Research topics



My expertise is in the areas of applied and computational mathematics, computational materials science, and computational physics. More generally, I am interested in the development of widely-applicable mathematical and algorithmic methodologies for the modeling of complex natural systems at the atomistic and mesoscopic levels, and their application to practical problems in fundamental and applied sciences and engineering. In particular, I work on developing algorithms that accelerate or systematically coarse-grain traditional methods such as Molecular Dynamics or (Kinetic) Monte Carlo, as well as multi-scale (hybrid) methods combining particle with stochastic (fluctuating) coarse-grained models. My present focus is on fluid dynamics at small scales, and in particular, fluctuating hydrodynamics. Visit the Computer Codes page for public-domain codes incorporating our work.

My research at the Courant Institute is funded by National Science Foundation grant NSF DMS-1115341, the Air Force's Young Investigator Research Program AFOSR YIP-2012, and the Department of Energy (DOE) Office of Science Early Career Research Program.

Specific details on several research subjects I have or am working on (with latest work first) can be found below. There are many interesting projects for undergraduate and graduate students in each of these areas, ranging from developing novel algorithms to applying computational methods to relevant problems in the sciences and engineering.


Hydrodynamics at Small Scales


This research focuses on fluid mechanics in regimes where thermal fluctuations are important. Notable examples include flows at micro and nano scales typical of new microfluidic, nanofluidic and microelectromechanical devices; biological systems such as lipid membranes, Brownian molecular motors, nanopores; as well as processes where the effect of fluctuations is amplified by strong non-equilibrium effects, such as combustion of lean flames, capillary dynamics, hydrodynamic instabilities, and others. Here is, for example, an animation showing the development of interface fluctuations (giant fluctuations) during diffusive mixing of two fluid in two dimensions, driven entirely by thermal fluctuations, that is, starting with no initial interface perturbation:

Mixing animation

My work in this field is joint with several collaborators, whose webpages contain additional information and publications of interest. In particular, I collaborate closely with John Bell (Lawrence Berkeley National Laboratory) and Alejandro Garcia (San Jose State University), Rafael Delgado-Buscalioni (UAM, Spain), Eric Vanden-Eijnden (Courant), and Boyce Griffith (NYU Medical School). Our work builds on the mature field of deterministic computational fluid dynamics, and combines fundamental cross-disciplinary investigations with development of extensible yet efficient parallel computer codes (in particular, our algorithms are incorporated into the BoxLib and IBAMR codes, and presently also extended to run on GPUs).

There are several main avenues of research, described next, see also these presentation slides.

Numerical schemes for fluctuating hydrodynamics

Thermal fluctuations can be included in the classical Navier-Stokes fluid equations through stochastic forcing terms that are essentially the divergence of a white-noise random field (stochastic flux), as first proposed by Landau and Lifshitz. The presence of non-trivial dynamics at all scales, as well as the necessity to maintain fluctuation-dissipation balance in spatio-temporal discretizations, makes the continuum stochastic partial differential equations of fluctuating hydrodynamics difficult to solve using existing approaches.

We have recently analyzed and implemented a specialized spatial discretization along with a three-stage stochastic Runge-Kutta temporal integrator for the single-fluid compressible fluctuating equations, as described in this presentation. Recently we have developing numerical methods for solving the fluctuating incompressible and low Mach number Navier-Stokes equations. This work is reported in these papers:

    1. "Temporal Integrators for Fluctuating Hydrodynamics", S. Delong and B. E. Griffith and E. Vanden-Eijnden and A. Donev, to appear in Phys. Rev. E, 2013 [arXiv:1212.1033].
    2.  "Staggered Schemes for Fluctuating Hydrodynamics", F. Balboa and J. Bell and R. Delgado-Buscalioni and A. Donev and T. Fai and B. Griffith and C. Peskin, submitted, 2011 [arXiv:1108.5188].
   3. "Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids", A. Donev and A. J. Nonaka and Y. Sun and T. Fai and A. L. Garcia and J. B. Bell, submitted to SIAM J. Multiscale Modeling and Simulation, 2013 [ArXiv:1212.2644].
  
Present work is focused on:
1. Developing algorithms for the fluctuating Low Mach number equations for multispecies fluid mixtures on adaptive Cartesian grids, see this presentation.
2. Accounting for phase separation via the stochastic Cahn-Hilliard equation and studying fluctuations at fluid-fluid interfaces.
3. Accounting for chemical reactions in complex fluid mixtures such as lean hydrogen fuels.

Brownian Dynamics via Fluctuating Hydrodynamics

The study of complex fluids such as micro- and nano-colloidal or polymeric solutions is an important application of fluctuating hydrodynamics. We have previously used a particle method to study the dynamics of polymer chains in solution. We are presently developing methods to directly couple immersed particles to a fluctuating fluid medium, based on existing Lattice-Boltzmann and Stochastic Immersed Boundary techniques, see this presentation. Initial work in this area is described here

   3. "Inertial Coupling Method for particles in an incompressible fluctuating fluid", F. Balboa Usabiaga and R. Delgado-Buscalioni and B. E. Griffith and A. Donev, submitted, 2013 [ArXiv:1212.6427], code available at https://code.google.com/p/fluam.

Present work is focused on:
1. Extending the temporal integrators to handle the overdamped (Brownian dynamics) limit.
2. Extending the physical fidelity of the blob models by adding stresslet terms.
3. Extending the work to rigid bodies. This work is in collaboration with the research group of Neelesh Patankar at Northwestern.
4. Understanding theoretical (applied stochastic analysis) issues related to the diffusive limit (in collaboration with Eric Vanden-Eijnden).

Particle methods for hydrodynamics

Thermal fluctuations can be included in fluid dynamics by explicitly accounting for the particle nature of matter using particle methods. It is also important to perform efficient particle calculations in order to assess and expand the range of validity of continuum approximations. Our research focuses on developing coarse-grained stochastic particle models that build upon the Direct Simulation Monte Carlo (DSMC) method. This class of methods are also related to the dissipative particle dynamics (DPD) and the multi-particle collision (also called stochastic rotation) dynamics techniques. Further details can be found in this presentation or the following papers:

    1. "A Thermodynamically-Consistent Non-Ideal Stochastic Hard Sphere Fluid", by A. Donev and A. L. Garcia and B. J. Alder, J. Stat. Mech., P11008, 2009 [arXiv:0908.0510].
    2. "Stochastic Hard-Sphere Dynamics for Hydrodynamics of Non-Ideal Fluids", by A. Donev, A. L. Garcia and B. J. Alder, Phys. Rev. Lett., 101:075902, 2008  [arXiv:0803.0359].  
    3. "Stochastic Event-Driven Molecular Dynamics", by A. Donev, A. L. Garcia and B. J. Alder, J. Comp. Phys., 227(4):2644-2665, 2008, [arXiv:0708.0251].

Our present research is focused on extending these types of algorithms to multi-species reactive mixtures, implementing the algorithms in a public domain code that we will release in the future, as well as parallelizing the particle algorithms.

Hybrid particle-continuum algorithms

Particle and continuum methods can be combined into hybrid multiscale methods that use the more expensive but accurate particle method only in regions where the continuum description fails or is difficult to implement (e.g., shocks, rarefication, singularities, near suspended structures). This can substantially lower the cost of particle methods while still keeping the advantages of particle methods in regions of interest.

Packings of
                      Ellipsoids


This image illustrates how three different aspects of my research are combined to study the dynamics of a polymer chain in shear flow. The chain itself is made up of hard disks (red, done using event-driven molecular dynamics) and suspended in a coarse-grained particle solvent (green, done using our I-DSMC algorithm), further embedded in a stochastic continuum fluid flow (purple arrows show the fluctuating velocities in our RK3D fluctuating hydrodynamics solver).

We have developed a bidirectional dynamic coupling between a stochastic particle fluid and a fluctuating continuum and demonstrated that thermal fluctuations have to be consistently included in the continuum component of hybrid calculations in order not to distort the thermal equilibrium in the particle solver. Here is a presentation (without the movies) on the subject, with details contained in this paper:

    1. "A hybrid particle-continuum method for hydrodynamics of complex fluids", by A. Donev and J. B. Bell and A. L. Garcia and B. J. Alder, SIAM J. Multiscale Modeling and Simulation 8(3):871-911, 2010 [arXiv:0910.3968].

There are severe time step restrictions inherent to both particle and hybrid calculations, stemming from the need to resolve the dynamics of the fluid particles. In order to extend the time step, one needs to avoid the particle representation in the solver, as we do in more recent work described above.


Event-Driven Asynchronous Algorithms


Rather broadly, I am interested in the development of efficient particle methods, specifically asynchronous event-driven Molecular Dynamics and Kinetic Monte Carlo methods. My research in particle packings, reaction-diffusion systems, and coarse-grained solvents, all use an event-driven framework to achieve substantial speedup over traditional time-driven methods, as described in this presentation and this review article:

    1. "Asynchronous Event-Driven Particle Algorithms", by A. Donev, SIMULATION: Transactions of the Society for Modeling and Simulation International, 85(4):229-242, 2009.

An important remaining challenge for future research is efficient parallelization of asynchronous event-driven algorithms.

Diffusion-Reaction Systems

Together with collaborators at Lawrence Livermore National Labs we developed an event-driven Kinetic Monte Carlo algorithm for diffusion-reaction systems that is far superior to traditional algorithms. This method has the potential to be applied in a variety of problems in material science and biology. Notably, we have used it to study radiation damage in metals, and the method has been implemented for biochemical reactive systems in the eGFRD code. Further details can be found in this presentation and these papers:

    1. "First-passage Kinetic Monte Carlo method", by T. Oppelstrup, V. V. Bulatov, A. Donev, M. H. Kalos, G. H. Gilmer and B. Sadigh, Phys. Rev. E, 80(6):066701, 2009  [arXiv:0905.3575].
    2. "A First-Passage Kinetic Monte Carlo Algorithm for Complex Diffusion-Reaction Systems", by A. Donev, V. V. Bulatov, T. Oppelstrup, G. H. Gilmer, B. Sadigh and M. H. Kalos, J. Comp. Phys., 229(9):3214-3236, 2010 [arXiv:0905.3576].

Future work will extend these methods to enable a mixed time-driven with event-driven framework that combines the advantages of each approach, without the inefficiency of time-driven algorithms or the complexity of purely event-driven handling.

Packing of Hard Particles


My Ph.D. dissertation (thesis) and my  Final Oral Exam (FPO) presentation (June 9th, 2006) give a rather extensive overview of my research on Jammed Packings of Hard Particles, performed under the supervision of Dr. Salvatore Torquato as part of the Complex Materials Theory Group. Here are some MNG and GIF animations related to my thesis research.


Packings of
                      Ellipsoids


You can find executable versions of my molecular dynamics packing codes along with some instructions here. These codes were used to make the packings of hard ellipsoids shown to the left on top of an experimental packing of M&M prolate ellipsoids.

I have also developed dynamic VRML models to render and animate packings of spheres and ellipsoids. These are barely documented but you can get them here if you are brave.

Rigidity Theory and Jamming

By using mathematical results from rigidity theory, in collaboration with Robert Connelly, we were able to develop a rigorous framework for defining and testing for jammed packings of hard spheres and ellipsoids, as described in these papers:

    1. "Underconstrained Jammed Packings of Hard Ellipsoids", by A. Donev, R. Connelly, F. H. Stillinger and S. Torquato, Phys. Rev. E, 75:051304, 2007 [cond-mat/0608334].
    2. "A  Linear Programming Algorithm to Test for Jamming in Hard-Sphere Packings", by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, J. Comp. Phys., 197(1):139-166, 2004.
    3. "Comment on "Jamming at zero temperature and zero applied stress: The epitome of disorder", by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Phys. Rev. E, 70:043301, 2004.
    4. "Jamming in Hard Sphere and Disk Packings", by A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, J. Appl. Phys., 95(3):989, 2004.
    5. "Breakdown of Elasticity Theory for Jammed Hard-Particle Packings: Conical Nonlinear Constitutive Theory", by S.Torquato, A. Donev, and F. H. Stillinger, Int. J. Solids Structures, 40(25):7143-7153, 2003.

Further work is necessary to develop codes that can analyze the rigidity properties of large-scale packings.

Packing of Hard Ellipsoids

We further developed sophisticated event-driven molecular dynamics algorithms and codes to generate jammed packings of hard spheres and also, for the first time, hard ellipsoids and hard super-ellipsoids, in two, three and, for spheres, in higher dimensions. Together  with the experimental research group of Paul Chaikin, we made some surprising discoveries about random packings of ellipsoids. Firstly, random packings of moderately aspherical ellipsoids are substantially denser that that of spheres, specifically, the density grows linearly with the aspect ratio, even though the packings were found to be hypostatic and not isostatic as commonly expected. An astounding packing fraction of 74% was found for a specific ellipsoid shape that we named ollipsoids (optimal ellipsoids). These results are detailed in the following publications:

    1. "Improving the Density of Jammed Disordered Packings using Ellipsoids" by A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato and P. M. Chaikin, Science, 303:990-993, 2004.
    2. "Neighbor List Collision-Driven Molecular Dynamics Simulation for Nonspherical Particles. I. Algorithmic Details II. Applications to Ellipses and Ellipsoids", by A. Donev, F. H. Stillinger, and S. Torquato, J. Comp. Phys, 202(2):737-764 (part I) and 202(2):765-793 (part II),  2005, [physics/0110034].
    4. "Experiments on Random Packings of Ellipsoids", W. Man, A. Donev , F. H.  Stillinger, M. T. Sullivan, W. B. Russel, D. Heeger , S. Inati, S. Torquato and P. M. Chaikin, Phys. Rev. Lett., 94:198001, 2005.

At the time we discovered random packings of ellipsoids as dense as the FCC hard-sphere crystal, it was thought that the FCC crystal structure and maximal volume fraction of 74% applies to ellipsoids as well. However, using hard-ellipsoid molecular dynamics and some geometry we soon discovered that there are denser ellipsoid crystals, as dense as 77% in volume fraction:

    1. "Unusually Dense Crystal Packings of Ellipsoids", by A. Donev, F. H. Stillinger, P. M. Chaikin and S. Torquato, Phys. Rev. Lett., 92:255506,  2004, [cond-mat/0110034].

Since then, this work has been extended to super-ellipsoids (generalized ellipsoids), which you can learn more about from the homepage of Dr. Salvatore Torquato.

Random Sphere Packings

We also performed detailed studies of large random packings of hard spheres in three and higher dimensions and found some surprising results, including that maximally random jammed packings of hard spheres are hyperinform (incompressible) and potentially exhibit long-ranged correlations, unlike hard-sphere fluids. Further details can be found in these papers:

    1. "Pair Correlation Function Characteristics of Nearly Jammed Disordered and Ordered Hard-Sphere Packings", by A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. E, 71:011105, 2005, [cond-mat/0408550].
    2. "Unexpected Density Fluctuations in Jammed Disordered Sphere Packings", by A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. Lett., 95:090604, 2005, [cond-mat/0506406].
    3. "Packing Hyperspheres in High-Dimensional Euclidean Spaces", by M. Skoge, A. Donev, F. H. Stillinger and S. Torquato, Phys. Rev. E, 74:041127, 2006 [ibid 75:029901, 2007], [cond-mat/0608362].

Glass Transition

One of the most fascinating open questions in condensed matter physics is the glass transition. Two central questions are whether there are thermodynamically-stable disordered (non-crystalline) solid phases and whether there is an ideal glass transition (vanishing configurational entropy in a disordered phase). By using our hard-sphere molecular dynamics algorithms we found convincing evidence that there can exist disordered solid phases and that there is no ideal glass transition in certain hard particle systems, as described in the following papers:

    1. "Do Binary Hard Disks Exhibit an Ideal Glass Transition?", by A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. Lett., 96:225502, 2006, [cond-mat/0603183].
    2. "Configurational Entropy of Binary Hard-Disk Glasses: Nonexistence of an Ideal Glass Transition", by A. Donev, F. H. Stillinger and S. Torquato,  J. Chem. Phys., 127:124509, 2007.
    3. "Calculating the Free Energy of Nearly Jammed Hard-Particle Packings Using Molecular Dynamics", by A. Donev, F. H. Stillinger, and S. Torquato, J. Comp. Phys., 225:509–527, 2007.
    4. "Tetratic Order in the Phase Behavior of a Hard-Rectangle System", by A. Donev, J. Burton, F. H. Stillinger, and S. Torquato, Phys. Rev. B, Vol. 73:054109, 2006, [cond-mat/0508550].

These results point to a kinetic origin of the glass transition, and further study is needed to understand the geometrical origin of the configurational trapping experienced near the kinetic glass transition.


Multifunctional Composites


During my Ph.D. studies I also joined an effort in the group of  Dr. Salvatore Torquato concerning the computational design of binary composites (mixtures of two materials) that optimize transport properties such as conductivity. This research led to the discovery that triply-periodic minimal surface structures are optimal for several transport properties, as illustrated in this figure for the Schwartz simple cubic surface bi-continuous structure:


Void
                      density


Void
                        density


Further details can be found in these papers and more recent publications by the Torquato group:

    1. "Multifunctional Optimal Composite Microstructures: Simultaneous Transport of Heat and Electricity", by S.Torquato, S. Hyun and A. Donev, Phys. Rev. Lett., 89(26):266601, 2002.
    2. "Manufacturable extremal low-dielectric, high-stiffness porous materials", S. Torquato, A. Donev, A. G. Evans, and C. J. Brinker, J. Appl. Phys., 97:124103, 2005.
    3. "Minimal Surfaces and Multifunctionality", by S.Torquato and A. Donev, Proceedings of the Royal Society of London: Mathematical, Physical and Engineering Sciences, 460(2047):1849 - 1856, 2004.
    4. "Optimal design of manufacturable three-dimensional composites with multifunctional characteristics", by S.Torquato, S. Hyun and A. Donev, J. Appl. Phys., 94(9):5748-5755, 2003.

Subsequent work in the group of Dr. Salvatore Torquato has also studied fluid permeability in such composite structures.