Boundary Conditions in Space and Time
Spatial Boundary Conditions
What spatial boundary conditions should we use for the simulations?
Boundary conditions strongly influence structure. Some possibilities:
- No boundaries: Consider a drop of N atoms in free space.
(For example a galaxy, a protein in vacuum, or a droplet)
How many are on the surface and how does that converge in N?
(The fraction of surface atoms is proportional to N -1/3 ).
If the droplet contains one million atoms and the surface layer is 5
atoms deep 25% of atoms are still on the surface. This is not a very
efficient way of sampling a bulk system. There are other problems as well.
With free boundary conditions one can only do simulations at zero pressure
and there is nothing to keep the atoms from evaporating. Also the
equilibration time can be much longer than it bulk since surface process
can be much slower.
- periodic boundary conditions: Suppose we want to simulate a
homogenous system such as a bulk liquid or solid. Then one should use
periodic boundary conditions (PBC). We can think of PBC as either:
- an infinite system which just happens to be periodic.
This picture is good for getting the total potential energy.
- Or as a system on a torus -good for understanding the dynamics
and the effects of the PBCs.
- Escher drawing of swans (PBC in 1D)
These are different ways of representing the system.
If a particle goes out one side of the box,
it should eventually be put back in the other side
(although sometimes one doesn't want to do this.)
How do we calculate distances and energies in periodic boundary conditions?
In the minimum image convention we use the nearest distance among
all the images. This implies that we must smoothly truncate the potential at L/2.
Otherwise energy will not be conserved as particles switch from one image to another.
See the pseudocode for force calculation in periodic boundary
conditions. We will discuss next week what happens to long range potentials in PBC.
It is possible to use any regular space filling lattice (Bravais) for boundary
conditions such as the truncated octahedron since it is more spherical.
This is also necessary when simulating non-cubic crystal structures.
It has been found that one hundred atoms in PBC can behave like an infinite
system in many respects. The corrections are order (1/N) and the coefficient
can be small. One cannot calculate long distance behavior (rL/2) or at large
times (the traversal time for a sound wave.) Angular momentum is no longer conserved.
Also PBC can influence the stability of phases. For example, PBC favor cubic lattice
structures that fit well into the box.
- Simulations on a sphere: The geometry is non-Cartesian
(there is a curvature to space) and calculating distances is more complicated.
The representation of crystals is a problem since there are only 5 perfect
lattices on a sphere with 4, 6, 8 12 and 20 particles.
- External potentials: An example would be a liquid in a tube.
- Mixed-boundary conditions: An example is PBC in the x and y
directions, and open boundaries in z direction. This is called a slab geometry.
Boundary Conditions in Time
How do we start up the system? What positions and velocities do we use?
In Molecular dynamics one normally works at constant energy.
Remember that in classical statistical mechanics that the average
kinetic energy is (3/2)kBT.
- If we start with random positions, the potential energy will be very high.
Hence when the systems equilibrates, the kinetic energy (hence temperature) will
be very high. The opposite extreme is to start from a perfect crystal and give it
some kinetic energy. But because the system is at the bottom of the well, it will
soak up the kinetic energy as it thermalizes (comes into equilbrium).
- A better approach is to start from the end of a thermalized run at a nearby
temperature or density.
- We can also dynamically adjust the kinetic energy (temperature) to get it
into the desired range. We will discuss temperature controls in more detail later.
In CLAMPS we have the QUENCH command which does this.
The best boundary conditions in both space and time critically depend on the
physics of the system. If you make the wrong choice, the simulation may no longer
be feasible!
Calendar
Sept. 4, 1999 D.D. Johnson