Is there a need for an extended phase definition for systems far from equilibrium?

3.1 First published definition

Some representative phase definitions for systems under equilibrium conditions found in the literature are discussed below. In his seminal work, Gibbs defines the term “phase” in relation to the thermodynamic equation of state, which reads [24].

This phase definition establishes a relationship between the concept of phase and thermodynamic state, and justifies the coexistence of phases at equilibrium. Therefore, the phase concept is only well defined at equilibrium, excluding any system outside equilibrium.

3.2 Some examples of highlighted definitions

In this case, the phase definition is centred on the separability of system parts. Let us consider a piece of clear plastic between two polarizers. When light passes through the material and the polarisers are homogeneous, it is observed that the system has no mechanically separable parts. Hence, a unique phase constitutes the system. Conversely, suppose a force (non-conservative variable) is applied to bend the plastic, and the bent piece is observed under polarised light. In that case, there are appreciable differences in the light pattern observed compared to the unaltered plastic piece. However, there is no possibility of separating any portions of the plastic mechanically by isolating sections with homogeneous light patterns. Neither is reversible for the bent plastic piece to “become part of another” phase. Hence, according to this phase definition, the bent plastic piece must be a single phase, which contradicts the previous reasoning. Since there is more than one phase in the bent plastic piece and such phases cannot be separated, this phase definition fails to describe this system.

This definition emphasises the importance of uniformity in determining the phases of a material. For instance, an unbent plastic sample mentioned in Definition 2, which exhibits uniformity in the polarised light pattern, would be considered a single phase. However, when the same plastic is bent, its optical properties become non-uniform due to the applied stress. Therefore, Definition 3 cannot describe this system under stress.

The key elements of this definition are homogeneity and defined boundaries. However, when we consider the analysis of the bent plastic piece presented in Definition 2, we see that homogeneity and defined boundaries are absent. Furthermore, the internal stress in the bent plastic piece relaxes slowly, leading to non-uniform pattern changes over time. Therefore, Definition 4 is inadequate to describe this system under stress. Similarly, when we look at the deformed plastic piece in the context of Definition 2, it is essential to note that a thermodynamic state is only well-defined at equilibrium. As a result, this definition fails to describe the piece mentioned above out of equilibrium.

This definition is closely related to an equation of state, like Gibbs’ definition. However, it is more detailed and explicitly describes the mathematical properties of this equation. It is important to note that this definition cannot describe out-of-equilibrium systems.

This phase definition accurately includes the analyticity of functions, resulting in a better and more complete definition of “phase” than those contained in the first five. However, other desirable mathematical requirements to reinforce the definition are absent, such as the convexity upward of the functions, ensuring thermodynamic stability. However, Definition 5 still considers the requirement of analyticity of functions. Since all these functions are defined at equilibrium, the phase coexistence or phase transitions of systems out of equilibrium continue to be ill-defined. Last phase definitions cannot be correctly applied to fluid systems subjected to shear flows out of equilibrium because of the transitions that non-conservative variables induce in such systems. With the definitions provided above, it is impossible to explain phase changes induced by a non-conservative variable, such as, for example, banded flow in complex systems subjected to shear [13], [15], [16], photo induced phase transitions in liquid crystals [29], [30], synthesis under external fields [31], among other cases with different driven external potentials.

At the present century many authors have reported phase transition beyond equilibrium, but they imply a phase definition similar to published recently by Dickman [18]:

And for phase coexistence Dickman [18] wrote:

“Now consider two systems in NESSs, subject to the same drive and external parameters, but with distinct, spatially uniform macroscopic properties. The systems represent coexisting phases if, when allowed to exchange energy or matter, the net flux of the quantity or quantities they may exchange is zero. Non equilibrium phase coexistence emerges spontaneously at phase separation, in which, varying some external parameter, a homogeneous phase becomes unstable, yielding a new stable steady state containing distinct macroscopic phases separated by a sharp interface. (The coexisting phases are clearly free to exchange particles and energy in this situation.)”

This definition could be applied to non-equilibrium systems and, although recently published, is analogous to definition 2, since it emphasises the importance of uniformity and reproducibility to determine the phases of a material, while it differs from Definitions 1 to 6 because the system must be under the influence of a variable that deviates from equilibrium. However, Definition 7 does not comment at all on the mathematical properties that it must fulfill.

5.1 Extended phase definition

5.2 Entropy production rate

The total differential of the function σ given in the proposed extended phase definition seems as

where P _i represents an extensive conservative variable, Q _i a conjugate conservative variable, J _α a flux, and X _α a driving thermodynamic force.

If the flux is a continuous function of the driving thermodynamic forces, the relationship between non-conservative variables and non-equilibrium fluxes can be expanded as a series using the Taylor’s theorem around the equilibrium (X _β  = 0) to obtain

where L α β ≡ ∂ J α / ∂ X β 0 and h α β X 1 , … , X ν = 1 2 ∑ δ ∫ 0 1 1 − t ∂ 2 J α ∂ X β ∂ X δ t X 1 , … , t X ν X δ d t with lim X 1 , … , X ν → 0 , … , 0 h α β X 1 , … , X ν = 0 . Notice that L _αβ and h _αβ may be also functions of the conjugate variables. The entropy source under non-equilibrium conditions reads.

here, the coefficients h α , β X 1 , … , X ν may not be positive defined, but always the entropy inequality σ ^s  > 0, therefore the matrix L ̂ , whose elements are L ̂ α , β = L α β + h α β X 1 , … , X ν , must be positive definite.

The latest phase definition should evolve towards a more comprehensive and accurate definition of the phase concept that encompasses systems that are not in a state of equilibrium.

Consider a faraway equilibrium process in a volume V at rest subjected to time-independent constraints at its surface. The total entropy S produced inside a system during a procedure taking place in volume V subjected to time-independent constraints can be written as

Considering that the thermodynamic forces X _α take the form of gradients of intensive variables X _α  = ∇Γ _α , the previous equation becomes

Taking the time derivative, using the Leibnitz rule (differentiation under the integral sign), and after rearranging, the equation for the rate of production of entropy follows

Therefore, the stability properties depend on the behaviour of this equation that further depends on the dynamical behaviour of the intensive variables.

6.1 Flow of complex fluids

A relatively simple theoretical model of complex fluids that qualitatively captures the key experimental observations has been proposed by Bautista et al. [14], [35]. It was derived using the extended irreversible thermodynamics formalism, that used extended space of parameters, here the stress tensor is the non-conserved variable. The Helmholtz free energy is a homogeneous function of first degree for all parameters, and follow all mathematical properties of the definition given in Section 5.1. This model predicts rather well steady states and transient nonlinear features of complex fluids that exhibit shear-banding flow and the shear-thinning shear-thickening transitions. The authors proposed a stress constitutive equation coupled with an evolution equation of fluidity and a constitutive equation for the mass flux. The fluidity is interpreted as a macroscopic measure of the internal structure of the fluid. The extended thermodynamic potential in this case is the extended Helmholtz free energy, given by

with the working variables: entropy (s), temperature (T), molar volume (v), hydrostatic pressure (P), elastic modulus at high frequencies (G ₀), the fluidity (φ), and the stress tensor ( σ ̲ ̲ ) , which is composed of shear stresses (σ ₁₂, σ ₁₃, and σ ₂₃), and normal stresses (σ _ii ). This potential is convex upward analytic in the conserved variables T, p, and in the non conserved ones, such as σ ̲ ̲ .

From dA, the evolution equations of the model are obtained [14]

Here, in the context of the BMP model, φ is the fluidity, while φ ₀ and φ _∞ are the fluidities at zero and high shear rates, respectively. λ is the relaxation time of the structure, k ₀ is a kinetic constant, ɛ is the transition shear thinning–thickening parameter, ϑ and μ are the shear-banding intensity parameters in the direction of gradient and in the vorticity direction. σ ̲ ̲ ▿ = d σ ̲ ̲ d t − L ̲ ̲ ⋅ σ ̲ ̲ + σ ̲ ̲ ⋅ L ̲ ̲ T is the convective derivative, L ̲ ̲ is the velocity gradient tensor, and D ̲ ̲ is the symmetric part of the rate of strain tensor.

Figure 1 depicts shear flow curves generated with the BMP model for a 20 wt% Pluronics P103/water system [16]. The curves plotted in a γ ̇ − σ − T graph resemble a P − V − T 3D diagram for equilibrium phase transitions. Near the non-equilibrium critical point, the orthogonal projection of the γ ̇ -T plane may be divided into two regions by a continuous curve, which we shall call the “phase boundary” or coexistence curve. For temperatures lower than the critical temperature T _c, there is an heterogeneous flow or two-phase region where bands at high ( γ ̇ C 1 ) and low shear rates ( γ ̇ C 2 ) coexist at steady-state and γ ̇ , which denotes the average shear rate of the fluid. Above T _c lies the homogeneous region. The phase boundary T γ ̇ is a convex upward analytic function of γ ̇ around the critical point except, perhaps, at γ ̇ = γ ̇ c , the unique point where T γ ̇ achieves its maximum. In the heterogeneous region, the free energy is the sum of free energies of the two bands, plus corrections due to surfaces separating the two phases and at the container walls. When the container is large, or there are a few bands or both, the surface free energy is negligible. Some models account for this contribution necessarily but imply non-linear terms.

Figure 1:

Scheme of a 3D rheological diagram for systems out of equilibrium (adapted from [16]).

Furthermore, in the orthogonal projection of the γ ̇ -σ plane, the steady-state of the two-phase region is obtained by a double tangent construction similar to Maxwell’s equal areas construction. For stress lower than the non-equilibrium critical point, the γ ̇ -σ plane may be divided into two regions by a continuous curve called the bands’ coexistence curve, which is a “phase boundary.” For shear stress lower than σ _c , there is an heterogeneous flow where bands at γ ̇ C 1 and γ ̇ C 2 coexist at steady-state, and γ ̇ denotes the average shear rate of the fluid. Above this σ lies the homogeneous region. Also, in the heterogeneous region, the free energy is the sum of free energies of the two bands, plus corrections due to surfaces separating the two phases and at the container walls. The phase boundary σ γ ̇ is a convex upward analytic function of γ ̇ around the critical point except, perhaps, at γ ̇ = γ ̇ c , the unique point where σ γ ̇ achieves its maximum.

It is noteworthy that the equilibrium thermodynamic properties of a system are entirely determined by the Helmholtz free energy per unit volume A ρ , T as a function of the density ρ and the temperature T, or another thermodynamic potential. As stated before, the necessary and sufficient condition to ensure their validity near the critical point is that the thermodynamic functions are analytical even near the critical point, as stated before. Additionally, they must satisfy the conditions of stability (convexity) [36], [37]. Hence, for non-equilibrium systems to comply with the above requirements, an extended free energy should be used since the non-conservative variables must be included. Note that in the biphasic region (heterogeneous), A ρ , T will be the sum of the free energies of the liquid and vapour phases present, plus the energies due to the surfaces that separate the two phases and on the walls of the container. If the container is not too small, the free energies on the surface are negligible. In addition, by applying the phase rule for non-equilibrium systems at steady state, Eq. (17), for two coexistence phases (π = 2) it is obtained that the number of degrees of freedom is three (f = 3), which is interpreted as an area in the non-equilibrium phase diagram in agree with experimental data. Since the number of species k is two (pluronics and water) and the number of driving forces is in this case one (the shear stress ν = 1). The degrees of freedom are associated with the temperature, the shear stress and the fraction of pluronics in the solution, that is constant and equal to 20 %wt for the particular case depicted in Figure 1. Each of the two coexistence phases are differentiated by a low shear rate ( γ ̇ C 1 ) and a high shear rate ( γ ̇ C 2 ) , respectively, that produce the shear-banding flow. On this case, Eqs. (15) describe the plateau for the same shear stress with two different shear rates, i.e. σ 12 γ ̇ C 1 = σ 12 γ ̇ C 2 = σ plateau .

6.2 Electrical system

In this case, the extended space of parameters is A , P , N i ; I ext , here the electric current, I _ext , is the non-conserved variable and its conjugated variable is the voltage, V. Also, the Helmholtz free energy

is a homogeneous function of first degree for all parameters, and follow all mathematical properties of definition (Section 5.1)

The organic salt θ-(BEDT-TTF)₂CsCo(SCN)₄ is a material composed of conductive layers of BEDT-TTF (bis(ethylenedithio) – Tetrathiafulvalene) and insulating layers of CsCo(SCN)₄ stacked alternatively along an axis. It presents a triangular BEDT-TTF network, where there is one hole for every two molecules, at low temperatures, due to Coulomb repulsion. Hence, charges in the system induce order. It exhibits a giant nonlinear resistance suggesting a coexistence of non-equilibrium phases, namely, a competition between different charge-induced order patterns [38]. Endo et al. [39] studied the electric field driven by a non-equilibrium phase transition and its critical phenomena, which explains this behaviour in the conductor θ-(BEDT-TTF)₂CsCo(SCN)₄ [38]. The conductivity of this conductor changes discontinuously when the electric field acting on the system varies continuously, that is, the sudden change in resistance evidences a phase transition driven by an electric current. Here, the non-conservative variable is the electric current. Endo et al. [39] found that the system undergoes the non-equilibrium phase transition due to the non-linear J-E characteristics. The authors produced a three-dimensional phase diagram for the non-equilibrium phase transition, examined critical phenomena, and calculated critical exponents of the experimental system and found them to agree with the Landau theory.

Figure 2 depicts experimental data of voltage against external current as a function of temperature for the organic salt θ-(BEDT-TTF)₂CsCo(SCN)₄. For this organic salt, the curves plotted in a graph, also resemble a P–V-T 3D diagram for equilibrium phase transitions, exhibiting a non-equilibrium critical point. Near this point an orthogonal projection of the T-V plane may be divided into two regions by a continuous curve, which we shall call the “phase coexistence line.” For temperatures lower than the critical temperature T _c ≈ 6.5 K, a two-phase region where a phase of high electric resistance becomes a phase of low electric resistance and vice verse. In the projection of the I–V plane, the current is multivalued, i., e., three values of current correspond a voltage value. Above T _c lies the homogeneous region. It is noteworthy that the equilibrium thermodynamic properties of a system are entirely determined by the Helmholtz free energy per unit volume as a function of the voltage, and current density and the temperature T, or another thermodynamic potential. By applying the phase rule for non-equilibrium systems, Eq. (17), here the number of species k is one (organic salt θ-(BEDT-TTF)₂CsCo(SCN)₄, the electric current is the unique driving force, so ν = 1; also there is a phase of low electric resistance coexisting with a high electric one, and the constant two refers to temperature and pressure. Hence, the number of degrees of freedom is two, interpreted as an area in the non-equilibrium phase diagram that ends at a critical point.

Figure 2:

Experimental data of voltage against external current as a function of temperature for the organic salt θ-(BEDT-TTF)₂CsCo(SCN)₄ [38].