Modified Fundamental Measure Theory Calculation process

16 Feb 2015

From $\omega$ to $n$

In FMT, Rosenfeld postulates that the excess Helmholtz energy functional of an inhomogeneous hard-sphere mixture, $F^{ex}$, can be expressed in the form of a weight density approximation

$$F^{ex}/k_BT = \int \Phi (\textbf{r}) d\textbf{r}$$

where $K_B$ is Boltzmann constant and $T$ is absolute temperature. The reduced excess Helmholtz energy density $\Phi$ is a function of only the weight averages of the density distribution function $\rho_i(\textbf{r}) $

$$n_{\alpha} (\textbf{r}) = \sum_i n_{\alpha ,i} (\textbf{r}) = \sum_i \int \rho_i (\textbf{r}') \omega_i^{(\alpha)} (\textbf{r} - \textbf{r}')d\textbf{r}'$$

The weight functions $\omega_i^{(\alpha)}(\textbf{r})$ characterize the gemoetry of a hard-sphere: two scalar functions are related to, respectively, the volume and the surface area

$$\omega_i^{(3)} (\textbf{r}) = \theta (\sigma_i/2 -r),$$ $$\omega_i^{(2)}(\textbf{r}) = \delta (\sigma_i/2 -r)$$

and a surface vector function characterize the variance across the particle surface

$$\omega_i^{(V2)}(\textbf{r}) = (\textbf{r}/r)\delta (\sigma_i/2 -r)$$

Three additional weight function characterizes thr variance across the particle surface

$$\omega_i^{(1)}(\textbf{r}) = \omega_i^{(2)}(\textbf{r})/(2\pi \sigma_i),$$ $$\omega_i^{(0)}(\textbf{r}) = \omega_i^{(2)}(\textbf{r})/(\pi \sigma_i^2),$$ $$\omega_i^{(V1)}(\textbf{r}) = \omega_i^{(V2)}(\textbf{r})/(2\pi \sigma_i)$$

$\sigma_i$ is the hard-sphere diameter, $\delta (r)$ is the Dirac delta function, and $\theta (r)$ is the Heaviside step function. The vector weighted densities, $\textbf{n}_{V1}$ and $\textbf{n}_{V2}$, vanish in the limit of a bulk fluid.

Hard-spheres in slit pores

We first apply the modified FMT to a one-component hard-sphere fluids confinde in slit pores. The external potential due to the slit walls is given by HS potential.

where $H$ stands for pore width, and $z$ is the distance from one of the slit walls. In this case, the weighted densities $n_2$ is

$$n_2(\textbf{r}) = \int \rho_i(\textbf{r'}) \omega^{(2)}(\textbf{r}-\textbf{r}')d\textbf{r'}$$

Now, we defined $r’’$:

$$\textbf{r}'' = \textbf{r}-\textbf{r}'$$

so we have:

$$n_{2} = -\int \rho_i (\textbf{r}-\textbf{r}'') \omega^{(2)}(\textbf{r}'')d\textbf{r}''$$ $$n_{2} = \int \rho_i (\textbf{r}-\textbf{r}'') \omega^{(2)}(\textbf{r}'')r''^2 dr'' d\cos \theta d \phi$$

We know that the $\rho$ is only connection with the $z$, We set $y = z + r’’ \cos \theta$ and $ \int d \phi = 2\pi $. So we have $r’’ d \cos \theta = d y$ :

$$n_2 = 2 \pi \int \rho_i (y) \omega^{(2)} (r'') r'' dr'' d y$$ $$n_2 = 2 \pi \int \rho_i (y) \delta (\sigma /2 - r'') r'' d r'' d y$$

Because $\int \delta (\sigma / 2 -r’’) r’’ dr’’= \sigma /2$

$$n_2=\pi \sigma \int \rho_i (y) d y$$

We set $r’’ \cos \theta = z’$, So $y = z + z’$:

$$n_2(z) = \pi \sigma \int_{-\sigma/2}^{\sigma/2} \rho_i (z+z') d z'$$

Now, we calculate the $n_{3}$

$$n_{3} = \int \rho_{i}( \mathbf{r'} ) \omega^{(3)} (\mathbf{r} - \mathbf{r'}) d \mathbf{r'}$$

Because, $\omega^{(3)}$ is the volume area:

$$\omega^{(3)}(\mathbf{r}) = \theta (\omega/2 - r)$$ $$n_3 = \int \rho (\mathbf{r'}) \omega^{(3)} (\mathbf{r} - \mathbf{r'}) d \mathbf{r'}$$ $$\label{eq:n_3-4} n_{3} = \int \rho(\mathbf{r'}) \theta (\sigma / 2 - r ) d \mathbf{r'}$$

So, we can rewrite $n_3$ :

$$\label{eq:n_3-5} n_{3} = \int_{V} \rho(\mathbf{r}) d \mathbf{r}$$ $$\label{eq:n_3-6} n_{3}(z) = \pi \int_{-\sigma/2}^{\sigma /2} \rho(z+z') \left[ \left(\sigma /2\right)^2 - (z')^2 \right] d z'$$

Finally, we calculate the $n_{V2}$:

$$\label{eq:n-v2-1} n_{V2} = \int \rho (\mathbf{r'}) \omega ^{(V2)} (\mathbf{r} - \mathbf{r'}) d \mathbf{r'}$$

Because $\omega^{V2} = (\mathbf{r}/r) \delta (\omega/2 - r) $, we can rewritte $n_3$ as

$$\label{eq:n-v2-2} n_{V2}(z) = \int_s e_{(\mathbf{r}-\mathbf{r'})} \rho(\mathbf{r'}) d \mathbf{r'}$$ $$\label{eq:n-v2-3} n_{V2}(z) = \mathbf{e_z} \int \rho (z+z') 2 \pi ( \sigma / 2 ) \sin \theta d\mathbf{z'}$$

We know $(\sigma / 2)\sin \theta$ is equal $z’$, so we have

$$n_{V2}(z) = 2 \pi \mathbf{e_z} \int \rho (z+z') z' d\mathbf{z'}$$

• Yu, Y. X.; Wu, J., Structures of hard-sphere fluids from a modified fundamental-measure theory. The Journal of Chemical Physics 2002, 117, 10156.