README.md

PyFMT

A python implementation of the Fundamental Measure Theory for hard-sphere mixture in classical Density Functional Theory

For a fluid composed by hard-spheres with temperature T, total volume V, and chemical potential of each species $\mu_i$ specified, the grand potential, $\Omega$, is written as

$$\Omega[{\rho_i(\boldsymbol{r})}] = F^\text{id}[{\rho_i(\boldsymbol{r})}] + F^\text{hs}[{\rho_i(\boldsymbol{r})}]+ \sum_i \int_{V} d \boldsymbol{r} [V_i^{(\text{ext})}(\boldsymbol{r})-\mu_i] \rho_i(\boldsymbol{r})$$

where $\rho_i(\boldsymbol{r})$ is the local density of the component i, and $V^\text{ext}_{i}$ is the external potential.

The ideal-gas contribution $F^\text{id}$ is given by the exact expression

$$F^\text{id}[{\rho_i (\boldsymbol{r})}] = k_B T\sum_i \int_{V} d\boldsymbol{r}\ \rho_i(\boldsymbol{r})[\ln(\rho_i (\boldsymbol{r})\Lambda_i^3)-1]$$

where $k_B$, and $\Lambda_i$ is the well-known thermal de Broglie wavelength of each component.

The hard-sphere contribution, $F^{\textrm{hs}}$, represents the hard-sphere exclusion volume correlation described by the fundamental measure theory (FMT) as

$$F^\text{hs}[{\rho_i (\boldsymbol{r})}] = k_B T\int_{V} d \boldsymbol{r}\ \Phi_\textrm{FMT}({ n_\alpha(\boldsymbol{r})})$$

where $ n_\alpha(\boldsymbol{r}) = \sum_i \int_{V} d \boldsymbol{s}\ \rho_i (\boldsymbol{s})\omega^{(\alpha)}_i(\boldsymbol{r}-\boldsymbol{s})$ are the weigthed densities given by the convolution with the weigth function $\omega^{(\alpha)}_i(\boldsymbol{r})$. The function $\Phi$ can be described using different formulations of the fundamental measure theory (FMT) as

• Rosenfeld Functional (RF) - Rosenfeld, Y., Phys. Rev. Lett. 63, 980–983 (1989)
• White Bear version I (WBI) - Yu, Y.-X. & Wu, J., J. Chem. Phys. 117, 10156–10164 (2002); Roth, R., Evans, R., Lang, A. & Kahl, G., J. Phys. Condens. Matter 14, 12063–12078 (2002)
• White Bear version II (WBII) - Hansen-Goos, H. & Roth, R. J., Phys. Condens. Matter 18, 8413–8425 (2006)

where [x] represents the implemented functionals.

The thermodynamic equilibrium is given by the functional derivative of the grand potential in the form

$$ \frac{\partial \Omega}{\partial \rho_i(\boldsymbol{r})} = k_B T \ln(\rho_i(\boldsymbol{r}) \Lambda_i^3) + \frac{\partial F^\text{hs}[\rho_j]}{\partial \rho_i(\boldsymbol{r})} +V_i^{(\text{ext})}(\boldsymbol{r})-\mu_i = 0$$

Examples

On the folder 'examples' you can find different applications of the FMT.

Hard-Sphere near a Hardwall

Fig.1 - The density profiles of a pure hard-sphere fluid at a planar hard wall with bulk packing fraction of η = 0.4257.	Fig.2 - The density profiles of a pure hard-sphere fluid at a planar hard wall with bulk packing fraction of η = 0.4783.

Hard-Sphere Mixture near a Hardwall

Fig.3 - Density profiles at a planar hard wall of the small spheres of a binary mixture with size ratio σ_b = 3σ_s and packing η = 0.39 and x1 = 0.25.	Fig.4 - Density profiles at a planar hard wall of the big spheres of a binary mixture with size ratio σ_b = 3σ_s and packing η = 0.39 and x1 = 0.25.

Hard-Sphere Radial Distribution Function

Fig.5 - The radial distribution function of a pure hard-sphere fluid with bulk density ρ_b = 0.2.	Fig.6 - The radial distribution function of a pure hard-sphere fluid with bulk density ρ_b = 0.9.