A model for the formation of foams

The Ashby-Gibson relation determining the mechanical properties of cellular materials depends on the relative density of the foam, i.e. the ratio of the density of the foam to the density of the material from which it is made. In this analysis we present a model to determine the relative density of a foam. The analysis focuses on the calculation of the gas fraction which we calculate by assuming that the foam forms from a liquid phase. This liquid phase (or matrix) allows the formation of nucleation sites which when there is a sudden release of external pressure causes the expansion of gas at the sites to form the cells in the foamed material. The ‘freezing’ of this process then allows the formation of the solid foam. The size of the critical nucleus r0 in the metastable state is determined by the external pressure and temperature and is the key quantity in the initial condition in the Rayleigh equation which controls the expansion of the nucleation site. The number of nucleation sites formed and the size from which they grow determines the gas fraction present and depends on the amount of the liquid phase. Thus controlling the pressure and temperature of the liquid will determine the gas fraction and thus the mechanical properties. The ‘freezing’ process will also contribute to the control of the gas fraction. The expansion of the nucleation sites is assumed to be an adiabatic expansion which creates a cooling effect so that the liquid glassifies. If the temperature falls below the glass transition temperature the expansion will stop and a solid foam will be formed