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Surface Energies? Continuum Molecular Dynamics?

I recently encountered the research of Phil Attard and others, in which the contact problem is solved, relaxing the restrictions of the traditional contact models (Hertz, JKR, DMT) and solving based on a formulation where the governing equations are derived using finite-range surface forces and calculated numerically, self-consistently.

I have enjoyed learning about such models and believe they may have practical application in nanomechanics. However, I find my breadth of my knowledge challenged when it comes to understanding the model's derivation. In especial, I do not really understand the origin of the repulsive pressure 

      p(h) = P_0 \exp(- \kappa h ),

and the adhesive pressure (taken from a Lennard-Jones continuum model)

        \frac{A}{6\pi h^3} \left[ \frac{z^6}{h^6}-1 \right],

where h is the separation between the surfaces and z is the equilibrium separation. (Less to the point, I think, A is Hammiker's constant and P0 and κ were not defined, but obviously contribute the shape of p(h).)

What potentials exist and for what materials and situations do they apply? How do continuum forms of these interaction models arise?  I have searched for information about this sort of modeling, but I have had little success finding information about it.

I am hoping someone here can direct me to a great primer on these sort of surface energy models or help to describe to me what I need to know about them.


The equations posted were taken from P Attard and JL Parker's 1992 paper "Deformation and adhesion of elastic bodies in contact".

I hope this helps a little. The Lennard-Jone (L-J) potential is not a continuum potential but rather a model of interaction between particles, so it is therefore only appropriate for particle methods, especially molecular dynamics (MD), or, depending on your frame of reference, meshless Lagrangian methods at the atomic length scale. L-J a specific progeny from a larger family of vein Mie potentials, where the attractive part (z^-6) corresponds to the van der Waals (vdW) equation of state and the repulsive part (z^-12) is adopted purely for mathematical convenience.

L-J is appropriate when considering atoms or small molecules and is the typical choice for first principles MD simulations; however, the number of particles required for systems of engineering interest are often prohibitively expensive to evaluate computationally (I am not an expert on this but I believe the current state-of-the-art limits using the largest clusters is still restricted to the low billions of atoms). Instead, you need larger representative elements, and other contact laws need to be used in that case.

Continuum equations derived from these potentials are tricky to construct and this constitutes a vigorous research area in itself. Often, a combined experimental and simulation approach is used where carefully constructed MD simulations are used to are combined with experimentally-derived parameters to develop these continuum (constitutive) equations. For many situations, though, analytically derived contact laws can suffice. The contact laws you mentioned in the post (JKR and DMT) are derived from analytical consideration of several particles in specific agglomerated geometries (spheres and planes) evaluated using vdW to determine constitutive equations for the pairwise agglomerate interaction and are therefore appropriate for modeling larger adhesive bodies (>nm). Strictly Hertzian contact is appropriate only for larger aggregates (>100's microns) where adhesion is not a significant influence.

A good overview of surface interactions at the scale level you are referring to can be found in Israelachvili's "Intermolecular and Surface Forces"

Thanks for your reply, it does help. I am getting Israelachvili's book, which hopefully will answer my questions.

I have learned a lot trying to understand this sort of modeling. I was naïvely hoping for a while there was a simple answer for me to understand the different surface energy forms for different materials, but in a way it is enough of an answer to know that there are no strong rules for choosing such forms.

You remark that "Continuum equations derived from these potentials are tricky to
construct and this constitutes a vigorous research area in itself." I am curious if you have any references or further explanation on this issue. I could find virtually no information on the Continuum L-J model, let alone any others, and Attard does not provide a reference. This seems like an extremely interesting (though irrelevant to my interests in the matter) part of this research.


Thanks again,


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