Cláudio L. N. Oliveira

Transport on exploding percolation clusters.

We propose a simple generalization of the explosive percolation process [Achlioptas et al., Science 323, 1453 (2009)], and investigate its structural and transport properties. In this model, at each step, a set of q unoccupied bonds is randomly chosen. Each of these bonds is then associated with a weight given by the product of the cluster sizes that they would potentially connect, and only that bond among the q set which has the smallest weight becomes occupied. Our results indicate that, at criticality, all finite-size scaling exponents for the spanning cluster, the conducting backbone, the cutting bonds, and the global conductance of the system, change continuously and significantly with q. Surprisingly, we also observe that systems with intermediate values of q display the worst conductive performance. This is explained by the strong inhibition of loops in the spanning cluster, resulting in a substantially smaller associated conducting backbone.

Fracturing Highly Disordered Materials

We investigate the role of disorder on the fracturing process of heterogeneous materials by means of a two-dimensional fuse network model. Our results in the extreme disorder limit reveal that the backbone of the fracture at collapse, namely, the subset of the largest fracture that effectively halts the global current, has a fractal dimension of 1.22 ± 0.01. This exponent value is compatible with the universality class of several other physical models, including optimal paths under strong disorder, disordered polymers, watersheds and optimal path cracks on uncorrelated substrates, hulls of explosive percolation clusters, and strands of invasion percolation fronts. Moreover, we find that the fractal dimension of the largest fracture under extreme disorder, d(f) = 1.86 ± 0.01, is outside the statistical error bar of standard percolation. This discrepancy is due to the appearance of trapped regions or cavities of all sizes that remain intact till the entire collapse of the fuse network, but are always accessible in the case of standard percolation. Finally, we quantify the role of disorder on the structure of the largest cluster, as well as on the backbone of the fracture, in terms of a distinctive transition from weak to strong disorder characterized by a new crossover exponent.

Oil displacement through a porous medium with a temperature gradient

We investigate the effect of a temperature gradient on oil recovery in a two-dimensional pore-network model. The oil viscosity depends on temperature as μ(o) [Please see text] e(B/T), where B is a physicochemical parameter, depending on the type of oil, and T is the temperature. A temperature gradient is applied across the medium in the flow direction. Initially, the porous medium is saturated with oil, and then another fluid is injected. We have considered two cases representing different injection strategies. In the first case, the invading fluid viscosity is constant (finite viscosity ratio), while in the second one, the invading fluid is inviscid (infinite viscosity ratio). Our results show that for the case of finite viscosity ratio, recovery increases with ΔT independent of strength or sign of the gradient. For an infinite viscosity ratio, a positive temperature gradient is necessary to enhance recovery. Moreover, we show that for ΔT>0, the percentage of oil recovery generally decreases (increases) with B for a finite (infinite) viscosity ratio. Finally, we also extend our results for infinite viscosity ratio to a three-dimensional porous media geometry.

INVASION PERCOLATION WITH A HARDENING INTERFACE UNDER GRAVITY

We propose a modified Invasion Percolation (IP) model to simulate the infiltration of glue into a porous medium under gravity in 2D. Initially, the medium is saturated with air and then invaded by a fluid that has a hardening effect taking place from the interface towards the interior by contact with the air. To take into account that interfacial hardening, we use an IP model where capillary pressures of the growth sites are increased with time. In our model, if a site stays for a certain time at interface, it becomes a hard site and cannot be invaded anymore. That represents the glue interface becoming hard due to exposition with the air. Buoyancy forces are included in this system through the Bond number which represents the competition between the hydrostatic and capillary forces. We then compare our results with results from literature of non-hardening fluids in each regime of Bond number. We see that the invasion patterns change strongly with hardening while the non-hardening behavior remains basically not affected.

Entropy Production and the Pressure-Volume Curve of the Lung.

We investigate analytically the production of entropy during a breathing cycle in healthy and diseased lungs. First, we calculate entropy production in healthy lungs by applying the laws of thermodynamics to the well-known transpulmonary pressure–volume (P–V) curves of the lung under the assumption that lung tissue behaves as an entropic spring similar to rubber. The bulk modulus, B, of the lung is also derived from these calculations. Second, we extend this approach to elastic recoil disorders of the lung such as occur in pulmonary fibrosis and emphysema. These diseases are characterized by particular alterations in the P–V relationship. For example, in fibrotic lungs B increases monotonically with disease progression, while in emphysema the opposite occurs. These diseases can thus be mimicked simply by making appropriate adjustments to the parameters of the P–V curve. Using Clausiuss formalism, we show that entropy production, ΔS, is related to the hysteresis area, ΔA, enclosed by the P–V curve during a breathing cycle, namely, ΔS=ΔA∕T, where T is the body temperature. Although ΔA is highly dependent on the disease, such formula applies to healthy as well as diseased lungs, regardless of the disease stage. Finally, we use an ansatz to predict analytically the entropy produced by the fibrotic and emphysematous lungs.

Subcritical fatigue in fuse networks

We obtain the Paris law of fatigue crack propagation in a fuse network model where the accumulated damage in each resistor increases with time as a power law of the local current amplitude. When a resistor reaches its fatigue threshold, it burns irreversibly. Over time, this drives cracks to grow until the system is fractured into two parts. We study the relation between the macroscopic exponent of the crack-growth rate ?entering the phenomenological Paris law? and the microscopic damage accumulation exponent, ?, under the influence of disorder. The way the jumps of the growing crack, ?a, and the waiting time between successive breaks, ?t, depend on the type of material, via ?, are also investigated. We find that the averages of these quantities, and , scale as power laws of the crack length a, and , where is the average rupture time. Strikingly, our results show, for small values of ?, a decrease in the exponent of the Paris law in comparison with the homogeneous case, leading to an increase in the lifetime of breaking materials. For the particular case of ??=?0, when fatigue is exclusively ruled by disorder, an analytical treatment confirms the results obtained by simulation.We obtain the Paris law of fatigue crack propagation in a diso r ered solid using a fuse network model where the accumulated damage in each resistor increases with time as a power law of the local current amplitude. When a resistor reaches its fatigue threshold, it burns irre ve sibly. Over time, this drives cracks to grow until the system is fractured in two parts. We study the relation be twe n the macroscopic exponent of the crack growth rate – entering the phenomenological Paris law – and t he microscopic damage-accumulation exponent, γ, under the influence of disorder. The way the jumps of the grow ing crack,∆a, and the waiting-time between successive breaks, ∆t, depend on the type of material, via γ, are also investigated. We find that the averages of these quantities, 〈∆a〉 and〈∆t〉/〈tr〉, scale as power laws of the crack length a, 〈∆a〉 ∝ aα and〈∆t〉/〈tr〉 ∝ a−β, where〈tr〉 is the average rupture time. Strikingly, our results show, f or small values ofγ, a decrease in the exponent of the Paris law in comparison with the homogeneous case, leading to an increase in the lifetime of breaking materials. For the particular case of γ = 0, when fatigue is exclusively ruled by disorder, an analyti cal treatment confirms the results obtained by simulation.

Crossover from mean-field to 2d Directed Percolation in the contact process

Abstract We study the contact process on spatially embedded networks, consisting of a regular square lattice with long-range connections. To generate the networks, we start from a square lattice and, to each node i we add a long-range connection to a node j , selected at random from all possible nodes, with a weight r i j − α per node. Extensive Monte Carlo simulations and a finite-size scaling analysis for different values of α reveal a crossover from the mean-field to 2 d Directed Percolation universality class with increasing α , in the range 3 α 4 .

Homeostatic maintenance via degradation and repair of elastic fibers under tension

Cellular maintenance of the extracellular matrix requires an effective regulation that balances enzymatic degradation with the repair of collagen fibrils and fibers. Here, we investigate the long-term maintenance of elastic fibers under tension combined with diffusion of general degradative and regenerative particles associated with digestion and repair processes. Computational results show that homeostatic fiber stiffness can be achieved by assuming that cells periodically probe fiber stiffness to adjust the production and release of degradative and regenerative particles. However, this mechanism is unable to maintain a homogeneous fiber. To account for axial homogeneity, we introduce a robust control mechanism that is locally governed by how the binding affinity of particles is modulated by mechanical forces applied to the ends of the fiber. This model predicts diameter variations along the fiber that are in agreement with the axial distribution of collagen fibril diameters obtained from scanning electron microscopic images of normal rat thoracic aorta. The model predictions match the experiments only when the applied force on the fiber is in the range where the variance of local stiffness along the fiber takes a minimum value. Our model thus predicts that the biophysical properties of the fibers play an important role in the long-term regulatory maintenance of these fibers.