Superfluid transition in liquid helium

# A thermodynamically consistent Ginzburg-Landau model for superfluid transition in liquid helium

Alessia Berti Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO), Italy  and  Valeria Berti Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy
###### Abstract.

In this paper we propose a thermodynamically consistent model for superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. The phase transition is described by means of an order parameter, according to the Ginzburg-Landau theory, emphasizing the analogies between superfluidity and superconductivity. The normal component of the velocity is assumed to be compressible and the usual phase diagram of liquid helium is recovered. Moreover, the continuity equation leads to a dependence between density and temperature in agreement with the experimental data.

AMS Classification: 82D50, 74A15, 82C26.

Keywords: Superfluids, second-order phase transitions, Ginzburg-Landau equation, thermodynamics.

## Introduction

The phenomenon of superfluidity occurs mainly in liquid helium below a characteristic temperature . Above , helium behaves like a conventional fluid with small viscosity. However, when the temperature is lowered below , liquid helium undergoes a phase transition characterized by the ability of the liquid to flow across narrow channels without apparent friction. Helium has two stable isotopes He and He that become superfluid at low temperatures. The most common isotope is He whose transition temperature, called the point, is about . The normal phase of He is called the He I-phase and the superfluid state is said He II.

In this paper we propose a phenomenological model to describe the phase transition in He. The first model to study the behavior of He was the two-fluid model, suggested by Tisza ([17]) and developed by Landau ([12]). According this theory, when the temperature is under , each particle of the fluid is endowed with two different excitations at the same instant: one of these is the superfluid velocity, denoted by , the other one is the normal velocity . The density of the fluid is the sum of a normal and a superfluid component

 ρ=ρn+ρs

and the total current density is given by

 j=ρnvn+ρsvs.

If the temperature overcomes the point, the density vanishes, so that liquid helium becomes a normal fluid.

The two-fluid model has been widely adopted to describe some phenomenological aspects of superfluidity, when the involved velocity of the fluid is quite small (see [16] and references therein). Besides the two-fluid model, some authors follow the one-fluid theory of liquid helium, which is based on the extended irreversible thermodynamics (see [13, 15] for instance). They analyze the behavior of liquid helium II considered as a unique substance obeying a suitable Navier-Stokes equation. More recently in [6] the author studies the phase transition between helium I and helium II in the framework of the Ginzburg-Landau theory, by considering this passage as a second order phase transition and introducing a scalar variable as order parameter such that represents the concentration of the superfluid phase. This point of view emphasizes also the analogies between superfluidity and superconductivity ([14, 16]). Indeed, as in the two-fluid model, the velocity of the fluid is due to a normal and a superfluid excitation; however the superfluid component is supposed to satisfy an evolution equation similar to the differential equation governing the motion of the superconducting electrons inside a superconductor ([5]).

In our paper we consider a generalization of this model by keeping into account variations of the mass density of the fluid and variations of the temperature. The main assumption, distinguishing our model by the one proposed in [5], is that the normal component is a compressible fluid. Accordingly, the pressure becomes a new variable of the problem and the divergence of the normal component satisfies a constitutive equation depending on the phase variable . When the fluid is in the normal state, the evolution equation for the normal component reduces to the Navier-Stokes equation.

The occurrence of the pressure in the phase equation allows us to recover in the phase diagram the line (line) separating the normal from the superfluid phase (see Fig. 1).

We prove that our model is consistent with thermodynamic principles, deducing the differential equation for the temperature from the energy balance law and proving that Clausius-Duhem inequality is satisfied. In particular we assume that the heat flux is the sum of two contributions: the first term is proportional to the gradient of the temperature (Fourier law), the other one is due to the superfluid transition since it involves the superfluid component and the phase variables . This point of view is not dissimilar from models based on the extended irreversible thermodynamics, which interpret the superfluid velocity as a kind of heat flux ([13, 15]).

In the last section of the paper we show that the differential system governing the evolution of the fluid can be written by means of a different set of variables, similar to the unknown fields used in the context of superconductivity ([7, 18]). Such a formulation could be useful in proving some analytical results concerning the well-posedness of the system.

## 1. The Ginzburg-Landau equation for the superfluid concentration

As known, the passage from the normal phase to the superfluid state is a second-order phase transition, since no latent heat is involved [14]. Therefore we propose a model to describe the phenomenon in the context of the Ginzburg-Landau theory. The first step is the identification of a suitable order parameter characterizing the state of the material. Here we introduce a scalar variable (phase-field), such that represents the concentration of the superfluid phase. Thus the values of are bounded in the interval with in the normal phase and in the superfluid regime. The variable provides a measure of the internal order structure of the material, since the superfluid phase is considered a more ”ordered” state than the normal one [12]. Accordingly, the differential equation governing the evolution of can be interpreted as a balance law on the internal order structure. For a general treatment of balance laws in continuous bodies with microstructure see [3, 4]. The interpretation of the Ginzburg-Landau equation as a balance equation has been proposed by Fried and Gurtin who introduce the notion of microforces ([10, 11]). In this context every change of the order parameter is related to the existence of microforces which expend power on the atomic configurations inside the material. A similar approach has been proposed in [5] where a balance of the internal order structure is postulated. The common idea of the two approaches is that, during the transition, in any sub-region of the body, the power expended on the atoms by the lattice is balanced by the power expended across the boundary by the configurations external but neighboring to and by the power expended by sources external to the body. We briefly recall the interpretation of the Ginzburg-Landau equation as a balance law, adopting the terminology of [5].

Let us consider a superfluid occupying a bounded subset with regular boundary , whose outward normal is denoted by . For any sub-body , we denote by the rate of absorption of the order structure per unit time, defined as

 (1.1) Si(A)=∫Aρkdv,

where is the mass density and is the internal specific structure order. Similarly, the external order structure is written in the form

 (1.2) Se(A)=∫∂Ap⋅nds+∫Aρσdv,

where the vector denotes the order structure flux coming from the boundary and is the structure order supply.

Hence, the order structure balance is expressed by the equality

 (1.3) ∫Aρkdv=∫∂Ap⋅nds+∫Aρσdv∀A⊂Ω.

In local form, the integral equality (1.3) leads to the equation

 (1.4) ρk=∇⋅p+ρσ.

Hence, the quantity can be assimilated to a vector stress and and to internal and external microforces distributed in the domain . As usual in phase transition problems, we assume . Moreover, the functions and are defined by means of the constitutive equations

 (1.5) k = τ˙φ+θλF′(φ)+mG′(φ) (1.6) p = ρκ2∇φ,

where the superposed dot stands for the time derivative, is a suitable coefficient depending on the variables that induce the transition, the potentials and characterize the order and the feature of the transition and are positive constants. Accordingly, the evolution equation for the order parameter reads

 (1.7) τρ˙φ=1κ2∇⋅(ρ∇φ)−ρθλF′(φ)−ρmG′(φ).

In steady and homogeneous conditions (i.e. and ) the solutions of (1.7) are the stationary points of the function

 W(φ)=θλF(φ)+mG(φ).

A typical choice adopted for second-order phase transitions is

 (1.8) F(φ)=φ44−φ22,G(φ)=φ22.

Such expressions are the same used in the classical Ginzburg-Landau theory of superconductivity ([9]). The functions and satisfy the following properties (see Fig. 2):

• the function admits its minimum value at when and at , , when ;

• as and as .

As a consequence the transition occurs when and identifies the normal phase.

Notice that there is no physical distinction between the positive and negative values of since the physical quantity is . As a consequence, we can consider as well as . In the latter case, we have to require that the potentials and are even functions.

Finally, we associate to equation (1.7) the usual Neumann boundary condition

 (1.9) p⋅n|∂Ω=0.

## 2. Evolution equation for the velocity

According to Landau’s viewpoint, we assume that each particle of the superfluid exhibits a normal and a superfluid excitation. Thus the velocity is given by the superposition of such flows. In particular, we let

 (2.1) v=(1−φ2)vn+φ2vs.

where , will be called normal and the superfluid components, respectively. This does not mean that superfluid is considered as a mixture of two fluids. Indeed, a particle of the material is endowed with two simultaneous excitations (normal and superfluid) when , while its velocity coincides with when . However, the expression (2.1) of the velocity is not in contrast with the traditional theories of superfluids which assume

 ρv=ρnvn+ρsvs,

when and are identified respectively with and .

The variables and are related by the continuity equation

 (2.2) ˙ρ+ρ∇⋅v=0.

By paralleling [6], the differential equations governing the evolution of the component are

 (2.3) ˙vs = −∇ϕs−∇×vn−ρφ2vs+∇θ (2.4) ∇⋅[ρφ2vs] = −τκ2ρφ2ϕs

where is a suitable scalar function referable to a “pressure” due to the superfluid component. It is worth noting that (2.3) and (2.4) are similar to the evolution equations governing the motion of superconducting electrons [9], emphasizing the evident analogies between superfluidity and superconductivity.

We associate to these equations the boundary condition

 (2.5) vs⋅n|∂Ω = 0

A further boundary condition on has to be prescribed. Our model allows us to choose such a condition in an arbitrary way, since, as we will see in Sect. 4, the condition (2.5) is sufficient to ensure the vanishing of the power flux at the boundary of the domain. In the superconductivity model it is assumed

 (2.6) (∇×vs)×n|∂Ω = ω,

where is a known function. Therefore we can assume (2.6) by analogy with that model.

For the velocity we propose the following equation

 (2.7) ρ(1−φ2)˙vn=∇×˙vs+ν∇(∇⋅vn)−∇p−1κ2∇⋅(∇φ⊗∇φ)−ρφ2∇θ+ρg,

where is the viscosity coefficient, is the pressure and denotes the external force density. By applying the curl operator to equation (2.3), we obtain

 ∇×˙vs=−∇×∇×vn−∇×[ρφ2vs].

A substitution into equation (2.7) leads to the generalizing Navier-Stokes equation

 ρ(1−φ2)˙vn+∇×∇×vn−ν∇(∇⋅vn)=−∇p−1κ2∇⋅(∇φ⊗∇φ)−ρφ2∇θ−∇×[ρφ2vs]+ρg.

We append to (2.7) the usual boundary condition:

 (2.8) vn|∂Ω=0.

## 3. Phase diagram

The differential equation of the phase variable is completed by the following constitutive choice:

 m=θ+λp+v2s−v2n,

 (3.1) τρ˙φ=1κ2∇⋅(ρ∇φ)−ρθλφ(φ2−1)−ρ(θ+λp+v2s−v2n)φ.

As pointed out in Sect.1, in homogeneous and steady conditions the material is in the normal state when , that is when

 (3.2) θ+λp+v2s−v2n>θλ

and the transition occurs when . Therefore the model is naturally able to account for the existence of a critical velocity (depending on the temperature and the pressure), above which superfluid properties disappear (see [16]).

The regions where is respectively grater and smaller than are the stability regions of the normal and superfluid phase and the curve represented by the equation separates such regions. In particular, if we consider the equilibrium states, i.e. then the curve is represented by the equation

 θ+λp=θλ,

which is a line with negative slope . This is good approximation of the line shown in the phase diagram of liquid helium and represented in Fig.1.

In a previous model for superfluidity ([6]) the coefficient was supposed to vanish. This corresponds to approximate the line with a vertical line. Moreover in such a model the normal component was assumed to be incompressible, so that

 ∇⋅vn=0.

In this paper, we consider a different case, by assuming that the normal component obeys the constraint

 (3.3) ∇⋅vn=λρφ˙φ

which generalizes incompressibility.

## 4. Heat equation and thermodynamics

In order to obtain the kinetic equation for the temperature, let us consider the first law of thermodynamics in the form ([8])

 (4.1) ρ˙E=Piφ+Pivs+ρh,

where is the total energy, are respectively the internal powers due to the order parameter and to the velocity and stands for the rate at which the heat is absorbed by the material. Hereafter we will consider some approximations of our model, valid in a neighborhood of the transition temperature. In particular, in the expression of the time derivative

 ˙φ=∂tφ+v⋅∇φ=∂tφ+vn⋅∇φ+φ2(vs−vn)⋅∇φ

we neglect last term, so that assumes the form

 ˙φ=∂tφ+vn⋅∇φ.

As a consequence

 (4.2) ∇˙φ=(∇φ)⋅+(∇vn)T∇φ,

where the superscript denotes the transpose of a tensor.

Multiplying equation (3.1) by and accounting for (3.3) and (4.2), we obtain the power balance related to , that is

 Piφ=Peφ

where the internal and external powers are given by

 (4.3) Piφ = ρddt[12κ2|∇φ|2+θλ(φ44−φ22)]+ρ(θ+v2s−v2n)φ˙φ −vn⋅∇p+τρ˙φ2+1κ2∇vn⋅(∇φ⊗∇φ), (4.4) Peφ = ∇⋅(1κ2ρ˙φ∇φ−pvn).

Similarly, by multiplying equation (2.3) by , we obtain

 |˙vs+∇ϕs−∇θ|2+vn⋅∇×˙vs+ρφ2vs⋅(˙vs+∇ϕs−∇θ) =−∇⋅(vn×˙vs+vn×∇ϕs−vn×∇θ).

We substitute the term with equation (2.7) and we take (2.4) into account. Thus, we obtain

 Pivs=Pevs

where the the internal and external powers due to the velocity are

 Pivs = |˙vs+∇ϕs−∇θ|2+12ρddt|vn|2−ρφ2vn⋅˙vn+ν|∇⋅vn|2+vn⋅∇p −1κ2∇vn⋅(∇φ⊗∇φ)+ρφ2vs⋅˙vs+τκ2ρφ2ϕ2s−ρφ2(vs−vn)⋅∇θ Pevs = −∇⋅[vn×˙vs+vn×∇ϕs−vn×∇θ−νvn∇⋅vn+1κ2(∇φ⊗∇φ)vn +ρφ2vsϕs]+ρg⋅vn.

It is worth noting that, as a consequence of the boundary conditions, after an integration on the domain the only contribution given by the external powers is due to the external source .

We assume that the total energy is written as

 (4.7) E=12κ2|∇φ|2+θλ(φ44−φ22)+e0(θ)+12φ2v2s+12(1−φ2)v2n,

where is a function depending only on the temperature. We identify the first three terms of (4.7) with the internal energy

 e=12κ2|∇φ|2+θλ(φ44−φ22)+e0(θ).

Last two terms of (4.7) involving the normal and the superfluid components of the velocity define the kinetic energy . Hence

 E=e+T.

Substitution of (4.3), (4) and (4.7) into (4.1) yields

 (4.8) ρh = ρe′0(θ)˙θ−ρθφ˙φ−τρ˙φ2−|˙vs+∇ϕs−∇θ|2 −ν|∇⋅vn|2−τκ2ρφ2ϕ2s+ρφ2(vs−vn)⋅∇θ.

The heat equation is given by

 (4.9) ρh=−∇⋅q+ρr,

where is the heat flux and is the heat supply. In this framework, the heat flux is assumed to satisfy the constitutive equation

 (4.10) q=−k0(θ)∇θ−ρφ2θ(vs−vn),

where denotes the thermal conductivity. Notice that, when the fluid is in the normal phase, i.e. , equation (4.10) reduces to the usual Fourier law. On the contrary, in the superfluid state, the superfluid component of the velocity is related to the heat flux inside the material.

By comparing (4.8) with (4.9), we obtain the evolution equation for the temperature, i.e.

 (4.11) ρe′0(θ)˙θ = ρθφ˙φ+τρ˙φ2+|˙vs+∇ϕs−∇θ|2+ν|∇⋅vn|2 +τκ2ρφ2ϕ2s+∇⋅[k0(θ)∇θ]+∇⋅[ρφ2(vs−vn)]θ+ρr.

Equation (4.11) is completed by the boundary condition

 q⋅n|∂Ω=0.

In view of the constitutive equation (4.10) and the boundary conditions (2.5) and (2.8), we conclude that

 (4.12) ∇θ⋅n|∂Ω=0.

Now we prove that our model is consistent with the second law of thermodynamics. We write the Clausius-Duhem inequality in the form

 ρ˙η≥−∇⋅(qθ)+ρrθ,

where is the entropy. We introduce the Helmholtz free energy density which is supposed to depend on . In view of (4.1) and (4.9), we deduce

 ρ˙Ψ+ρη˙θ+ρ˙T≤Piφ+Pivs−1θq⋅∇θ.

Accordingly, we have

 ρ(∂θΨ+η)˙θ+ρ[∂φΨ−θλφ(φ2−1)−θφ]˙φ+ρ[∂∇φΨ−12κ2∇φ]⋅∇˙φ −τρ˙φ2−|˙vs+∇ϕs−∇θ|2−ν|∇⋅vn|2−τκ2ρφ2ϕ2s−k0(θ)θ|∇θ|2≤0.

In virtue of the arbitrariness of , the functions are compatible with the second law of thermodynamics if assume non-negative values, and the free energy satisfies the following conditions:

 ∂θΨ=−η,∂φΨ=θλφ(φ2−1)+θφ,∂∇φΨ=12κ2∇φ.

Hence,

 (4.13)

where depends only on the temperature. From the relation it follows that

 Ψ′0(θ)=Ψ0(θ)−e0(θ).

Finally, we conclude this section by proving that the passage from the normal phase to the superfluid one is a second-order transition since no latent heat is involved. Indeed, from (4.13), it follows that the entropy assumes the form

 η=−∂θΨ=−12φ2−Ψ′0(θ)

and the latent heat is given by

 L=θ[η(θλ,φ0(θλ))−η(θλ,0)],

where is the minimum of the function and , characterize the pure phases (see Fig.2). Since when , we have that .

## 5. The differential system and gauge invariance

Collecting the equations of motion we write the system of equations:

 (5.1) τρ˙φ = 1κ2∇⋅(ρ∇φ)−ρθλφ(φ2−1)−ρ(θ+λp+v2s−v2n)φ. (5.2) ∇⋅[ρφ2vs] = −τκ2ρφ2ϕs (5.3) ˙vs = −∇ϕs−∇×vn−ρφ2vs+∇θ (5.4) ρ(1−φ2)˙vn = −∇×∇×vn+ν∇(∇⋅vn)−∇p−1κ2∇⋅(∇φ⊗∇φ) −ρφ2∇θ−∇×[ρφ2vs]+ρg. (5.5) ∇⋅vn = λρφ˙φ (5.6) ˙ρ = −ρ∇⋅[(1−φ2)vn+φ2vs] (5.7) ρe′0(θ)˙θ = ρθφ˙φ+τρ˙φ2+|˙vs+∇ϕs−∇θ|2+ν|∇⋅vn|2 +τκ2ρφ2ϕ2s+∇⋅[k0(θ)∇θ]+∇⋅[ρφ2(vs−vn)]θ+ρr,

in the unknowns .

As we have pointed in Section 2, the model we propose is similar to the Ginzburg-Landau model of superconductivity. In order to stress this analogy, following [2], we introduce the transformation:

 (φ,vs,ϕs)⟷(ψ,A,ϕ)

where

 φ=ψe−iχ,vs=A−1κ∇χ,ϕs=ϕ+1κ˙χ,

is an arbitrary scalar function and denotes the imaginary unit.

Our aim is to write equations (5.1)-(5.7) by means of the variables . Multiplying equation (5.1) by , we obtain

 (5.8) τρ˙ψ = (iτρφ˙χ+1κ2ρΔφ+1κ2∇ρ⋅∇φ)eiχ−ρψ[θλ(|ψ|2−1)+θ+λp−v2n] −ρψ(|A|2−2κA⋅∇χ+1κ2|∇χ|2).

Dividing (5.2) by and substituting the expressions of and , we deduce

 (5.9) τρφ˙χ = −1κφA⋅∇ρ+1κ2φ∇ρ⋅∇χ−2κρA⋅∇φ+2κ2ρ∇φ⋅∇χ −1κρφ∇⋅A+1κ2ρφΔχ−τκρφϕ.

We substitute equation (5.9) into (5.8). Moreover, the identities

 ∇ψ = (∇φ+iφ∇χ)eiχ, Δψ = (Δφ+2i∇φ⋅∇χ−φ|∇χ|2+iφΔχ)eiχ,

 τρ˙ψ = 1κ2ρΔψ+1κ2∇ψ⋅∇ρ−iκψA⋅∇ρ−2iκρA⋅∇ψ −iκρψ∇⋅A−iτκρψϕ−ρψ[θλ(|ψ|2−1)+θ+λp−v2n]−ρψ|A|2.

In addition, it is easy to prove the relations

 ˙vs+∇ϕs = ˙A+∇ϕ φ2vs = −|ψ|2A+i2κ(ψ∇ψ∗−ψ∗∇ψ) φ˙φ = 12(ψ˙ψ∗+ψ∗˙ψ) ˙φ2+κ2φ2ϕ2s = |˙ψ|2+κ2φ2ϕ2+2κϕφ2˙χ = |˙ψ|2+κ2|ψ|2ϕ2−iκϕ(˙ψψ∗−ψ˙ψ∗) ∇φ⊗∇φ = 14|ψ|2(ψ∗∇ψ+ψ∇ψ∗)⊗(ψ∗∇ψ+ψ∇ψ∗)

where denotes the conjugate of . Accordingly, the equations (5.1)-(5.7) transform into

 τρ˙ψ = 1κ2ρΔψ−iκψA⋅∇ρ+1κ2∇ψ⋅∇ρ−2iκρA⋅∇ψ −iκρψ∇⋅A−iτκρψϕ−ρψ[θλ(|ψ|2−1)+θ+λp−v2n]−ρψ|A|2 ˙A = −∇ϕ−∇×vn+|ψ|2A−i2κ(ψ∇ψ∗−ψ∗∇ψ)+∇θ ρ(1−|ψ|2)˙vn = −∇×∇×vn+ν∇(∇⋅vn)−∇p−14κ2∇⋅[1|ψ|2(ψ∗∇ψ+ψ∇ψ∗)⊗(ψ∗∇ψ+ψ∇ψ∗)] −ρ|ψ|2∇θ−∇×[−ρ|ψ|2A+i2κρ(ψ∇ψ∗−ψ∗∇ψ)]+ρg ∇⋅vn = λ2ρ(ψ˙ψ∗+ψ∗˙ψ) ˙ρ = −ρ∇⋅[(1−|ψ|2)vn−|ψ|2A+i2κ(ψ∇ψ∗−ψ∗∇ψ)] ρe′0(θ)˙θ = 12ρθ(ψ˙ψ∗+ψ∗˙ψ)+τρ|˙ψ|2+τρκ2|ψ|2ϕ2−iτκρϕ(˙ψψ∗−ψ˙ψ∗) +|˙A+∇ϕ−∇θ|2+ν|∇⋅vn|2+∇⋅[k0(θ)∇θ] +∇⋅[ρ|ψ|2vn−ρ|ψ|2A+i2κρ(ψ∇ψ∗−ψ∗∇ψ)]θ+ρr.

It is worth noting that the previous equations are independent of , which can be chosen arbitrarily. This formulation allows a more direct comparison with the Ginzburg-Landau model of superconductivity, where the choice of the variables is crucial in order to prove well-posedness results for the differential system.

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