A particle-pad interaction is critical to the particle scale models considering that the
function of the pad is to trap the abrasive particles and transmit load to the particle-wafer
interface. Cook's model [24]suggested a close-packing of spherical abrasives onto the
pad. It is assumed that the wafer and pad are separated completely by the abrasives and
no direct pad-wafer contact exists. The force applied on a single abrasive under these
assumptions is given by
F = Pr2 (2 15)
ki
where P is the polishing pressure, r the abrasive size and kl the particle fill fraction on the
pad. This particle-pad interaction model has been integrated into the material removal
model of Cook. It is also used by Ahmadi and Xia [85] to evaluate the force on a particle
in their case of a hard-pad and a larger concentration of abrasive particles. Later, Zhao
and Shi [90] proposed that when the pad is soft enough, the abrasive particles will be
embedded deeply into the pad and the force from the wafer is supported by the pad and
abrasives together. This idea has been applied by Luo and Domfeld [27, 91, 92] and Fu et
al. [93] in their model. Luo and Dornfeld's model [27, 91, 92] suggested that this force is
proportional to the contact pressure multiplied by the abrasive size by assuming that the
abrasives are closely packed to each other and these closely packed abrasives are
enclosed by the pads so that the effective contact area between wafer and pad is equal to
that without abrasives. Moreover, the size of the abrasives that may be captured by the
pad is assumed to be a function of abrasive size distribution and pad properties. Their
model, however, neglects the lower sizes of the particle size distribution in material
removal calculation. Fu et al. [93] later assumed that the abrasives are dispersed evenly
over the pad surface and used a beam model to evaluate the wafer-pad direct contact