In 2007 and during his postdoctoral work, Professor Hernandez-Ortiz and professors Juan de Pablo and Michael D. Graham developed the General Geometry Ewald-like Method (GGEM). This method is an efficient O(N) technique to calculate long-range interactions, i.e. hydrodynamic interactions (HI) and electrostatics, in any geometry. The GGEM has been used to study the effects on confinement in flowing polymer solutions, the self-assembly of beta-peptides, the dynamics of self-propelled particles, among other applications.

The original GGEM consider point forces, for the case of hydrodynamics, and point charges or dipoles, for the case of electrostatics. It is a Green's function based method. The following routines are available for use after e-mail request to Prof. Juan P. Hernandez-Ortiz, Prof. Michael D. Graham or Prof. Juan J. de Pablo:

Periodic Systems:

1. Stokeslet

2. Regularized Stokeslet

3. Electrostatics for point charges

4. Electrostatics for point dipoles

Slit – two parallel walls:

1. Stokeslet with non-slip boundary conditions at walls

2. Regularized Stokeslet with non-slip boundary conditions at walls

3. Electrostatics for point charges with homogeneous Dirichlet boundary conditions at walls

4. Stokeslet and Regularized Stokeslet with zero velocity gradient at walls (SOON)

5. Electrostatics with homogeneous Neumann boundary conditions at walls (SOON)

Complex geometries FEM-GGEM:

1. Stokeslet with non-slip boundary conditions at walls: rectangular channel, grooved walls, rectangular pore and rectangular multi-pore channels.

2. Electrostatics with Dirichlet or Neumann boundary conditions at walls (SOON)

Currently, there are two generalizations to the GGEM method: the inclusion of lubrication (SD-GGEM, where

SD denotes Stokesian Dynamics) and the coupled solution of Nernst-Planck equation (NP-GGEM). SD-GGEM is been developed in collaboration with Dr. Sam Anekal (from Graham's group) and it is ready for periodic and slit geometries. It is the first method to simulate confined suspensions. The NP-GGEM is been developed in Prof. Hernandez-Ortiz group in collaboration with Professors de Pablo and Graham; it allows simulations of macro-ions within a charged solvent. After publication these routines will be also available for use:

SD-GGEM: (SOON) to be publish in J. Fluid Mech.

1. Periodic domains

2. Slit with non-slip boundary conditions at walls

NP-GGEM: (SOON) to be publish in J. Chem. Phys.

1. Periodic domains

2. Complex geometries with both type of boundary conditions for charges and non-slip for velocities, all solved with FEM.

These projects are in collaboration with Prof. Juan de Pablo and Prof. Michael D. Graham from the Department of Chemical and Biological Engineering at the University of Wisconsin-Madison. They are founded through the UW-Madison Nanoscale Science and Engineering Center (NSEC) from the National Science Foundation (NSF).

Confined Polymer Solutions

The dynamics of polymer solutions, driven by flow or electric fields, in a confined geometry is a fundamental and important research topic underlying many applications in nano- and micro-engineering. With the development of new fabrication techniques micro- and nano-fluidic devices are used to separate and manipulate biological polymers like DNA and poly-peptides, creating novel techniques of gene mapping and DNA separation or hybridization.

Our intention is to study the effects of confinement on flowing polymer solutions at finite concentrations considering full hydrodynamic interactions. We are able to simulate range of concentrations from infinitely dilute to 10-30% overlap concentration. We found that the shear-induced migration is affected by concentration. In addtion, for non-smooth channels we found significant differences as the concentration is increased.

Steady-state l-phage DNA chain center-of-mass distributions, for an infinitely dilute concentration: equilibrium (left) and flowing solution (top wall is moving to the right) at Wi=20: free-draining (middle) and with hydrodynamic interactions (right).

Steady-state l-phage DNA chain center-of-mass distributions, for a concentration of 12% overlap at Wi=20: free-draining (left) and with hydrodynamic interactions (right).

Liquid Crystals

In recent years, nematic liquid crystals have been employed extensively in detection of targeted biological entities, whereby surface events, such as binding of proteins, viruses and microbes, cause a local change in liquid crystal orientation. These orientational changes are amplified over several thousand molecular lengths through the emergence of mesoscopic defects that are easily detected using optical microscopy. Applications to date have relied on optical images of final states, and have therefore been limited to static information. The dynamics or evolution of a sensor could potentially provide a wealth of information about analytes of interest, but to extract this information, one must first develop a realistic model for the dynamics of confined liquid crystals

Our intention is to solve a detailed molecular model of liquid crystal dynamics on a model sensor, and to investigate the differences in defect relaxation that arise when effects of hydrodynamic interactions (HI) are considered. The dynamic equations for the liquid crystal dynamics are based on the model of Stark and Lubensky.

Currently, we are investigating the dynamic of nano-particles within a confined liquid crystal. This project is in collaboration with Prof. Juan de Pablo and Prof. Nicholas Abbott from the Department of Chemical and Biological Engineering at the University of Wisconsin-Madison. They are founded through the UW-Madison Materials Research Science and Engineering Center (MRSEC) from the National Science Foundation (NSF).

Self-Propelled particles

The collective behavior of individual agents is important for a wide range of disciplines. In particular, the collective behavior of swimming microorganisms has received much recent attention. Not only is this system an example of a general phenomena, but also has important implications for motion of cells and organisms at small scales and ability to use it to perform important tasks. Specifically for swimming microorganisms, it has been postulated that long-ranged hydrodynamic interactions play a key role in the collective behavior, and the role of these interactions has been examined. Computer simulations can and have played an important role in this understanding because the physics of hydrodynamic interactions can be included without other physics as a way of testing whether they are sufficient to cause the behavior seen in experiments. Simulations have shown that hydrodynamic interactions are sufficient to cause the same qualitative behavior seen in experiments

In this research we focus on the role that confinement plays in the collective behavior. Many of the experiments looking at the collective behavior of swimming bacteria have boundaries nearby. These boundaries will alter the hydrodynamic interactions. Therefore, studies which aim to understand how the hydrodynamic interactions lead to collective behavior need to understand the effect of the walls. The presence of the walls could play a number of important roles in collective behavior. The hydrodynamic interactions of a swimmer with the walls leads to a nonuniform concentration of swimmers within the domain. The walls also alter the hydrodynamic interaction between the organisms, which in our simulations is responsible for the collective behavior. Finally, the walls introduce another length scale into the problem which could prohibit collective structures at larger length scales. It is important to understand these effects not only to correctly interpret simulations and experiments but also to include the effect of boundaries into designs of new devices.

Snapshot of the fluid velocity field, generated due to the swimmers, at a concentration of 10% overlap. This is a top view of a slit geometry.

This project is in collaboration with Prof. Michael D. Graham from the Department of Chemical and Biological Engineering at the University of Wisconsin-Madison

Rare Events far from Equilibrium

Transition path sampling (TPS) and its variations, such as transition interface sampling (TIS), forward flux sampling (FFS), are now well establish methodologies to study rare events and to calculate reaction rate constants. The main advantage of these methods is that they do not require the specification of a reaction coordinate. Rare events are fluctuation-driven processes which occur infrequently. There are several natural processes that can be classified as rare events: chemical reactions, nucleation of crystals, protein agglomeration, etc. Technically, a process is defined as rare event if the waiting time between events is orders of magnitude longer than the time scale of the event if self. Therefore, pure dynamical simulations, like molecular dynamics or Brownian dynamics, are highly inefficient.

TPS finds transitions path-ways for rare events. It is applicable to both equilibrium and non-equilibrium systems. However, it presents a restriction: the time during a transition from a stable state A to a stable state B must be smaller than the characteristic time spend in each stable state A or B. In other words, the transition must occur very rapidly. TIS and/or FFS are methodologies (described below) that overcomes this limitation. Again, these methods do not require the specification of a reaction coordinate, and transition paths are generated through a series of interfaces in the phase space, which prevents restrictions in the length and size of the paths.

Combining a FFS methodology with Brownian Dynamics (BD), we investigate the hydrodynamic interactions effects on complex fluids transitions. We start with simple polymer translocation through pores and we will proceed to study DNA hybridization, liquid crystal defects among others.

This project is in collaboration with Prof. Juan de Pablo from the Department of Chemical and Biological Engineering at the University of Wisconsin-Madison. They are founded through the UW-Madison Materials Research Science and Engineering Center (MRSEC) from the National Science Foundation (NSF).

Other Projects

· Molecular models of VIRUS, DNA encapsulation and RNA

· Confined fiber suspensions: fiber-matrix separation

· Modeling and Simulation of elastomeric foams

· Modeling high populations of confined bacteria: Chemotaxis

· Energy savings on close heat-exchanger fluid systems: drag reduction

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Date last modified: 27-July-2008

Universidad Nacional de Colombia,Sede Medellin