rates of all processes are equal, gives where x ∗ is the equilibrium value of x , etc. Given a value for the concentration in the bulk solution, these relations determine the equilibrium values of all of the other quantities as well as imposing the consistency relation Supersaturation at the surface is defined as log ( x/x ∗ ). To illustrate the general behavior of this model, we have solved for the time-dependence of the various quantities for the case that a = a ′ = 0 . 04 k$_{0}$ , k$_{1}$ = 0 . 02 k$_{0}$ , k ′ 1 = k ′ 2 = 0 . 004 k$_{0}$ , k ′ 0 = 0 . 5 k$_{0}$ , k$_{2}$ = 0 . 01 k$_{0}$ , N = 2 with initial conditions x$_{0}$ (0) = 1, and x (0) = y (0) = c (0) = 0. Here the original time t is normalized to k$_{0}$t . The value of k ′ 0 reflects the idea that once adsorbed, a molecule spends on average a long time on the surface before being desorbed back to the bulk. The choice k$_{2}$ < k$_{1}$ amounts to assuming that the activation barrier for aggregation alone is larger than that for adsorption and aggregation. Aside from these assumptions and the physical requirements that the rates at which molecules join a cluster, k$_{1}$ and k$_{2}$ , are greater than the rates for the reverse processes and the detailed balance condition, Eq.8, these values are somewhat flexible. Given an initially flat surface, c$_{N}$ (0) = 0, islands will form causing the mass of the surface to increase until, at long times, equilibrium is reached. At intermediate times, a more-or-less steady growth regime is reached, see Fig. 1, corresponding to a plateau in the growth velocity, dc$_{N}$ dt , which defines a quasi-steady state growth velocity as shown in Fig. 2. Taking the maximum growth velocity as a measure of the typical velocity at intermediate times, one again finds a crossover between small x and large x regimes as illustrated in Figure 3. 7 ax$_{0}$ = a $^{′}$x∗ (7) k$_{0}$x ∗ = k ′ $_{0}$y ∗ k ′ $_{1}$c ∗ N = k$_{1}$x $^{∗}$y ∗ N − 1 k$_{2}$y ∗ N = k ′ $_{2}$c ∗ N k ′ $_{0}$k$_{1}$ k ′ 2 = k$_{0}$k ′ $_{1}$k$_{2}$ . (8)