Classical Density Functional Theory

In this section, the implementation of the core expressions in classical density functional theory is explained.

Euler-Lagrange equation

The fundamental expression in classical density functional theory is the relation between the grand potential :math:`\Omega` and the intrinsic Helmholtz energy :math:`F`.

What makes this expression so appealing is that the intrinsic Helmholtz energy does only depend on the temperature :math:`T` and the density profiles :math:`\rho_i(r)` of the system and not on the external potential :math:`V_i^\mathrm{ext}(r)`.

For a given temperature :math:`T`, chemical potentials :math:`\mu` and external potentials :math:`V^\mathrm{ext}(r)` the grand potential reaches a minimum at equilibrium. Mathematically this condition can be written as

where :math:`F_{\rho_i}(r)=\left.\frac{\delta F}{\delta\rho_i(r)}\right|_T` is short for the functional derivative of the intrinsic Helmholtz energy. In this context, eq. (1) is commonly referred to as the Euler-Lagrange equation, an implicit nonlinear integral equation which needs to be solved for the density profiles of the system.

For a homogeneous (bulk) system, :math:`V^\mathrm{ext}(r)=0` and we get

which can be inserted into (1) to give

Spherical molecules

In the simplest case, the molecules under considerations can be described as spherical. Then the Helmholtz energy can be split in to an ideal and a residual part:

The functional derivatives for an inhomogeneous and a bulk system follow as

Using these expressions in eq. (2) and solving for the density results in

which is the common form of the Euler-Lagrange equation for spherical molecules.

Homosegmented chains

For chain molecules that do not resolve individual segments (essentially the PC-SAFT Helmholtz energy functional) a chain contribution is introduced as

Here, $m_i$ is the chain length, $y_{ii}$ the cavity correlation function at contact in the reference fluid, and $\lambda_i$ a weighted density.

The presence of $\rho(r)$ in the logarithm poses numerical problems. Therefore, it is convenient to rearrange the expression as

Then the total Helmholtz energy

can be rearranged to

The functional derivatives are then similar to the spherical case

and lead to a slightly modified Euler-Lagrange equation

Heterosegmented chains

Thex expressions are more complex for models in which density profiles of individual segments are considered. A derivation is given in the appendix of ***. The resulting Euler-Lagrange equation is given as

with

The index $\alpha$ is used for every segment on component $i$, $\alpha'$ refers to all segments bonded to segment $\alpha$ and $\alpha''$ to all segments bonded to $\alpha'$.

For bulk systems the expressions simplify to

which shows that, by construction, the density of every segment on a molecule is identical in a bulk system. The index $\gamma$ refers to all segments on moecule $i$. The expressions can be combined in a similar way as for the molecular DFT:

At this point it can be numerically useful to redistribute the bulk contributions back into the bond integrals

Combined expression

To avoid having multiple implementations of the central part of the DFT code, the different descriptions of molecules can be combined in a single version of the Euler-Lagrange equation:

If molecules consist of single (possibly non-spherical) segments, the Euler-Lagrange equation simplifies to that of the homosegmented chains shown above. For heterosegmented chains, the correct expression is obtained by setting $m_\alpha=1$.