# Modified Fundamental Measure Theory Calculation process

## 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

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 $\rho_i(\textbf{r})$

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

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

Three additional weight function characterizes thr variance across the particle surface

$\sigma_i$ 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

Now, we defined $r’’$:

so we have:

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 $r’’ d \cos \theta = d y$ :

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

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

Now, we calculate the $n_{3}$

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

So, we can rewrite $n_3$ :

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

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

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

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