3、周期性边界条件

Reduced unit----continue It should be pointed that all the reduced units in the previous two slides are based on LennardJones interaction potential. For hard sphere fluid, the reduced units are obtained using kBT to replace .
V (R) Vbonded Vnon bonded
Bonded potential
Bond angle-bend
Torsion(rotate-along-bond)
Bond-strentch
Angle-bend potential
The deviation of angles from their reference values is frequently described using a Hooke’s law or harmonic potential:
Calculate rij
2
( / rij )6
1.7.1.5 Potentials for ions
A simple approach to construct potentials for ions is to supplement one of the above pair potentials with the Coulomb charge-charge interaction:
External potential Pair potential about 90% Three body contribution In FCC crystal, up to 10%
eff eff V v1 (ri ) v2 (ri , r j ) v2 (ri , rj ) v(rij ) v(r) i i j i
i 1
i i j i i j i k j i
Analysis of the Potential Energy
V v1 (ri ) v2 (ri , rj ) v3 (ri , r j , rk )
i i j i i j i k j i
H (q, p) K (p) V (q)
Hamiltonian
N
Kinetic energy
Potential energy
V v1 (ri ) v2 (ri , rj ) v3 (ri , r j , rk ) K pi2 / 2 m i
K V ( ) ( 0 ) 2 angles 2
Bond-stretch potential
V (l )
1, 2 pairs
2 K ( l l ) l 0
Torsion potential
Torsional potentials are almost always expressed as a cosine series expansion.
The reduced units for other properties such as chemical potential and heat capacity can be deduced from the units given in this section. Especially, for electrolyte solution, we can use
1.7 Model System and Interaction Potential
In most of this course, the microscopic of a system may be specified in terms of the position and momenta of a constituent set of particles. In this case the rapid motion of the electrons have been averaged out.
v( r )

is the diameter of hard spheres
r
(r ) v (r ) 0 (r )
HS

1.7.1.2 Square-Well potential
SW v ( r ) 0
v( r )
(r ) ( r ) (r )
v (r ) 4 ( / r ) ( / r )
LJ 12
For argon:
150 100

6

/ k 120 K
0.34 nm
0.3 0.4 0.5 0.6 0.7 0.8
v(r)/k (K)
50 0 -50 -100 -150
r (nm)
Codes for calculating the total potential
Effective pair potential
1.7.1 Effective pair potential for spherical molecules
For the purposes of investigating general properties of liquids and for comparison with theory, highly idealized pair potentials may be of value. In this section , I will introduce hard-sphere, square-well, Yukawa and Lennard-Jones potentials, etc. • 1.7.1.1 Hard-sphere potential
W/(m K) Pa s m2/s
Test of unit (For example, viscosity):
m2 [*] Pa s 1/2 (J kg) 2 m N m 2 s 1 2 1/ 2 (N m N s / m)

-0.5 -1.0 -1.5
r3/
Yukwa potential can 4 also be used to model plasma, colloidal particles, and some electrical interactions.
1.7.1.4 Lennard-Jones potential
r r r /
v( r )
2.0 1.5 1.0 0.5 0.0
When 1.8 , it can be used to model Ar reasonably well.
0 1 2
1.7.2 potential for macromolecules
The energy, V, is a function of the atomic positions, R, of all the atoms in the system, these are usually expressed in term of Cartesian coordinates. The value of the energy is calculated as a sum of internal, or bonded terms , which describe the bonds, angles and bond rotations in a molecule, and a sum of external or nonbonded terms. These terms account for interactions between nonbonded atoms or atoms separated by 3 or more covalent bonds.
V ( ) K 1 cos(n ) 1, 4 pairs 2
‘barrier’ height
The number of minimum points in the function as the bond is rotated through 2.
Determines where the torsion angle passes through its minimum value.
lB /
e
2
Surface charge density:
4 0 r kBT
Q* Q 2 / e
1.9 Simulation Box and Its Boundary Conditions
Computer simulations are usually performed on a small number of molecules, 10<N<10,000. The time taken for a double loop used to evaluate the forces and potential energy is proportional to N 2. Whether or not the cube is surrounded by a containing wall, molecules on the surface will experience quite different forces from molecules in the bulk. It is essential to propose proper methods to overcome the problem of surface effects.
1.8 Reduced Units
Why use reduced unit?