Chapter 4
Gibbs-Ensemble Monte Carlo Simulations of Phase Equilibria in Supercritical Fluid Mixtures
Downloaded by UNIV ILLINOIS URBANA on June 1, 2013 | http://pubs.acs.org Publication Date: August 29, 1989 | doi: 10.1021/bk-1989-0406.ch004
A. Z. Panagiotopoulos School of Chemical Engineering, Cornell University, Ithaca, NY 14853-5201
A novel technique for molecular-level computer simulations, the Gibbs-ensemble Monte Carlo methodology, is applied to the calculation of phase equilibria in supercritical fluid systems. The Gibbs method is based on performing a simulation in two regions in a way that ensures that the criteria for equilibrium between coexisting phases are satisfied in a statistical sense. Lennard-Jones intermolecular potentials are used in the calculations, with parameters obtained from pure component thermodynamic properties to represent carbon dioxide, acetone and water. Calculated binary phase diagrams are in good agreement with experimental data. Ternary calcula tions are only in qualitative agreement with experiment. Additional results obtained by varying the intermolecular potential parameters for the unlike pair interactions illustrate the effect of intermolecular forces on phase behavior. Thermodynamic modelling o f phase e q u i l i b r i a i s an important part i n the successful development and operation o f p h y s i c a l separation processes. The p r e d i c t i o n o f phase e q u i l i b r i a f o r s u p e r c r i t i c a l f l u i d extraction applications presents s p e c i a l challenges, not shared by more conventional separation systems. The challenges a r i s e as a r e s u l t of operation at high, and highly v a r i a b l e , pressures and the inherently asymmetric character of the mixtures involved that contain components of very d i f f e r e n t size or v o l a t i l i t y . A s u b s t a n t i a l e f f o r t i n recent years has been devoted to the development of improved equations-of-state and mixing rules that can describe the often complex phase e q u i l i b r i a encountered i n s u p e r c r i t i c a l e x t r a c t i o n operations. While much progress has been made, e x i s t i n g models allow interpolations and moderate extrapolations of experimental data and cannot be used i n the absence of experimental information. An a l t e r n a t i v e to macroscopic, phenomenological models i s provided by molecular simulation methods which provide a d i r e c t l i n k between the intermolecular forces and macroscopic phase behavior. 0097-6156/89AM06-€039$06.00/0 ο 1989 American Chemical Society
In Supercritical Fluid Science and Technology; Johnston, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.
Downloaded by UNIV ILLINOIS URBANA on June 1, 2013 | http://pubs.acs.org Publication Date: August 29, 1989 | doi: 10.1021/bk-1989-0406.ch004
40
SUPERCRITICAL FLUID SCIENCE AND
TECHNOLOGY
Computer simulation techniques have been used since t h e i r inception for the c a l c u l a t i o n of basic thermodynamic and s t r u c t u r a l properties of l i q u i d s (1). While i t i s r e l a t i v e l y easy to c a l c u l a t e simple configurational properties such as the energy and pressure of a f l u i d , quantities that are r e l a t e d to the entropy or free energy have been considered very d i f f i c u l t to obtain from simulation. In recent years, improved techniques for the estimation of free energies have appeared (a recent review i s given i n 2). The Wldom test p a r t i c l e method Q ) i s the most widely used method, and recent developments ( D e i t r i c k et al., J.Chem. Phys. i n press) have pointed to possible s i g n i f i c a n t improvements i n the accuracy and speed of the technique. Several successful predictions of phase e q u i l i b r i a of mixtures have been reported for atomic (4) and molecular f l u i d s (S) using t h i s method. Molecular simulation r e s u l t s for the s o l u b i l i t y of s o l i d s i n s u p e r c r i t i c a l f l u i d s have appeared i n (6). The computer time requirements for these c a l c u l a t i o n s are high because of the need to perform a large number of simulations at d i f f e r e n t densities and compositions, a number which increases r a p i d l y with the number of components i n a mixture. The semigrand ensemble approach developed by Kofke and Glandt (2) eliminates this l a s t problem but s t i l l requires a series of simulations for the determination of a single phase coexistence point. The recently proposed Gibbs-ensemble Monte Carlo simulation method (8), i s a s i g n i f i c a n t improvement r e l a t i v e to previously described methods, as i t always requires only a single simulation per coexistence point.
The Gibbs Method Basic Concepts. The methodology for the determination of phase e q u i l i b r i a i n mixtures using the Gibbs method has been presented i n d e t a i l elsewhere ( 8 , 9 , 1 0 ) . The essence of the technique i s to perform a simulation i n two d i s t i n c t regions (e.g. a l i q u i d and a gas region), each i n periodic boundary conditions with images of i t s e l f . Three types of perturbation are performed i n a way that ensures that the conditions f o r phase equilibrium between the phases are s a t i s f i e d i n a s t a t i s t i c a l sense: (a) displacements within each region, considered separately, to s a t i s f y i n t e r n a l e q u i l i b r a t i o n (b) volume rearrange ments, to s a t i s f y equality of pressures and (c) p a r t i c l e transfers between the two regions, to s a t i s f y the condition of equality of chem i c a l p o t e n t i a l s . The corresponding c r i t e r i a f o r the acceptance of each move are: (a) p a r t i c l e displacements ?
m o v 9
-
min^ 1 , exp(-0AE) j
(1)
where ΔΕ i s the conf igurational energy change r e s u l t i n g from the t r i a l displacement. (b)
volume change steps In the cons tant-WT ensemble the volume changes i n the two regions are equal and opposite, so that t o t a l system volume i s conserved:
In Supercritical Fluid Science and Technology; Johnston, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.
4.
PANAGIOTOPOULOS
P
Monte Carlo Simulations ofPhase Equilibria
- m i n l l , e x p l -β I Δ Ε + Δ Ε 1
v o l
1 1
- î^kTin
~JJJ
—-N^kTin
(2)
In the cons tant-NPT ensemble the two regions undergo independent volume changes:
P
r ς r - m i n Ι,βχρ -β ΔΕ + Δ Ε ν V L 1
v o l
Downloaded by UNIV ILLINOIS URBANA on June 1, 2013 | http://pubs.acs.org Publication Date: August 29, 1989 | doi: 10.1021/bk-1989-0406.ch004
+ (c)
I
Ι Σ
V +AV - N*kTin yl
I
II
V +AV
11
11
Ν kTin yll
PiAV^AV ) j j j 11
