Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
line are given below. We give the line internal energy, line internal energy density, Gibbs equation, Gibbs-Duhem’s equation, contact line Gibbs energy density and the contact line Helmholtz energy density:
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(3.36) |
3.5Defining local equilibrium
We have so far considered global and local thermodynamic relations for heterogeneous systems that are in global equilibrium. We shall now move one step further and introduce the assumption of local equilibrium. For a volume element in the homogeneous phase i, we say, following Fitts [33], Bedeaux and Kjelstrup [64] and Ortiz and Sengers [112], that there is local equilibrium when the thermodynamic relations (3.25)–(3.27) are valid:
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(3.37) |
The position and time dependence of all the quantities are now explicitly indicated. We used that Deq is independent of the position in electroneutral systems. The subscript eq of the displacement field indicates that this is the value of the displacement field when the system is in global equilibrium, with entropy density, concentrations and polarization density constant and equal to the values of the volume element. The actual displacement field in the volume element is not equal to Deq(t) when the system is not in global equilibrium. Local equilibrium is defined in the same manner for the o-phase.
The use of the word local in this context means that the quantities are averages over sensibly chosen volume elements and time intervals. By a sensibly chosen volume element, we mean a volume element that is large compared to microscopic distances, and small compared to the distance over which the averaged quantities vary. A sensibly chosen time interval is long compared to microscopic times, but small compared to the time rate of variation of the averaged quantities. The choices of the size of the volume element and the time interval are not independent. With a slowly varying process, the volume element may be chosen small, and the time interval large. These statements can be made precise by non-equilibrium molecular dynamics simulations, see Chapter 22.
For a surface element, we say that there is local equilibrium when the thermodynamic relations (3.29)–(3.31) are valid:
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(3.38) |
The intensive thermodynamic variables for the surface, indicated by superscript s, are derivatives:
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(3.39) |
The temperature and chemical potentials, defined in this manner, depend only on the surface excess variables, not on bulk variables. By introducing these definitions, we therefore allow for the possibility that the surface has a different temperature and/or chemical potentials than the adjacent homogeneous systems have. The equilibrium displacement field for the surface is defined by
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(3.40) |
The actual displacement field D(t) which is externally controlled, and which is also independent of x, is not equal to Deq(t). In the surface, a suitable surface element and time interval must be chosen for the averaging procedure, when thermodynamic quantities are determined.
For a contact line element, we say that there is local equilibrium when the thermodynamic relations (3.33)–(3.35) are valid:
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(3.41) |
The intensive thermodynamic variables for the contact line, indicated by superscript c, are derivatives:
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(3.42) |
The temperature and chemical potentials, defined in this manner, depend only on the excess variables for the contact line, not on bulk or surface variables. By introducing these definitions, we therefore allow for the possibility that the contact line has a different temperature and/or chemical potentials, than the adjacent homogeneous phases and surfaces. Along the contact line, suitable line elements and time intervals must be chosen for the averaging procedure, when the thermodynamic quantities are determined.
The assumption of local equilibrium, as formulated above, does not imply that there is local chemical equilibrium [23, 71]. According to the original formulation of Prigogine [113], Gibbs equation remains valid for a large class of irreversible processes, provided that the Maxwell distribution of molecular velocities are perturbed only slightly. The class includes also chemical reactions slow enough, to not disturb the equilibrium form of the distribution to an appreciable extent.
The thermodynamic variables are, with the above relations, position dependent. In particular, the thermodynamic variables for the surface depend on the position along the surface. Most excess densities depend on the choice of the dividing surface. The surface tension of a flat surface, however, does not depend on the position of the surface when the system is in mechanical equilibrium.
An essential and surprising aspect of the local equilibrium assumption for the surface and the adjacent homogeneous phases is, that the temperature and chemical potentials on both sides of the surface may differ, not only from each other, but also from the values found for the surface.
The surface temperature represents the thermodynamic state of the surface. This temperature depends in principle on the choice of the dividing surface, but we have found that this dependence, both in molecular dynamics simulations [42] and in the dynamic van der Waals square gradient model [110], is negligible. The surface temperature can be calculated in molecular dynamics simulations, assuming the validity of the equipartition principle. We can then find the surface temperature from the kinetic energy. This is discussed in Chapter 22. It is difficult to imagine a direct measurement of the surface temperature. A direct measurement is hampered by the thickness of the thermocouple leads, and the positioning of the thermocouple. The description of the liquid–vapor interface, using a dynamic extension of the van der Waals model, is discussed in Chapter 23.
Whether the non-equilibrium surface as described by excess densities is an autonomous or self-contained thermodynamic system or not is an important question. Our results [42, 110] support the validity of this property, see Chapters 22 and 23. Its validity is crucial for the treatment given in this book. It has been defended by some authors [97, 98] and rejected by others [99]. Molecular dynamics simulations support the validity of the assumption of local equilibrium [68–70, 114, 115]. This is discussed in Chapter 22 for homogeneous phases as well as for surfaces.
Given that there is very often local equilibrium in systems with large gradients, one may ask: Can we distinguish between volume elements that belong