Constants and conversions
Real units to dimensionless units
- Boltzmann constant: \( k_B = 1.3806 \times 10^{-23} \text{J/K} = 8.314 \, \text{J/mol/K} = 1.987 \, \text{cal/mol/K} \)
- Thermal energy at 298K is: \( k_B = 8.314 \text{J/mol/K} \cdot 298\text{K} = 2478 \, \text{J/mol} = 4.114 \cdot 10^{-21} \, \text{J} = 0.59 \, \text{kcal/mol} \)
- Van der Waals strength: \( 0.4 - 4.0 \, \text{kJ/mol} = 0.1 - 1.0 \, \text{kcal/mol} \) same order as \( k_BT \), range 0.3 - 0.6 nm
- Coulombic repulsion: the Bjerrum length \( l_B = 0.7 \, \text{nm} \) in water (dielectric constant, \( \epsilon = 80 \)), which means electrostatic energy becomes equal \( k_B T \) at about 0.7 nm. In a highly ionic medium, \( l_B \) would be reduced.
- Hydrogen bonding (electrostatic attraction): 1 - 5 kcal/mol -> about 2 - 10 times stronger than \( k_B T \), range = 0.3 nm
Reduced time unit
Usually in a simulation the time unit is defined based on the length scale, mass and energy scale: \[ \tau = \sigma \sqrt{\frac{m}{\epsilon}}, \] where- \(m = 10^{-21} \, \text{kg}\)
- \(\epsilon = 50 \, \text{kJ/mol} = 50 \cdot 10^{+3}/6.023\cdot 10^{+23} = 8.3 \cdot 10^{-20} \, \text{Nm} = 8.3 \cdot 10^{-20} \text{kg m}^2/\text{s}^2 \)
- \( \sigma = 5 \cdot 10^{-9} \, \text{m} \) (typical diameter of a nanoparticle)
Energy scale
1 kcal/mol = 0.0433634 eV; then 1 (kcal/mol)/e = 0.0433634 V = 43.3634 mV
1 \( k_BT \) (at T = 300K) = 0.59 kcal/mol; then 1 kT/e = 0.59 kcal/mol/e = 0.59 * 43.3634 mV = 25.584406 mV
At T = 298K, an interaction energy of 50 kJ/mol corresponds to a reduced energy of \( \epsilon/k_BT = 50 \cdot 10^{3} J/(6.023 \cdot 10^{23}) / (4.114 \cdot 10^{-21} J ) = 20.0 \), where \( N_A = 6.023 \cdot 10^{23} \) is the Avogadro's number.