## Abstract

Thanks to an expansion with respect to densities of energy, mass and entropy, we discuss the concept of *thermocapillary fluid* for inhomogeneous fluids. The non-convex state law valid for homogeneous fluids is modified by adding terms taking account of the gradients of these densities. This seems more realistic than Cahn and Hilliard’s model which uses a density expansion in mass-density gradient only. Indeed, through liquid–vapour interfaces, realistic potentials in molecular theories show that entropy density and temperature do not vary with the mass density as it would do in bulk phases. In this paper, we prove using a rescaling process near the critical point, that liquid–vapour interfaces behave essentially in the same way as in Cahn and Hilliard’s model.

## 1. Introduction

Phase separation between liquid and vapour is due to the fact that density of internal energy (i.e. internal energy per unit volume) *ε*_{0}(*ρ*,*η*) of homogeneous fluids is a non-convex function of mass density *ρ* and entropy density *η*. At a given temperature *T*_{0}, this non-convexity property is related with the non-monotony of thermodynamical pressure *P*(*ρ*,*T*_{0}).

The reader may be accustomed to use specific quantities *α*=*ε*/*ρ*, *s*=*η*/*ρ* and *v*=1/*ρ* instead of volume densities. Indeed, the non-convexity property of *ε*_{0} is equivalent to the non-convexity of *α* as a function of *s* and *v*. In this paper, in accordance with Cahn–Hilliard standard presentation, we privilege volume densities.

In continuum mechanics, the simplest model for describing inhomogeneous fluids inside interfacial layers considers an internal-energy density *ε* as the sum of two terms: the first one previously defined as *ε*_{0}(*ρ*,*η*), corresponds to the fluid with an uniform composition equal to its local one, and the second one associated with the non-uniformity of the fluid is approximated by a gradient expansion,
*m* is a coefficient assumed to be independent of *ρ*, *η* and *grad* *ρ*. This form of internal energy density can be deduced from molecular mean-field theory where the molecules are modelled as hard spheres submitted to Lennard–Jones potentials [1,2].

This energy has been introduced by van der Waals [3] and is widely used in the literature [4–8]. This model, nowadays known as Cahn–Hilliard fluid model, describes interfaces as diffuse layers. The mass density profile connecting liquid to vapour becomes a smooth function.

The model has been widely used for describing micro-droplets [9,10], contact-lines [11–14], nanofluidics [15–17], thin films [18], vegetal biology [19,20]. It has been extended to more complex situations, e.g. in fluid mixtures, porous materials, thanks to the so-called second-gradient theory [21,22] which models the behaviour of strongly inhomogeneous media [23–28].

It has been noticed that, at equilibrium, expression (1.1) for the energy density yields an uniform temperature *T*_{0} everywhere in inhomogeneous fluids
*η* and *ε* can be written in term of *ρ*, only. The points (*ρ*,*η*,*ε*) representing phase states lie on curve *T*=*T*_{0} and such a model inevitably leads to monotonic variations of all densities [1]. Original assumption (1.1) of van der Waals which uses long-ranged but weak attractive forces is not exact for more realistic intermolecular potentials [29–31]. Aside from the question of accuracy, there are qualitative features like non-monotonic behaviours in transition layers, especially in systems of more than one component, that require two or more independently varying densities (entropy included, see ch. 3 of [32]). For these reasons, model (1.1) has been extended in [32,33] by taking into account not only the strong mass density variations through interfacial layers but also the strong variations of entropy associated with latent-heat of phase changes. In ([32], ch. 3 and 9), Rowlinson & Widom noticed that *T*=*T*_{0} is not exact through liquid–vapour interfaces and they introduced an energy arising from the mean-field theory and depending on densities *ρ* and *η* and also on the gradients of these densities; furthermore, they said that *near the critical point, a gradient expansion typically truncated in second order, is most likely to be successful and perhaps even quantitatively accurate*. This extension has been called *thermocapillary fluid model* in [33] and used in different physical situations when the temperature varies in strongly inhomogeneous parts of complex media [33–37].

Near a single-fluid critical point, the mean-field molecular theory yields an approximate but realistic behaviour [32,38]. In mean-field theory, the differences of thermodynamical quantities between liquid and vapour phases are expressed in power laws of the difference between temperature and critical temperature. Transformations from liquid to vapour are associated with the second-order phase transitions and the mass density difference between the two phases goes to zero as the temperature is converging to the critical one. The same phenomenon holds true for the latent-heat of phase transition and for the difference of entropy densities between liquid and vapour phases.

In this paper, we neglect gravity and we use a slightly more general model. We consider state laws which link densities *ε*,*ρ*,*η* and their gradients. We derive the liquid–vapour equilibrium equations of non-homogeneous fluids. As, at equilibrium, a given total mass of the fluid in a fixed domain maximizes its total entropy while its total energy remains constant, the problem can be studied in a variational framework.

We make explicit a polynomial expansion of the homogeneous state law near the critical point. In convenient units, we obtain a generic expression depending only on a unique parameter *χ*.

We introduce a small parameter *κ* which measures the distance of the considered equilibrium state to the critical point. Using a rescaling process near the critical point, we obtain mass and temperature profiles through the liquid–vapour interface. The magnitude orders with respect to *κ* of mass, entropy, temperature are analysed. The variations of temperature and entropy density inside the interfacial layer appear to be of an order less than the variation of mass density. Consequently, neglecting these variations is well founded and justifies the utilization of Cahn–Hilliard’s model near the critical point and indeed we prove that the mass density profile converges towards the classical profile obtained by using the Cahn–Hilliard model which does not take account of variations of entropy density. A conclusion highlights these facts.

## 2. Equations of equilibrium

### (a) Preliminaries

When homogeneous simple fluids are considered, a state law
*ε*, entropy density *η* and mass density *ρ*. This local law is generally made explicit under the form

However, when the state of the material endows strong spatial variations of the thermodynamical variables—as it is the case near a liquid–vapour interface—the locality of the state law has to be questioned. This is what we do in this paper by considering a general law of the type
^{1} :
*η*, *ε* and *ρ*. Generalization (2.1) is widely studied [3,6] in the particular case *D*_{0}=*E*_{0}=*F*_{0}=*G*_{0}=*H*_{0}=0 in (2.2). This special case coincides with the well-known model of Cahn–Hilliard’s fluids [6].

In our framework, we still call temperature, chemical potential, thermodynamical pressure the quantities

### (b) The variational method

The total mass and the total energy of an isolated and fixed domain *dx* is the volume element. They remain constant during the evolution of the system towards equilibrium. The equilibrium is reached when the total entropy
*μ*_{0} are constant Lagrange multipliers (*T*_{0} has the physical dimension of a temperature while *μ*_{0} has the physical dimension of a chemical potential). This equation is valid for all variations (*δε*,*δη*,*δρ*) compatible with the state law, i.e. *Λ* (with no physical dimension) and write that
*δε*,*δη*,*δρ*). This equation reads

## 3. Thermodynamical potentials near a critical point

Let (*ε*_{c},*η*_{c},*ρ*_{c}) be an admissible homogeneous state. Then,
*P*_{c}, *T*_{c}, *μ*_{c} be the associated thermodynamical quantities. At point (*ε*_{c},*η*_{c},*ρ*_{c}), we assume that *ε*_{c},*η*_{c},*ρ*_{c}), it is natural to make a change of variables in order to work in the vicinity of zero; we set

It is clear that maximizing *T*, *μ* by the quantities derived from *C*_{0},…,*H*_{0}) have also to be modified but it is not worth writing the expressions of the new constants *C*_{0},…,*H*_{0}), *ε*_{c}, *ρ*_{c}, *T*_{c}, *μ*_{c} and *a*_{c}. We have *a*_{c} in the change of variables, we have also
*η*_{c},*ρ*_{c}) in the space (*η*,*ρ*). Indeed

Now, we assume that (*ε*_{c},*η*_{c},*ρ*_{c}) corresponds to the critical point of

The critical conditions state that, at *fixed* critical temperature *a*_{02}=*a*_{03}=0. Let us now go a bit further in the expansions of *a*_{12} and *a*_{04} like *a*_{20} do not vanish, we get
*a*_{04}=1 and an entropy unit such that *a*_{12}=1. We denote *a*_{20} in such an unit system. We finally get

From now on, we study the equilibrium of two phases by assuming that

## 4. Integration of equations in planar interfaces

We consider a planar interface and assume that all fields depend only on transverse space-variable *z*. We denote *φ*′ the derivative of any field *φ* with respect to *z*.

### (a) System of equilibrium equations

System of equilibrium equations (2.4) completed by the state law reads in term of new

Multiplying the three first equations, respectively, by *z*, leads to

In the bulk, the fields become constant and the equilibrium equations lead to

We denote by superscripts ^{+} and ^{−} the values of the fields in the two bulk phases. From (3.5)–(3.7) we deduce the equalities of thermodynamical quantities *κ* measures the distance from the critical point. Using again

### (b) The rescaling process

In the view of equations (4.7)–(4.9), the values of *κ*,

Using (3.6) and (4.12), we obtain that the temperature through the interface is constant at the first order:

However, the second equation of system (4.11) gives a more accurate information about the temperature profile through the interface; indeed, at order *κ*,

Note that in equation (4.18) the variation of the temperature across the interface is no more monotonic (figure 3). Moreover, the variation of temperature *κ* and is negligible with respect to the variation of

## 5. Surface tension

Surface tension *σ* of a plane liquid–vapour interface corresponds to the excess of free energy *κ*, we obtain
*κ*^{4} would come from (i) the perturbation of system (4.12) by taking into account the coupling term

## 6. Conclusion

We have obtained the mass density and temperature profiles through an interface near the critical point. Our results present some similarities with the ones obtained in [24] for fluid mixtures where two mass densities have the role played here by mass and entropy densities. The differences lie in the fact that we are not here impelled to deal with combinations of densities and also in the fact that the notion of critical point is more complex in the case of a mixture where non-monotonic profiles can be obtained at the leading order.

We have introduced a state law in which all gradients are considered with respect to mass, entropy and *energy* densities. At our knowledge, it is the first time that this though natural assumption is used. In this framework, we confirm the conjecture made by Rowlinson & Widom [32] that, near the critical point, the variations of temperature inside the interfacial layer are negligible. This result is mainly due to the fact that the variations of entropy density are negligible with respect to the variations of mass density.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

H.G. and P.S. conceived the mathematical model, interpreted the results and together wrote the paper.

## Competing interests

We have no competing interests.

## Funding

This work was supported by CNRS.

## Acknowledgements

P.S. thanks the Laboratoire de Mécanique et d’Acoustique (Marseille) for its hospitality.

## Footnotes

- Received March 31, 2017.
- Accepted July 11, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.