Difference between revisions of "Energy, force, and stiffness"

Latest revision as of 15:41, 19 January 2024

Energy, force, and stiffness.
These are derivatives of each other like so:.

• stiffness * path = force
• force * path = energy
• Integration: stiffness => force => energy

Or, the other way around, antiderviatives. Thus we have:

• energy / path = force
• force / path = stiffness
• Differentiation: energy => force => stiffness

(wiki-TODO: see page associated discussion page)
(wiki-TODO: add table of example values)

Models for chemical bonds

There are several models approximating the behaviour of chemical bonds in a mass and spring model.

• Lennard Jones Potential {wikitodo|add image of the potential - quantity vs distance}
• …

These is by far not as accurate as quantum mechanical modelling, but depending on the problem at hand this can more than suffice.

From this energy curve a force curve and a stiffness curved can be derived by taking the first and second spacial derivative. Note that in 3D this would give a force vector field and a stiffness tensor field.

From the original and the derive curves special values can be read out.

Special values

Comparison of all three properties There are special characteristic values that can be taken from the quantity vs distance curves. These are:

• a bonds total energy (enthalpy) -- (equivalent to bonds toughness)
• a bonds absolute tensile stress and point of absolute tensile strength
• a bonds maximum stiffness and point of maximum stiffness

For gaining a better intuitive feeling about how the strong covalent bonds, the much weaker Van der Waals force (per equal area), and other forces compare to each other it might be useful to look at all three of the aspects Energy force and stiffness. See main article: Comparing reversible energetic bonds

About the historically caused focus on bond energies rather than forces

There's a historically caused focus on frequencies (and proportional energies) rather than forces (and stiffnesses). Frequencies is what was first accessible to experiment via optical spectroscopy. And to this day (state 2020) energies and frequencies associated with inter-atomic bonds are still usually easier to measure directly than forces. Direct measurements of forces can be done via pulling experiments with sharp scanning probe microscope tips, but it's still very hard to single out the effect of a single bond.
See main article: Energies and frequencies.

Also till now bond forces where rarely needed whereas bond energies very often. Thus bond enthalpy tables (useful e.g. for calculating the amount of energy stored in hydrocarbon fuels) are a thing that can be easily found. Whereas tables for maximum tensile strength of bonds are not common to find. (wiki-TODO: add image of a bond enthalpy table)

When doing mechanical engineering near the atomic scale forces of individual bonds become highly relevant though.

Instead of direct measurements the force-over-pulling-distance behaviour of single bonds is usually gained From theoretically calculated energies for different fixed stretching states of the bond and then approximated by more or less well matching phenomenological models.

Involved Math

For each enforced (non-equilibrium) stationary stretching distance there is a an iterative quantum mechanical calculation needed that is refining base approximations. Here's a brief overview over some of the mathematical tools that are involved:

• The mathematical base shape of the orbitals of single atoms can be gained by analytical solutions of the Schrödinger wave equation. E.g. the shape of s-orbitals, p-orbitals and their linear combinations (aka hybridizations). Except for hydrogen only this is quite wrong though due to inner electrons shielding the part of the the positive charge of the atoms core, changing the shape of the funnel like electrostatic potential hugely.
• To get the shape of the orbitals to better match the electrostatic potential the Gram–Schmidt process [1] can be used.

• Once hybrid orbitals of different atoms get overlapped to form molecular orbitals it can become harder to adhere to the parity rules of fermions (to adhere to the pauli-principle) that says that two electrons can never be in the same quantum state in all of their quantum numbers simultaneously. The Slater determinant [2] can be used to get a valid electron configuration which can be fed as initial state into the Hartree–Fock method, a variational method that through iteration eventually leads to a self consistent field [3]. Limiting is that the results (that is resulting energies for the given enforced bonding separations) can only get optimal within what the limits of the parameterization for the orbitals allow. There are more approximations involved usually causing a smaller error.

• For mechanical the purpose of analysis of mechanical applications extremely accurate results for energies are not needed so many of the approximations involved have errors small enough to ignore.

• A more accurate but also more computationally intense alternative is density functional theory [4]. For strongly isolated single covalent bonds the additional computational cost is very much manageable though.

Related

• Comparison of mechanical character of different bonds types

• Covalent bond
• Van der Waals force

• Intuitive feel
• Pages with math

• Stiffness, Superelasticity, Emulated elasticity
• Lower stiffness of smaller machinery
• Same relative deflections across scales

• Molecular dynamics implementation cheat sheet