[ x-12 - x-6]
[ x-12 - x-6]= maximal well-depth of the attractive part
of the potential. It is related to the cohesive energy.
/kB is about 120 K for liquid Argon.
Warning: these are not properties arising
from isolated pairs of Argon atoms.)
It is worth mentioning that the L-J potentials has been used
to study dynamics in super-cooled liquids, in an effort to understand
glass transitions, to study packing of hard-spheres of varying radii,
and many other situations. Regarding L-J, as well as many of those
to follow, there is a nice review of classical systems if interested
(see, Caccamo, "Integral Equation Theory Description of Phase Equilibria in
Classical Fluids," Physics Reports 274, 1 (1996)).
Hard spheres are obviously short-ranged and non-bonding, leading
only to entropic behavior.
Complexity from Simple Potentials -
Recall that for even a nearest-neighbor Ising model the
statistical mechanics and phase transformation can be very interesting
depending on the type of Bravais lattice simulated. In other words,
you do not need complexity of potential to get complex phase behavior.
Examples of highly complex, so-called "Devil's Staircase", ordering is
found using the simple AFM Next-Nearest Neighbor Interaction (ANNI) Model.
Lattice Gas Models - For Lattice Problems, the entropy
(or number of accessible configurations for the lattice) is not simple to
enumerate or calculate. Hence, textbooks usually take the simple approach
of ignoring all but the large so-called point entropy, i.e.
c*ln(c) where is the concentration or probability of lattice
occupation. This leads to the well-known lattice-gas model.
Because of its simplicity, there are many approaches in the
literature trying to model real materials which try to place the
effect of the higher-order entropic contribution elsewhere
and keep only the point entropy.
This, in general, is just a parametric approach to a difficult
problem, CALPHAD type calculations fall into this category.
Quantum-based Potentials -
For more recent modelling of alloys from first-principles, a
general many-body potential (as given on the first page)
is constructed via fits to a database of electronic-structure
(quantum mechanical) calculations for a number of alloy "configurations".
These configurations depend on concentration of the alloying elements
and are just a collection of decorations (or arrangments)
on the lattice, so as to incorporate more information of the
various alloy stoichiometries and phases.
The idea here is to incorporate QM energies but perform the
statistical mechanics of the alloy classically (a mixed Quantum
and classic model). Importantly, the
fits to obtain the potential are not unique, of course. Changing the
database may lead to very different "interactions" but to very similar
energetics, which is all that is required for statistical mechanics.
For a binary alloy alone, one may require 10-30 mixing energies for
a single phase of an alloy to obtain a "robust" potential, meaning
the parameters do not change dramatically with increasing the
fit database. For a ternary alloy, this can go up by a factor of 10
(or more) because already the system has become extremely complex and
it has a very large parameter space.
Sum Rules - As discussed in class, one must always keep in mind that
independent degrees of freedom are the only variables that can
be simulated. Hence, in a canonical ensemble, there is always (N-1) D.O.F.
in the classical alloy problem because the composition of N elements must
sum to 1, i.e. we conserve particle number. But, this is just a particular
example of a more general expectation.
Aug 1998 by D.D. Johnson Sept. 4 1999 by D.D. Johnson
Morse: E[e-2a(r-r0)
- 2e-a(r-r0)]
The Morse potential offers the same properties as Lennard-Jones,
however, it allows more intermediate range of interaction and it is
a more bonding-type potential. Most elements, esp. metals, that form
solids in the Periodic Table can be fit with this type of potential,
at least locally near their equilibrium lattice constant, by choosing the
lattice constant, s, the bulk moduli, and the cohesive energy.
This is the basis for the so-called Rose Equation of State and
the way the Effective Medium Theory choses to reduce parameters
in the potential.
Double Yukawa: (E/x)[e-a(x-1)
- e-b(x-1)]
Yukawa: (E/x)e-a(x-1)
Here x=r/s just as for Lennard-Jones.
For those in Physics, you may have come across this potential in
regards to nuclear physics, in fact interactions within the nulceus, a
field in which Yukawa was honored with a Nobel Prize. Nonetheless,
this is a potential with a functional form which has some appealing
properties (see Caccamo, for example).
For Soft-sphere-like interactions, the Single-Yukawa offers advantages
over the Soft-sphere potential for analytic investigations of
thermodynamics. In addition, it can be shown that the Double-Yukawa
form adequately reproduces the L-J spatial behavior, however, it
offers the ability to perform useful analytic proceedures to
study free-energy variationally in some restricted systems
(e.g., "Variational Theory of Phase Separation in Binary Liquids,"
Foiles and Ashcroft, J. Chem. Phys. 75 3594 (1981)).
Most the rest of these are briefly commented on in the text.
Hard-Sphere: V(r)= infinity for r<s and
V(r)=0 for r>s.
This "wall-like" potential is very useful in some contexts.
Also a systems of hard spheres is a very nice idealized problem.
And, Percus and Yevik offered an exact solution of the thermodynamic properties
for this potential, obtained using Laplace Transforms, which use exponentials
like exp(-ar) rather than exp(ikr) like Fourier Transforms. In fact,
the Hard-Sphere free energy is purely entropic, because of the potential.
Later Liebowitz (1964) provided exact solution for a systems with varying
hard-sphere radii, i.e. a mixed system, such as Ar-Kr. The Hard-Sphere
pair-correlation functions have also been shown to be exact for systems
with potentials not too different from Hard-Sphere (Foiles and Ashcroft,
Phys. Rev. A 24, 424 (1981)).
Soft-Sphere: x-n
Again, x=r/s and n can vary: n ˜ 1 is soft and the larger n the harder
the repulsion. Also, this offers the useful analytic form for some
mathematic investigations into properties.
Square-Well: V(r)= inf for r<s1,
V(r)=-E for s1<r>s2, V(r)=0 for r>s2.
While this potential at first sight may seem strange, it can be quite
useful in the study of polymer fluids. There is a strong hard-core
repulsion with an attraction only in the vicinity of the sphere.
Polymers with large central units with functional groups attached
around its circumference interact in a way very much like this rather
simplistic form.
Ionic: qi qj/r
With charge q, there may be strong attraction or repulsion
depending on the sign of the charge. This form can be added
to other functional forms so as to introduced the effect of
charging, or polarization.
Lattice Potentials (such as for Lattice Gas or Ising Model Monte Carlo)
For modeling phase stability of alloys and, in some cases, e.g.,
molecular solids and liquid crystals, a lattice simulation can
be performed. For the well-known Ising model, the form
Vij= Jij with thermodynamic (spin) variables of
si which can be either ±1. Here, J can be (Ferromagentic)
FM-like interactions (tending to cluster like spins together),
or Anti-FM (tending to order spins periodically...up...dn...up...dn).
There are examples of these in 2-d in the LINK page.