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Theory: thermodynamic properties-osmotic coefficient

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Calculation of the osmotic coefficient is relatively involved and, therefore, it is discussed separately in this text.

Osmotic coefficient Φ – general theory

Osmotic coefficient Φ is measure of the activity of water (solvent) in a solution.

\displaystyle \Phi = -\frac{n_w}{\nu_s n_s} \ln a_w = -\frac{\ln a_w}{\nu_s \frac{n_s}{n_w} \frac{M_w}{M_w}} = -\frac{\ln a_w}{\nu_s m_s M_w}

n_w and n_s are molar amounts of solvent and solute in the solution, M_w is solvent molar mass, m_s molality of the solution and a_w the solvent activity. The value of \nu_s indicates, how many components contributes the solute to the solution after dissolution (non-dissociable species 1, uni-univalent electrolytes 2, uni-bivalent electrolytes 3, …)

For practical reasons, it’s better to use Φ instead of activity (although it’s easy to convert between these quantities), since with osmotic coefficient small numbers are avoided. For example, for 0.1M NaCl solution Φ=0.9324 and a=1.000234.

Statistical thermodynacal expression for Φ (Krienke, Barthel, Kunz: Physical chemistry of electrolyte solutions, p. 134) can be written as

\displaystyle\Phi = \frac{p_{\mathrm{osm}}}{\rho kT} = 1-\frac{1}{6 \rho kT} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty r \frac{\partial U_{ij}(r)}{\partial r} g_{ij}(r) 4\pi r^2 dr

It, therefore, depends on the total number density ρ, number densities of individual components, pair correlation function g(r) and differential of the total pair potential U(r) with respect to distance. Some rearrangements yield

\displaystyle\Phi = 1 + \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty -\frac{1}{kT} \frac{\partial U_{ij}(r)}{\partial r} g_{ij}(r) r^3 dr

Total pair potential can be written as a sum of contributions of n potentials (rule of pair additivity) and the expression then reads

\displaystyle\Phi = 1 + \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty -\frac{1}{kT} \sum_n \frac{\partial U_{ij}^n(r)}{\partial r} g_{ij}(r) r^3 dr =\\= 1 + \sum_n \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty -\frac{1}{kT} \frac{\partial U_{ij}^n(r)}{\partial r} g_{ij}(r) r^3 dr = 1 + \sum_n \phi^n

defining the contribution \phi^n of a potential n to the total osmotic coefficient Φ.

This idea can be evolved further by working with the pair correlation function. From the definitions discussed previously, a pair correlation function can be written in the HNC approximation as

\displaystyle g_{ij}(r) = \exp \left( \Gamma_{ij}(r)  + \sum\limits_n -\frac{U_{ij}^n(r)}{kT} \right) = \exp(\Gamma_{ij}(r)) \prod\limits_n \exp \left( -\frac{U_{ij}^n(r)}{kT} \right) = \xi_{ij}(r) \prod\limits_n g_{ij}^n(r)

and in the PY approximation as

\displaystyle g_{ij}(r) = \exp \left( \sum\limits_n -\frac{U_{ij}^n(r)}{kT} \right) (1 + \Gamma_{ij}(r)) =  \prod\limits_n \exp \left( -\frac{U_{ij}^n(r)}{kT} \right) (1+\Gamma_{ij}(r)) = \xi_{ij}(r) \prod\limits_n g_{ij}^n(r)

introducing pair correlation functions for a given potential n

\displaystyle g_{ij}^n = \exp \left(-\frac{U_{ij}^n(r)}{kT}\right)

and a function ξ(r) taking the Γ(r) into account in a transparent way (it’s convenient).When we then note further that

\displaystyle \frac{\partial g_{ij}^n(r)}{\partial r} = g_{ij}^n(r) \frac{\partial}{\partial r} \left( -\frac{U_{ij}^n(r)}{kT}\right )

we can write the expression for the contribution of a potential to the total osmotic coefficient in the form involving derivative of the pair correlation function.

\displaystyle \phi^n = \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty \frac{\partial}{\partial r} \left(-\frac{U_{ij}^n(r)}{kT} \right) g_{ij}^n(r) \xi_{ij}(r) \left(\prod_{m \neq n}g_{ij}^m(r)\right) r^3 dr =\\= \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty \frac{\partial g_{ij}^n(r)}{\partial r} \xi_{ij}(r) \left(\prod_{m \neq n}g_{ij}^m(r)\right) r^3 dr

Whether the approach involving derivative of the interaction potential or the approach involving derivative of the pair correlation function is used depends on the interaction potential itself. In some cases, the second possibility is better, since g(r) has smaller dynamical range and its derivative is more numerically stable. Also, in some cases (HS), g(r) allows for analytical evaluation of the integral, whereas U(r) doesn’t.

In the following text, individual contributions to the total osmotic coefficient will be discussed (for all potential types supported by pyOZ).

Osmotic coefficient – hard spheres potential

For hard spheres, the approach using the derivative of the pair correlation functions is more suitable. Pair correlation function of hard spheres is the Heaviside step function \mathcal{H}(r-\sigma_{ij}), whose derivative is the Dirac delta function \delta(r-\sigma_{ij}). Then, using properties of the δ-function, the contribution to the osmotic coefficient from hard spheres can be calcualted.

\displaystyle \phi^{\mathrm{HS}} = \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty \frac{\partial g_{ij}^{\mathrm{HS}}(r)}{\partial r} \xi_{ij}(r) \left(\prod_{m \neq HS}g_{ij}^m(r)\right) r^3 dr =\\=  \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty \delta(r-\sigma_{ij}) \xi_{ij}(r) \left(\prod_{m \neq HS}g_{ij}^m(r)\right) r^3 dr =\\= \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \xi_{ij}(\sigma_{ij}) \left(\prod_{m \neq HS}g_{ij}^m(\sigma_{ij})\right) \sigma_{ij}^3

Osmotic coefficient – Coulomb potential

Evaluation using the derivative of the interaction potential is used, since it is known analytically.

\displaystyle\phi^{\mathrm{Coul}} = \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty -\frac{1}{kT} \frac{\partial U_{ij}^{\mathrm{Coul}}(r)}{\partial r} g_{ij}(r) r^3 dr =\\= \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty \frac{U_{ij}^{\mathrm{Coul}(r)}}{kT}  g_{ij}(r)  r^2 dr

Osmotic coefficient – Lennard-Jones potential

Evaluation using the derivative of the interaction potential is used, since it is known analytically.

\displaystyle\phi^{\mathrm{LJ}} = \frac{2 \pi}{3 \rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty -\frac{1}{kT} \frac{\partial U_{ij}^{\mathrm{LJ}}(r)}{\partial r} g_{ij}(r) r^3 dr =\\= \frac{16 \pi}{\rho} \sum_i \sum_j \rho_i \rho_j \int\limits_0^\infty \frac{\epsilon_{ij}}{kT}  g_{ij}(r) \left[ 2\left(\frac{\sigma_{ij}}{r}\right)^{12} -\left(\frac{\sigma_{ij}}{r}\right)^6\right] r^2 dr

Osmotic coefficient – potential of mean force

For PMF, both approaches to the osmotic coefficient are applicable. Both potential and pair correlation function have to be evaluated numerically. pyOZ evaluates the contribution using the numerical derivative of the pair correlation function at the moment, since it provides the best results due to smaller dynamical range of g(r).


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