Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a characteristic time $t_c$ at which a dynamical transition occurs with a power law divergence (with the exponent $z$) of the average cluster size. Close to $t_c$, the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the cluster growth near $t_c$ also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length $\xi$ of the clusters increses and diverges at $t_c$ as $\xi \sim (t_c-t)^{-\nu}$, with $\nu = 0.83 \pm 0.06$. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at $t_c$ with an effective dimensionality $d_f \sim 1.85 \pm 0.05$. These values are independent (within the limit of statistical error) of the concentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem. The values of $\nu$ and $d_f$ supports the scaling relation $z=\nu d_f$ with the value of $z$ obtained before.Comment: A single ps file (6 figures included), 12 pages, to appear in Physica

Is it really possible to grow isotropic on-lattice diffusion-limited aggregates?

In a recent paper (Bogoyavlenskiy V A 2002 \JPA \textbf{35} 2533), an algorithm aiming to generate isotropic clusters of the on-lattice diffusion-limited aggregation (DLA) model was proposed. The procedure consists of aggregation probabilities proportional to the squared number of occupied sites ($k^2$). In the present work, we analyzed this algorithm using the noise reduced version of the DLA model and large scale simulations. In the noiseless limit, instead of isotropic patterns, a $45^\circ$ ($30^\circ$) rotation in the anisotropy directions of the clusters grown on square (triangular) lattices was observed. A generalized algorithm, in which the aggregation probability is proportional to $k^\nu$, was proposed. The exponent $\nu$ has a nonuniversal critical value $\nu_c$, for which the patterns generated in the noiseless limit exhibit the original (axial) anisotropy for $\nu<\nu_c$ and the rotated one (diagonal) for $\nu>\nu_c$. The values $\nu_c = 1.395\pm0.005$ and $\nu_c = 0.82\pm 0.01$ were found for square and triangular lattices, respectively. Moreover, large scale simulations show that there are a nontrivial relation between noise reduction and anisotropy direction. The case $\nu=2$ (\bogo's rule) is an example where the patterns exhibit the axial anisotropy for small and the diagonal one for large noise reduction.Comment: 12 pages, 8 figure

Metagenomic analyses of marine new production under elevated CO2 conditions

A mesocosm experiment was carried out in a Norwegian fjord near Bergen in May 2006, with the main objective being the study of the effects of increasing concentrations of atmospheric CO2 (and associated effects such as increased acidification) on blooms of natural marine coastal plankton. Three mesocosms were bubbled with CO2(g) to achieve a high (~700ppm) CO2 concentration (pH ~7.8) to simulate predicted future conditions as a result of rising atmospheric CO2 concentrations. Another three mesocosms were treated as controls and bubbled with ambient air to represent a near pre-industrial scenario (atmospheric CO2 concentration ~300ppm, surface seawater pH ~8.15). Blooms in the mesocosms were stimulated by the addition of nutrients at a near-Redfield ratio ([N:P] ≈ [16:1]), and scientific measurements and analyses were carried out over the course of the blooms for approximately one month. Of particular interest in this study were the autotrophic plankton. The diversity and activities of these microorganisms under the two treatments was therefore investigated. By designing and using new degenerate primers specifically targeting ‘Green-type’ (Form IA and IB), ‘Red-type’ (Form IC and ID) and Form II RuBisCO, analysis of primary producers was carried out using PCR and either gDNA or cDNA (mRNA) templates from key time points spanning the complete duration of the blooms throughout the mesocosm experiment. Over 1250 novel RuBisCO large subunit sequences have been fully annotated and deposited in the NCBI GenBank® database. These sequences revealed distinct changes in the diversity of primary producers both over the courses of the blooms and between treatments. Particularly striking was the effect of acidification on the community structure of the eukaryotic picoplankton, Prasinophytes. A clade of prasinophytes closely related to Micromonas pusilla showed a distinct preference for the high CO2 conditions; a laboratory-based experiment confirmed the high tolerance of Micromonas pusilla to lower pH. Conversely, a clade related to Bathycoccus prasinos was almost entirely excluded from the high CO2 treatments. Clades of form II RuBisCO-containing dinoflagellates were also abundant throughout the experiment in both treatments. The high similarity of some of these clades to the toxin-producing species Heterocapsa triquetra and Gonyaulax polyedra, and apparent high tolerance of some clades to high CO2 conditions, is perhaps cause for concern in a high CO2 world and demands further research. In parallel with the RubisCO work, new primers were designed that target the gene encoding the Fe protein of nitrogenase (NifH). 82 Bergen genomic nifH sequences have been annotated and submitted to GenBank®. These sequences include those from organisms related to Alpha, Beta, and Gammaproteobacteria, and Cluster II and Cluster III sequences that align most closely with anaerobic Bacteria, Gram positive, and/or sulphur-reducing Bacteria. The biggest surprise, however, was the apparent abundance and significance of a Rhodobacter sphaeroides-like microorganism throughout the duration of the experiment in both treatments. Whilst this clade was unsurprisingly absent in the RuBisCO cDNA libraries, all but two of 128 nifH cDNA clones analysed were identical to the gene from Rhodobacter sphaeroides. This shows that this clade was potentially fixing N2 throughout the entire experiment, even in the presence of combined N added to both sets of mesocosms at the start of the experiment. A group of Rhodobacter sphaeroides-like microorganisms present at Bergen may therefore have been an unexpected source of new N during the experiment and contributed to the maintenance of the mesocosm communities as nutrients became depleted. One organism dominated the autotrophic communities after the blooms in both treatments. Synechococcus spp. Form IA rbcL clones most closely related to the coastal strain Synechococcus sp. strain CC9902 were recovered throughout the experiment but were particularly numerous toward the end of the experiment and dominated the “Green-type” libraries at this time. Initially, rbcL clones from these cyanobacteria were mostly derived from the ambient CO2 mesocosms but were equally distributed between treatments by the end of the experiment. This suggests that cyanobacteria related to strain CC9902 may be less tolerant of elevated CO2 (which was greatest at the beginning rather than the end of the experiment). However, despite the mesocosms being Pi-limited at the end of the experiment, several Synechococcus species (including those related to strain CC9902 and another coastal strain, CC9311) thrived. Following on from this observation, Pi uptake and assimilation mechanisms in a Synechococcus species were investigated in the laboratory. This led to the sequencing and characterisation of a pstS gene from the marine cyanobacterium Synechococcus sp. WH 8103. Unlike conventional pstS, it was discovered that the pstS II gene in this organism is constitutively expressed and unresponsive to or only weakly regulated by Pi supply. The use of PstS/pstS as a marker for P-limitation in natural samples, therefore, should be interpreted with caution

Aggregation in a mixture of Brownian and ballistic wandering particles

In this paper, we analyze the scaling properties of a model that has as limiting cases the diffusion-limited aggregation (DLA) and the ballistic aggregation (BA) models. This model allows us to control the radial and angular scaling of the patterns, as well as, their gap distributions. The particles added to the cluster can follow either ballistic trajectories, with probability $P_{ba}$, or random ones, with probability $P_{rw}=1-P_{ba}$. The patterns were characterized through several quantities, including those related to the radial and angular scaling. The fractal dimension as a function of $P_{ba}$ continuously increases from $d_f\approx 1.72$ (DLA dimensionality) for $P_{ba}=0$ to $d_f\approx 2$ (BA dimensionality) for $P_{ba}=1$. However, the lacunarity and the active zone width exhibt a distinct behavior: they are convex functions of $P_{ba}$ with a maximum at $P_{ba}\approx1/2$. Through the analysis of the angular correlation function, we found that the difference between the radial and angular exponents decreases continuously with increasing $P_{ba}$ and rapidly vanishes for $P_{ba}>1/2$, in agreement with recent results concerning the asymptotic scaling of DLA clusters.Comment: 7 pages, 6 figures. accepted for publication on PR

Anisotropic Diffusion Limited Aggregation

Using stochastic conformal mappings we study the effects of anisotropic perturbations on diffusion limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map $m$ preferred directions for growth of angular width $\sigma$ to a distribution on the unit circle which is a periodic function with $m$ peaks in $[-\pi, \pi)$ such that the width $\sigma$ of each peak scales as $\sigma \sim 1/\sqrt{k}$, where $k$ defines the ``strength'' of anisotropy along any of the $m$ chosen directions. The two parameters $(m,k)$ map out a parameter space of perturbations that allows a continuous transition from DLA (for $m=0$ or $k=0$) to $m$ needle-like fingers as $k \to \infty$. We show that at fixed $m$ the effective fractal dimension of the clusters $D(m,k)$ obtained from mass-radius scaling decreases with increasing $k$ from $D_{DLA} \simeq 1.71$ to a value bounded from below by $D_{min} = 3/2$. Scaling arguments suggest a specific form for the dependence of the fractal dimension $D(m,k)$ on $k$ for large $k$, form which compares favorably with numerical results.Comment: 6 pages, 4 figures, submitted to Phys. Rev.

Non-universal dynamics of dimer growing interfaces

A finite temperature version of body-centered solid-on-solid growth models involving attachment and detachment of dimers is discussed in 1+1 dimensions. The dynamic exponent of the growing interface is studied numerically via the spectrum gap of the underlying evolution operator. The finite size scaling of the latter is found to be affected by a standard surface tension term on which the growth rates depend. This non-universal aspect is also corroborated by the growth behavior observed in large scale simulations. By contrast, the roughening exponent remains robust over wide temperature ranges.Comment: 11 pages, 7 figures. v2 with some slight correction

Dynamics of Fluctuation Dominated Phase Ordering: Hard-core Passive Sliders on a Fluctuating Surface

We study the dynamics of a system of hard-core particles sliding downwards on a one dimensional fluctuating interface, which in a special case can be mapped to the problem of a passive scalar advected by a Burgers fluid. Driven by the surface fluctuations, the particles show a tendency to cluster, but the hard-core interaction prevents collapse. We use numerical simulations to measure the auto-correlation function in steady state and in the aging regime, and space-time correlation functions in steady state. We have also calculated these quantities analytically in a related surface model. The steady state auto-correlation is a scaling function of t/L^z, where L is the system size and z the dynamic exponent. Starting from a finite intercept, the scaling function decays with a cusp, in the small argument limit. The finite value of the intercept indicates the existence of long range order in the system. The space-time correlation, which is a function of r/L and t/L^z, is non-monotonic in t for fixed r. The aging auto-correlation is a scaling function of t_1 and t_2 where t_1 is the waiting time and t_2 the time difference. This scaling function decays as a power law for t_2 \gg t_1; for t_1 \gg t_2, it decays with a cusp as in steady state. To reconcile the occurrence of strong fluctuations in the steady state with the fact of an ordered state, we measured the distribution function of the length of the largest cluster. This shows that fluctuations never destroy ordering, but rather the system meanders from one ordered configuration to another on a relatively rapid time scale

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

On the multifractal statistics of the local order parameter at random critical points : application to wetting transitions with disorder

Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig (A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)) on the case of diluted two-dimensional Potts model, the moments $\bar{\rho^q(r)}$ of the local order parameter $\rho(r)$ scale with a set $x(q)$ of non-trivial exponents $x(q) \neq q x(1)$. In this paper, we revisit these ideas to incorporate more recent findings: (i) whenever a multifractal measure $w(r)$ normalized over space $\sum_r w(r)=1$ occurs in a random system, it is crucial to distinguish between the typical values and the disorder averaged values of the generalized moments $Y_q =\sum_r w^q(r)$, since they may scale with different generalized dimensions $D(q)$ and $\tilde D(q)$ (ii) as discovered by Wiseman and Domany (S. Wiseman and E. Domany, Phys Rev E {\bf 52}, 3469 (1995)), the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponent $c=1.5$ (marginal disorder) and $c=1.75$ (relevant disorder). Finally, we argue that the presence of finite Griffiths ordered clusters at criticality determines the asymptotic value $x(q \to \infty) =d$ and the minimal value $\alpha_{min}=D(q \to \infty)=d-x(1)$ of the typical multifractal spectrum $f(\alpha)$.Comment: 17 pages, 20 figure

The effect of self-affine fractal roughness of wires on atom chips

Atom chips use current flowing in lithographically patterned wires to produce microscopic magnetic traps for atoms. The density distribution of a trapped cold atom cloud reveals disorder in the trapping potential, which results from meandering current flow in the wire. Roughness in the edges of the wire is usually the main cause of this behaviour. Here, we point out that the edges of microfabricated wires normally exhibit self-affine roughness. We investigate the consequences of this for disorder in atom traps. In particular, we consider how closely the trap can approach the wire when there is a maximum allowable strength of the disorder. We comment on the role of roughness in future atom--surface interaction experiments.Comment: 7 pages, 7 figure