h is represented by an equation of the form: (x/r)n = kt (1) Where x is the neck radius at time t and r is the radius of the particle. The value of exponent n depends on the sintering mechanism: for visco-plastic flow n = 2. The value of k depends on the surface energy, g - the magnitude of an applied force and viscosity - m. In the Frenkel equation [3], which considers surface energy only is given by: (x/r)2 = 3gt/2rm (2)Rumpf [4] derived an expression which included the sintering effect of an applied load, F, and in which the proportion constant for the first term was found to be 8/5 for hydrodynamic load instead of the 3/2 found by Frenkel: (x/r)2 = [(8g/5r)+(2F/5pr2)]t/m (3)Assuming the stress to be evenly distributed over all particle contacts, the above equation may be rewritten using Rumpf's equation, in terms of stress, s1, applied to an assembly of spherical particles of porosity e:(x/r)2 = [(8g/5r)+(8es1/5p(1-e))]t/m (4)Both Equations (3) and (4) can be used in practical situations in which the applied load is significant. They also can be used to analyze data to obtain an apparent value of the viscosity and its variation with temperature. In this case the importance of the first term may be negligible compared to that of the second term.In real systems, however, the particles are neither perfectly smooth spheres nor are they monodisperse. Taking this into consideration, Equation (4) is probably best written in the form: (x/r)2 = k1t/m (5)Where k1 is a factor dependant on both material properties and environmental conditions. Furthermore, in glassy systems the flow is unlikely to be Newtonian in essence. Hence, in practice, the viscosity value must be regarded as a value used to fit the data. An Arrhenius relationship of viscosity and temperature is often taken as: ...
