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Modeling Interfaces with Fluid Phase

It seemed necessary to reflect on interface models with at least one fluid phase. The concept and the thermodynamic and mechanical modeling of interfaces are partially revisited. The balance laws and constitutive relations are recalled, and various examples are given to support the remarks. Some questions and perspectives are also proposed.

List of symbols

1.1. The concept of an interface

The general concept of an interface assumes contact or separation between material objects. While the concept of an interface in physics may refer to situations of a very different nature, there are also commonalities between these. Among these commonalities, we will study surface quantities and balance laws in particular, i.e. equations with partial derivatives that connect these quantities, in interaction with those in the media in contact.

Under certain conditions, using simplifying hypotheses, it is possible to establish balance laws for interfaces using physico-chemical quantities - constituent mass, total mass, momentum, energy, entropy - in a unique form, as is done for continuous 3D media (whether fluid or solid). We will see that this requires a small-scale internal exploration of the interface throughout its thickness, and an integration of the results obtained in the normal direction.

1.1.1. Interface in physics and geometric surfaces

The term "interface" refers to a separation surface. However, while the term "surface" has a precise meaning in mathematics, being a 2D manifold endowed with geometric properties, the definition of discontinuity surfaces must be refined by mechanics, more generally, in physics or chemistry.

Figure 1.1. Interfacial layer and interface. a) Gray rectangular zone obtained by dilating the X-axis of the gray part of the figure, b) the sudden growth is replaced by continuous growth between f- and f+

Thus, as soon as this interface presents internal physical properties such as surface tension, or when it modifies the exchanges between the media that it separates, or again, when it is the site of production of different natures, it is no longer a simple separation surface.

On the contrary, there is the question of the scale of observation. The interface appears as a discontinuity surface on a macroscopic scale, but may become a continuous medium on a smaller scale. Moreover, if we observe it on an even smaller scale, the continuity gives way to atomic and molecular discontinuities and their constituents, the elementary particles (Rocard 1933; Roussel 2016).

For a mechanical physicist, the macroscopic description is required to model a problem. However, the mechanical physicist must sometimes go down to a very small scale within the objects to understand their behavior.

This is true for both interfaces and bulks. The mechanical physicist and thermodynamician willingly explore the molecular level to understand the behavior of gases, liquids or solids. The application of the laws of mechanics to objects at a molecular level makes it possible to establish macroscopic constitutive laws of mechanics for continuous media through the statistical theories that use approximations. These scientists commonly use Boltzmann's equation, Fermi's theory and the BBGKY hierarchy. The experimental measurements provide the data that allows them to use the modeling techniques developed from these laws.

It is the same for interfaces. These interfaces can be considered as surfaces with physical properties on a macroscopic scale, but if we examine them on a smaller scale, using powerful microscopes, we will see many differences. Whether or not we go as far as the molecular level, there is indeed a thickness to what appears to be a simple geometric surface. Thus, it is useful to talk about interfacial thickness and an interfacial zone - or an interfacial layer - when we study the interior of an interface in detail (Gatignol and Prud'homme 2001). This exploration will be carried out on a single scale, that of thickness, i.e. in the direction normal to the surface. Thus, the scales in the other directions that are tangential to the surface are left unchanged.

Figure 1.1 shows the classic case of an interfacial zone of very small thickness d, compared to the characteristic length L of the other observed physical phenomena.

The concept of a continuous medium then comes into play. It is not always necessary to go as far down as the molecular level (which, by definition, is not very continuous) to model what happens in an interfacial zone. For example, capillary tension can be explained by considering that the interfacial zone is a continuous medium with properties that can be described from the second gradient theory. This does not mean that we should not go to an even smaller scale to determine the values of internal coefficients, but it is not necessary if we are only interested in the form of constitutive laws.

This is also true for volume. Indeed, the laws derived from the first gradient theory lead to the expression of the stress tensor. However, this first gradient is not sufficient to reveal interfacial behaviors, especially capillary tension. For this, we must go to the second gradient.

It should be recalled here that the virtual power method applied to continuous media postulates that the virtual power of internal efforts exerted in a control volume (), is the volume integral of a function p(i) * that is a linear form of virtual speed V * and its derivatives (Germain 1987). In the first gradient theory, we are limited to: p(i) * = A · V * +B : ? ? V *, where A and B are first-order tensor coefficients. In the second gradient theory, we add a term with a third-order tensor coefficient, such that the virtual power of volume is written as: p(i) * = A · V * + B : ?? V * + C??? (??V *).

1.1.2. The concept of equilibrium in the domain of interfaces

Phase separations can be observed at thermodynamic equilibrium at rest. For example, an interface between a liquid and a gas can present itself as being at equilibrium. We then consider its capillary tension, as well as the equilibrium of the liquid in the presence of its vapor. Similarly, thermodynamic equilibrium is a reference for solid-liquid systems, as well as for solid phases when they are multiple. The presence of stresses that are external to the system, of a mechanical or thermal nature, can lead to lesser or greater deviations from the equilibrium. The concepts of a local state and local equilibrium are used to study these systems, which are the domain of thermodynamics of irreversible processes (Defay and Prigogine 1946; Defay 1949; Prigogine and Defay 1949; Defay et al. 1972). However, there are systems with interfaces - which we will call generalized interfaces - that offer no reference to the equilibrium state. This is the case with thin flames or shock waves, for instance, which can be localized as discontinuity surfaces at a macroscopic level. However, these interfaces can also be studied when not in thermodynamic equilibrium by starting with the concept of a local equilibrium. Although these situations do not necessarily require interface laws, it is satisfying to observe the similarity in the analysis and writing between these and the case of phase separation interfaces that are out of equilibrium.

1.2. Some examples of interfaces

1.2.1. Fluid phase change and separation interface

This is the separation between a liquid and a gas - which can be or contain the vapor of the liquid - or between two immiscible liquids at rest.

Figure 1.2 illustrates a classic situation where the separation surface is curved because of surface tension (or capillary tension) in the presence of a cylindrical wall that constitutes a limiting condition for the interface. The meniscus is the line of contact.

Figure 1.2. Capillary interface. Capillary surface at rest between a liquid and a gas above it, both contained in a graduated glass tube. For a color version of this figure, see www.iste.co.uk/prudhomme/fluid1.zip

On the other hand, a moving interface with deformation, resulting from a high-velocity, annular injection of a liquid in a gas, reveals unstable motion (see (Marmottant and Villermaux 2004, Figure 7, p.79)). Apart from the capillary interface, the presence of liquid digitations (also called fibers, filaments or ligaments) can be observed.

1.2.2. Solidification interface

Solidification interfaces are the site of surface tensions. The phase field method is an original and productive method to study the oriented growth of eutectics (Karma and Sarkissian 1996).

Figure 1.3. Eutectic colonies in CBr4-C2Cl6-naphthalene, V = 31 µm s-1 (Akamatsu and Faivre 2000, Figure 1.2, p. 3758)

The phase field methods make it easier to resolve interface problems, by more or less artificially smoothening the crossing of discontinuities. In their recent form, these methods make it possible to take crystalline...