{c) equilibrium enamel surface for dentifrice 'C' Figure 6 Effect of dentifrice on etched enamel surfaces Facing page 400

MEASUREMENT AND INTERPRETATION OF DENTIFRICE ABRASIVENESS 401 concentration. Under these circumstances each particle's contribution to the total wear is directly proportional to the fraction of the load it carries. If one of the sliding surfaces is replaced by a brush, the individual fibres will transmit a constant fraction of the load which is independent of the presence of any trapped abrasive material between the opposing surfaces. Assuming that the fibres themselves cause little or no wear to the opposite surface, the wear rate will become sensitive to particle concen- tration and will depend on the probability of trapping one or more particles within some 'region of influence' surrounding each fibre tip. The density of particles in a dentifrice slurry is extremely high, but because of the small diameter of the fibre the trapping probability is low and may be derived from the Poisson approximation to the binomial theorem. The general term of this approximation is e-nvnrvr , r! where n----population density of the abrasive particles, v=volume of the 'region of influence', and r=number of particles trapped by a fibre. If one makes the further simplification that the wear rate will be in- dependent of the number of abrasive particles trapped by each abrasively loaded fibre, then the wear rate will be determined by an expression of the form wN(1- e-nv), where e -nv represents the first term of the Poisson approximation, namely the probability of trapping no particles (r----O), w the intrinsic wear rate of an abrasively loaded fibre and N the total number of brush fibres. The assumption that the wear rate of an abrasively loaded fibre will be largely independent of the number of abrasive particles trapped is justified on the basis of the load-sharing property of an individual fibre. As the population density of the abrasive particles will be proportional to the dentifrice concentration (vol/vol), C, the relationship between wear rate, dW/dS, and dentifrice concentration can be expressed in the form dW/dS=a(1 -e-[ IC) where a represents the maximum wear rate possible with all the fibres loaded in an abrasive capacity and [I a measure of the ratio of the trapped volume of slurry at the fibre tip and the volume of an abrasive particle. Although the trapped volume will not remain perfectly constant throughout a brush stroke, an approximate value is given by the volume of slurry entrained by a fibre and the tooth surface when the two are in contact. The boundary of the trapped volume will be defined by the