We know (for instance : https://en.wikipedia.org/wiki/Poisson%27s_ratio ) that the Poisson ratio is $\nu=1/2-Y/(6B)$, with $Y$ the Young modulus and $B$ the bulk modulus.

Let's assume we have a energy density $f(r)$ that depends on the distance $r$ between 2 particles of the solid. It can thus also be written as depending on the volume fraction of the solid $\phi$, or on the volumic mass $\rho$ for a homogeneous solid, with a relation : $\phi=\phi(r)$, $\rho=\rho(r)$...

The potential we're talking about has a minimum at $r_0$ (or $\phi_0$, $\rho_0$), where $f'(r)=0$.

The energy for a volume $v$ will be $F(v,r)=v f(r)$. For a cubic cristal, $v=r_0^3$.

If we assume free stress boundary condition, the osmotic pressure is zero, and $f=\rho df/d\rho=0$.

The Young modulus

If we deform a small volume $r_0^3$ Young modulus will write, (see http://www.animations.physics.unsw.edu.au/jw/elasticity.htm for instance) : $Y=r_0^2\frac{d^2f}{dr^2}\Big|_{r=r_0}$.

The Bulk modulus

Similarly, one can compute the bulk modulus for a simple cubic cristal (see for instance https://eis.hu.edu.jo/ACUploads/10010/Cohesive%20energy-2.pdf page 7) : $B=\frac{r_0^2}{9}\frac{d^2f}{dr^2}\Big|_{r=r_0}$

The Poisson's ratio

It comes out $\nu=-1$ !!!! What's the error ???

  • $\begingroup$ Why do you think there is an error? $\endgroup$ – Jon Custer Nov 26 '18 at 15:12
  • $\begingroup$ Well $\nu=-1$ is the lowest possible value for Poisson's ratio with some weird mechanical properties ! It sounds strange that a "cubic cristal" share those properties. Moreover, this result doesn't depend on a specific potential, so it's true also for a Lennard-Jones potential in a cristal atom, or for a colloidal aggregate with a cubic structure. I'm taking always the cubic structure since it's the easiest, but the result is not really different for any other common structure... $\endgroup$ – J.A Nov 26 '18 at 15:16
  • $\begingroup$ Well, it is still a possible value. Now, you should google "poisson ratio lennard jones solid" and look through some of the papers and see how others have gotten to an expression for Poisson's ratio. $\endgroup$ – Jon Custer Nov 26 '18 at 15:25
  • $\begingroup$ That's what I did, but I didn't find any theoretical derivation, even now it's a quite simple calculus. As I wrote, some people compute the Young modulus and others the Bulk modulus, or make assumptions and afterwards calculations. Some papers also make more a numerical computations. Anyway my result is not consistent... $\endgroup$ – J.A Nov 26 '18 at 15:30
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    $\begingroup$ If you consider that hand-wavy estimates of stiffness based on the potential energy curvature and the geometry of a toy model are good only for (say) order-of-magnitude predictions, then the question becomes moot. I would say you've (re)discovered that these models don't handle lateral atomic motion very accurately. $\endgroup$ – Chemomechanics Nov 26 '18 at 17:02

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