Description of Inhomogeneous Systems in a Thermodynamic Framework
Appendix
Description of inhomogeneous systems in a thermodynamic framework
In the following we refer to a confined system with a fluid component A at constant temperature. The confinement is caused by a solid, rigid structure made from component B. In confined thermodynamics all quantities are in general functions of the spatial coordinate . Thus, the density of amount is to be replaced by , the molar Helmholtz energy by and the pressure by . While arbitrary situations lead to a position dependent chemical potential , an equilibrium state is distinguished by a constant value for all . For further use, we note that the following relation holds
(1)
For a state this expression simplifies to
(2)
The way how to formulate all expressions is given for a binary system in a previous paper
and in the supporting information therein[1]. For isothermal situations and for given confinement a state is defined by the distribution of the density of amount
The following quantities can be evaluated
- the amount
- the Helmholtz energy
- the grand potential
where the volume refers to the range occupied by the fluid component A.
The above relation between pressure , molar Helmholtz energy , density of amount and chemical potential leads to
(3)
which means that knowledge of three of the quantities n, , , determines the fourth.
Formally, one can define the molar Gibbs energy as and evaluate the total Gibbs energy . For a state one obtains the simple expression or . In the following, we will not make use of this quantity.
Defining metrics for states by introducing dissimilarity / distance
At given temperature and for given confinement, a state is defined by its density distribution. In contrast to standard thermodynamics two states with the same amount n in a given volume can still have different density distributions and . We introduce a measure for the dissimilarity between two states
(4)
D vanishes if the distributions are identical. Otherwise we obtain a finite positive value. The upper limit is unity. This can be verified, if one takes as a state with a density pertaining to condensed matter and as a state with extremely low density in the entire volume. The limit for turns out to yield D=1.
Further, we can introduce a distance between two states in the following way. At constant temperature the density distributions and determine the related values of amount, chemical potential, grand potential and Helmholtz energy, i.e. , , , and , , , . Two identical states coincide with respect to all four quantities. Two different states must differ with respect to at least one quantity. As mentioned above, knowledge of three quantities determines the fourth. Thus, it is sufficient to employ only three of these quantities in order to define a measure for the distance d and it could be defined via . However, this definition would be impractical since it would depend on the units chosen. Therefore, it would be advantageous to normalize the differences to reference values n, and . In the presence of a hysteretic loop we propose to adopt the variation of the respective quantity within the loop as reference. This leads to our working formula for measuring the distance between two states
(5)
The two quantities D and d are related in the following way:
- may be correct (and most likely is correct) but is not proven
A finite value of the distance between isothermal states indicates that the states have different density distributions and, thus, are not identical. Dissimilarity D and distance d between any two states in a system can be evaluated. We will make use of these definitions in order to introduce metrics in the three dimensional (n, μ, Ω)-space for the states encountered in a system.
[1] Harald Morgner, J. Phys. Chem. C 114 (2010) 8877-83