부산대학교 도서관

The canonical steady-state (s-s) particle coalescence model (PCM) inone dimension is solved for the cluster mass distribution. In thismodel single particles are fed into the system and no clusters areremoved. The increase in the total cluster concentration is balancedby the continual coalescence of clusters. The concentration ofclusters of any fixed mass reaches a steady state but the large-masstail of the cluster mass distribution evolves forever. More precisely,the point clusters hop randomly from bond to bond on a linear lattice.Two clusters aggregate when they hop onto the same bond. A $(k+p)$-mer,i.e. a cluster of $(k+p)$ particles, is formed if a $k$-mer occupyingone bond aggregates with a $p$-mer. The hopping rate of the $k$-mersis $2\alpha$ hops per unit of time, independently of $k$. Monomers areadded to the lattice at random at the rate $\beta$ per unit time per bond.Let the probability $P^N_{ij}=$Prob\{exactly $N$ particles occupy thebonds $(ij)$\}. The aggregation of point clusters is described bythe equations $(d/dt)P^0_k=2\alpha (P^0_{k-1}-2P^0_k+P^0_{k+1})-\betakP^0_k$, and $(d/dt)P^N_k=2\alpha(P^N_{k-1}-2P^N_k+P^N_{k+1})+\beta kP^{N-1}_k-\beta kP^N_k$. In the continuous limit, i.e. when thelattice spacing $\Delta\to 0$ and the diffusion constant for individual$k$-mers is $D=\alpha\Delta^2$, the steady-state continuous systemequations are $0=2D(d^2/dx^2)P^0(x)-\beta^*xP^0(x)$, and $0=2D(d^2/dx^2)P^N(x)+\beta^*xP^{N-1}(x)-\beta^*xP^N(x)$, where the feed rate per unitdistance, $\beta^*=\beta/\Delta$, remains constant. The aggregating systemreaches a steady state in which $P^N(x)$ is constant.