COSMO-RS

1. The liquid state is incompressible
2. All parts of the molecular surfaces can be in contact with each other
3. Only pairwise interactions of molecular surface patches are allowed

As long as the above assumptions hold, the chemical potential μ in solution can be calculated from the interaction energies of pairwise surface contacts.

Within the basic formulation of COSMO-RS, interaction terms depend on the screening charge density σ. Each molecule and mixture can be represented by the histogram p(σ), the so-called σ-profile. The σ-profile of a mixture is the weighted sum of the profiles of all its components. Using the interaction energy E_int(σ,σ') and the σ-profile of the solvent p(σ'), the chemical potential μ_s(σ) of a surface piece with screening charge σ is determined as:

${\displaystyle \mu _{s}(\sigma )=-kT\ln \int p_{s}(\sigma ')e^{-{\frac {E_{int}(\sigma ,\sigma ')-\mu _{s}(\sigma ')}{kT}}}d\sigma '}$

Due to the fact that μ_s(σ) is present on both sides of the equation, it needs to be solved iteratively. By combining the above equation with p^x(σ) for a solute x, and adding the σ-independent combinatorial and dispersive contributions, the chemical potential for a solute X in a solvent S results in:

${\displaystyle \mu _{s}^{x}=\mu _{comb}^{x}+E_{disp}+\int p^{x}(\sigma )\mu _{s}(\sigma )d\sigma }$

In analogy to activity coefficient models used in chemical engineering, such as NRTL, UNIQUAC or UNIFAC, the final chemical potential can be split into a combinatorial and a residual (non ideal) contribution. The interaction energies E_int(σ,σ') of two surface pieces are the crucial part for the final performance of the method and different formulations are used within the various implementations. In addition to the liquid phase terms a chemical potential estimate for the ideal gas phase μ_gas has been added to COSMO-RS to enable the prediction of vapor pressure, free energy of solvation and related quantities.

The residual part is the sum of three different contributions, where E_misfit and E_hb are part of E_int and E_disp is added directly to the chemical potential.

${\displaystyle E_{misfit}(\sigma )={\frac {\alpha }{2}}(\sigma +\sigma ')^{2}}$

In the E_misfit expression α is an adjustable parameter and σ and σ' refer to the screening charge densities of the two surface patches in contact. This term has been labeled "misfit" energy, because it results from the mismatch of the charged surface pieces in contact. It represents the Coulomb interaction relative to the state in a perfect conductor. A molecule in a perfect conductor (COSMO state) is perfectly shielded electronically; each charge on the molecular surface is shielded by a charge of the same size but of opposite sign. If the conductor is replaced by surface pieces of contacting molecules the screening of the surface will not be perfect any more. Hence an interaction energy from this misfit of σ on the surface patches will arise.

${\displaystyle E_{hb}(\sigma )=c_{hb}(T)\max[0,\sigma _{acc}-\sigma _{hb}]\min[0,\sigma _{don}+\sigma _{hb}]}$

In the E_hb expression σ_acc and σ_don are the screening charge densities of the hydrogen bond acceptor and donor respectively. The hydrogen bonding threshold σ_hb and the prefactor c_hb are adjustable parameters. The max[] and min[] construction ensures that the screening charge densities of the acceptor and donor exceeds the threshold for hydrogen bonding.

${\displaystyle E_{disp}=\sum _{k}\gamma _{k}A_{k}}$

The COSMO-RS dispersion energy of a solute depends on an element (k) specific prefactor γ and the amount of exposed surface A of this element. It is not part of the interaction energy but enters the chemical potential directly.

Though the use of quantum chemistry reduces the need for adjustable parameters, some fitting to experimental data is inevitable. The basic parameters are α, c_hb, σ_hb as used in the interaction energies, and one general parameter for the effective contact area. In addition, one adjustable van der Waals parameter γ per element is required. All parameters either are general or element specific, which is a distinctive feature of COSMO-RS as compared to group contribution methods like UNIFAC.