The evaporation rate of a single droplet is derived from mass conservation equation and simplified by assuming that the temperature inside the droplet reaches steady state condition. Therefore, a pseudo-steady state evaporation is proposed [15] . The solvent evaporation rate is determined by the decreasing droplet mass and can be formulated as:

where is the diffusion coefficient of solvent vapor in air, is Avogadro’s constant (6.0221413 × 10²³ 1/mole), is the molecular weight of water, and and are the vapor concentrations at the droplet surface and in the surrounding air, respectively [15] . Initial mass is determined from CFD modeling of spray for a droplet at the substrate. This value considers the mass of water inside the droplet. In the specific experiment considered for validation, the precursor comprised 1% ZnCl₂ (for ZnO single oxide) and 5% ZnCl₂ + 95% In(NO₃)₃ (for ZnO + In₂O₃ mixed oxide) of the total mass of droplet. Therefore, the mass of precursor can be neglected inconsideration of the total mass at this stage. The parameter in Equation (1) is the Knudsen correction which allows for the effect of transport when the size of droplet is on the order of the mean free path of molecules in air (λ~68 nm):

where is the droplet diameter.

The vapor concentration can be calculated as:

where is the pressure and is the mole fraction of solvent (water). The parameter can be defined as [15] :

Since, and are dependent on surrounding air pressure and temperature, the temperature and pressure are determined by CFD modeling of spray using the FLUENT software package which was developed in previous papers [13] [20] . The substrate temperature is fixed at 400˚C as a reference.

By solving Equation (1) in MATLAB, the final mass inside the droplet can be determined. This value can be used to calculate the initial droplet diameter and the concentration of initial precursor for the next stage.

The pyrolysis of the precipitate starts before the precipitate reaches the substrate and nucleation and growth of thin films on the substrate occurs later. Subsequently, a continuous thin layer of metal oxide is synthesized by the growth of the nuclei. The desired condition is for the droplet to approach the substrate when the solvent has been largely removed [21] .

A chemical reaction (thermal decomposition) occurs during spray pyrolysis in addition to the physical pro- cesses of evaporation and drying [15] . The chemical reaction for the system considered can be formulated as:

Since the decomposition rates of precursors are determined by the heating rate, a thermal analysis is performed at different heating rates to obtain the kinetic reaction equation as a function of activation energy [15] . Therefore, the general kinetic equation can be expressed as [15] :

where is the reaction rate, is gas constant (8.3144621 J/mol∙K), is the fraction of precursor

reacted [ZnCl₂ and/or In(NO₃)₃], and is the function of fraction reacted which depends on the mechanism of the decomposition reaction. This function represents the theoretical kinetic equation of the decomposition mechanism and it can be defined by normal grain growth model: where is a function of reaction order [15] . In Equation (8), is the pre-exponential factor and is the activation energy which can be determined from the Arrhenius kinetic reaction equation. By solving the differential Equation (8), the final can be found which is the fraction of the precursor reacted at the final stage.

The final diameter after chemical reaction is derived from the one-droplet to one-particle principle in spray pyrolysis [15] thus:

where is the final particle diameter, is initial droplet diameter, is concentration of initial precursor, is molecular weight of the generated particle, and is the density of the generated particle (ZnO in single oxide and ZnO + In₂O₃ in mixed oxide). Since only a fraction of precursor is involved in the chemical reaction at this stage, the final droplet diameter needs to be modified accordingly. Therefore, the final droplet size is corrected using the factor. In effect, when, the final is equal to the calculated diameter, and when, there is no particle created and.

The growth rate in a spray deposition process is linearly dependent on spray time and logarithmically dependent on the substrate temperature [16] . The experimental results of the study are in good agreement with the Arrhenius expression. Therefore the rate can be represented by:

where is the film thickness, is the growth rate coefficient, is temperature, is time and is the Boltzmann constant (=1.3806488‰ × 10⁻²³ m²∙kg/s²∙K). Based on the experimental results, the following parameters were used to solve Equation (10):, , ,. Since the amount of indium nitrate in the mixture is less than 20%, the same growth rate coefficient can be used for the mixed oxide cases.

The initial film thickness is derived from the reaction stage. Specifically, the final droplet diameter at that stage is assumed to be the same as the initial film thickness, assuming a spherical particle shape. The final film thickness is predicted subsequently by solving Equation (10). It should be remarked that since the precursor solution used for the spray is typically dilute, the above model is expected to be valid for both single and mixed oxides.

An experimental study was conducted to validate the modeling results. In the experiment, the precursor solution is sprayed onto a heated alumina substrate using spray pyrolysis. The precursor solution consists of a mixture of ZnCl₂ and In(NO₃)₃ dissolved in water for synthesis of ZnO + In₂O₃ and only ZnCl₂ for ZnO single oxide. The optimum processing conditions are strongly dependent on the composition of the precursor solution.

The deposited single and mixed oxide thin films are subsequently characterized by Scanning Electron Microscopy (SEM) for the particle size and film thickness. SEM combines high spatial resolution with a wide field of view to improve accuracy [22] . Image processing techniques are applied on the SEM micrographs to determine particle size distribution. Specifically, the mean diameter of each particle is obtained from which the average particle size is determined for all particle dimensions. The mean area of particles is calculated by considering the total number of particles and the space intervals, thus [23] :

where is the actual mean area, is the assumed mean area, is the class interval between particles, d is deviation of midpoint from assumed mean, and is total number of particles considered. Table 2 summarizes the input data used for the experiments.

Figure 3 shows the predicted variation of ZnO film thickness with deposition duration. The film initially grows

rapidly and after a certain duration (~10 s), becomes gradual. The result illustrates the role of mass diffusion in the growth of crystalline structures. This observed trend may be attributed to space limitation on the substrate surface and pressure from adjacent growing particles which subsequently limit the growth rate. At long time duration, the growth rate is limited by diffusion at a constant level in the consensus of previous studies [24] .

Figure 4 shows the predicted variation of ZnO film thickness with substrate temperature. Increasing the substrate temperature enhances the growth of particles on the surface. This trend is consistent with the direct influence of temperature on both diffusion and chemical reaction. It should be noted that the threshold deposition time of 10 s of previous Figure 3 which produced a film thickness of 272 nm corresponds to a temperature of 400˚C in Figure 4.

Table 3 compares the film thickness of ZnO films obtained from modeling and experiments at two substrate temperatures. The experiment was performed with deposition duration of 10 s which is the threshold deposition time obtained from the model (Figure 3). The model predicts the film thickness on the same order as the experiment and the accuracy is quite good (<3% error) at the higher temperature of 400˚C. The improved model accuracy at higher temperature is attributed to the heat loss in the experiment which impedes film growth. Some of the heat loss is compensated at the higher temperatures. These results are also consistent with Figure 3 and Figure 4. Thus the proposed model has the potential to successfully predict the optimum processing parameters for the film synthesis.

Figures 5(a)-(c) show the SEM micrographs of single ZnO oxides processed at defined temperature and different concentrations. At low concentration (C = 0.1 mol/lit) (Figure 5(a)), small spherical crystallites are formed that agglomerate at the surface in the shape of powder. The average particle size is 112 nm which is in the consensus of previous studies [25] -[27] .

The properties of particles change by increasing the concentration. The grain size increases with increase in the amount of precursor dissolved in solution in Figure 5(b) as was also observed in previous studies [28] [29] . The film is mostly homogenous with average nanoparticle size of about 119 nm. The particles exhibit the

xagonal flake morphology similar to a previous study [30] . Upon further increase in concentration (C = 0.3 mol/lit), the density and size of particles are increased to an average size of 233 nm. In this case, the crystals are plate-like and the sides of the walls are grown packed together (Figure 5(c)). A similar hexagonal wurtzie structure has also been observed at high concentration [1] .

Figure 6(a) and Figure 6(b) show the SEM micrographs of ZnO + In₂O₃ mixed oxide thin films on Al₂O₃ substrate at T = 400˚C composed of 25% of ZnO mixed with 75% of In₂O₃ (Figure 6(a)), and 80% of ZnO mixed with 20% of In₂O₃ (Figure 6(b)). At lower composition ratio of ZnO (Figure 6(a)), the growth of particle side walls results in the formation of nano tubes [28] [31] . By increasing the amount of zinc (Figure 6(b)), crystallization is enhanced and a well-structured thin film is synthesized. In this case, there is a better chance to produce ZnO crystals or rods [28] . This result confirms that the overall particle size is increased when the concentration of In₂O₃ is decreased. The observed trend may be attributed to the fact that indium ions limit the growth of ZnO particles on the surface. The grain size has been observed to decrease by increasing the indium doping [32] . This is attributed to the stresses applied by the mixture which limits the growth of grain size.

The results show that addition of In₂O₃ component to ZnO results in the growth of particles over a wider size range as was also observed in a previous study [33] . Thus the particles are deposited with less homogeneity compared to the single oxide ZnO. A comparison of Figure 6(a) and Figure 6(b) also shows that the film is more homogenous when ZnO is the dominant precursor. At low concentration of ZnO, the structure of the mixed oxide is similar to that obtained for single oxide ZnO in Figure 5(c). However the average particle size obtained (~136 nm) is less than the single oxide. The value predicted from the mathematical model for the mixed oxide of

Figure 6(a) is 209 nm which is lower than the corresponding value for ZnO deposition. This value is however still in the range of the measured particle size. The predicted particle size is 244 nm at the higher concentration of ZnO (Figure 6(b)). Table 4 summarizes the values of the average particle sizes obtained from modeling and experiments.

The agreement between the experimental measurements and modeling is generally acceptable and the results are on the same order of magnitude. The agreement is particularly good when the deposited film is completely well-structured and the crystal shapes are fully formed as in cases 3 and 5 for which the maximum error is less than 14% in case 3.