of the chemical potential with respect to pressure, multiplied by the difference in pressure from the reference state. ρl and ρs
are densities, whereas ps and pl are the pressure in the solid and in the liquid droplet respectively.
The equation can be rearranged to
−𝜟𝒇𝒖𝒔𝑯𝒎
𝑻∗ (𝑻 − 𝑻∗) +𝑴
𝝆𝒍(𝒑𝒍− 𝒑∗) −𝑴
𝝆𝒔(𝒑𝒔− 𝒑∗) = 𝟎 Equation 4
From the Laplace equation on pressure, ps and pl can be calculated by knowing the gas pressure, the surface tension with the gas and the radius of curvature. The equation can be rearranged to
𝜟𝒇𝒖𝒔𝑯𝒎
The mass of the liquid and the solid are assumed to be the same. It follows that the curvature of the liquid and the solid particles are related by ρl Vl = ρs Vs, where Vi is the volume of a sphere with radius ri. One can find a relation between rs and rl. By inserting it, the final equation will be
𝟏 − (𝑻𝒇𝒖𝒔)𝒓 respectively. The melting point decreases with decreasing size of a particle. Figure 69 shows an example for gold nanoparticles. The effect of the size starts becoming relevant in the order of magnitude of 10 nm. A particle of 1 nm diameter has reduced its melting point by more than 400 degrees. The trends of variation depend on every material, but this theory is applicable for all materials, even when solids and liquid differ in composition and amounts.
Figure 69: Decrease in melting temperature of a spherical gold particle as a function of its radius [80].
87
2. Experimental method
The main aim of this work is to investigate the type and structure of the condensates and to find the variables affecting their features. To do so, we have to produce a SiO(g)-CO(g) mixture, make it condensate, isolate the condensation products and analyze them. SiO2-SiC and Si-SiO2 pellets are the raw materials used for SiO(g) and CO(g) production. Two different equipments, differing in scale, are used to produce the condensates. Inert gases such as Ar(g) or He(g) could be added to tune the SiO(g) concentration and the gas velocity. Substrates of carbon, quartz or SiC collected the condensing gas.
After the experiments, the condensates microstructure and composition will be analyzed with different characterization techniques. In addition, the crucibles are excavated, to record the points where clogging occurs.
Samples of condensates are extracted during excavation, and analyzed with optical microscopy, SEM, TEM, XPS and XRD. The analysis is compared to condensates samples extracted during excavation or operation. Finally, from the condensate weight and the experimental conditions, a kinetic model calculates the reaction rate and the activation energy for one of the condensation reactions.
A. Experimental method overview
Figure 70 resumes the experimental parameters analyzed during the research. The main thought of the setup is that the SiO-CO gas mixture is the raw material for the research. The gas could be produced from two mixes of powders, a Si-SiO2 or a SiO2-SiC mixture. According to the size of the two setups chosen, the amount of pellets was either 20 g or 200 g. The other parameters related to the pellets were kept constant, to avoid variations in the gas production reaction kinetics. For example, size, composition, raw materials, pellets size and calcination were kept constant for each batch.
The other parameters analyzed in this work are the gas temperature, the holding time, the substrate composition and its size, and the added inert gas flow. The gas temperature was fixed for each experiment at a constant value of either 1890, 1900, 2000 or 2200°C. The target gas temperature was kept at times between 10 minutes and 4 hours. The graphite of the crucible walls, together with SiC and quartz particles, were chosen as surfaces for condensate deposition. Different substrate sizes may influence the effect of clogging given by condensation. The substrates particle sizes chosen were 3-5, 5-8, 3-10, 8-14 and 12-20 mm for SiC, and 3-5, 5-8 mm for Quartz A. The partial pressure of SiO(g) and CO(g) could be tuned by adding an inert gas. He(g) or Ar(g) were added at different flows, to tune the gas speed and the partial pressure of SiO(g). Some experiments were performed also without adding any gas, as this would be similar to the industrial process.
Figure 71 shows a sketch of how each experiment was handled. The final weight of condensates, the initial moles of gas and the temperature measurements through the condensation chamber are measured. They were used to compute the temperature of formation, the partial pressures of SiO(g) in the system, and as basis for calculation of reaction rate and kinetic constants.
Characterization techniques are used to investigate the features of different condensates. The main techniques used for compositional purposes were EPMA, EDS, EDX and XPS. EPMA and XPS are quantitative, whereas EDS and EDX are semi-quantitative. Microstructure was inquired by TEM and SEM, with the help of FIB-preparation when needed. SiC polytypes were found by peak fitting of XRD spectra.
Samples from industrial plants were collected from other projects or previous works. The samples were analyzed in SEM and EDS. The places from which the samples were collected are the following:
- REC Solar (Kristiansand, Norway), Furnace 11, samples extracted during operation in 2019.
- Wacker Chemicals (Kyrksæterøra, Norway), Furnace 1, excavated in 2016 [55]
- Wacker Chemicals (Kyrksæterøra, Norway), Furnace 4, excavated in 2016 [55]
- Elkem Salten (Straumen, Norway), Furnace 2, excavated in 2018. [81]
Figure 70: Experimental variables analyzed during the experiments: substrate composition and size, gas temperature, pellets composition, time exposure and inert gas flow added.
89 Figure 71: Work structure of experimental method and characterization.
B. Pellets characterization
SiO
2-SiC pellets
The raw materials are commercial quartz provided by Elkem ASA, which will be called Quartz B, and SiC from Washington Mills. The composition of the starting SiC was provided by Washington Mills, whereas the Quartz B analysis were performed by SINTEF Molab by using a Tiger 4 kW X-ray spectrometer. By XRD and XRF analysis
(Table 6), it was found that the quartz percentage in Quartz B is ≈99%. SiC powders are at least 95% pure. The XRD spectrum is shown in Appendix H.
Table 6: Composition of starting materials detected by XRD (Quartz B) and XRF (SiC)
Composition (wt. %) Quartz B Composition (wt. %) SiC
SiO2 98.9 SiC >95
Fe2O3 0.58 SiO2 1.6
Al2O3 0.17 Al 0.05
K2O 0.02 Fe 1.8
CaO 0.02
Most of the SiC powder size distribution ranges between 0.1 and 10 µm, whereas SiO2 particles are mostly between 1 and 100 µm size. Figure 72 compares the powder diameters of SiO2 and SiC. The average particle size for SiC is 0.953 µm. Quartz B powders were ground by using a Retsch RS200 grinding machine, equipped with a tungsten carbide grinding disc. The grinding time is 70 seconds, and the rotation speed is set at 1200 rpm. The mean particle dimension is reduced to 10.39 µm.
Figure 72: Particles size distribution of SiO2 and SiC powders in the SiO2-SiC pellets.
SEM analysis of pellets showed the final aspect of the particle size distribution (Figure 73). As demonstrated by the particle size distribution test, some large SiO2 particles are present in the sample. SiC surrounds them effectively and fills the voids together with small SiO2 particles.
91 Figure 73: Pellets characterization. The red rectangle corresponds to the zoomed area of the picture on the right. Zones with SiC are lighter than zones with silica.
The powders were mixed for two hours starting from a molar ratio SiO2 : SiC = 2:1 (75 wt. % SiO2 and 25 wt. % SiC), and then pelletized at room temperature, with water as binder. The pellets final diameter ranges between 1 and 2 mm. The pellets underwent drying for 6 hours at 120°C and calcination for 30 minutes at 1200°C. Before feeding the pellets into the experiments, a 600-µm size sieve separated the fines. Figure 74 illustrates the final aspect of the pellets.
Figure 74: Calcined SiO2-SiC pellets.