Supporting information or Appendix for
‚Computer simulation on static and dynamic properties during transient sorption of fluids in mesoporous materials’
Chemical potentials for inhomogeneous binary system with components A and B
In an inhomogeneous binary system, Helmholtz energy depends on the local densities and on the convoluted densities. Thus, the molar Helmholtz energy at position
depends on the densities of both components , and on the convoluted densities
and
and
The total Helmholtz energy of the system is evaluated via
where.
According to standard thermodynamics, the equilibrium state of a system depends on the experimental boundary conditions. If volume, temperature and amounts nA, nB are kept constant, then the equilibrium state of the system is realized, if the Helmholtz energy takes on a minimum while varying the distributions, . If this situation is realized, variations of the densities of both components, and in consequence of the convoluted densities do not lead to a variation of the Helmholtz energy. The corresponding variation of must vanish.
(A.1)
The variations of the densities must fulfill the conditions
and (A.2)
in order to keep the amounts nA, nB constant.
From the above definitions of the convoluted densities we obtain the expressions for the variation of the convoluted densities
and
and
Inserting these expressions into eq.(A.1) yields
(A.3)
In the terms with twofold integration we change the sequence of integration and exchange the names of and to get
(A.4)
Based on eq. (A.2), it is obvious that the validity of eq.(A.4) for all possible variations of the local densities and requires that the expressions in square brackets must be constant. Thus, these expressions are the suitable choice for defining the chemical potentials in binary inhomogeneous systems
As only the internal energy depends on the convoluted densities we can write
(A.5)
For atomic compounds A, B and for isothermal application we can replace the molar internal energy by the expression from eq.(12)
(A.6)
This leads to the expressions to be convoluted in eq. A.5
Using these expressions the chemical potentials take on the form
and
With we obtain
(A.7)
With the expression from (A.6) we get a fairly simple expression for
which allows to evaluate the chemical potential mA as
(A.8)
The chemical potential of component B is likewise evaluated to give
This is the general treatment for an inhomogeneous binary fluid system.
Defintion of pressure in inhomogeneous binary system
The relation between the pressure and the interface tension will be needed in order to adjust a parameter that controls the range of the interaction potential between the atoms of the fluid A. In the section “Parametrization of the model system” this parameter is introduced as aAA.
We consider here a binary system in equilibrium with a flat interface, i.e. with a variation of the densities rA(z), rB(z) only in z-direction. The amounts of both components are
and
and the total free energy of the system is evaluated via the molar Helmholtz energy Am
We are seeking an expression for the pressure in the inhomogeneous system which allows evaluating the interface tension via the relation
The natural definition of the interface tension is the free energy increase per area due to the interface formation. Thus, we have to compare the free energy of the inhomogeneous system as given above with the free energy of the same amount of the components A, B in the corresponding homogeneous system. In phase 1 of the homogeneous system we have the amounts nA1 and nB1 and the densities rA1 and rB1 , in phase 2 we have the quantities nA2 and nB2 and the densities rA2 and rB2. The extensions in z-direction of the phases are given by
and
In general the sum Dz1+Dz2 is not identical to the system length z2-z1. The expression
measures how much the size of the hypothetical homogeneous reference system is larger (Dz>0) or smaller (Dz<0) than the inhomogeneous system. In case of Dz ¹ 0 the expansion work constitutes part of the energy needed to form the interface. As the inhomogeneous system and the reference system do not have a common system size, but a common external pressure, one might feel motivated to evaluate the difference between inhomogeneous and homogeneous system via the Gibbs energy rather than via the Helmholtz energy. However, we face the problem that the Gibbs energy needs for its definition the pressure which we are about to define for the inhomogeneous system. At present we are able to give a consistent definition of the molar Helmholtz energy and of the chemical potential in the inhomogeneous system, but not of the Gibbs energy.
The difference in system size leads to expansion work. We can either add the expansion work -P×Dz to the energy of the inhomogeneous system or subtract it from the energy of the homogeneous reference system. We choose the latter option.
In order to evaluate the interface energy we have to subtract from the free energy of the inhomogeneous system
the free energy of the homogeneous system diminished by the expansion work
The free energy of the homogeneous system is given by
This can be expanded to
where we use the abbreviations and . This expression refers to the homogeneous system and, thus, we can make use of the standard thermodynamic relations. We obtain
where and . We convert this expression into
In equilibrium the chemical potentials must be identical in both phases, thus and . Further, the pressure must be identical in both phases which leads to
The total free energy of the homogeneous system simplifies to
Now, we develop the expression for the total free energy of the inhomogeneous system
Making use of the chemical potential defined to be constant in equilibrium in the inhomogeneous system we can write
In the integrands we formerly retain the z-dependence of the chemical potentials.
Now, we can proceed to the calculation of the interface energy via
Inserting the expressions derived above we obtain
Finally, we obtain
Now, we see that the definition of the pressure in the inhomogeneous system via
(A.9)
leads to the expected expression for the interface tension