Common Potential Energy Functions of Separation Distance

1
Common Potential Energy Functions of Separation
Distance

• The Potential Energy function describes the
  energy of a particular state.
• When given as a function of the separation
  between two bodies it is related to the force
  acting between the two bodies.
• Correct model for U(r) depends upon what balls
  and springs represent.

r12
M1
M2
2
Common Potential Energy Functions of Separation
Distance

• For molecular models Balls represent nuclei and
  Springs represent either bonds or weaker
  interatomic potentials (coulombic attraction,
  hydrogen bonds ,Van der Waals attraction etc.)

3
Common Potential Energy Functions of Separation
Distance

• In meso-scale PMMA case what are Balls and
  Springs?

4
Common Potential Energy Functions of Separation
Distance

• In meso-scale PMMA case what are Balls and
  Springs?

5
Common Potential Energy Functions of Separation
Distance

• In meso-scale PMMA case what are Balls and
  Springs?

6
Common Potential Energy Functions of Separation
Distance

• In meso-scale PMMA case what are Balls and
  Springs?
• do not represent a single covalent
  bond.

7
Lennard-Jones

• Originally developed to describe force between
  non-bonded molecules.
• Attraction term is the Van
  der Waals potential of interacting dipoles.
• Note that dipoles may only exist temporarily as
  is the case for nonpolar atoms/molecules such as
  Helium and Methane.
• Repulsive term is supposedly due to repulsion of
  overlapping orbitals. But this functional form
  should be exponential (see Morse Potential).
• s is effective Van der Waals radius of molecule

8
Lennard-Jones Potential between non-bonded
molecules

• Lennard-Jones Parameters for typical non polar
  molecules.
• Values can be found for gaseous molecules in
  physical chemistry text books.
• Used in calculating second virial coefficient
  perturbation of ideal gas law.

9
Estimating Lennard-Jones between molecules

• s is simply Van der Waals radius of molecules.
• Can be estimated using bond additive methods
• eg Zhou Fast calculation of van der Waals volume
  as a sum of atomic and bond contributions and its
  application to drug compounds. Zhao YH, Abraham
  MH, Zissimos, AM. J Org Chem. 2003 Sep
  1968(19)7368-73
• Or easily calculated using available quantum or
  molecular modeling software
• eg MOPAC, MM2 etc.
• e can be estimated using dipole moments of
  molecules and Gibbs Ensemble formula.
• For dislike molecules use Lorentz-Berthelot
  combination rules.

10
Using Lennard-Jones Potential to describe bond
between two atoms.

• s and e can be estimated from Average bond
  length rbond and bond dissociation energy Ebond.
• Typical values for rbond and Ebond can be found
  in most Organic chemistry texts. Specific
  calculation requires quantum calculation eg MOPAC
  etc.
• Assume F(r) 0 at rrbond and
  U(rrbond) U(r8)Ebond

11
Shifted-Lennard-Jones

• Potential Function
• f(r) chosen so both Force and Potential Energy is
  zero at rrcutoff
• Typically f(r) arb with rcutoff 2.5s
• Permits clipping of U(r) at short distance
  without discontinuity in force.
• Used for molecular modeling
• Calibration is similar to Lennard-Jones Potential

12
Morse Potential

• Leonard-Jones is not good for real bonds. Does
  not correctly represent both bonding energy Ebond
  and vibrational strength k (eg
  U(r)0.5k(r-r0)2 ) at bottom of well.
• Better simple potential is Morse potential.
• Parameter a controls strength of vibration at
  bottom of well.
• Found from vibrational spectrum frequency of
  bond.

13
Proposal of Ways to Estimate U(r) for Mesoscale
non-bonded PMMA

• Interaction potential between non-bonded groups
  determined from Lennard-Jones potential between
  two methy-methacrylate monomers or model monomer
  units.
• Use MOPAC etc. to determine molecular radius and
  molecular dipole moment.
• Confirm values using macroscopic experiments from
  literature if available. melting point, boiling
  point, density, Gas/Liquid virial coefficients,
  yadda yadda ...

14
Proposal of Ways to Estimate U(r) for Mesoscale
bonds in PMMA

• Canned Molecular modeling (MM2, Charmm, Amber,
  Alchemy etc.) and semi-empirical quantum programs
  (MOPAC etc.) can generate reasonably accurate
  potential energies for large molecules.
• They typically do this for a given conformations
• They also can minimize Energy to find lowest
  energy conformation.
• Proposal
• Determine lowest energy conformation for 3-5 MW
  PMMA.
• Stretch chain along some coordinate r and
  calculate new U(r).

15
Proposal of Ways to Estimate U(r) for Mesoscale
bonds in PMMA

• Potential Problems
• Coordinate r is not a simple coordinate.
• It probably does not involve stretching of
  covalent bonds but rather twisting of bonds.
• How to get r?
• Many different conformations are sampled at each
  r this introduce an Entropy contribution
• Also in statistical mechanics sense U(r) really
  should be the Free Energy and not potential
  energy.

16
Entropic Springs

• Systems that have degeneracies along a coordinate
  transformation possess entropic contributions to
  Force field.
• Classic example are isolated polymer chains in
  perfect solvent.
• Because the solvent is perfect the DU with any
  conformational change is zero.
• However there is a resistive force to stretching
  the chain away from ltR2gt
• This force is due to the lower entropy (fewer
  possible conformations) in the streched position
  than in the coiled position.