Purdue_symbol Purdue University
School of Science

John H. Cushman

                                                    University Distinguished Professor of Earth and Atmospheric Science
Professor of Mathematics
                 Office Address: Math Sciences Building, Rm 816 and Civil Engineering Biulding Rm 3223
Mailing Address: Purdue University
Earth and Atmospheric Sciences
550 Stadium Mall Drive
West Lafayette, IN 47907-2051
Phone: (765) 494-8040 or (765) 494-3258
E-mail: jcushman@math.purdue.edu

Curriculum Vitae / Research Group/  Selected Publications /

Prof. John Cushman
          New arrival
The focus of Professor Cushman's research is the physics of fluids in porous media, over time/space scales ranging from picoseconds/angstroms to years/miles. Problems of special interest are (i) species separation and phase change in micropores, (ii) dispersion in media with continuously evolving heterogeneity, (iii) swelling colloidal systems, (iv) reservoir-scale dispersion of environmental contaminants in natural geological media, (v) transconjugation of genes between microbes their evolution in the environment, and (vi) developing theories for the evolution of earths plates. Some past and present examples are presented below.
Species separation and phase change in micropores Consider a fluid contained in a pore when that pore is only a few fluid-molecular diameters wide in at least one dimension. Such fluids are of importance in condensed matter physics (model systems for the study of critical phenomena), in biology (protein folding and transport through membranes), in engineering and materials science (nanotechnologies), and in environmental science (chemical adsorption on soil colloids). Computational statistical mechanical experiments carried out by Cushman's group enhance our understanding of such fluids. Even a fluid as simple as a rare gas mixture displays an extremely rich and anomalous behavior when confined to a structured planar system of width on the order of a few fluid-molecular diameters. The fluid's phase diagram is changed, its transport coefficients are radically altered from those in the fluids bulk phase, and it becomes inhomogeneous and anisotropic. The properties of the fluid depend in a complex way on the initial structure of the liquid, the structure and commensurability of the confining walls, the wall-fluid interaction, the separation of the walls, asperities within the pore walls, and, if the pore-fluid is in equilibrium with its bulk-phase, then the pore-fluid depends strongly on the bulk-phase composition. 

See animation: Capillary Snapoff

If a porous medium looks inhomogeneous at every scale on which it can be viewed, then it is said to have continuously evolving heterogeneity. Many natural geologic media, and more generally fractal porous media, are of this category. By using nonequilibrium statistical mechanics, Cushman's group developed general theories of conservative chemical transport in this type of system. The theories are non-Markovian, but they reduce to their appropriate Fickian counterparts in the asymptotic limits. Interestingly, these theories can be applied to turbulent bulk-phase dispersion as well as to fluids in porous media. 

Dispersion in media with continuously evolving heterogeneity

colloidal systems
Swelling porous media include many natural soils, baked foodstuffs (chips, cookies, pasta, breads), many drug delivery substrates, and body tissues. Cushman's group has provided the first correct derivations and statements of Darcy's and Fick's laws for such systems. The group showed that contrary to classical belief, flow in swelling systems is not driven by gradients in pressure and external fields (e.g. gravity) alone, but is also driven by changes in Helmholz free energy with volume fraction (the "interaction" potential). This result is of major significance in problems of drying that involve crust formation in soils and food polymers. The Cushman group also provided rational definitions for the nonequilibrium capillary and swelling (disjoining) pressures in such systems. Nonequilibrium swelling pressure gives rise to the well-known exponential swelling law when applied at equilibrium. Most recently Cushman's group has developed theories for coupled heat and mass transfer and swelling systems which include electroquasistatic effects.

Large-scale heterogeneities in an aquifers hydraulic and chemical character play a fundamental role in the evolution of contaminants in the environment. The work of Cushman's group shows that uncertainty in the parameters that characterize an aquifer give rise to spatially and temporally nonlocal constitutive laws for chemical transport. When computationally implemented, these laws often lead to different conclusions regarding groundwater contamination scenarios than those commonly employed by litigators and by the Environmental Protection Agency when enforcing environmental regulations.

Coming soon: Animation 

Reservoir-scale dispersion of
environmental contaminants in
natural geologic media

       Tranconjugation of  genes   between microbes and their
            evolution in the environment 



Our group focus is on horizontal transfer of extra chromosomal DNA between microbes via mobile elements (plasmids or other transposable elements). Specifically we are interested in visualization of the process on the micron scale, it's evolution in the environment and in developing a mechanistic model to predict the evolution of the genes in the recipient population. 

Plate tectonics has changed the way earth scientists look at the surface of the Earth. It has recently become clear that there are broad zones of deformation within what were once thought to be the earths rigid plates. Our group is focusing on finding theories of such deformation, and more generally of the formation and evolution of the plates themselves. We are attacking this problem with numerical models based on micromorphic continuum theories, and a generalized "statistical mechanics" of the lithosphere and mantle.

Developing theories of the evolution of the earths plates

Tracking the paths of microbes
The ability to characterize the Lagrangian trajectories will allow for the incorporation of microbial motility into models of porous media, but first Lagrangian trajectories of real experiments must be gathered. The video below was created from a video by Howard Berg by using particle tracking velocimetry to trace the path of the microbes. Several different points on each microbe were used. This is a first trial in gathering Lagrangian trajectories of microbes. Particle tracking velocimetry was also used in the experiments related to the general theories for inhomogeneous systems.
Click here to see the video

  • Selected Publications
    • Park*, M. and J. H. Cushman (2006) On upscaling operator-stable Levy motions in fractal porous media. J. Comp. Phys. 217:159-165.
    • Park*, M., N. Kleinfelter* and J. H. Cushman (2006) Renormalizing chaotic dynamics in fractal porous media with application to microbe motility.  Geophysical Research Letters, 33, L01401.
    • Park*, M., N. Kleinfelter* and J. H. Cushman (2005) Scaling laws and Fokker-Planck equations for 3-dimensional porous media with fractal mesoscale.  SIAM Multiscale Modeling and Simulation, 4(4): 1233-1244.
    • Park*, M., N. Kleinfelter* and J. H. Cushman (2005) Scaling laws and dispersion equations for Levy particles in 1-dimensional fractal porous media., Physical Review E, 72, 056305: 1-7.
    • Kleinfelter*, N., M. Moroni*, and J. H. Cushman (2005) Application of a finite-size Lyapunov exponent to particle tracking velocimetry in fluid mechanics experiments. Physical Review E, 72, 056306: 1-12.
    • Cushman, J. H., M. Park*, N. Kleinfelter*, and M. Moroni* (2005) Super-diffusion via Levy Lagrangian velocity processes. Geophysical Research Letters, 32 (19) L19816: 1-4.
    • Axtell*, N.K., M. Park* and J.H. Cushman (2005) Micromorphic fluid in an elastic porous body: Blood flow in tissues.  Int. J. Multiscale Computational Engineering. 3(1).
    • Cushman, J. H., L.S. Bennethum* and P.P. Singh* (2004) Toward rational design of drug delivery substrates: I. Mixture theory for two-scale biocompatible polymers.  SIAM J.  Multiscale Modeling and Simulation 2(2): 302-334.
    • Cushman, J. H., P.P. Singh* and L.S. Bennethum* (2004) Toward rational design of drug delivery substrates: II.  Mixture theory for three-scale biocompatible polymers and a computational example. SIAM J.  Multiscale Modeling and Simulation 2(2): 335-357.
    • Singh*, P.P., D. Maier, J.H. Cushman and O.H. Campanella (2004) Effect of viscoelastic relaxation on fluid transport in foods.  Part II: Imbibition and drying of seeds.  J. Math. Biol. 49: 1-19.
    • Singh*, P.O., D.E. Maier, J.H. Cushman, K. Haghighi and C. Corvalan (2004) Effect of viscoelastic relaxation on fluid transport in foods.  Part I: Solution of the general transport equation.  J. Math. Biol. 49: 20-34.
    • Cushman, J.H. and J.E. Curry* (2004) The complex behavior of simple fluids in nanoscale restricted geometrics.  In Dynamics and Friction in Sub-micron Scale Confined Systems (ed) Y. Braiman, ACS Symposium Series 882: 157.
    • Su*, Z., J.E. Curry* and J.H. Cushman (2003) Tunable diffusion of OMCTS and cyclohexane monolayers in mica slit pores.  J. Chem. Phys. 118:1417-1422.
    • Cushman, J.H., L.S. Bennethum* and B.X. Hu* (2002) A primer on upscaling methods for porous media.  Adv. Water Resour. 25:1043-1067.
    • Stroud*, W., J. Curry*, and J.H. Cushman (2001) Capillary condensation and snapoff in nanoscale contacts.  Langmuir 17(3):688-698.
    • Cushman, J.H. and M. Moroni* (2001) Statistical mechanics with 3D-PTV experiments in the study of anomalous dispersion: Part I. Theory.  Phys. Fluids 13(1):75-80.
    • Moroni*, M. and J.H. Cushman (2001) Statistical mechanics with 3D-PTV experiments in the study of anomalous dispersion: Part II. Experiment.  Phys. Fluids 13(1):81-91.
    • Murad*, M.A. and J.H. Cushman (2000) Thermomechanical theories for swelling porous media with microstructure.  Int. J. Eng. Sci. 38:517-564.
    • Bennethum*, L.S. and J.H. Cushman (1996) Multiscale hybrid mixture theory for swelling systems.  Part I: Balance laws.  Int. J. Eng. Sci. 34(2):125-145.
    • Bennethum*, L.S. and J.H. Cushman (1996) Multiscale hybrid mixture theory for swelling systems.  Part II: Constitutive theory.  Int. J. Eng. Sci. 34(2):147-169.
    • Curry*, J. and J.H. Cushman (1995) Nanophase coexistence and sieving in binary mixtures confined between corrugated walls.  J. Chem. Phys. 103:2132-2139.
    • Curry*, J. and J.H. Cushman (1995) Binary mixtures of simple fluids in structured slit-micropores.  Mol. Phys. 85(1):173-192.
    • Curry*, J.E., Zhang*, F., J.H. Cushman, M. Schoen*, D.J. Diestler (1994) Transiently coexisting nanophases in ultrathin films confined between corrugated walls.  J. Chem. Phys. 101(12):10824-10832.
    • Schoen*, M., D.J. Diestler, J.H. Cushman (1994) Stratification-induced order-disorder phase transitions in molecularly thin confined films.  J. Chem. Phys. 101(8):6865-6873.
    • Schoen*, M., D.J. Diestler, J.H. Cushman (1994) Fluids in Micropores IV.  The behavior of molecularly thin confined films in the Grand Isostress Ensemble.  J. Chem. Phys. 100(10):7707-7717.
    • Diestler, D.J., M. Schoen*, J.H. Cushman (1993) On thermodynamic stability of confined thin films under shear.  Science 262:545-547.
    • Lee*, K.-K., F.-W. Deng*, J.H. Cushman (1993) Multiscale adaptive estimation of the conductivity field using head and tracer data.  Stoch. Hydrol. Hyd. 7(1):66-82.
    • Schoen*, M., D.J. Diestler, J.H. Cushman (1993) Isostress-isostrain ensemble Monte Carlo simulation of second-order phase transitions in a confined monolayer fluid.  Molecular Physics 78:1097-1115.
    • Cushman, J.H. (1990) Molecular Scale Lubrication, Nature 347(6290):227-228.
    • Schoen*, M., C.L. Rhykerd*, Jr., D.J. Diestler, and J.H. Cushman (1989) Shear Forces in Molecularly Thin Films, Science 245:1223-1225.
    • Schoen*, M., J.H. Cushman, D.J. Diestler, and C.L. Rhykerd*, Jr. (1988) Fluids in Micropores, II.  Self-Diffusion of a Simple Classical Fluid in a Slit-Pore, J. Chem. Phys. 88(2):1394-1406.
    • Rhykerd*, Jr., C.L., M. Schoen*, D.J. Diestler, and J.H. Cushman (1987), Epitaxy in Simple Classical Fluids in Micropores and Near Solid Surfaces, Nature 330(3):461-463.

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