Ajuste de modelos

For a single solid mass, there can be said to be three different values o f density. These are defined as the ‘true density’ which is equal to the mass divided by the volume excluding open and closed pores; the ‘apparent density’ which takes the volume excluding open pores but including closed pores; and the ‘effective density’ which considers the volume including both open and closed pores. Thus, the true density represents the density o f the material, irrespective o f the structure. Since open pores are connected to the external surface and closed pores are sealed off, the method o f measuring the volume would determine which density is being considered. Apparent density would apply if the particle was immersed in a fluid that penetrated only the open pores, and the effective density would apply when the surrounding fluid did not

enter open pores i.e. the external surface would be considered as the boundary o f the particle.

The density o f pellets may be influenced by the individual components o f the formulation; for example increased density could be achieved by including a dense material such as barium sulphate as a filler [Boutell 1995]. Several steps in the manufacturing process could also lead to consolidation o f the initial wet mix i.e. the pressure exerted on pushing the mass through a die during the extrusion process or through a sieve/mesh when granulation takes place. The spheronisation process also results in densification o f the extrudate, and thus any change in formulation or production may affect the final density characteristics o f the product.

An homogenous and reproducible density from batch to batch is desirable since knowledge o f this property o f the pellets is used in various further processes. These include determining the batch size during the coating process, selecting an appropriate volume dosator for the required dose o f pellets during capsule filling, and porosity calculations. Any variation in density could lead to inaccurate dosing or drug release. In the situation where more than one type o f pellet is mixed together, differing densities could lead to segregation o f the mixture resulting in an inhomogenous mix. Bulk density is a further density characteristic which relates to the whole mass o f pellets. The volume used in its calculation is the volume that the pellet bed occupies, including the volume o f voids, when packed into a container such as a measuring cylinder. Thus bulk density is related to the pellets’ packing properties which is influenced by the size o f the pellets, as well as their shape [Fiske et al 1994]. Pitkin and Carstensen [1990] showed that there was a linear relationship between bed porosity and shape factor for glass spheres and cylinders. When investigating bulk density, often the tapped density is determined too. This involves using a mechanical tapping device to tap the measuring cylinder once the initial untapped volume o f the sample has been measured, and recording any decrease in volume. In similar experiments with powders, Kawakita’s equation has been applied to study the compression o f powder beds when tapped [Kawakita and Lüdde 1970; Yamashiro et al 1983]. The equation defines the relationship between the number o f taps and the degree o f volume reduction produced when powder solids are tapped. Similar studies with pellets could indicate the compressibility and packing ability o f pellet beds.

As mentioned previously, the true density o f a material is purely the ratio o f its mass to the volume occupied by it, excluding any pores in the volume measured. Thus the accuracy o f a true density value depends on the method used to determine the volume o f the sample. For a non-porous material the volume may be determined by fluid displacement as long as there is no interaction between the sample and fluid. However most solid particles do contain pores or surface features which would not be penetrated by a displaced fluid, in which case a gas may be used instead. Commercially there are many pycnometers available for density determinations, the simple ones just using air displacement, and others using gases such as helium which has the advantages o f being inert and having small atoms [Lowell and Shields 1984]. The apparatus is able to purge the system with the gas and control the pressures within the system using modem pressure transducers. With a measure o f the volume o f the sample, it is a simple matter to find its mass and thus calculate its density (density = mass/volume).

However, if the fluid being used is not able to penetrate any closed pores and cracks within the sample then the measured volume would be greater than the actual, and thus the density value would be lowered; this is known as the apparent density value. In the case o f mercury pycnometry, where the penetration fluid has a high contact angle and thus does not penetrate small open pores at ambient pressures, the volume measured includes both open and closed pores leading to a lower density value. This is termed the effective density o f the sample.