## Reduced Units in Tramonto

Reduced units serve two purposes in the molecular theory calculations performed by Tramonto. First, reduced units help to normalize calculations keeping many terms O(1) so that the stability of the nonlinear solve is optimized. Second, from a corresponding states perspective, the results of a calculation in reduced units can be translated to any physical system. Consider a liquid-vapor equilibria calculation. For a single component fluid there is only one possible value for the critical point in reduced units. The molecular energy interaction parameters that are suitable for a given real fluid can be set by converting the theoretical critical point in reduced units to real units using data from the real triple point (temperature and pressure) of the fluid of interest. This is one simple way to set the parameters for a simple single site coarse grained model of some real fluid.

Specific dimensionless groups (reduced units) used in the Tramonto code (some input others output parameters) are provided below. Note that all output from the Tramonto code is reported in dimensionless units even when the input is provided in real dimensions (e.g. Kelvin, Angstroms, etc.). Furthermore note that Tramonto currently solves only for 3-dimensional systems. However, when symmetries are present, the numerical problem may be reduced to a 1-dimensional or 2-dimensional calculation.

• Particle size (input): &sigma/&sigmaref where &sigma is a particle diameter and &sigmaref is a reference distance. Typically, &sigmaref will be the particle diameter of the first species in a given problem so that the reduced value for that species will be unity.
• Distances (input/output): All distance parameters (e.g. mesh size, mesh spacing, particle diameters) are reduced as Length/&sigmaref
• Density (input/output): &rho&sigma3 where &sigma is the reference distance typically set to the fluid particle diameter as defined above.
• Energy interaction parameters (input): &epsilon/kT where k is the Boltzmann constant (1.380658x10-23J/K), and T is the temperature. Note that in computations, this reduced quatnity defines both the energies and the temperature.
• Pressure (output and bulk thermodynamics): p&sigma3/kT
• Chemical potential (output and bulk thermodynamics): &mu/kT
• Electrostatic potentials (input/output): &Phi e/kT where e is the elementary charge (1.60217733 x10-23C).
• Adsorption (output): Adsorptions (both total &Gamma and excess &Gammaex) are reported per unit area if the input parameter "Lper_area" is set to TRUE. In this case, the reduced parameter in the output file will be &Gamma&sigma2/A. In 2D or 3D calculations, there is the option to set Lper_area=FALSE. In this case, the reported parameter will be &Gamma&sigma/L Adsorption per unit length, L, for 2D calculations or &Gamma (total number of particles for 3D problems).
• Force (output): Force is typically reported per unit area if the input parameter "Lper_area" is set to TRUE. In this case, the reduced parameter in the output file will be f&sigma3/kT. In 2D or 3D calculations, there is the option to set Lper_area=FALSE. In this case, the reported parameter will be f&sigma2/kT (force per unit length for 2D cases) or f&sigma/kT (total force for 3D cases).
• Energy (output): Free Energies (both total, &Omega, and surface free energy &Omegas) are also reported per unit surface area if the input parameter "Lper_area" is set to TRUE. In this case, the output energy=&Omega&sigma2/AkT where \$A\$ is a surface area. In 2D or 3D calculations, there is the option to set Lper_area=FALSE. In this case, the reported energy will be energy=&Omega&sigma/LkT (free energy per unit length,L, for 2D problems) or &Omega/kT& (total free energy for 3D problems).