Particle Survival and Polydispersity in Aggregation

We study the probability, $P_S(t)$, of a cluster to remain intact in one-dimensional cluster-cluster aggregation when the cluster diffusion coefficient scales with size as $D(s) \sim s^\gamma$. $P_S(t)$ exhibits a stretched exponential decay for $\gamma < 0$ and the power-laws $t^{-3/2}$ for $\gamma=0$, and $t^{-2/(2-\gamma)}$ for $0<\gamma<2$. A random walk picture explains the discontinuous and non-monotonic behavior of the exponent. The decay of $P_S(t)$ determines the polydispersity exponent, $\tau$, which describes the size distribution for small clusters. Surprisingly, $\tau(\gamma)$ is a constant $\tau = 0$ for $0<\gamma<2$.Comment: submitted to Europhysics Letter

No self-similar aggregates with sedimentation

Two-dimensional cluster-cluster aggregation is studied when clusters move both diffusively and sediment with a size dependent velocity. Sedimentation breaks the rotational symmetry and the ensuing clusters are not self-similar fractals: the mean cluster width perpendicular to the field direction grows faster than the height. The mean width exhibits power-law scaling with respect to the cluster size, ~ s^{l_x}, l_x = 0.61 +- 0.01, but the mean height does not. The clusters tend to become elongated in the sedimentation direction and the ratio of the single particle sedimentation velocity to single particle diffusivity controls the degree of orientation. These results are obtained using a simulation method, which becomes the more efficient the larger the moving clusters are.Comment: 10 pages, 10 figure

Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation

The persistence probability, $P_C(t)$, of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. In the mean-field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For $\gamma \ge 0$ the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For $\gamma < 0$ the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory. The decay of persistence determines the small size tail of the cluster size distribution. For $0 < \gamma < 2$ the distribution is flat and, surprisingly, independent of $\gamma$.Comment: 11 pages, 6 figures, RevTeX4, submitted to Phys. Rev.

The electric field close to an undulating interface

The electric potential close to a boundary between two dielectric material layers reflects the geometry of such an interface. The local variations arise from the combination of material parameters and from the nature of the inhomogeneity. Here, the arising electric field is considered for both a sinusoidally varying boundary and for a “rough,” Gaussian test case. We discuss the applicability of a one-dimensional model with the varying layer thickness as a parameter and the generic scaling of the results. As an application we consider the effect of paper roughness on toner transfer in electrophotographic printing.Peer reviewe

Persistence in Cluster--Cluster Aggregation

Persistence is considered in diffusion--limited cluster--cluster aggregation, in one dimension and when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. The empty and filled site persistences are defined as the probabilities, that a site has been either empty or covered by a cluster all the time whereas the cluster persistence gives the probability of a cluster to remain intact. The filled site one is nonuniversal. The empty site and cluster persistences are found to be universal, as supported by analytical arguments and simulations. The empty site case decays algebraically with the exponent $\theta_E = 2/(2 - \gamma)$. The cluster persistence is related to the small $s$ behavior of the cluster size distribution and behaves also algebraically for $0 \le \gamma < 2$ while for $\gamma < 0$ the behavior is stretched exponential. In the scaling limit $t \to \infty$ and $K(t) \to \infty$ with $t/K(t)$ fixed the distribution of intervals of size $k$ between persistent regions scales as $n(k;t) = K^{-2} f(k/K)$, where $K(t) \sim t^\theta$ is the average interval size and $f(y) = e^{-y}$. For finite $t$ the scaling is poor for $k \ll t^z$, due to the insufficient separation of the two length scales: the distances between clusters, $t^z$, and that between persistent regions, $t^\theta$. For the size distribution of persistent regions the time and size dependences separate, the latter being independent of the diffusion exponent $\gamma$ but depending on the initial cluster size distribution.Comment: 14 pages, 12 figures, RevTeX, submitted to Phys. Rev.

Coarsening of Sand Ripples in Mass Transfer Models with Extinction

Coarsening of sand ripples is studied in a one-dimensional stochastic model, where neighboring ripples exchange mass with algebraic rates, $\Gamma(m) \sim m^\gamma$, and ripples of zero mass are removed from the system. For $\gamma < 0$ ripples vanish through rare fluctuations and the average ripples mass grows as \avem(t) \sim -\gamma^{-1} \ln (t). Temporal correlations decay as $t^{-1/2}$ or $t^{-2/3}$ depending on the symmetry of the mass transfer, and asymptotically the system is characterized by a product measure. The stationary ripple mass distribution is obtained exactly. For $\gamma > 0$ ripple evolution is linearly unstable, and the noise in the dynamics is irrelevant. For $\gamma = 1$ the problem is solved on the mean field level, but the mean-field theory does not adequately describe the full behavior of the coarsening. In particular, it fails to account for the numerically observed universality with respect to the initial ripple size distribution. The results are not restricted to sand ripple evolution since the model can be mapped to zero range processes, urn models, exclusion processes, and cluster-cluster aggregation.Comment: 10 pages, 8 figures, RevTeX4, submitted to Phys. Rev.

Porous structure of thick fiber webs

The bulk properties and stochastic pore geometry of finite-thickness fiber webs are studied using a realistic model for the sedimentation of flexible fibers [K. J. Niskanen and M. J. Alava, Phys. Rev. Lett. 73, 3475 (1994)]. The resulting web structure is controlled by a dimensionless number F=Tfwf/tf, where Tf is fiber flexibility, wf fiber width, and tf fiber thickness. The fiber length (≫wf,tf) is irrelevant. With increasing coverage c̄, a crossover occurs at c̄=c0≈1+2F from a vacancy-controlled two-dimensional (2D) structure to a pore-controlled 3D structure. The 3D structures are isomorphic in that the pore dimensions are exponentially distributed, with the decay rate dependent only on F.Peer reviewe

Failure of planar fiber networks

We study the failure of planar random fiber networks with computer simulations. The networks are grown by adding flexible fibers one by one on a growing deposit [K. J. Niskanen and M. J. Alava, Phys. Rev. Lett. 73, 3475 (1994)], a process yielding realistic three dimensional network structures. The network thus obtained is mapped to an electrical analogue of the elastic problem, namely to a random fuse network with separate bond elements for the fiber-to-fiber contacts. The conductivity of the contacts (corresponding to the efficiency of stress transfer between fibers) is adjustable. We construct a simple effective medium theory for the current distribution and conductivity of the networks as a function of intra-fiber current transfer efficiency. This analysis compares favorably with the computed conductivity and with the fracture properties of fiber networks with varying fiber flexibility and network thickness. The failure characteristics are shown to obey scaling behavior, as expected of a disordered brittlematerial, which is explained by the high current end of the current distribution saturating in thick enough networks. For bond breaking, fracture load and strain can be estimated with the effective medium theory. For fiber breaking, we find the counter-intuitive result that failure is more likely to nucleate far from surfaces, as the stress is transmitted more effectively to the fibers in the interior.Peer reviewe

Dynamic Scaling in One-Dimensional Cluster-Cluster Aggregation

We study the dynamic scaling properties of an aggregation model in which particles obey both diffusive and driven ballistic dynamics. The diffusion constant and the velocity of a cluster of size $s$ follow $D(s) \sim s^\gamma$ and $v(s) \sim s^\delta$, respectively. We determine the dynamic exponent and the phase diagram for the asymptotic aggregation behavior in one dimension in the presence of mixed dynamics. The asymptotic dynamics is dominated by the process that has the largest dynamic exponent with a crossover that is located at $\delta = \gamma - 1$. The cluster size distributions scale similarly in all cases but the scaling function depends continuously on $\gamma$ and $\delta$. For the purely diffusive case the scaling function has a transition from exponential to algebraic behavior at small argument values as $\gamma$ changes sign whereas in the drift dominated case the scaling function decays always exponentially.Comment: 6 pages, 6 figures, RevTeX, submitted to Phys. Rev.

Persistence properties of a system of coagulating and annihilating random walkers

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In 1-dimension, the system models the zero temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d < 2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) ~ m^(z/d) t^(-theta), with theta=d Q + Q(Q-1/2) epsilon + O(epsilon^2), z=(2Q-1) epsilon + (2 Q-1) (Q-1)(1/2+A Q) epsilon^2 +O(epsilon^3), where Q=(q-1)/q, epsilon =2-d and A =-0.006. M(t) is shown to scale as t^(d/2-delta), where delta = d (1 -Q)+(Q-1)(Q-1/2) epsilon+ O(epsilon^2). In two dimensions, we show that P(m,t) ~ ln(t)^(Q(3-2Q)) ln(m)^((2Q-1)^2) t^(-2Q) for m << t^(2 Q-1). The 1-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.Comment: 12 pages, revtex, 5 figure