Models of Crystallisation in Evaporating Droplets
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ABSTRACT The spray drying of a droplet containing a substance in solution can produce solid particles with a variety of final shapes: hollow, punctured, squashed, as well as solid spheres. The geometry affects the properties of the product. Models are presented here which describe the processes of solvent evaporation and solute crystallisation as drying takes place. The formation of a crust on the surface of the droplet is addressed. It is proposed that such a crust with a thickness of two crystallite diameters can develop into dry hollow shell. Some example calculations of the spray drying of droplets of sodium chloride solution are described.
INTRODUCTION Aerosol processing is an attractive option for the production of finely divided powders if the product is sensitive to high temperatures or stresses, which would make mechanical grinding unsuitable. The idea is to generate droplets containing the required material in solution, and then to dry the particles until the solvent is completely removed, leaving a solid residue. The process is known as spray drying, and is used extensively in the pharmaceutical and food industries [1]. One drawback is that control over the geometry of the final powder can be poor. A spherical particle may form, but it is also possible that a thin shell or crust might be produced, which could later fracture into mis-shapen fragments [3]. This is undesirable since particle shape can affect the flow and packing properties of the powder. On the other hand, strong, hollow particles can be exploited, for example for a low density powder [2]. This paper is an attempt to model the coupled processes of crystal nucleation and growth, and to consider criteria for the formation of hollow shells.
SOLVENT EVAPORATION The two important factors affecting evaporation are the effect of solute on the solvent evaporation rate, and the heating of the the droplet during drying [5, 6, 7]. Solving the diffusion equation for vapour outside a spherical droplet of radius r0 , the evolution in droplet radius is given by dro D (pP)q - P() dt roc•, where c, is the mass concentration of solvent per unit volume of solution at the droplet surface, and DR is the vapour diffusion coefficient in the carrier gas. Peq and pe. are the vapour densities at the droplet surface and at infinity, respectively. Pq is taken to be the
equilibrium density for the current droplet surface temperature and composition. Small corrections can be made to eq. (1) corresponding to the temperature dependence of D, and Stefan flow [7], but these will be ignored here. The effect of solute on the evaporation rate is expressed in terms of the solvent activity a: Peq = apreP where pP-epis the vapour density 637 Mat. Res. Soc. Symp. Proc. Vol. 398 @1996 Materials Research Society
in equilibrium with a droplet of pure solvent. The surface temperature is calculated by solving the thermal diffusion equation in the gas phase around the droplet. Assuming diffusion of heat within the droplet is rapid, the droplet can be characterised by a single tempe
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