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A field of material particles vs. a field of markers

Zhigang Suo's picture

In continuum mechanics, it is a common practice to view a body as a field of material particles, so that the continuum mechanics is phrased as an algorithm to determine the function x(X, t), where X is the name of a particle, and x is the place of the particle at time t.

It seems to me this practice is only sensible if you can identify material particles. For example, when a crystal lattice undergoes an elastic deformation, we can regard a collection of atoms as a material particle. By contrast, if the crystal creeps and atoms diffuse around, the notion of material particles becomes questionable. In such a case, any collection of atoms will not stay together for very long for you to identify them collectively as one material particle.

In the literature of diffusion, there has been a long tradition to invoke markers, both in experiments and in theories. In this paper (Suo, a continuum theory that couples creep and self-diffusion, Journal of Applied Mechanics 71, 646-651, 2004), I describe the use of markers in formulating a theory. A preprint of this paper is attached with this post.

I'd be interested in your experience with markers or similar ideas.

Abstract-In a single-component material, a chemical potential gradient or an electron wind drives self-diffusion.  The diffusion flux may have a divergence, which deforms the material.  We formulate coupled differential equations for convection velocity and chemical potential.  When the diffusion flux is divergence-free, the theory decouples into Stokes's theory for creep and Herring's theory for self-diffusion.  A length emerges from the coupled theory to characterize the relative rate of diffusion and creep.  For a flow in a film driven by a stress gradient, creep dominates in thick films, and self-diffusion dominates in thin films.  Depending on the film thickness, either stress-driven creep or stress-driven diffusion prevails to counterbalance electromigration. The transition occurs when the film thickness is comparable to the characteristic length of the material. 


         Self-diffusion can generate stress in a single-component material.  During deposition, for example, a thin film sometimes develops a compressive stress.  One possible mechanism has to do with the injection of atoms into the film [1].  Impinging atoms may not have enough time to find equilibrium positions on the film surface, and may diffuse into the film.  Oxidation leads to analogous phenomena.  For some materials, during oxidation, atoms may emit from the materials, causing tension in the materials [2].  For other materials, notably silicon, atoms may inject into the materials, causing compression in the materials [3].  Electromigration provides yet another example.  When electrons flow in a metal, the impact of electrons on atoms motivates atoms to diffuse, generating tension upstream and compression downstream [4].    

         The stress generated in the material depends on the deformation mechanism of the material.  Only elastic property enters the consideration if inelastic deformation (i.e., creep) is either so slow as negligible, or so fast as to relax the stress field locally to a hydrostatic state.  For electromigration along a thin line, encapsulated in a stiff dielectric, it was thought that local stress relaxes to a hydrostatic state long before diffusion along the line reaches a steady state [5,6].  Experiments, however, have shown large deviatoric stresses [7].  Indeed, the initial discovery of electromigration-induced stress was made in a wide aluminum film, which could only sustain in-plane stresses [4]. 

         This paper formulates a theory to couple self-diffusion and creep in single-component materials.  The new theory will contain Stokes's creep and Herring's diffusion as special cases.  Stokes's creep, as formulated in fluid mechanics, describes a velocity field and a pressure field; it neglects self-diffusion.  Herring's theory [8] for self-diffusion is in terms of the chemical potential, a scalar; it makes no attempt to equilibrate stress tensor field or maintain kinematic compatibility. 

            Our theory parallels that of nonreciprocal diffusion in multi-component solid solusions (i.e., the Kirkendall effect) due to Darken [9] and Stephenson [10], and extends our previous one-dimensional theory [11].  The divergence in the diffusion flux must have a compensating divergence in the convection velocity.  Our theory builds on this kinematic constraint.  Sections 2-4 describe the kinematics, energetics, and kinetics of the theory.  Section 5 gives the coupled partial differential equations for the velocity field and the chemical potential field, and identifies the characteristic length in the theory.  Sections 6 discusses examples of flows driven by stress gradient, electron wind, and atomic injection.  Stress gradient-driven channel flow is dominated by creep in thick channels, and by self-diffusion in thin channels.  Section 7 discusses an anisotropic rule to place diffusion flux divergence as strain-rates in various directions.

PDF icon 156 JAM-04-1037 Suo.pdf151.96 KB


Henry Tan's picture

In numeric simulations, there are similar problems: to choose Lagrangian description (using material particles), or Eulerian descriptions (using markers). The difficulties encountered in structure/fluid coupling, and also in atomistic/continuum coupling, come from the interactions between two views of the deformation field.

A smarter solution may adopt both views: using markers as a computation background, and using material particles to record the motion.

Zhigang Suo's picture

Dear Henry:  I'm not sure if you and I are talking about the same issue.  The issue I had in mind is that, in some physical situations, we cannot identify material particles.  An example is when you have a solid solution, and atoms can change neighbors by diffusion.  While you may identify each atom, you cannot identify a collection of atoms as a material particle, because atoms do not stick together long enough.  In the paper attached with my post, I reviewed attempts to introduce markers.  It is possible to formulate a continuum mechanics to evolve the positions of the markers.

In experiments, markers are small and inert particles, such as oxide particles in a solid solution of a few kinds of metal atoms.   The markers are disconnected with one another, so that two markers can drift apart as additional atoms diffuse in between them.

In this respect, makers are different from a grid usually drawn in textbooks.  Should we draw a grid in a solid solution, the grid will soon break up into little pieces, as atoms diffuse around.  By contrast, if we draw a grid in a cross-linked polymer, such as a rubber, the grid will remain intact after distortion.

Henry Tan's picture

Dear Zhigang, I have a question:

Is it possible to extend marker descriptions to dynamic problems, where the mass term is involved?

For material particle description, dynamics can be easily handled since material particles contain masses by themselves.

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