1 Introduction
Theoretical electrochemistry aims to describe and predict behavior of chemically reacting systems of charged substances. The modeling methods vary according to the characteristic times, lengths and details of the observed electrochemical systems. This paper aims to develop a hierarchy of continuum models on different levels of description using the framework of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC)
[
1‐
4].
Let us first briefly recall GENERIC. Consider an isolated system described by state variables
\({\mathbf {x}}\). The state variables can be for instance position and momentum of a particle, field of probability density on phase space, fields of density and momentum density, electromagnetic fields, etc. Evolution of functionals
\(F({\mathbf {x}})\) of the state variables is then expressed as
$$\begin{aligned} {\dot{F}}({\mathbf {x}}) = \Big \{ F,E \Big \} + \left\langle \frac{\delta F}{\delta {\mathbf {x}}}, \frac{\delta \varXi }{\delta {\mathbf {x}}^*} \Bigg |_{{\mathbf {x}}^*=\frac{\delta S}{\delta {\mathbf {x}}}}\right\rangle , \end{aligned}$$
(1)
where the former term on the right-hand side stands for a Poisson bracket of the functional
F and the energy
E while the latter for scalar product of gradient of
F and gradient of the dissipation potential
\(\varXi \). Conjugate variables (derivatives of the entropy
S in the entropic representation) are denoted by
\({\mathbf {x}}^*\). The Poisson bracket is antisymmetric, which leads to automatic energy conservation, and satisfies Jacobi identity, which expresses intrinsic compatibility of the reversible evolution. The irreversible term yields a generalized gradient flow driven by gradient of entropy and ensures the second law of thermodynamics. Many successful models in non-equilibrium thermodynamics have been formulated in the GENERIC structure (
1), and many new thermodynamically consistent models have been obtained by seeking that structure; see, e.g.,
[
3,
4].
We shall start the present work by recalling the semi-direct product coupling of reversible fluid mechanics and electrodynamics in vacuum
[
5]. In other words, we let the electrodynamics be advected by fluid mechanics as in
[
4]. To go beyond, we shall add also the field of polarization density and a canonically coupled momentum of polarization. It is important to express the behavior of dipole moment of molecules in interaction with electromagnetic field and the overall motion. To this picture, we shall add magnetization (the famous Landau & Lifshitz model) advected by fluid mechanics. This way we shall build a hierarchy of levels of description with appropriate Poisson brackets expressing kinematics on the levels; see Fig.
1.
Subsequently, we shall introduce dissipation on the most detailed levels of description, we shall present reductions to less detailed (lower) levels down to the level of mechanical equilibrium. On this level, the evolution is governed by the generalized Poisson–Nernst–Planck equations. We believe that such a complete and geometric picture of continuum thermodynamics of matter coupled with electrodynamics (including polarization and magnetization) was missing in the literature.
1.1 Mathematical aspects, terminology and further remarks
A description in terms of fields describing physical quantities, e.g., mass density and magnetic field, is developed throughout the manuscript. A mathematically rigorous formulation of the Poisson brackets and the dissipation potentials for the fields would require deployment of an advanced functional analysis; see, e.g.,
[
6]. We opt-out from discussing the setting of the function spaces, convergence of the integrals and other mathematical aspects that are not pertinent to the discussed physics at this stage of the development.
Only the bulk equations are investigated in this manuscript, and all boundary terms are thus neglected in the calculations below. This means that the considered systems are assumed to be either boundary-less, isolated or (infinitely) large with a reasonable decay of the fields at infinity. This is a standard assumption in geometric field theories, see e.g.
[
3,
4,
6].
A level of description is given by a set of variables. If two sets of variables, belonging to the respective levels of description, can be ordered by inclusion, then we say that the level of description with smaller set of variables is less detailed level of description and vice versa.
This brings us to an another interpretation of the multiscale description not necessarily connected to different space-time scales. For instance, in the Grad hierarchy
[
7,
8], where kinetic moments are considered
[
9], the scale (or rather level of description) is given by the number of moments considered. The advantage of the latter meaning is that it is independent of the particular non-objective and observer-dependent spatiotemporal scale.
The term
reversible is used through out the article in connection with the structure of the evolution equations in the sense of the time-reversal transformation
[
10]. The Hamiltonian systems may often be cast in to the first-order, symmetric hyperbolic partial differential equation form
[
11]. Assuming smooth initial data, solutions of the hyperbolic systems may be constructed for short times using the Cauchy–Kowalevskaya theorem
[
12]. Such solutions are then short-time reversible in the usual sense of partial differential equations.
1.2 Notation \((\partial _t{\mathbf {x}})_\text {rev/irr}\) and \({\mathbf {x}}^\dagger \), \({\mathbf {x}}^*\)
A problem under consideration usually admits multiple equivalent choices of state variables. Let us denote the family of the admissible, equivalent, levels of description as
\({\mathbb {X}}\). For a given level of description, characterized by state variables
\({\mathbf {x}}\in {\mathbb {X}}\), the presented theory aims to describe evolution of functionals depending on
\({\mathbf {x}}\), i.e., we are interested in
\(\dot{F}({\mathbf {x}})\), see (
1). Therefore, functional derivatives
$$\begin{aligned} \left( \frac{\delta F}{\delta {\mathbf {x}}}\right) \equiv F_{\mathbf {x}}~ \end{aligned}$$
(2)
are often used in this manuscript.
The part of the evolution generated by the Poisson bracket
\(\{\cdot ,\cdot \}\) and the energy
\(E({\mathbf {x}})\), see (
1), will be called
Hamiltonian or
reversible and denoted as
$$\begin{aligned} \left\{ F,E\right\} = \left( \dot{F} \right) _\text {rev} = \left( \partial _t F\right) _\text {rev}. \end{aligned}$$
(3)
The functional derivative of the energy w.r.t. variable
\({\mathbf {x}}\), also called the
energy-conjugate to the variable
\({\mathbf {x}}\), will be briefly denoted by
\(\dagger \) in the superscript, i.e.,
\(E_{{\mathbf {x}}} = {\mathbf {x}}^\dagger \). Poisson brackets are usually conveniently expressed in the
energetic representation, i.e., when entropy density
s is amongst the state variables,
\((s, \varvec{\xi })\in {\mathbb {X}}\), where
\(\varvec{\xi }\) denotes the state variables other than the entropy density.
The part of the evolution generated by the dissipation potential
\(\varXi \) and the entropy
\(S({\mathbf {x}})\), see (
1), will be called
irreversible and denoted as
$$\begin{aligned} \left\langle F_{\mathbf {x}}, \varXi _{{\mathbf {x}}^*}|_{{\mathbf {x}}^*=S_{\mathbf {x}}}\right\rangle = \left( \dot{F} \right) _\text {irr} = \left( \partial _t F\right) _\text {irr}. \end{aligned}$$
(4)
The symbol
\({\mathbf {x}}^*\), called
entropy-conjugate, denotes the variables on which the dissipation potential
\(\varXi ({\mathbf {x}}^*)\) depends. The notation
\({\mathbf {x}}^*\) is going to be overloaded in Sect.
3 because it will be also used to denote the value of the entropy derivative, i.e.,
\({\mathbf {x}}^* = S_{\mathbf {x}}\). Since the irreversible evolution is expressed in terms of the derivatives of the dissipation potential w.r.t.
\({\mathbf {x}}^*\), presence of energy density
e amongst the variables makes it simple to control the energy conservation, i.e.,
\((e,\varvec{\xi })\in {\mathbb {X}}\). Such set of variables is called
entropic representation
[
13].
The relations between the energy conjugates in the energetic representation
\((s^\dagger , \varvec{\xi }^\dagger )\) and the entropy conjugate in the entropic representation
\((e^*, \varvec{\xi }^*)\) are explained in
[
4,
14].
The overall evolution of a functional
F is then composed from the reversible and irreversible parts,
$$\begin{aligned} \partial _t F = \left( \partial _t F\right) _\text {rev} + \left( \partial _t F\right) _\text {irr}. \end{aligned}$$
(5)
Symbols
\((\partial _t F)_\text {rev/irr}\) be interpreted as the reversible and irreversible (w.r.t TRT) parts of the right-hand side of the evolution equations, or as the reversible and irreversible vector fields; see, e.g.,
[
10].
4 Discussion
W. Dreyer, C. Guhlke and R. Müller in
[
67] published a comprehensive analysis of fluid mixtures coupled with electromagnetic fields, including polarization and magnetization which will be further referred as the DGM approach. Their treatment of surfaces, c.f.
[
68], as independent thermodynamic systems interacting with the bulk, being beyond the scope of the presented work, have elucidated many electrochemical problems using non-equilibrium thermodynamics, for instance a unified theory of the Helmholtz and Stern layers, a derivation of Butler–Volmer equations, or useful asymptotic techniques, see
[
67,
69‐
71,
71,
72]. Since our goal is in close relation to that works, let us compare the two approaches in detail.
An important conceptual difference between the two approaches lies in the treatment of the state variables, i.e., the levels of the description. In our approach, the level of description is always defined first by declaring the set of state variables. Energy and entropy of the system can then, in principle, depend on all the state variables. Each state variable has its evolution given by the GENERIC equation (
1), where the reversible and the irreversible parts of the evolution are separated. In contrast, the DGM approach develops a system of general balance equations which outlines the relations between the physical quantities and their fluxes and productions. The dependence of the fluxes and production upon the variables is determined using the entropy principle.
The DGM approach is formulated in an
inertial frame of reference for which holds that (i) mass center not subjected to external forces moves with a constant velocity and (ii) the Lorentz–Maxwell–aether (or material) relations
41 are valid. This is similar, although not equivalent, to our construction. The choice of the energy in DGM,
$$\begin{aligned} e^\text {D} = \frac{{\mathbf {u}}^2}{2\rho } +\frac{\varvec{{\mathcal {D}}}\cdot {\mathbf {E}}}{2} +\frac{{\mathbf {B}}\cdot {\mathbf {H}}}{2} +\epsilon ^\text {D} \end{aligned}$$
(117)
is analogous to the energy
42. Moreover, the formulation of the Maxwell equation in DGM is, in vacuum, formally equivalent to our approach.
In the non-vacuum case, the total charge is for both formulations given by
\({{\,\mathrm{div}\,}}\varvec{{\mathcal {D}}}\). The
\({{\,\mathrm{div}\,}}{\mathbf {P}}\) also bears the same meaning, i.e., the density of the bounded charge. The DGM evolution of the polarization
\({\mathbf {P}}\) is given as formal solution to the bound charge balance. It reads
$$\begin{aligned} \text{(DGM) }\quad \partial _t P^i ={}&-\varepsilon _{ijk} \partial _j M_k^\text {D} + v_i \partial _j P^j + J_i^\text {P}, \end{aligned}$$
(118)
where
\({\mathbf {M}}^\text {D}\) is the
Lorentz magnetization and
\(J^\text {P}\) denotes the non-convective part of the polarization current, cf. equation (
79f). The DGM formulation contains no balance equation concerning
\({\mathbf {M}}^\text {D}\), an evolution equation is later on found for
\(\widehat{{\mathbf {M}}}^\text {D} = {\mathbf {M}}^\text {D} + {\mathbf {v}}\times {\mathbf {P}}\) as a consequence of the closure. In contrast, the polarization bracket (
45) contains the polarization momentum
\(\varvec{\pi }\) as a variable and therefore implies its reversible evolution. The polarization momentum
\(\varvec{\pi }\) can be projected to a reduced variable
\(\varvec{{\mathcal {M}}} = {{\,\mathrm{curl}\,}}\varvec{\pi }\), would lead to the appearance of
\({{\,\mathrm{curl}\,}}\varvec{{\mathcal {M}}}^\dagger \) in the equation for polarization. This is perhaps the closest the two approaches can get in this respect.
In DGM, the balance equations for the densities of the partial masses
\(\rho _\alpha \), mass momentum
\({\mathbf {u}}\), total momentum
\({\mathbf {u}}+\varvec{{\mathcal {D}}}\times {\mathbf {B}}\), total energy
\(e^\text {D}\), internal energy
\(\epsilon ^\text {D}\), total charge
\(\rho _\text {f} - {{\,\mathrm{div}\,}}{\mathbf {P}}\), bound charge
\(-{{\,\mathrm{div}\,}}{\mathbf {P}}\), flux of the magnetic field, and the entropy
\(s^\text {D}\) are formulated, see
[
67], which is translated in our approach to the respective evolution equations. In particular, the total energy and the total momentum
\({\mathbf {u}}+ \varvec{{\mathcal {D}}}\times {\mathbf {B}}\) are, in the absence of external forces, conserved quantities in DGM as in our approach. Coupling between the charged fluid and electromagnetic field is given by the choice of Lorentz’s force
[
67, Eqn. 36a]:
$$\begin{aligned} k_i = \partial _j {\mathcal {D}}^j E_i + \varepsilon _{ijk} \left( v_j \partial _l {\mathcal {D}}_l + J^\text {F}_j + J^\text {P}_j\right) B^k \end{aligned}$$
(119)
as a source term in the mass momentum balance and by the choice of Joule heating
$$\begin{aligned} \pi = (v_i\partial _j {\mathcal {D}}^j + J_i^\text {F} + J_i^\text {P}) E_i \end{aligned}$$
(120)
as a source term in the internal energy balance, cf. Appendix
D. Symbols
\(J^\mathrm {F}\) denotes the non-convective part of the free charge current. This is similar to our formula (
60), at least in the absence of irreversible terms.
As it is shown in Appendix
D, the presented coupling between the motion of mass and the electromagnetic field can be formulated using a similar choice of Joule’s heating, see (
138), and restricts the form of the total momentum to (
142). The dissipation potentials for the irreversible, conductive, currents are formulated in terms of total momentum
\({\widehat{{\mathbf {m}}}}\), so that
\(\varvec{{\mathcal {D}}}^\dagger \) is the co-moving electric field. Hence, the Lorentz force acting upon the conductive currents can be found when the dissipation potentials are transformed into the variables with the mass momentum
\({\mathbf {u}}\).
The choice of Lorentz’s force (
119) tells that the electric field acts upon the total charge and the magnetic field upon convective current density,
\({\mathbf {J}}^\mathrm {F}\) and
\({\mathbf {J}}^\text {P}\) within the DGM theory. This is equivalent for the total charge and the conductive free charge
\({\mathbf {J}}^\mathrm {F}\). The difference appears for
\({\mathbf {J}}^\text {P}\), since it is not purely irreversible and contains reversible terms in the sense of the time-reversal transform.
Our evolution equation for polarization consists of reversible and irreversible parts. The former is given by Eq. (
80f), and the terms of the right-hand side containing velocity be rewritten as the Lie derivative of
\({\mathbf {P}}\),
\({\mathcal {L}}_{\mathbf {v}}{\mathbf {P}}\). Polarization is thus simply advected by the fluid (apart from interacting with the electromagnetic fields and apart from relaxation processes). On the other hand, polarization is not simply advected in DGM as the equation for
\({\mathbf {P}}\) contains other velocity-dependent terms therein. This is a difference between our approach and DGM.
Authors of DGM assumed that the entropy density
\(s^\text {D}\) depends only on a specific subset of the variables
$$\begin{aligned} s^\text {D}(\epsilon ^\text {D} + {\widehat{{\mathbf {M}}}}^\text {D}\cdot {\mathbf {B}}, \rho _\alpha , {\mathbf {P}}, {\widehat{{\mathbf {M}}}}^\text {D}) \implies \frac{1}{T^\text {D}}=\frac{\partial s^\text {D}}{\partial (\epsilon ^\text {D} + {\widehat{{\mathbf {M}}}}^\text {D}\cdot {\mathbf {B}})}~, \end{aligned}$$
(121)
thus obtaining a specific definition of the temperature
\(T^\text {D}\), see
[
67, Eqn. 48a, 49a]
4. This choice allowed them to find a reasonably simple closure of the equations using the entropy principle, c.f.
[
73]. On the other hand, temperature is in our approach defined as derivative of the internal energy density with respect to entropy as is usual
[
36].
Let us now focus on some features of the closure, especially, the non-convective flux of the bound charge that. They derived the following evolution of the polarization
\({\mathbf {P}}\) and magnetization
\({\widehat{{\mathbf {M}}}}^\text {D}\),
$$\begin{aligned} \partial _t P^i&= \underbrace{ -v_j\partial _j P^i + \frac{1}{2}P^j\left( \partial _j v_i - \partial _i v_j \right) + \frac{1}{\tau ^\text {P}}\left( T^\text {D}\frac{\partial s^\text {D}}{\partial P^i}+ E_i + \varepsilon _{ijk}v_j B^k \right) }_{={J}^\text {P}_i - \varepsilon _{ijk}\partial _j M_k^\text {D} + {v}_i\partial _j P^j}~, \end{aligned}$$
(122a)
$$\begin{aligned} \partial _t {{\widehat{M}}}_i^\text {D}&= -v_j\partial _j {{\widehat{M}}}_i^\text {D} + \frac{1}{2}\widehat{M}_j^\text {D}\left( \partial _j v_i - \partial _i v_j \right) + \frac{1}{\tau ^\text {M}} \left( T^\text {D}\frac{\partial s^\text {D}}{{{\widehat{M}}}_i^\text {D}} + B^i \right) ~, \end{aligned}$$
(122b)
respectively. The phenomenological coefficients
\(\tau ^\text {P}\) and
\(\tau ^\text {M}\) are relaxation times of polarization and magnetization, respectively. The relaxation of the polarization, i.e., the last bracket in (
122a), is equivalent to the dissipation derived using the DynMaxEnt principle given by potential
\(\varXi ^{{\mathbf {P}}}\), see (
87). The conductive flux of free charge
\({\mathbf {J}}^\text {F}\) is equivalent to the one generated by the dissipation potential
\(\varXi ^\text {D}\) in (
110c), which also holds for the mass diffusion fluxes.
When the relaxation time of polarization
\(\tau ^\text {P}\) can be considered large, the dissipative part in (
122a) can be neglected. Eventually, the reversible part of the bound charge evolution in a volume
V for the two theories read
$$\begin{aligned} \text{(DGM) }\quad \partial _t \int _V\partial _i P^i&= \int _{\partial V} \left( -v_j\partial _j P^i + \frac{1}{2}P^j\left( \partial _j v_i - \partial _i v_j\right) \right) \nu _i~, \end{aligned}$$
(123)
$$\begin{aligned} \text{(our) }\quad \partial _t \int _V \partial _i P^i&= \int _{\partial V} \left( \pi ^{\dagger i} - v^i\partial _j P^j \right) \nu _i. \end{aligned}$$
(124)
The net bound charge in a fixed volume
V can thus be reversibly changed by convection or by change of
\(\varvec{\pi }^\dagger \). In contrast to DGM, the bound charge is advected like a scalar field in our approach, which corresponds with that polarization is Lie-dragged in our approach.
Also the evolution of magnetization in DGM, Eq. (
122b), is different from our equation (
80h). In our case the evolution of magnetization caused by fluid motion is again just the Lie drag. Apart from that, there is a contribution from inertia of the magnetization itself, as in the Landau–Lifshitz model, which brings interaction with the magnetic field when energy depends on both magnetization and magnetic field.
Let us consider the volume and shear viscosity to be vanishing, the stress tensor presented in
[
67, Eqn 62] reads
5$$\begin{aligned} \text{(DGM) }\quad \sigma _{ij}&= p(T^\text {D}, \rho _\alpha , \widehat{{\mathbf {M}}}^\text {D}, {\mathbf {P}})\delta _{ij} +\left( {\widehat{M}}^\text {D}_k B_k + {\widehat{E}}_k P_k \right) \delta ^{ij}\nonumber \\&\quad +\frac{1}{2}\left( {\widehat{E}}_i P_j + P_i{\widehat{E}}_j \right) -\frac{1}{2}\left( {\widehat{M}}^\text {D}_i B_j + B_i{\widehat{M}}^\text {D}_j \right) \end{aligned}$$
(125)
here
\(\widehat{{\mathbf {E}}} = {\mathbf {E}}+ {\mathbf {v}}\times {\mathbf {B}}\). The dependence of the stress tensor
\(\sigma \) on the material properties linked to the polarization
\({\mathbf {P}}\) and magnetization
\({\widehat{{\mathbf {M}}}}^\text {D}\) is provided by the isotropic part
\(p\delta _{ij}\) the rest of the tensor is linear in
\({\mathbf {P}}\) and
\(\widehat{{\mathbf {M}}}^\text {D}\), independent of the choice of the energy. An energy weakly non-local in polarization or magnetization, see, e.g., (
62) and (
75), leads in the here presented treatment to a structurally different stress since a non-symmetric components can appear in the evolution equation for the mass momentum
\({\mathbf {u}}\); see (
66) or (
77). The stress in DGM is different from our formulas for stress, mainly in the off-diagonal part.
The Onsager-Casimir reciprocal relations
[
36,
74,
75] are one of the corner-stones of non-equilibrium thermodynamics. Roughly speaking, they say that state variables with the same parity are coupled through an operator symmetric with respect to the simultaneous transposition and time-reversal while variables with opposite parities are coupled through an antisymmetric operator. They are automatically satisfied within the GENERIC framework in a generalized sense (beyond near equilibrium)
[
3,
4,
10]. In
[
67] they seem to be satisfied as well, but parities of the state variables are determined from the power of seconds in the units of the variables instead of the time-reversal transformation, which is not invariant with respect to changing the physical units. Although the results seem to be all right, one should be careful in principle, perhaps using a precise definition of affine and vector spaces
[
76,
77] or the definition based on projections from more detailed levels
[
4].
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.