# impossible Poisson's ratio of Lennard-Jones potential type solid (cubic cristal for example), or any potential depending on $r$

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 ???

• Why do you think there is an error? – Jon Custer Nov 26 '18 at 15:12
• 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... – J.A Nov 26 '18 at 15:16
• 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. – Jon Custer Nov 26 '18 at 15:25
• 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... – J.A Nov 26 '18 at 15:30
• 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. – Chemomechanics Nov 26 '18 at 17:02