NSF-PREM Sponsored Research

Simulations of material instabilities

Cavitation type instabilities In a sequence of two papers, Negrón-Marrero and Sivaloganathan (2011, 2012) introduced the concept of volume derivative to study the initiation of fractures in materials. The volume derivative measures the amount of energy per unit volume required to open a hole (cavitation) of unit size in a given material. They showed that the boundary of the region in strain space for the initiation of cavitation is characterized as the zero level set of the volume derivative. Computing the zero level set of the volume derivative requires evaluating it many times which is a very time consuming process. (An evaluation of the volume derivative requires the solution of the (nonlinear) equations of 3d elasticity subject to a certain volume constraint.) We developed a numerical scheme based on a penalty multiplier technique to evaluate the volume derivative very efficiently. Our next goal is to develop a web based application that can be used remotely by researchers to study the critical surface or boundary for cavitation for specific materials based on the numerical schemes that we have developed to compute the volume derivative.

Surface wrinkling and violation of the complementing condition The complementing condition (CC) for a boundary value problem (BVP) is an algebraic compatibility requirement between a linear differential operator and the corresponding boundary operators. Violations of the CC have been associated to surface wrinkling instabilities. If the BVP under consideration depends on a load parameter, then wrinkling is associated with the existence of increasingly ``oscillatory" solutions corresponding to values of the load parameter accumulating at a particular load. This motivated the following conjecture or question: are violations of the CC equivalent to the accumulation of highly oscillatory solutions of a given BVP? If so, one could characterize wrinkling instabilities by an algebraic criterion. In Negron-Marrero and Montes-Pizarro (2011, 2012) the authors show that the answer to this question is affirmative for compressions (either axial or lateral) of cylindrical structures composed of a Hadamard--Green type material.

Minimum energy configurations on finite molecular arrays Recently we have begun to study the problem of cavitation in fluids but at the molecular level. It has been observed in laboratory experiments and molecular dynamics simulations that as the density of a fluid decreases (keeping the temperature constant), there exists a certain threshold density such that if the density of the fluid decreases further, then cavities or “bubbles” appear in the fluid. We are working on a model to measure or to predict where this cavities are located on a given molecular dynamics simulation. We are currently studying finite molecular arrays an characterizing whether or not such inhomogeneous configurations are stable as the density becomes small. Also we are employing techniques based on Voronoi diagrams and Delaunay triangulations to measure local densities in a given molecular array. This work has been done with the graduate student Melissa Lopez (UPRRP) as part of her master thesis. In the figure we show a volumetric sketch of a density function for a certain molecular array based on a Delaunay triangulations. Regions in blue correspond to low density regions where it is more likely that some kind of hole or bubble has formed. At a more theoretical level, we have studied the stability of arrays of three particles or molecules under a Lennard-Jones type potential. In particular we look for energy minimizing configurations of the array under the constraint of fixed area for the array. We found that a certain relation between the area of the array and the inter molecular potential must be satisfied in order for the equilateral trianglar configuration to be a minimizer of the potential energy functional. In the figure we show a numerical bifurcation diagram for a specific potential showing the different possible configurations for the energy minimizers and their stability as the area parameter in the constraint changes.

Computer simulations of Drosophila wavefronts

In early embryos of many species mitosis progresses as a wavefront through the embryo. Such signaling is generally assumed to be biochemical in nature, but there is also a strong mechanical component due to the displacements of the chromosomes during division. A. Liu’s group (UPENN) proposed a model based in mechanical signaling given by the following heat-type equation: Γ ut=div(C ∇u)+f(u). We developed software to perform (time dependent) computer simulations of wavefronts in Drosophila embryos solving this equation (in curvilinear coordinates) over a shell. (See figure to the right.) We are currently revising the proposed model as it does not describe the dynamics of the problem as observed on actual experiments. The revised model is based on a nonlinear damped wave equation with coefficients and damping terms depending on the nuclei density. Preliminary simulations of a one dimensional version of this model where performed with the assistance of the student J. Velázquez (UPRH) and where reported in the 8th PREM Annual Symposium of 2014. The figure below show some of the preliminary results. The speed profile predicted by the model for the wave, as the different cycles of the nuclei division progress, qualitatively agrees with the one observed in laboratory experiments. Our next goal is to perform computer simulations with the revised model over a shell.

Ion transport through carbon activated EDLC's

Electrical Double Layer Capacitors (EDLC), also known as supercapacitors, are electric devices with high energy storage. They consist of two layers of an electrode divided by a membrane permeated in an electrolyte. It has been observed experimentally that as the voltage in a super-capacitor is increased, then for a certain threshold voltage, the capacitance increases abruptly. To study this phenomenon, with the assistance of the student Aixa De Jesus (UPRH), we have performed molecular dynamics simulations using the Wolffia software to model the interaction between the ions in the electrolyte and the electrode. In this model the geometry of the activated carbon is simulated using CNT, graphene and carbon atoms. As electrolyte we used sulphuric acid. The diameter of the pores tapers from 1.6 to 0.6 nm. The model predicts ion transport into the activated carbon pores. We still have to measure the capacitance of the system and study its dependence on the porous size (radius, length and tapering), weight percent of the sulfuric acid as well as other types of electrolyte. We are also interested in performing simulations of the time dependent process of charge and discharge of capacitors (general geometry) in order to understand better the performance of these devices in terms of their different components.

Actuators

On-going efforts to produce novel materials with improved or augmented functionalities as sensors and actuators frequently focus in smart materials. A novel actuation mechanism, actuation by irradiation from a source, has been reported for polymer-carbon nanotube composites. In a fist stage of this project we considered a simple model for photo-actuation proposed by Ahir and Terentjev based on the theory of linear elasticity. The basic assumption in their model is that a given carbon nanotube contracts by a certain ratio under the influence of an external energy source. The critical value of pre–strain is the value of the (macro) strain at which the stress–strain curves for the composite, with and without an external energy source, intersect. In laboratory experiments with various polymer–nanotube composites at different concentrations of nanotubes, it has been observed that this critical pre–strain value is approximately 10%. The model of Ahir and Terentjev predicts that for a critical pre–strain value of 10%, the average contraction per nanotube should be around 20%. However, laboratory data reveals that the actual critical pre–strain value is achieved with only a 1-2% average contraction per nanotube. With a slight change in the way the projection of a given nanotube deformation onto the principal axis of strain is computed, we improved in Negron-Marrero and Campo (2011) on Ahir and Terentjev results to a 10–13% nanotube deformation for a 10% critical pre–strain value. On another related project we considered composite fibers combining MWCNT and Sodyum dodecyl sulfate (SDS) that have been produced in the Physics UPRH laboratory by the electro-spinning technique. These fibers appear to exhibit photo-actuation properties when exposed to an external light source. In De Jesus and Negron-Marrero (2013) we employed a model from nonlinear elasticity to describe the deformation of a given fiber and for computing the forces and internal torques that induce the photo-actuation effect. We designed an inverse method in which starting from the known deformation of the fiber, obtained from digitized pictures of the bended fiber, the corresponding forces and torques are calculated.