Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
for the different contributions to the polarization. We shall not make the formulae more cumbersome by distinguishing these different contributions.
The surface tension is, like the pressure, a function of T, μj, and Deq. The most directly observable effect of the surface tension is found when the interface between two phases is curved. Consider for instance a spherical bubble in a liquid. The pressure inside the bubble is higher than the pressure outside. The same is true for a liquid droplet in air. The pressure difference, called the capillary pressure, is given in terms of the surface tension by 2γ/R, where R is the radius of the droplet. The capillary pressure is the reason why water rises in a capillary. The following exercise demonstrates that the energy represented by a surface can be comparable to the energy of a homogeneous phase.
Exercise 3.1.1.Use the expression for the capillary pressure to derive a formula for the rise of liquid in a capillary as function of the diameter d of the capillary. The liquid wets the surface. Calculate the capillary rise of water in a capillary with a diameter of a micron. The surface tension of water is γw = 75 × 10−3N/m.
•Solution: If the diameter of the capillary is not too large, and when the liquid wets the surface, the interface between the water and the air is a half sphere with radius d/2. This gives a capillary pressure of 4γ/d. This would cause an under-pressure in the liquid below the surface. In order to have a normal atmospheric pressure at the bottom of the capillary, the water rises until the weight of the column gives the capillary pressure. This implies that π(d/2)2hgρ/π(d/2)2 = hgρ = 4γ/d where h is the height of the column, ρ is the density in kg/m3 of water and g is the acceleration of gravity. The capillary rise is therefore h = 4γ/dgρ. In a capillary with a diameter of a micron, the capillary rise of water is 30 m.
In this book, we restrict ourselves to surfaces that are flat. The equilibrium pressure is therefore always constant throughout the system, pi = po = p.
The Gibbs equation (3.9) gives the energy of both homogeneous phases plus the surface. We need also equations for the separate homogeneous phases and for the surface alone. With Eqs. (3.1)–(3.3), we have for phase i:
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(3.11) |
This is the Gibbs equation for phase i. We integrate the equation for constant composition, temperature, pressure and displacement field. This gives
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(3.12) |
The energy is a state function, so the result is generally valid (cf. Eq. (3.10)). By differentiating this expression and subtracting Eq. (3.11), we obtain Gibbs–Duhem’s equation:
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(3.13) |
Similar equations can be written for phase o.
The Gibbs equation for the surface is likewise:
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(3.14) |
By integration for constant surface tension, temperature, composition and displacement field, we obtain
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(3.15) |
Gibbs–Duhem’s equation for the surface follows by differentiating this equation and subtracting Eq. (3.14):
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(3.16) |
The thermodynamic relations for the autonomous surface, Eqs. (3.14)–(3.16), are identical in form to the thermodynamic relations for the homogeneous phases, cf. Eqs. (3.11)–(3.13).
Global equilibrium in a heterogeneous, polarizable system in a displacement field can now be defined. The chemical potentials and the temperature are constant throughout the system
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(3.17) |
Furthermore, the normal component of the pressure and of the displacement field are the same in the adjacent homogeneous phases:
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(3.18) |
For curved surfaces, the normal component of the pressure is not constant and we refer to Blokhuis, Bedeaux and Groenewold [101–106].
The surface energy and therefore also the surface entropy and surface mole numbers have so far been defined from the total values minus the bulk values. We shall see in Sec. 3.3, how more direct definitions can be formulated.
3.2The contact line in global equilibrium
Consider three phases of a multi-component system in equilibrium with each other. The three surfaces separating these phases come together in a contact line. For simplicity, we assume that the contact line is not polarizable. In Sec. 3.1, we have seen that the thermodynamic relations for an autonomous surface has the same form as for homogeneous phases. The same is true for the autonomous three-phase contact line. The Gibbs equation for the contact line is as follows:
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(3.19) |
Superscript c indicates a contribution of the contact line, γc is the line tension and L is the length of the line. The curvature of the line is assumed to be negligible, so that there are no contributions to dUc from changes in the curvature. By integration for constant line tension, temperature and composition, we obtain
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(3.20) |
The line energy and therefore also the line entropy and the line mole numbers have here been defined as the total values minus the bulk and the surface values.
Gibbs–Duhem’s equation for the line follows by differentiating this equation and subtracting Eq. (3.19):
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(3.21) |
When the contact line is part of a heterogeneous system in global equilibrium, the chemical potentials and the temperature are constant throughout the system. For the contact line,