softmatter:Simulation Variables/Units
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Reduced units
Dimensionless variables, or reduced units, are typically utilized in simulation. That is, quantities such as temperature, density, pressure, etc. are expressed in terms of convenient units of energy, length, and mass. One natural and convenient (but not unique) choice of units is
- Unit of energy, ε
- Unit of length, σ
- Unit of mass of a particle, M
Other relevant units can be obtained from combinations of these units. For example,
- Unit of time, τ = σ(M / ε)1 / 2
- Unit of temperature, [T] = ε / kBT
Here kB is Boltzmann's constant. All simulation variables can then be written in reduced units by dividing by the appropriate basic unit. Examples are given in the table below.
| symbol | meaning | definition |
|---|---|---|
| r* | dimensionless distance | r / σ |
| E* | dimensionless energy | E / ε |
| T* | dimensionless temperature | kT / ε |
| ρ * | dimensionless number density | ρσ3 |
| U* | dimensionless internal energy | U / ε |
| t* | dimensionless time | t / [σ(M / ε)0.5] |
| v* | dimensionless velocity | v / (ε / M)0.5 |
| F* | dimensionless force | Fσ / ε |
| P* | dimensionless pressure | Pσ3 / ε |
| D* | dimensionless self diffusion coefficient | D / [σ(ε / M)0.5] |
Rationale
The reason behind using reduced units is that infinitely many combinations of density, temperature, particle diameter, mass, energy, etc. all correspond to the same state in reduced units. This is known as the law of corresponding states. For example, a simulation using the Lennard-Jones Potential at reduced units of ρ * = 0.5 and T * = 0.5 corresponds to both Argon at a state point characterized by T = 60K, ρ = 840kg / m3and Xenon at a state point characterized by T=112K and ρ = 1617kg / m3.[1] Without the use of reduced units, certain universal features of fluids, such as the equivalence of these different systems at these two state points, would be missed.
There is another, practical, reason for using reduced units[1]. In an actual simulation run with SI units, the absolute numerical values of the computed quantites are typically very small or very large compared to one. When several such quantities are multiplied using standard floating point multiplication, there is a risk that, at some stage of the calculation, a result will create an overflow or underflow. In reduced units, however, nearly all quantities of interest are of low order, typically between, say, 0.001 and 1000. Thus if, during a simulation, a very large or very small number suddenly results (e.g. of the order of Avogadro's number or larger), then that should trigger the suspicion of an error in the calculation. With non-reduced units, this same error might be more difficult to detect.
Of course, simulation results obtained in reduced units can always be converted back to real (e.g. SI) units, using the above table and values for the basic units (m, ε, etc.) for the system of interest. Examples of basic units for various substances are given in the table below {2}.
| substance | ε / kB(K) | σ(Angstroms) | |||||
|---|---|---|---|---|---|---|---|
| Acetone | 560.2 | 4.600 | |||||
| Acetylene | 231.8 | 4.033 | |||||
| Air | 78.6 | 3.711 | |||||
| Ammonia | 558.3 | 2.900 | |||||
| Argon | 119.8 | 3.405 | *these values from page 42, Ref. [1] | ||||
| Benzene | 412.3 | 5.349 | |||||
| Bromine | 507.9 | 4.296 | |||||
| n-butane | 310 | 5.339 | |||||
| i-butane | 313 | 5.341 | |||||
| Carbon dioxide | 195.2 | 3.941 | |||||
| Carbon disulfide | 467 | 4.483 | |||||
| Carbon monoxide | 91.7 | 3.690 | |||||
| Carbon tetrachloride | 322.7 | 5.947 | |||||
| Carbonyl sulfide | 336 | 4.130 | |||||
| Chlorine | 316 | 4.217 | |||||
| Chloform | 340.2 | 5.389 | |||||
| Cyanogen | 348.6 | 4.361 | |||||
| Cyclohexane | 297.1 | 6.182 | |||||
| Cyclopropane | 248.9 | 4.807 | |||||
| Ethane | 215.7 | 4.443 | |||||
| Ethanol | 362.6 | 4.530 | |||||
| Ethylene | 224.7 | 4.163 | |||||
| Fluorine | 112.6 | 3.357 | |||||
| Helium | 10.22 | 2.551 | |||||
| n-hexane | 339.3 | 5.949 | |||||
| Hydogen | 59.7 | 2.827 | |||||
| Hydrogen cyanide | 569.1 | 3.630 | |||||
| Hydrogen chloride | 344.7 | 3.339 | |||||
| Hydrogen iodide | 288.7 | 4.211 | |||||
| Hydrogen sulfide | 301.1 | 3.623 | |||||
| Iodine | 474.2 | 5.160 | |||||
| Krypton | 178.9 | 3.655 | |||||
| Methane | 148.6 | 3.758 | |||||
| Methanol | 481.8 | 3.626 | |||||
| Methylene chloride | 356.3 | 4.898 | |||||
| Methyl choride | 350 | 4.182 | |||||
| Mercury | 750 | 2.969 | |||||
| Neon | 32.8 | 2.820 | |||||
| Nitric oxide | 116.7 | 3.492 | |||||
| Nitrogen | 71.4 | 3.798 | |||||
| Nitrous oxide | 232.4 | 3.828 | |||||
| Oxygen | 106.7 | 3.467 | |||||
| n-Pentane | 341.1 | 5.784 | |||||
| Propane | 237.1 | 5.118 | |||||
| n-Propyl alcohol | 576.7 | 4.549 | |||||
| Propylene | 298.9 | 4.678 | |||||
| Sulfur dioxide | 335.4 | 4.112 | |||||
| Water | 809.1 | 2.641 |
Example
For Argon, M = 0.03994 kg/mol (from the periodic table of elements). Looking at the table above, we see that ε / kB = 119.8 K and σ = 3.405 Angstrom . From these three basic units we can convert reduced units into SI units for Argon:
temperature: T* = 1 corresponds to T=119.8 K
density: ρ * = 1 corresponds to ρ = 0.02533 molecules/angstrom3 = 1680 kg / m3
time: Δt * = 0.005 corresponds to Δt = 10.9 fs
pressure: P* = 1 corresponds to P = 41.9 MPa
Important Note on Density
In simulation, density typically refers to number density, which is the number of particles per volume. When dealing with reduced units, one must convert the real density into molecules/volume by either:
- dividing by the mass of a molecule when given mass/volume.
- multiplying by Avogadro's number when given moles/volume.
References
[1] D. Frenkel and B. Smit, Understanding molecular simulation : from algorithms to applications, 2002.
[2] Table reproduced from J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liquids, New York, Wiley, 1954.
