Phase transitions in systems with aggregation and shattering

We consider a system of clusters made of elementary building blocks, monomers, and evolving via collisions between diffusing monomers and immobile composite clusters. In our model, the cluster-monomer collision can lead to the attachment of the monomer to the cluster (addition process) or to the total breakup of the cluster (shattering process). A phase transition, separating qualitatively different behaviors, occurs when the probability of shattering events exceeds a certain threshold. The novel feature of the phase transition is the dramatic dependence on the initial conditions.

Figure 1

(a) Evolution of the monomer density $c_1\left(t\right)$ and total cluster density $N\left(t\right)$ for the case of constant kernels and the monodisperse initial condition when $\lambda =0.2$. The system approaches a final state $c_1(t=\infty )$ and $N(t=\infty )$ which differs from the equilibrium state (16) and (17). (b) Asymptotic cluster size distribution $c_k(t=\infty )$ dramatically differs from the equilibrium distribution (16).

Figure 2

The monomer density $c_1\left(\tau \right)$ remains positive when the shattering rate satisfies $\lambda >{\lambda }_c\approx 0.27846$. Shown (bottom to top) is the monomer density for $\lambda =0,{\lambda }_c,2/3$. Inset: when physical time diverges, $t\to \infty$, the auxiliary time for $\lambda <{\lambda }_c$, remains finite, $\tau \to {\tau }_{\max }\left(\lambda \right)$.

Figure 3

The final density of clusters $N_{\infty }\left(\lambda \right)$ is a decreasing function of $\lambda$ in the range $0\le \lambda \le {\lambda }_c\approx 0.27846$; it undergoes a first order transition at ${\lambda }_c$, and becomes an increasing function of $\lambda$ in the range $\lambda >{\lambda }_c$ where $N_{\infty }=N^{*}=\lambda /(1+\lambda )$.

Figure 4

The discontinuous jump in the cluster size distribution from $C_k\left({\lambda }_c\right)$ to $C_k({\lambda }_c+0)$, given by Eq. (16) at the critical point $\lambda ={\lambda }_c$. Inset: the final density of monomers $C_1\left(\lambda \right)$ vanishes in the range $0\le \lambda \le {\lambda }_c\approx 0.278465$, undergoes a first order transition at ${\lambda }_c$, and becomes an increasing function of $\lambda$ in the supercritical region $\lambda >{\lambda }_c$ where $C_1={\lambda }^2/{(1+\lambda )}^2$.

Figure 5

Phase trajectories for the total density $N\left(t\right)$ and the density of monomers $c_1\left(t\right)$ for different initial conditions. The region marked by yellow corresponds to the supercritical regime, where all trajectories terminate at the universal point, $N^{*}=\lambda /(1+\lambda ),C_1^{*}={\lambda }^2/{(1+\lambda )}^2$. The curved part of the boundary of the yellow domain corresponds to the critical regime. For the initial conditions, located outside the yellow domain (and satisfying $c_{1,0}\le N_0)$ the subcritical regime is realized; the trajectories terminate in this case on the axis $c_1=0$.

Figure 6

The 3D plot of the critical shattering ${\lambda }_c$ as the function of the initial conditions $N_0,c_{1,0}$ for the model with constant kinetic rates (4). The domain inside the depicted surface corresponds to the supercritical behavior of the system; Fig. 5 provides the cross sections of this surface for particular values of $\lambda$.

Figure 7

For the case of linear kernels, $A_k=k$ and $S_k=\lambda k$, the final density of monomers undergoes a first order phase transition at $\lambda ={\lambda }_c=\frac{1}{2}$ in the situation when initially almost all clusters are dimers. The final density of monomers is given by (44) in this case.

Figure 8

For the case of the linear kernels, $A_k=k$ and $S_k=\lambda k$, the discontinuous jump in the cluster size distribution from $C_k\left({\lambda }_c\right)$ to $C_k({\lambda }_c+0)$ [see Eq. (38)] happens for monodisperse initial conditions at the critical point ${\lambda }_c=0.16773277...$. Inset: the final density of monomers $C_1\left(\lambda \right)$ vanishes in the range $0\le \lambda \le {\lambda }_c$ and undergoes a first order transition at ${\lambda }_c$. In the supercritical region $\lambda >{\lambda }_c,C_1$ is given by Eq. (38).

Figure 9

The boundaries of the supercritical region in the phase plane $\left(c_{1,0},c_{2,0}\right)$ for different values of $c_{3,0}$. For each contour, with the value of $c_{3,0}$ indicated on the plot, the supercritical region lies inside the contour. The initial densities $c_{k,0}=0$ for $k\ge 5$ and $c_{4,0}=\left(1-c_{1,0}-2c_{2,0}-3c_{3,0}\right)/4$ have been chosen. The rate kernels are $A_k=k,S_k=\lambda k$, and $\lambda =0.16773277...$ (which corresponds to ${\lambda }_c$ for the monodisperse initial condition).

Figure 10

Evolution of the monomer density $c_1\left(\tau \right)$ for supercritical, critical, and subcritical regimes for the monodisperse initial condition. Shown are results for the model with kernels $A_k=\sqrt{k}$ and $S_k=\lambda \sqrt{k}$. The critical shattering rate in this case is ${\lambda }_c=0.2277920103...$.

Figure 11

The dependence of the critical shattering rate ${\lambda }_c$ on the exponent $s$ in the case of the monodisperse initial condition. The critical rate ${\lambda }_c$ separates the supercritical evolution regime $\left(\lambda >{\lambda }_c\right)$ and subcritical regime $\left(\lambda <{\lambda }_c\right)$.

Figure 12

The boundaries of the supercritical region in the phase plane $\left(c_{1,0},c_{2,0}\right)$ for different values of $c_{3,0}$. Shown are results for the model with kernels $A_k=\sqrt{k}$ and $S_k=\lambda \sqrt{k}$ in the case when the initial densities are $c_{k,0}=0$ for $k\ge 5$ and $c_{4,0}=\left(1-c_{1,0}-2c_{2,0}-3c_{3,0}\right)/4$. The shattering rate is $\lambda =0.2277920103...$ (it corresponds to ${\lambda }_c$ for the monodisperse initial condition).