(166i) Modelling of Interfacial Tension and Adsorption of Inhomogeneous Systems with Classical Density Functional Theory

Modelling of
Interfacial Tension and Adsorption of Inhomogeneous Systems with Classical
Density Functional Theory

E. L. Camacho Vergara, X. Liang, G. M. Kontogeorgis

Center
for Energy Resources Engineering (CERE), Department of Chemical and Biochemical
Engineering, Technical University of Denmark, Søltofts Plads 229, Kgs. Lyngby
2800, Denmark.

Classical
Density Functional Theory (DFT) has shown to be a consistent framework for the study
of surface thermodynamics, being used for the calculation of interfacial
tensions and adsorption, to the calculation of properties of colloidal systems
and self-assembly [1,2]. Classical DFT finds the local density profile of an
inhomogeneous fluid system in the presence of fluid-fluid interfaces or
fluid-solid surfaces, that later can be used for the calculation of interfacial
tensions between fluid phases, or adsorption of fluids on the walls of an
adsorbent.

Classical
DFT is based on statistical mechanics concepts used in molecular simulations,
and with the definition of a Helmholtz free energy functional of the particle density,
it allows the implementation of the already well-known knowledge of bulk
equations of state into the study of inhomogeneous fluids. This locates
classical DFT in an advantageous position, as it allows the determination of density
profiles and surface properties with accuracy similar to molecular simulations,
but with only requiring a modest amount of time characteristic of the
conventional models, such as Multicomponent Potential Adsorption Theory [3] (MPTA)
and Density Gradient Theory [4] (DGT), for adsorption and surface tension
respectively. The Helmholtz free energy functional is defined using a
perturbation approach over the Helmholtz free energy of the inhomogeneous hard
sphere as the reference state. The free energy functional contains the
information of the particle interactions of the inhomogeneous fluid, such as
dispersion, chain formation, association, polar and electrostatic interactions,
which in the bulk limit reduces to the Helmholtz free energy of the homogeneous
fluid.

In
this work, the Perturbed Chain Statistical Association Fluid Theory [5]
(PC-SAFT) equation of state is used to define the Helmholtz free energy
functional within the classical DFT framework. Which enables the important
feature that it is not required to fit new pure component parameters and only
those from bulk fluids are used. The PC-SAFT DFT implementation is used for the
calculation of adsorption and surface tension of pure components and
multicomponent systems of non-associating and associating fluids of interest
for the chemical and petroleum industries. The objective is to test the
performance, accuracy and limits of the results obtained with classical DFT with
the already implemented models by our research group for interfacial tensions
and adsorption, DGT and MPTA, respectively. 

Examples
of the results obtained with classical DFT and PC-SAFT for the calculation of
adsorption isotherms of methane, nitrogen and carbon dioxide on an activated
carbon in a slit-like pore are shown in Figure 1. This calculation uses the
pure component parameters of methane, nitrogen and carbon dioxide used in the bulk
PC-SAFT equation of state [5]. In Figure 2 are shown the particle density
profiles at different temperatures for the vapor-liquid interfacce of methane. As
it can be seen, as the temperature reaches the critical temperature the
interface begins to vanish.

Figure 1. Adsorption of methane, nitrogen and carbon dioxide on
activated carbon [6] calculated with classic DFT with the PC-SAFT equation of
state at 318 K.

Figure 2. Particle density profiles of the vapor-liquid interface
of methane at different temperatures calculated with classical DFT and PC-SAFT.

[1]  J. Wu, “Density Functional Theory for Chemical Engineering: From
Capillarity to Soft Materials,” AIChE Journal, vol. 52, no. 3, pp. 1169-1193,
2005.

[2]  C. P. Emborsky, Z. Feng, K. R. Cox, W. G. Chapman, “Recent
Advances in Classical Density Functional Theory for Associating and Polyatomic
Molecules,” Fluid Phase Equilibria, vol. 306, pp. 15-30, 2011.

[3]  M. A.
Monsalvo, A. A. Shapiro, “Study of high-pressure adsorption from supercritical
fluids by the potential theory,” Fluid Phase Equilibria, vol. 283, pp. 56-64,
2009

[4]  X. Liang, M. L. Michelsen, G. M. Kontogeorgis, “A Density Gradient
Based Method for Surface Tension Calculations,” Fluid Phase Equilibria, vol. 428,
pp. 153-163, 2016.

[5]  J. Gross and G. Sadowski, “Perturbed-Chain SAFT: An Equation of
State Based on a Perturbation Theory for Chain Molecules," Industrial
& Engineering Chemistry Research, vol. 40, no. 4, pp. 1244-1260, 2001.

[6]  M.
Sudibandriyo, Z. Pan, J. E. Fitzgerald, R. L. Robinson, and K. A. Gasem,
“Adsorption of methane, nitrogen, carbon dioxide, and their binary mixtures on
dry activated carbon at 318.2 K and pressures up to 13.6 MPa," Langmuir,
vol. 19, no. 13, pp. 5323-5331, 2003.