Research Article A general theory for coupled chemo‑electro‑thermo‑mechanical heterogeneous system Zhen‑Bang Kuang1 Received: 7 April 2020 / Accepted: 4 November 2020 / Published online: 12 December 2020 © Springer Nature Switzerland AG 2020 Abstract Many transport and rate processes in chemical, physical, mineral, material and biological felds are controlled by the coupled chemo-electro-thermo-mechanical (CETM) process. Though many literatures discussed these coupled problems, but a unifed rigorous theory and a unifed method based on the chemical thermodynamics are lacked. In this paper on the basis of electrochemistry, the non-equilibrium thermodynamics and modern continuum mechanics we modify some previous theories and give a general theory including mass conservation equation, the electric charge conservation equation, complete energy conservation equation, entropy equation, evolution equations and the complete governing equations of these couple CETM systems. An extension of Nernst–Planck equation is derived for the CETM system. This theory gives a theoretical foundation and a universal method to improve and develop engineering theories, especially for the gradual failure components and cells. In appendix we also discuss the interdifusion problems in solids with vacancies shortly as a complement of the continuum difusion. Keywords Energy equation · Chemo-electro-thermo-mechanical systems · Entropy equation and entropy production rate · Gibbs equation · Evolution equation · Governing equation 1 Introduction are for electrically neutral system. However, in engineer- ing a system may be worked under electromagnetic feld, In chemical, physical, material and biological systems, which may be externally imposed or internally created, many transport and rate processes are controlled by the or both. In a thermoelectric material there exist Seebeck, coupled chemo-electro-thermo-mechanical interaction. Peltier and Thomson phenomena [3, 9]. Ionic and mixed We shall abbreviate a system with coupled Chemo-Elec- ionic–electronic devices, such as solid oxide fuel cells, oxy- tro-Thermo-Mechanical interaction as CETM system. Simi- gen pumps and hydrogen production, have gained many larly, a system with coupled Chemo-Thermo-Mechanical applications [10–13]. For expansive media including clays, interaction is a CTM system and a system with coupled shales, polymers gels, corneal endothelium, immature Chemo-Electro-Mechanical interaction is a CEM system. articular cartilage and connective biological tissues, elec- All these systems are complex thermodynamic systems. trochemical interaction are also typical [14–16]. In theories A general theory for CTM system has been discussed in of these chemo-electro-mechanical (CEM) system the tem- papers [1, 2], where a complete energy conservation equa- perature efects are neglected [9–16]. Especially when one tion, an appropriate entropy equation and the govern- discusses the gradual failure of components and cells the ing equation system have been given and modifed the temperature efects may be important. So integrating the previous theories [3–8]. Most of these previous theories efects of the electromagnetic felds into the CTM system * Zhen-Bang Kuang, [email protected] | 1School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China. SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 and integrating the efects of the temperature felds into to denote the serial number of the species and the serial the CEM system are necessary. Though in [3] the CETM number of the chemical reaction and the summation nota- system without chemical reaction had been discussed, tion is written in evidence, as shown in Eq. (1); but the sub- but its results are still left improvement. Therefore the scripts are used to components of a vector or tensor and CETM systems are worth to study. For a CETM system the the summation rule for the repeated indices is used. mass equation, energy equation, entropy equation and Let (k) , c(k) , v(k) , (jk) , �(jk)�̇ (j) and Θ̇ (k) be the partial den- momentum equation can be studied unitedly, but the sity or the apparent density, the mass fraction, the velocity, electric felds are produced due to various reasons and the reaction rate, the mass production rate per volume in jth every case should be researched independently. This situ- chemical reaction and in all chemical reactions of the species ation is analogous to the forces in the mechanical action. k respectively; v = u̇ and u are the barycentric velocity and The complete coupling efects of heat, difusion, chemi- the mechanical displacement vector of a representative ele- cal efect and electromagnetic feld are fully considered ment respectively, J(k) is the mass difusion fow of the species on a unifed thermodynamic foundation and a theoretical k . The mass conservation equation of the species k is [1–4]: frame of governing equations are given. An appropriate ⋅ d() �() complete governing equation system is the foundation for �ċ (k) = −∇ ⋅ J(k) + Θ̇ (k), () = = + v ⋅ ∇() solving engineering problems. The electrochemistry, the dt �t ��(k) non-equilibrium thermodynamics and modern continuum = −∇ ⋅ �(k)v(k) + Θ̇ (k); ̇� + �∇ ⋅ v = 0 mechanics allow us to construct an efcient theory for this �t N N coupling problems. �(k) c(k) = , � = �(k), c(k) = 1 We discuss an open system consisted of N (charged � k=1 k=1 (2) M or uncharged) species with total mass , total volume N N V and total density . According to the engineering cus- J(k) = �(k) v(k) − v , �v = �(k)v(k), J(k) = 0 tom we use the partial density or the mass concentration k=1 k=1 (k) (k) = M ∕V of a species k , rather than the molar con- L N (k) (k) (k) (k) (k) ̇ (k) (jk) ̇ (j) ̇ (k) centration C = ∕MM , where M and MM , are the Θ = � � , Θ = 0; k = 1, 2, ⋅⋅⋅, N total mass in V and the molecular weight of the species j=1 k=1 k respectively. where ∇ is the Eulerian gradient operator. Equation (2) shows that the mass fraction rate ċ , the mass fow J(k) and ̇ (k) 2 The mass and electric charge conservation the total mass production rate Θ of species k subject to equations the equation N N N 2.1 The mass conservation equation ċ (k) = 0, J(k) = 0, Θ̇ (k) = 0 (3) k=1 k=1 k=1 The mass conservation equation of the species k is the In the appendix, one will see that the interdifusion in same for CETM and CTM systems because the macroscopic noncontinuum solids with vacancies [21, 22] has diferent mass is independent to the electromagnetic feld. For easy mechanism with the above theory and discussed shortly to read the mass conservation equation given in literatures as the complement of the difusion phenomena in con- [1–4] is repeated here. tinuum media. Let a CETM system be consisted of N = N1 + N2 species N N M with 1 reactants and 2 products, the total mass , total 2.2 The electric charge conservation equation volume V , total density and chemical reaction number L . The jth chemical reaction equation can be written as [1, According to literature [3] one defne 2, 17–20] N N N I = (k)z(k)v(k) = I + i; I = zv, i = z(k)J(k) (jk)B(k) = 0, j = 1, 2 … , L 0 0 (1) k=1 k=1 k=1 (4) N N −1 (k) (k) (k) (k) where B(k) is the chemical formula of species k , (jk) is the z = z = c z k=1 k=1 stoichiometric constant of a species k in the chemical reac- tion j , which is positive if species k is a product and nega- where z(k) is the charge per mass of component k , z is the (m) th tive if k is a reactant. If B is not appear in the j chemical total charge per mass of the system, I is the total electric reaction, then (jm) = 0 . In this paper, we use superscript Vol:.(1234567890) SN Applied Sciences (2020) 2:2185 | https://doi.org/10.1007/s42452-020-03842-4 Research Article current density, I0 is the convective current density and i From Eq. (9) it is known that a part of electromagnetic is the conductive current density. Using Eq. (2) the electric work, zE ⋅ v − i ⋅ (v × B) , is changed to the kinetic energy. charge conservation equation is According to the Maxwell equation the balance law of electromagnetic energy [3, 23–26] is N N N (k) (k) ⋅ (k) ̇ (k) (k) ⋅ ̇ (k) (k) �ż = � ċ z = −∇ J + Θ z = −∇ i + Θ z Φ D B k=1 k=1 k=1 = E ⋅ + H ⋅ = −∇ ⋅ (E × H) − I ⋅ E (10) (5) t t t where Φ is the energy density stored in the electro-mag- netic feld, E × H is the Poynting vector of energy fow, I ⋅ E 3 The momentum and energy conservation is the work on matter supplied by the electro-magnetic equations in a CETM open system feld. In I ⋅ E the part i ⋅ E produced by conductive current i is changed to Joule heat [26] and constitutes a part of the 3.1 The momentum conservation equation internal energy, but the part I0 ⋅ E , produced by convective current I0 which moves with the Centroid of medium, is The electromagnetic force f (em) applied on a moving not related to internal energy or Joule heat, but is changed charge is given by Lorentz force law, so the momentum to the kinetic energy of the system (see Eq. 9). equation in CETM system is [23–26] The energy equation in a CETM open system for a vis- cous fuid is ∇ ⋅ + � f (m) + f (em) = �v̇ = �ü , U̇ + K̇ = Ẇ + Q̇ + �̇ + �̇ N (6) (em) (k) (k) (k) 1 ⋅ �f = � z E + v × B = �zE + I × B U̇ = �udV̇ , K̇ = � (v ⋅ v) dV k=1 V V 2 where (k)z(k) E + v(k) × B is the Lorentz force per volume Ẇ = � f (m) + f (em) ⋅ vdV + p ⋅ vda acting on species k , f (m) is the mechanical body force per V a volume, is the stress tensor of a representative element, Q̇ =− q ⋅ nda + (i ⋅ E)dV + −�̇ dV (11) E is electric feld intensity, B is the magnetic induction.