Fluctuation Solution Theory Properties from Molecular Simulation - Fluctuation Theory of Solutions - page 157

0.25

0.2

0.15

0.1

0.05

0

-0.05

0 0.2 0.4 0.6 0.8 1

Mole Fraction Water, x 1

FIGURE 6.8 Isothermal compressibilities for the mixture water (1)/t-butanol (2). Results
from our Verlet (◻), truncation ( ○ ) (Results from S. Weerasinghe and P. E. Smith, 2003b,
Kirkwood-Buff Derived Force Field for Mixtures of Acetone and Water, Journal of Chemical
Physics , 118, 10663), and distance-shifting (Results from B. Hess and N. F. A. van der Vegt,
2009, Cation Specific Binding with Protein Surface Charges, Proceedings of the National
Academy of Sciences of the United States of America , 106, 13296) (Δ) methods compared
with values obtained from the fluctuation formula (⎯x⎯).

varied in the interval for R lim from 1.0 to 1.5 nm. With the distance-shifting method,
the scaling factors α ij were evaluated from the calculated RDFs with the parameter
R = 2.0 nm. Numerical integration of the rescaled RDFs did not converge within the
sampling range, so the integrals of the rescaled TCFs were evaluated by truncation
using intervals of 1.4-1.9 nm. The truncation radii employed for integration of the
rescaled TCFs were larger than those used with the simple truncation approach since
the rescaled TCFs probably were more accurate than the original TCFs for large r , as
discussed in Section 6.3. For comparison, isothermal compressibilities were evalu-
ated via the fluctuations of the simulation box volume. The results are shown in
Figure 6.8. The Verlet method reproduced the fluctuation formula results to within
5%, while the simple truncation and distance-shifting methods were greatly in error.
In fact, distance shifting yielded negative compressibilities. It is likely that reliable
results by these methods require simulations of larger systems.
In order to validate the partial molar volumes obtained by the different integra-
tion methods, the excess molecular volume, V E , was evaluated for the simulations at
each composition according to

E o o

Vx () = Vx () − xV − xV (6.47)

m 1 m 1 11 22

where V m ( x 1 ) denotes the average molecular volume obtained at the composition x 1 ,
the mole fraction of water. Also, V 1 o and V 2 o denote the average molar volumes of
the corresponding pure components, obtained from separate simulations. The poly-
nomial model of Handa and Benson (1979),

E (model) =

)

2

(6.48)

V x xa ax + ( − xaxx ) + ( −

m 12 0 1 2 1 22 1