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Our current research is centered around operator-theoretic approach to analysis of nonlinear dynamical systems, applications in microfluidics and (bio)-nanotechnology. The research topics can be grouped as follows: |
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Mixing and separation in fluids across the scales with applications ranging
from microfluidic phenomena to oceanographic flows. | |
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Nano and micro-scale particle dynamics induced by dielectrophoresis and
other electrokinetic phenomena, with applications to biotechnology. | |
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Multiscale dynamics of the Atomic Force Microscope, including interactions with biomolecules. | |
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Dynamical systems theory of complex systems, including large-scale networked systems. |
In each of these topics, the research is characterized by pursuit of the key physical phenomena in a device or system followed by the abstraction of the mathematical problem (or problems) associated with it. We close the loop by applying the solution of the mathematical problem to explain the physical phenomena or design new concepts based on which devices can be built or improved. |
An example of this strategy, within the context of the topic above, is our treatment of optimal mixing protocols for superposition of shear flows [6] that lead to the design of a micromixing device based on composition of shear flows [21, 4] (see figure; the width of the main channel on the left is 200¹m; a conceptual model composed of two parabolic shear flows is shown on the right in the figure) that in turn led to new mathematical problems on characterization of multiscale nature of mixing [9]. Solution of these problems produced new mixing protocols - and we are doing experiments on these currently. |
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From the mathematical perspective, the research on the micromixer lead to search for new mixing measures (Kolmogorov-Sinai entropy and the multiscale ”Mixing Variance Coefficient”) and the focus on dynamical properties of iteration of aperiodic maps, in contrast with the standard focus of dynamical systems theory on iterations of periodic maps. An example of this is our work on aperiodic perturbations of the Standard Map in [19]. We are extending our study of the micromixer and the associated mathematical issues to study conformation rates of DNA in the flow produced by the micromixer, in colaboration with Professor Mabuchi’s group at the Physics Department at Caltech. |
Another aspect of the research on mixing deals with the application of local geometric perturbation theory for weakly three-dimensional flows for characterization of global mixing in the flow [11, 17]. These types of flows are interesting in the context of geophysical applications, since most of the large-scale flows in the atmosphere and oceans are weakly three - dimensional. They are also interesting from the perspective of microscale flows, since, due to the low Reynolds numbers in such flows, it is difficult to generate strong three-dimensional motions in flat microchannels [18]. |
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Our research on nano- and micron-size electromagnetic field-driven particle motion in liquid environments started by studying the geometry shown in figure. Particles suspended in the liquid move due to the dielectrophoretic force that is proportional to their volume and the square of the gradient of the electric field. Thus, the more non-uniform the electric field, the larger the force on the particle. We derived electromagnetic fields for several electrode configurations of interest in [5]. |
| Since dielectrophoretic forces depend on the non-uniformity of the applied electric field, and not on the charge of the particle (i.e. the particle can be neutral), studies of this kind are of interest for bioparticle separation technologies. Besides dielectrophoretic forces, and viscous forces, particles are typically affected by fluid motion induced e.g. by nonuniform heating of the liquid (the electrothermal effect). |
| O current research in this direction includes studying effects of fluid motion on particle dielectrophoresis (partly in collaboration with Professor Oreste Piro’s group, Physics Department, Institut Mediterrani d’Estudis Avan¸cats, Palma de Mallorca, Spain). We have already discovered that flow effects allow for the formation of recirculation zones (akin to Stommel plankton recirculation zones in oceanographic Langmuir cells) thus possibly impairing dielectrophoretic effects. The associated theory can in turn be used to design efficient ways of controlling micro/nanoparticle motion in liquids. The theory is based on methods for controllability of nonlinear systems described in [13]. Using these types of methods proved useful in optimal control of vortex dynamics - or more generally control of Hamiltonian systems [20, 19]. |
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The Atomic Force Microscope (AFM) is an oscillator capable of picoNewton force resolution measurement, whose basic schematic is shown in the figure. The oscilator consists of a cantilever with a massive tip whose sharp side is at nanometer scale. The measurement of deflection of the cantilever tip is obtained by a laser beam reflected off the top surface of the cantilever, collected by a photodiode. |
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Due
to the nonlinear, attractive-repulsive nature of the interaction between
the sample and the cantilever tip in dynamic atomic force microscopy,
issues in nonlinear dynamics and control have attracted a lot of attention for
AFM operation in air. Our own research concentrated on modelling of cantilevers
in air using Lennard-Jones potential interaction between the sample
and the tip [1, 2, 3]. The nonlinearity of the forces between the tip and the
sample has profound consequences: the device exhibits bistability, complicated
bifurcation scenarios are present, and chaotic motion can occur. In our
future research in this direction we will study operation of microcantilevers
in liquid. This is of importance for imaging soft biological samples, such as
cells, and presents a whole new set of issues for nonlinear dynamics studies.
The model of coupled cantilever-fluid system is a set of macro- (continuum)
-scale partial differential equations with one boundary condition specified by
the nano-scale interaction of the tip with the sample. The theory that we
will be developing will be tested in the experiments of Professor Meinhart’s
group at the university of California, Santa Barbara. |
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Our research has interdisciplinary flavor, not only due to the various physical phenomena that we need to address to solve a given problem, but also due to the variety of mathematical tools used. In a single problem such as ”the characterization and control of mixing” we employ tools of geometric perturbation theory, ergodic theory, probability theory, and control theory. In addition, in research on complex networked systems we introduce graphtheoretic methods and employ arguments based on linear (transfer) operator theory. All of these are used to provide predictions for various experimental settings. In fact, many of the projects we pursue are in direct concurrent collaboration with experimentalists interested in testing some of the predictions of the theory [17, 16] or are tested after publication of the theory [7, 12, 18]. Some of the dynamical systems theory that we have developed for comparison of dynamics of complex physical systems [15] has been successfully used in the industrial context of the United Technologies Research Center - for identification of the nature of combustion instabilities. |
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Bibliography |
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[1] M. Ashab, M. V. Salapaka, M. Dahleh, and I. Mezi´c. Dynamical analysis
and control of microcantilevers. Automatica, 35:1663–1670, 1999. |
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