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Revised association contribution (#150)
* More generic association models * Rewrite association contribution to be more generic * Michael L. Michelsen to the rescue once again * Add hard-sphere theory guide * update parameter files * cleanup * add changelog
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‎CHANGELOG.md‎

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and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
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## [Unreleased]
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### Changed
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- Changed the internal implementation of the association contribution to accomodate more general association schemes. [#150](https://github.com/feos-org/feos/pull/150)
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- To comply with the new association implementation, the default values of `na` and `nb` are now `0` rather than `1`. Parameter files have been adapted accordingly. [#150](https://github.com/feos-org/feos/pull/150)
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## [0.4.3] - 2023-03-20
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- Python only: Release the changes introduced in `feos-core` 0.4.2.

‎docs/theory/models/FeOs_Association.png‎

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‎docs/theory/models/association.md‎

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# Association
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The Helmholtz contribution due to short range attractive interaction ("association") in SAFT models can be conveniently expressed as
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$$\frac{A^\mathrm{assoc}}{k_\mathrm{B}TV}=\sum_\alpha\rho_\alpha\left(\ln X_\alpha-\frac{X_\alpha}{2}+\frac{1}{2}\right)$$
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Here, $\alpha$ is the index of all distinguishable **association sites** in the system. The site density $\rho_\alpha$ is the density of the segment or molecule on which the association site is located times the multiplicity of the site. The fraction of non-bonded association sites $X_\alpha$ is calculated implicitly from
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$$\frac{1}{X_\alpha}=1+\sum_\beta\rho_\beta X_\beta\Delta^{\alpha\beta}$$ (eqn:x_assoc)
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where $\Delta^{\alpha\beta}$ is the association strength between sites $\alpha$ and $\beta$. The exact expression for the association strength varies between different SAFT versions. We implement the expressions used in PC-SAFT but a generalization to other models is straightforward.
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The number of degrees of freedom in the association strength matrix $\Delta$ is vast and for practical purposes it is useful to reduce the number of non-zero elements in $\Delta$. In $\text{FeO}_\text{s}$ we use 3 kinds of association sites: $A$, $B$ and $C$. Association sites of kind $A$ only associate with $B$ and vice versa, whereas association sites of kind $C$ only associate with other sites of kind $C$. By sorting the sites by their kind, the entire $\Delta$ matrix can be reduced to two smaller matrices: $\Delta^{AB}$ and $\Delta^{CC}$.
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```{image} FeOs_Association.png
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:alt: Association strength matrix
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:class: bg-primary
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:width: 300px
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:align: center
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```
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In practice, the $AB$ association can be linked to hydrogen bonding sites. The $CC$ association is less widely used but implemented to ensure that all the association schemes defined in [Huang and Radosz 1990](https://pubs.acs.org/doi/10.1021/ie00107a014) are covered.
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## Calculation of the fraction of non-bonded association sites
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The algorithm to solve the fraction of non-bonded sites for general association schemes follows [Michelsen 2006](https://pubs.acs.org/doi/full/10.1021/ie060029x). The problem is rewritten as an optimization problem with gradients
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$$g_\alpha=\frac{1}{X_\alpha}-1-\sum_\beta\rho_\beta X_\beta\Delta^{\alpha\beta}$$
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The analytic Hessian is
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$$H_{\alpha\beta}=-\frac{\delta_{\alpha\beta}}{X_\alpha^2}-\rho_\beta\Delta^{\alpha\beta}$$
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with the Kronecker delta $\delta_{\alpha\beta}$. However, [Michelsen 2006](https://pubs.acs.org/doi/full/10.1021/ie060029x) shows that it is beneficial to use
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$$\hat{H}_{\alpha\beta}=-\frac{\delta_{\alpha\beta}}{X_\alpha}\left(1+\sum_\beta\rho_\beta X_\beta\Delta^{\alpha\beta}\right)-\rho_\beta\Delta^{\alpha\beta}$$
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instead. $X_\alpha$ can then be solved robustly using a Newton algorithm with a check that ensures that the values of $X_\alpha$ remain positive. With the split in $AB$ and $CC$ association the two kinds different versions could be solved separately from each other. This is currently not implemented, because use cases are rare to nonexistent and the benefit is small.
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A drastic improvement in performance, however, can be achieved by solving eq. {eq}`eqn:x_assoc` analytically for simple cases. If there is only one $A$ and one $B$ site the corresponding fractions of non-bonded association sites can be calculated from
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$$X_A=\frac{2}{\sqrt{\left(1+\left(\rho_A-\rho_B\right)\Delta^{AB}\right)^2+4\rho_B\Delta^{AB}}+1+\left(\rho_B-\rho_A\right)\Delta^{AB}}$$
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$$X_B=\frac{2}{\sqrt{\left(1+\left(\rho_A-\rho_B\right)\Delta^{AB}\right)^2+4\rho_B\Delta^{AB}}+1+\left(\rho_A-\rho_B\right)\Delta^{AB}}$$
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The specific form of the expressions (with the square root in the denominator) is particularly robust for cases where $\Delta$ and/or $\rho$ are small or even 0.
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Analogously, for a single $C$ site, the expression becomes
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$$X_C=\frac{2}{\sqrt{1+4\rho_C\Delta^{CC}}+1}$$
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## PC-SAFT expression for the association strength
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In $\text{FeO}_\text{s}$ every association site $\alpha$ is parametrized with the dimensionless association volume $\kappa^\alpha$ and the association energy $\varepsilon^\alpha$. The association strength between sites $\alpha$ and $\beta$ is then calculated from
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$$\Delta^{\alpha\beta}=\left(\frac{1}{1-\zeta_3}+\frac{\frac{3}{2}d_{ij}\zeta_2}{\left(1-\zeta_3\right)^2}+\frac{\frac{1}{2}d_{ij}^2\zeta_2^2}{\left(1-\zeta_3\right)^3}\right)\sqrt{\sigma_i^3\kappa^\alpha\sigma_j^3\kappa^\beta}\left(e^{\frac{\varepsilon^\alpha+\varepsilon^\beta}{2k_\mathrm{B}T}}-1\right)$$
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with
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$$d_{ij}=\frac{2d_id_j}{d_i+d_j},~~~~d_k=\sigma_k\left(1-0.12e^{\frac{-3\varepsilon_k}{k_\mathrm{B}T}}\right)$$
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and
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$$\zeta_2=\frac{\pi}{6}\sum_k\rho_km_kd_k^2,~~~~\zeta_3=\frac{\pi}{6}\sum_k\rho_km_kd_k^3$$
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The indices $i$, $j$ and $k$ are used to index molecules or segments rather than sites. $i$ and $j$ in particular refer to the molecule or segment that contain site $\alpha$ or $\beta$ respectively.
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On their own the parameters $\kappa^\alpha$ and $\varepsilon^\beta$ have no physical meaning. For a pure system of self-associating molecules it is therefore natural to define $\kappa^A=\kappa^B\equiv\kappa^{A_iB_i}$ and $\varepsilon^A=\varepsilon^B\equiv\varepsilon^{A_iB_i}$ where $\kappa^{A_iB_i}$ and $\varepsilon^{A_iB_i}$ are now parameters describing the molecule rather than individual association sites. However, for systems that are not self-associating, the more generic parametrization is required.
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## Helmholtz energy functional
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The Helmholtz energy contribution proposed by [Yu and Wu 2002](https://aip.scitation.org/doi/abs/10.1063/1.1463435) is used to model association in inhomogeneous systems. It uses the same weighted densities that are used in [Fundamental Measure Theory](hard_spheres) (the White-Bear version). The Helmholtz energy density is then calculated from
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$$\beta f=\sum_\alpha\frac{n_0^\alpha\xi_i}{m_i}\left(\ln X_\alpha-\frac{X_\alpha}{2}+\frac{1}{2}\right)$$
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with the equations for the fraction of non-bonded association sites adapted to
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$$\frac{1}{X_\alpha}=1+\sum_\beta\frac{n_0^\beta\xi_j}{m_j}X_\beta\Delta^{\alpha\beta}$$
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The association strength, again using the PC-SAFT parametrization, is
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$$\Delta^{\alpha\beta}=\left(\frac{1}{1-n_3}+\frac{\frac{1}{4}d_{ij}n_2\xi}{\left(1-n_3\right)^2}+\frac{\frac{1}{72}d_{ij}^2n_2^2\xi}{\left(1-n_3\right)^3}\right)\sqrt{\sigma_i^3\kappa^\alpha\sigma_j^3\kappa^\beta}\left(e^{\frac{\varepsilon^\alpha+\varepsilon^\beta}{2k_\mathrm{B}T}}-1\right)$$
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The quantities $\xi$ and $\xi_i$ were introduced to account for the effect of inhomogeneity and are defined as
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$$\xi=1-\frac{\vec{n}_2\cdot\vec{n}_2}{n_2^2},~~~~\xi_i=1-\frac{\vec{n}_2^i\cdot\vec{n}_2^i}{{n_2^i}^2}$$

‎docs/theory/models/hard_spheres.md‎

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# Hard spheres
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$\text{FeO}_\text{s}$ provides an implementation of the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state ([Boublík, 1970](https://doi.org/10.1063/1.1673824), [Mansoori et al., 1971](https://doi.org/10.1063/1.1675048)) for hard-sphere mixtures which is often used as reference contribution in SAFT equations of state. The implementation is generalized to allow the description of non-sperical or fused-sphere reference fluids.
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The reduced Helmholtz energy density is calculated according to
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$$\frac{\beta A}{V}=\frac{6}{\pi}\left(\frac{3\zeta_1\zeta_2}{1-\zeta_3}+\frac{\zeta_2^3}{\zeta_3\left(1-\zeta_3\right)^2}+\left(\frac{\zeta_2^3}{\zeta_3^2}-\zeta_0\right)\ln\left(1-\zeta_3\right)\right)$$
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with the packing fractions
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$$\zeta_k=\frac{\pi}{6}\sum_\alpha C_{k,\alpha}\rho_\alpha d_\alpha^k,~~~~~~~~k=0\ldots 3$$
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The geometry coefficients $C_{k,\alpha}$ and the segment diameters $d_\alpha$ depend on the context in which the model is used. The following table shows how the expression can be reused in various models. For details on the fused-sphere chain model, check the [repository](https://github.com/feos-org/feos-fused-chains) or the [publication](https://doi.org/10.1103/PhysRevE.105.034110).
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||Hard spheres|PC-SAFT|Fused-sphere chains|
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|-|:-:|:-:|:-:|
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|$d_\alpha$|$\sigma_\alpha$|$\sigma_\alpha\left(1-0.12e^{\frac{-3\varepsilon_\alpha}{k_\mathrm{B}T}}\right)$|$\sigma_\alpha$|
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|$C_{0,\alpha}$|$1$|$m_\alpha$|$1$|
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|$C_{1,\alpha}$|$1$|$m_\alpha$|$A_\alpha^*$|
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|$C_{1,\alpha}$|$1$|$m_\alpha$|$A_\alpha^*$|
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|$C_{1,\alpha}$|$1$|$m_\alpha$|$V_\alpha^*$|
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## Fundamental measure theory
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An model for inhomogeneous mixtues of hard spheres is provided by fundamental measure theory (FMT, [Rosenfeld, 1989](https://doi.org/10.1103/PhysRevLett.63.980)). Different variants have been proposed of which only those that are consistent with the BMCSL equation of state in the homogeneous limit are currently considered in $\text{FeO}_\text{s}$ (exluding, e.g., the original Rosenfeld and White-Bear II variants).
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The Helmholtz energy density is calculated according to
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$$\beta f=-n_0\ln\left(1-n_3\right)+\frac{n_{12}}{1-n_3}+\frac{1}{36\pi}n_2n_{22}f_3(n_3)$$
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The expressions for $n_{12}$ and $n_{22}$ depend on the FMT version.
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|version|$n_{12}$|$n_{22}$|references|
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|-|:-:|:-:|:-:|
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|WhiteBear|$n_1n_2-\vec n_1\cdot\vec n_2$|$n_2^2-3\vec n_2\cdot\vec n_2$|[Roth et al., 2002](https://doi.org/10.1088/0953-8984/14/46/313), [Yu and Wu, 2002](https://doi.org/10.1063/1.1520530)|
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|KierlikRosinberg|$n_1n_2$|$n_2^2$|[Kierlik and Rosinberg, 1990](https://doi.org/10.1103/PhysRevA.42.3382)|
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|AntiSymWhiteBear|$n_1n_2-\vec n_1\cdot\vec n_2$|$n_2^2\left(1-\frac{\vec n_2\cdot\vec n_2}{n_2^2}\right)^3$|[Rosenfeld et al., 1997](https://doi.org/10.1103/PhysRevE.55.4245), [Kessler et al., 2021](https://doi.org/10.1016/j.micromeso.2021.111263)|
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For small $n_3$, the value of $f(n_3)$ numerically diverges. Therefore, it is approximated with a Taylor expansion.
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$$f_3=\begin{cases}\frac{n_3+\left(1-n_3\right)^2\ln\left(1-n_3\right)}{n_3^2\left(1-n_3\right)^2}&\text{if }n_3>10^{-5}\\
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\frac{3}{2}+\frac{8}{3}n_3+\frac{15}{4}n_3^2+\frac{24}{5}n_3^3+\frac{35}{6}n_3^4&\text{else}\end{cases}$$
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The weighted densities $n_k(\mathbf{r})$ are calculated by convolving the density profiles $\rho_\alpha(\mathbf{r})$ with weight functions $\omega_k^\alpha(\mathbf{r})$
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$$n_k(\mathbf{r})=\sum_\alpha n_k^\alpha(\mathbf{r})=\sum_\alpha\int\rho_\alpha(\mathbf{r}')\omega_k^\alpha(\mathbf{r}-\mathbf{r}')\mathrm{d}\mathbf{r}'$$
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which differ between the different FMT versions.
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||WhiteBear/AntiSymWhiteBear|KierlikRosinberg|
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|-|:-:|:-:|
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|$\omega_0^\alpha(\mathbf{r})$|$\frac{C_{0,\alpha}}{\pi\sigma_\alpha^2}\,\delta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|$C_{0,\alpha}\left(-\frac{1}{8\pi}\,\delta''\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)+\frac{1}{2\pi\|\mathbf{r}\|}\,\delta'\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)\right)$|
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|$\omega_1^\alpha(\mathbf{r})$|$\frac{C_{1,\alpha}}{2\pi\sigma_\alpha}\,\delta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|$\frac{C_{1,\alpha}}{8\pi}\,\delta'\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|
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|$\omega_2^\alpha(\mathbf{r})$|$C_{2,\alpha}\,\delta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|$C_{2,\alpha}\,\delta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|
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|$\omega_3^\alpha(\mathbf{r})$|$C_{3,\alpha}\,\Theta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|$C_{3,\alpha}\,\Theta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|
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|$\vec\omega_1^\alpha(\mathbf{r})$|$C_{3,\alpha}\frac{\mathbf{r}}{2\pi\sigma_\alpha\|\mathbf{r}\|}\,\delta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|-|
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|$\vec\omega_2^\alpha(\mathbf{r})$|$C_{3,\alpha}\frac{\mathbf{r}}{\|\mathbf{r}\|}\,\delta\!\left(\frac{d_\alpha}{2}-\|\mathbf{r}\|\right)$|-|

‎docs/theory/models/index.md‎

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```{eval-rst}
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.. toctree::
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:maxdepth: 1
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hard_spheres
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association
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```

‎parameters/pcsaft/gross2002.json‎

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‎parameters/pcsaft/loetgeringlin2015_homo.json‎

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