Transient behaviour of surface supersaturation caused by an abrupt change in bulk supersaturation

The transient behaviour of the surface supersaturation, caused by an abrupt (stepped) change in bulk supersaturation from one constant value σ0 to another σ1, is studied with the use of the surface diffusion theory of Burton, Cabrera and Frank (BCF). The presented analytical solution of the time-dependent BCF equation takes into account a phenomenological coefficient Λs describing the kinetics of growth unit exchange between the step edge and the adsorbed layer. It is shown that the solution can be approximated by a simple expression, which enables an easy estimate of the time ts required to attain the steady-state value of the surface supersaturation. The time ts is independent of the difference between values σ0 and σ1 but it increases with increasing coefficient Λs or increasing interstep distance. On the other hand, the average rate of change in the surface supersaturation during the transient state is determined neither by the coefficient Λs nor by the interstep distance but it is proportional to the difference between values of σ0 and σ1 of bulk supersaturation.