Dynamical scaling and the growth of diffusion-limited aggregates

The validity of dynamical scaling for the growth of aggregates in a bath of particles of concentration c0 is investigated. A length scale ξ∼c0−1/(d−D) is introduced which governs the crossover from a fractal of dimension D to compact aggregate behavior. For static scaling a variable x=R/ξ is introduced where R is the radius of gyration of the aggregate. The cluster size N is found to vary as N(c0,R)=(ξ/a)DxDg(x), where a is the particle size and the static structure function is well represented by g(x)=(A+Bx)d−D. To consider dynamical scaling the asymptotic dynamics are first examined. For R≪ξ the radius of gyration and cluster size are respectively found to grow with time as R(c0,t)∼(c0t)1/(2+D−d) and N(c0,t)∼(c0t)D/(2+D−d). While for R≫ξ one finds R(c0,t)∼c01/(d−D)t and N(c0,t)∼c01+d/(d−D)td. If a scaled time variable τ=Dft/ξ2 is introduced where Df is a microscopic diffusion constant, then the previous asymptotic results can be incorporated into dynamical scaling forms for R(c0,t) and N(c0,t). These are R(c0,t)/ξ=x=τ1/(2+D−d)f1(τ) where f1(τ) is well represented by the form f1(τ)=[(A1+B1τ)]1−1/(D+2−d); while N(c0,t)=(ξ/a)DτD/(2+D−d) f2(τ) where f2(τ) can be written [A2+B2τ]d−D/(2+D−d). Using these results all concentration c0 and time t regimes can be investigated. Dynamical scaling is found to hold within the errors inherent in computer simulations due to depletion effects of finite lattices.