The below description is a simplified overview of the COSMO-RS version published in 1998.
Assumptions • The liquid state is incompressible • All parts of the molecular surfaces can be in contact with each other • 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.
COSMO-RS equations 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 Eint(σ,σ') and the σ-profile of the solvent p(σ'), the chemical potential μs(σ) of a surface piece with screening charge σ is determined as: {{center | \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 μs(σ) is present on both sides of the equation, it needs to be solved iteratively. By combining the above equation with px(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: {{center | \mu^x_s =\mu^x_{comb}+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 Eint(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 μgas has been added to COSMO-RS to enable the prediction of vapor pressure, free energy of solvation and related quantities.
Interaction energy (Residual) The residual part is the sum of three different contributions, where Emisfit and Ehb are part of Eint and Edisp is added directly to the chemical potential.
Electrostatic interaction {{center | E_{misfit} (\sigma)=\frac{\alpha}{2}(\sigma+\sigma')^2}} In the Emisfit 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.
Hydrogen bonding energy {{center | E_{hb} (\sigma)=c_{hb}(T)\max[0,\sigma_{acc}-\sigma_{hb}] \min[0,\sigma_{don}+\sigma_{hb}]}} In the Ehb expression σacc and σdon are the screening charge densities of the
hydrogen bond acceptor and donor respectively. The hydrogen bonding threshold σhb and the prefactor chb 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.
Dispersion (van der Waals energy) {{center | 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.
Parameters Though the use of quantum chemistry reduces the need for adjustable parameters, some fitting to experimental data is inevitable. The basic parameters are α, chb, σhb 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. ==Implementations==