1. Introduction
With a rich history spanning over a century, traditional cryogenic refrigeration technology, known as vapor compression, has grappled with persistent challenges, including energy inefficiency, spatial requirements, and the use of environmentally harmful chemicals [1]. Emerging technologies are on the brink of revolutionizing temperature control, particularly in the realm of heating and cooling, especially around room temperature. An alternative to conventional technology lies in caloric effects, offering the potential for diverse applications, from thermal sensors and cooling devices to precise temperature maintenance for a wide range of microchip-based electronic applications [2].
Caloric effects are defined as the reversible alteration of a material's temperature (ΔT, adiabatic temperature change) or entropy (ΔS, isothermal entropy change) in response to external stimuli. The family of caloric effects can be specifically categorized into four distinct types: electrocaloric effect (ECE), magnetocaloric effect (MCE), elastocaloric effect (eCE), and barocaloric effect (bCE). The ECE and MCE operate in a similar fashion, involving the alignment of electric dipoles or magnetic moments through the application of electric or magnetic fields [3, 4]. For eCE and bCE, mechanical force configurations are considered mechanocaloric effects, in which materials are subjected to uniaxial or triaxial stress, with the distinction in the latter case being a volume change in the unit cell accompanied by lattice distortion [5, 6].
In addition to the development of mono-caloric effects, the concept of multi-caloric effects emerged in 2010 through the work of Mañosa et al [7, 8]. This concept introduces the utilization of multiple types of stimuli sources, enabling a combination of external fields to be applied simultaneously or sequentially to a specific caloric material. Over time, the range of multi-caloric materials has expanded and developed, encompassing configurations that range from single-phase to composite materials, with a primary focus on mechanical, electric, or magnetic fields [9, 10]. Multi-caloric effects present a promising avenue for harnessing the manipulation of temperature and entropy within a material, potentially leading to more significant caloric effects.
Recent efforts have been dedicated to experimental frameworks that bring multi-caloric effects to the forefront. However, from a theoretical perspective, the emphasis has largely been on describing mono-caloric effects, lacking the necessary depth for a comprehensive understanding of multi-caloric effects induced by the application of external fields and their cross-synergistic responses [11, 12]. In this paper, we present a comprehensive thermodynamic theory of multi-caloric effects utilizing an exterior derivative approach. This approach is aimed at providing a generalized presentation of traditional Maxwell relations, applicable to both simple and complex systems featuring an arbitrary combination of stimuli sources. Importantly, the proposed presentation for a variety of multi-caloric phenomena is concise and elegant, highlighting its universality and conformity with general principles.
2. Maxwell relations
According to the first law of thermodynamics, the internal energy U has a generalized differential form1 ), it yields2 ), let's obtain3 ) can be expanded by the selected independent variables, which is written as6 ), the expansion of equation (5 ) is reorganized to combine its similar terms, obtaining the following three parts, which are denoted as I (equation (7a )), II (equation (8a )), and III (equation (9a )):5 ) is equal to zero, I + II + III = 0. That is, all the coefficients of I, II, and III at the same time must be zero, namely equations (7b )–(9b ):7a )–(9a ), it can be seen the number of the relational expression with a form such as equation (7b ) is $\tfrac{M{\rm{\cdot }}(M-1)}{2}$ and those for equations (8b ) and equation (9b ) are $M{\rm{\cdot }}(N-M)$ and $\tfrac{(N-M){\rm{\cdot }}(N-M-1)}{2},$ respectively. Finally, we have all the relations as follows
$\begin{eqnarray}{\rm{d}}U=\displaystyle \sum _{i=1}^{N}{Y}_{i}{\rm{\cdot }}{\rm{d}}{X}_{i}.\end{eqnarray}$
Xi and Yi are considered as a pair of conjugate physical quantities, which are generalized coordinates and generalized forces respectively, along with the suffix to be i = 1, 2, …, N. For a specific system, the internal energy allows N degrees of freedom that are not less than 2, that is, N ≥ 2. Operating an exterior derivative on equation ( $\begin{eqnarray}{\rm{dd}}U=\displaystyle \sum _{i=1}^{N}{\rm{d}}{Y}_{i}\wedge {\rm{d}}{X}_{i}=0.\end{eqnarray}$
Describing a thermodynamic system with N degrees of freedom, that is N of sets of conjugate physical quantities, it is selected as independent variables of the thermodynamic system to be M of generalized coordinates Xi (i = 1, 2, … , M) and N − M of generalized forces Yi (i = M + 1, M + 2, … , N), where M and N satisfy 0 ≤ M ≤ N. Following the above form, to rearrange the summation terms in equation ( $\begin{eqnarray}\displaystyle \sum _{i=1}^{M}{\rm{d}}{Y}_{i}\wedge {\rm{d}}{X}_{i}+\displaystyle \sum _{i=M+1}^{N}{\rm{d}}{Y}_{i}\wedge {\rm{d}}{X}_{i}=0.\end{eqnarray}$
Below, the set of thermodynamic independent variables is denoted for brevity as $\begin{eqnarray}{\mathscr{A}}\triangleq \left\{{X}_{1},{X}_{2},\cdots ,{X}_{M}{\rm{;}}{Y}_{M+1},{Y}_{M+2},\cdots ,{Y}_{N}\right\}.\end{eqnarray}$
The derivatives with respect to Xi and Yi in equation ( $\begin{eqnarray*}\displaystyle \sum _{i=1}^{M}\left[\displaystyle \sum _{j=1}^{M}{\left(\frac{\partial {Y}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}{\rm{d}}{X}_{j}+\displaystyle \sum _{j=M+1}^{N}{\left(\frac{\partial {Y}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}{\rm{d}}{Y}_{j}\right]\wedge {\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray*}+\,\displaystyle \sum _{i=M+1}^{N}{\rm{d}}{Y}_{i}\wedge \left[\displaystyle \sum _{j=1}^{M}{\left(\frac{\partial {X}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}{\rm{d}}{X}_{j}\right.\end{eqnarray*}$
$\begin{eqnarray}\left.+\,\displaystyle \sum _{j=M+1}^{N}{\left(\frac{\partial {X}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}{\rm{d}}{Y}_{j}\right]=0.\end{eqnarray}$
As has been known, the operation on exterior derivatives has a nature of anti-commutativity, which is satisfying $\begin{eqnarray}{\rm{d}}{X}_{i}\wedge {\rm{d}}{X}_{j}=\left\{\begin{array}{cc}-{\rm{d}}{X}_{j}\wedge {\rm{d}}{X}_{i} & \left(i\ne j\right)\\ 0 & \left(i=j\right)\end{array}\right..\end{eqnarray}$
Using equation ( $\begin{eqnarray*}I=\displaystyle \sum _{i=1}^{M}\displaystyle \sum _{\begin{array}{c}j=1\\ i\ne j\end{array}}^{M}{\left(\frac{\partial {Y}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}{\rm{d}}{X}_{j}\wedge {\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray}=\displaystyle \sum _{\begin{array}{c}i,j=1\\ i\lt j\end{array}}^{M}\left[{\left(\frac{\partial {Y}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}-{\left(\frac{\partial {Y}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}}\right]{\rm{d}}{X}_{j}\wedge {\rm{d}}{X}_{i},\end{eqnarray}$
$\begin{eqnarray*}{II}=\displaystyle \sum _{i=1}^{M}\displaystyle \sum _{j=M+1}^{N}{\left(\frac{\partial {Y}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}{\rm{d}}{Y}_{j}\wedge {\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray*}+\,\displaystyle \sum _{i=M+1}^{N}\displaystyle \sum _{j=1}^{M}{\left(\frac{\partial {X}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}{\rm{d}}{Y}_{i}\wedge {\rm{d}}{X}_{j}\end{eqnarray*}$
$\begin{eqnarray*}=\,\displaystyle \sum _{\begin{array}{c}1\leqslant i\leqslant M\\ M+1\leqslant j\leqslant N\end{array}}{\left(\frac{\partial {Y}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}{\rm{d}}{Y}_{j}\wedge {\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray*}+\,\displaystyle \sum _{\begin{array}{c}1\leqslant i\leqslant M\\ M+1\leqslant j\leqslant N\end{array}}{\left(\frac{\partial {X}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}}{\rm{d}}{Y}_{j}\wedge {\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray}=\displaystyle \sum _{\begin{array}{c}1\leqslant i\leqslant M\\ M+1\leqslant j\leqslant N\end{array}}\left[{\left(\frac{\partial {Y}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}+{\left(\frac{\partial {X}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}}\right]{\rm{d}}{Y}_{j}\wedge {\rm{d}}{X}_{i},\end{eqnarray}$
$\begin{eqnarray*}{III}=\displaystyle \sum _{i=M+1}^{N}\displaystyle \sum _{\begin{array}{c}j=M+1\\ i\ne j\end{array}}^{N}{\left(\frac{\partial {X}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}{\rm{d}}{Y}_{i}\wedge {\rm{d}}{Y}_{j}\end{eqnarray*}$
$\begin{eqnarray}=\displaystyle \sum _{\begin{array}{c}i,j=M+1\\ i\lt j\end{array}}^{N}\left[{\left(\frac{\partial {X}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}-{\left(\frac{\partial {X}_{j}}{\partial {Y}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{i}}\right]{\rm{d}}{Y}_{i}\wedge {\rm{d}}{Y}_{j}.\end{eqnarray}$
Also note that the right end of equation ( $\begin{eqnarray}{\left(\frac{\partial {Y}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}-{\left(\frac{\partial {Y}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}}=0,\end{eqnarray}$
$\begin{eqnarray}{\left(\frac{\partial {Y}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}+{\left(\frac{\partial {X}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}}=0,\end{eqnarray}$
$\begin{eqnarray}{\left(\frac{\partial {X}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}-{\left(\frac{\partial {X}_{j}}{\partial {Y}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{i}}=0.\end{eqnarray}$
According to the summation terms in equation ( $\begin{eqnarray}\left\{\begin{array}{cc}{\left(\frac{\partial {Y}_{i}}{\partial {X}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{j}}={\left(\frac{\partial {Y}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}} & \left({X}_{i}\ne {X}_{j}\right)\\ {\left(\frac{\partial {X}_{j}}{\partial {X}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{X}_{i}}=-{\left(\frac{\partial {Y}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}} & \left(i\ne j\right)\\ {\left(\frac{\partial {X}_{i}}{\partial {Y}_{j}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{j}}={\left(\frac{\partial {X}_{j}}{\partial {Y}_{i}}\right)}_{{\mathscr{A}}{\rm{\backslash }}{Y}_{i}} & \left({Y}_{i}\ne {Y}_{j}\right)\end{array}\right..\end{eqnarray}$
Up to this point, we successfully developed a generalized and universal presentation of multi-field thermodynamics, known as Maxwell relations, to describe the various multi-caloric phenomena by using the exterior derivative methods and theories. Note equation (10 ), which classifies Maxwell's relations into three categories, specifically in the fact that main independent variables serve as the denominators at both ends of the equations. For the first formula of equation (10 ), the main independent variables are generalized coordinate variables (i.e., Xi, Xj), and for its third formula, they are generalized force variables (i.e., Yi, Yj). While for the second formula of equation (10 ), it is both Xi and Yj that are the main independent variables. According to results of permutation and combination, there is a total number of $4{{\rm{C}}}_{{\rm{N}}}^{2}$ for all the Maxwell's relations in a specific system with N degrees of freedom. Examples and applications of equation (10 ) will be further discussed in this article.
3. Examples
3.1. Simple system
According to equation (10 ), here we apply to a thermodynamic system with 2 degrees of freedom, whose internal energy is
$\begin{eqnarray}{\rm{d}}U=\displaystyle \sum _{i=1}^{2}{Y}_{i}{\rm{\cdot }}{\rm{d}}{X}_{i}={Y}_{1}{\rm{d}}{X}_{1}+{Y}_{2}{\rm{d}}{X}_{2}.\end{eqnarray}$
When N = 2, there are two sets of conjugate quantities X1, Y1, and X2, Y2. There are 2 × 2 = 4 combined selections as for the main independent variables, which is ${\mathscr{A}}$ = {X1,X2} or {X1,Y2} or {Y1,X2} or {Y1,Y2}. Then, four of Maxwell's relations are as follows, corresponding to table 1(a).Table 1. (a). Maxwell's relations for a system with 2 degrees of freedom. (b) Maxwell's relations for a p−V system. |
| ${X}_{2}$ | ${Y}_{2}$ | |
|---|---|---|
| (a) | ||
| ${X}_{1}$ | ${\mathscr{A}}=\left\{{X}_{1},{X}_{2}\right\}$ | ${\mathscr{A}}=\left\{{X}_{1},{Y}_{2}\right\}$ |
| ${\left(\frac{\partial {Y}_{2}}{\partial {X}_{1}}\right)}_{{X}_{2}}={\left(\frac{\partial {Y}_{1}}{\partial {X}_{2}}\right)}_{{X}_{1}}$ | ${\left(\frac{\partial {X}_{2}}{\partial {X}_{1}}\right)}_{{Y}_{2}}=-{\left(\frac{\partial {Y}_{1}}{\partial {Y}_{2}}\right)}_{{X}_{1}}$ | |
| ${Y}_{1}$ | ${\mathscr{A}}=\left\{{Y}_{1},{X}_{2}\right\}$ | ${\mathscr{A}}=\left\{{Y}_{1},{Y}_{2}\right\}$ |
| ${\left(\frac{\partial {Y}_{2}}{\partial {Y}_{1}}\right)}_{{X}_{2}}=-{\left(\frac{\partial {X}_{1}}{\partial {X}_{2}}\right)}_{{Y}_{1}}$ | ${\left(\frac{\partial {X}_{2}}{\partial {Y}_{1}}\right)}_{{Y}_{2}}={\left(\frac{\partial {X}_{1}}{\partial {Y}_{2}}\right)}_{{Y}_{1}}$ | |
| (b) | ||
| $V$ | $p$ | |
| $S$ | ${\mathscr{A}}=\left\{S,V\right\}$ | ${\mathscr{A}}=\left\{S,p\right\}$ |
| ${\left(\frac{\partial p}{\partial S}\right)}_{V}=-{\left(\frac{\partial T}{\partial V}\right)}_{S}$ | ${\left(\frac{\partial V}{\partial S}\right)}_{p}={\left(\frac{\partial T}{\partial p}\right)}_{S}$ | |
| $T$ | ${\mathscr{A}}=\left\{T,V\right\}$ | ${\mathscr{A}}=\left\{T,p\right\}$ |
| ${\left(\frac{\partial p}{\partial T}\right)}_{V}={\left(\frac{\partial S}{\partial V}\right)}_{T}$ | ${\left(\frac{\partial V}{\partial T}\right)}_{p}=-{\left(\frac{\partial S}{\partial p}\right)}_{T}$ | |
The above system, in fact, is the simplest and most common p−V thermodynamic system, whose internal energy is
$\begin{eqnarray}{\rm{d}}U={Y}_{1}{\rm{d}}{X}_{1}+{Y}_{2}{\rm{d}}{X}_{2}=T{\rm{d}}S-p{\rm{d}}V.\end{eqnarray}$
That is, X1→S, Y1→T, X2→V, Y2→−p, thus being the most well-known Maxwell's relations. See table 1(b).3.2. Electromagnetic system
Equation (10 ) provides a general and universal presentation of Maxwell relations that can be generalized into complex systems, along with much more degrees of freedom. For example, when N = 4, the internal energy of a system involving electric fields and magnetic fields is10 ), it is easy to have all the Maxwell's relations, as listed in table 2.
$\begin{eqnarray}{\rm{d}}U=\displaystyle \sum _{i=1}^{4}{Y}_{i}{\rm{\cdot }}{\rm{d}}{X}_{i}=T{\rm{d}}S+\sigma {\rm{d}}\varepsilon +E{\rm{d}}P+{\mu }_{0}H{\rm{d}}M.\end{eqnarray}$
As for the system, there are four sets of conjugate quantities as well as $4{{\rm{C}}}_{4}^{2}$ = 24 combined selections of the main independent variables. Based on equation (Table 2. Maxwell's relations for the system by equation ( |
| ${X}_{2}\to \varepsilon $ | ${Y}_{2}\to \sigma $ | ${X}_{3}\to P$ | ${Y}_{3}\to E$ | ${X}_{4}\to M$ | ${Y}_{4}\to H$ | |
|---|---|---|---|---|---|---|
| ${X}_{1}$ $\downarrow \,$ | ${\mathscr{A}}=\left\{S,\varepsilon ,P{or}E,M{or}H\right\}$ | ${\mathscr{A}}=\left\{S,\sigma ,P{or}E,M{or}H\right\}$ | ${\mathscr{A}}=\left\{S,P,\mathrm{\varepsilon \; or\; \sigma },M{or}H\right\}$ | ${\mathscr{A}}=\left\{S,E,\mathrm{\varepsilon \; or\; \sigma },M{or}H\right\}$ | ${\mathscr{A}}=\left\{S,M,\mathrm{\varepsilon \; or\; \sigma },P{or}E\right\}$ | ${\mathscr{A}}=\left\{S,H,\mathrm{\varepsilon \; or\; \sigma },P{or}E\right\}$ |
| $S$ | ${\left(\frac{\partial \sigma }{\partial S}\right)}_{{\mathscr{A}}{\rm{\backslash }}S}={\left(\frac{\partial T}{\partial \varepsilon }\right)}_{{\mathscr{A}}{\rm{\backslash }}\varepsilon }$ | ${\left(\frac{\partial \varepsilon }{\partial S}\right)}_{{\mathscr{A}}{\rm{\backslash }}S}=-{\left(\frac{\partial T}{\partial \sigma }\right)}_{{\mathscr{A}}{\rm{\backslash }}\sigma }$ | ${\left(\frac{\partial E}{\partial S}\right)}_{{\mathscr{A}}{\rm{\backslash }}S}={\left(\frac{\partial T}{\partial P}\right)}_{{\mathscr{A}}{\rm{\backslash }}P}$ | ${\left(\frac{\partial P}{\partial S}\right)}_{{\mathscr{A}}{\rm{\backslash }}S}=-{\left(\frac{\partial T}{\partial E}\right)}_{{\mathscr{A}}{\rm{\backslash }}E}$ | ${\mu }_{0}{\left(\frac{\partial H}{\partial S}\right)}_{{\mathscr{A}}{\rm{\backslash }}S}={\left(\frac{\partial T}{\partial M}\right)}_{{\mathscr{A}}{\rm{\backslash }}M}$ | ${\left(\frac{\partial M}{\partial S}\right)}_{{\mathscr{A}}{\rm{\backslash }}S}=-\frac{1}{{\mu }_{0}}{\left(\frac{\partial T}{\partial H}\right)}_{{\mathscr{A}}{\rm{\backslash }}H}$ |
| ${Y}_{1}{\rm{\ }}\downarrow {\rm{\ }}$ | ${\mathscr{A}}=\left\{T,\varepsilon ,P{or}E,M{or}H\right\}$ | ${\mathscr{A}}=\left\{T,\sigma ,P{or}E,M{or}H\right\}$ | ${\mathscr{A}}=\left\{T,P,\mathrm{\varepsilon \; or\; \sigma },M{or}H\right\}$ | ${\mathscr{A}}=\left\{T,E,\mathrm{\varepsilon \; or\; \sigma },M{or}H\right\}$ | ${\mathscr{A}}=\left\{T,M,\mathrm{\varepsilon \; or\; \sigma },P{or}E\right\}$ | ${\mathscr{A}}=\left\{T,H,\mathrm{\varepsilon \; or\; \sigma },P{or}E\right\}$ |
| $T$ | ${\left(\frac{\partial \sigma }{\partial T}\right)}_{{\mathscr{A}}{\rm{\backslash }}T}=-{\left(\frac{\partial S}{\partial \varepsilon }\right)}_{{\mathscr{A}}{\rm{\backslash }}\varepsilon }$ | ${\left(\frac{\partial \varepsilon }{\partial T}\right)}_{{\mathscr{A}}{\rm{\backslash }}T}={\left(\frac{\partial S}{\partial \sigma }\right)}_{{\mathscr{A}}{\rm{\backslash }}\sigma }$ | ${\left(\frac{\partial E}{\partial T}\right)}_{{\mathscr{A}}{\rm{\backslash }}T}=-{\left(\frac{\partial S}{\partial P}\right)}_{{\mathscr{A}}{\rm{\backslash }}P}$ | ${\left(\frac{\partial P}{\partial T}\right)}_{{\mathscr{A}}{\rm{\backslash }}T}={\left(\frac{\partial S}{\partial E}\right)}_{{\mathscr{A}}{\rm{\backslash }}E}$ | ${\mu }_{0}{\left(\frac{\partial H}{\partial T}\right)}_{{\mathscr{A}}{\rm{\backslash }}T}=-{\left(\frac{\partial S}{\partial M}\right)}_{{\mathscr{A}}{\rm{\backslash }}M}$ | ${\left(\frac{\partial M}{\partial T}\right)}_{{\mathscr{A}}{\rm{\backslash }}T}=\frac{1}{{\mu }_{0}}{\left(\frac{\partial S}{\partial H}\right)}_{{\mathscr{A}}{\rm{\backslash }}H}$ |
| ${X}_{2}\,\downarrow $ | ${\mathscr{A}}=\left\{\varepsilon ,P,S{or}T,M{or}H\right\}$ | ${\mathscr{A}}=\left\{\varepsilon ,E,S{or}T,M{or}H\right\}$ | ${\mathscr{A}}=\left\{\varepsilon ,M,S{or}T,P{or}E\right\}$ | ${\mathscr{A}}=\left\{\varepsilon ,H,S{or}T,P{or}E\right\}$ | ||
| $\varepsilon $ | ${\left(\frac{\partial E}{\partial \varepsilon }\right)}_{{\mathscr{A}}{\rm{\backslash }}\varepsilon }={\left(\frac{\partial \sigma }{\partial P}\right)}_{{\mathscr{A}}{\rm{\backslash }}P}$ | ${\left(\frac{\partial P}{\partial \varepsilon }\right)}_{{\mathscr{A}}{\rm{\backslash }}\varepsilon }=-{\left(\frac{\partial \sigma }{\partial E}\right)}_{{\mathscr{A}}{\rm{\backslash }}E}$ | ${\mu }_{0}{\left(\frac{\partial H}{\partial \varepsilon }\right)}_{{\mathscr{A}}{\rm{\backslash }}\varepsilon }={\left(\frac{\partial \sigma }{\partial M}\right)}_{{\mathscr{A}}{\rm{\backslash }}M}$ | ${\left(\frac{\partial M}{\partial \varepsilon }\right)}_{{\mathscr{A}}{\rm{\backslash }}\varepsilon }=-\frac{1}{{\mu }_{0}}{\left(\frac{\partial \sigma }{\partial H}\right)}_{{\mathscr{A}}{\rm{\backslash }}H}$ | ||
| ${Y}_{2}$ $\downarrow \,$ | ${\mathscr{A}}=\left\{\sigma ,P,S{or}T,M{or}H\right\}$ | ${\mathscr{A}}=\left\{\sigma ,E,S{or}T,M{or}H\right\}$ | ${\mathscr{A}}=\left\{\sigma ,M,S{or}T,P{or}E\right\}$ | ${\mathscr{A}}=\left\{\sigma ,H,S{or}T,P{or}E\right\}$ | ||
| $\sigma $ | ${\left(\frac{\partial E}{\partial \sigma }\right)}_{{\mathscr{A}}{\rm{\backslash }}\sigma }=-{\left(\frac{\partial \varepsilon }{\partial P}\right)}_{{\mathscr{A}}{\rm{\backslash }}P}$ | ${\left(\frac{\partial P}{\partial \sigma }\right)}_{{\mathscr{A}}{\rm{\backslash }}\sigma }={\left(\frac{\partial \varepsilon }{\partial E}\right)}_{{\mathscr{A}}{\rm{\backslash }}E}$ | ${\mu }_{0}{\left(\frac{\partial H}{\partial \sigma }\right)}_{{\mathscr{A}}{\rm{\backslash }}\sigma }=-{\left(\frac{\partial \varepsilon }{\partial M}\right)}_{{\mathscr{A}}{\rm{\backslash }}M}$ | ${\left(\frac{\partial M}{\partial \sigma }\right)}_{{\mathscr{A}}{\rm{\backslash }}\sigma }=\frac{1}{{\mu }_{0}}{\left(\frac{\partial \varepsilon }{\partial H}\right)}_{{\mathscr{A}}{\rm{\backslash }}H}$ | ||
| ${X}_{3}\,\downarrow \,$ | ${\mathscr{A}}=\left\{P,M,S{or}T,\mathrm{\varepsilon \; or\; \sigma }\right\}$ | ${\mathscr{A}}=\left\{P,H,S{or}T,\mathrm{\varepsilon \; or\; \sigma }\right\}$ | ||||
| $P$ | ${\mu }_{0}{\left(\frac{\partial H}{\partial P}\right)}_{{\mathscr{A}}{\rm{\backslash }}P}={\left(\frac{\partial E}{\partial M}\right)}_{{\mathscr{A}}{\rm{\backslash }}M}$ | ${\left(\frac{\partial M}{\partial P}\right)}_{{\mathscr{A}}{\rm{\backslash }}P}=-\frac{1}{{\mu }_{0}}{\left(\frac{\partial E}{\partial H}\right)}_{{\mathscr{A}}{\rm{\backslash }}H}$ | ||||
| ${Y}_{3}$ $\downarrow \,$ | ${\mathscr{A}}=\left\{E,M,S{or}T,\mathrm{\varepsilon \; or\; \sigma }\right\}$ | ${\mathscr{A}}=\left\{E,H,S{or}T,\mathrm{\varepsilon \; or\; \sigma }\right\}$ | ||||
| $E$ | ${\mu }_{0}{\left(\frac{\partial H}{\partial E}\right)}_{{\mathscr{A}}{\rm{\backslash }}E}=-{\left(\frac{\partial P}{\partial M}\right)}_{{\mathscr{A}}{\rm{\backslash }}M}$ | ${\left(\frac{\partial M}{\partial E}\right)}_{{\mathscr{A}}{\rm{\backslash }}E}=\frac{1}{{\mu }_{0}}{\left(\frac{\partial P}{\partial H}\right)}_{{\mathscr{A}}{\rm{\backslash }}H}$ |
4. Isothermal entropy change
Herein, we will take advantage of Maxwell's relations in equation (10 ) to obtain the presentation of isothermal entropy change (ΔS)T (Part 4) and adiabatic temperature change (ΔT)S (Part 5). As known, temperature T and entropy S are a pair of conjugate physical quantities, and it is reasonable to assume that S is taken as a function of T:
$\begin{eqnarray}S=S\left(T{\rm{;}}{X}_{1},{X}_{2},\cdots ,{X}_{{M}_{1}}{\rm{;}}{Y}_{{M}_{1}+1},{Y}_{{M}_{1}+2},\cdots ,{Y}_{N-1}\right).\end{eqnarray}$
Let us study the differential form of entropy dS, expanded by independent variables $\begin{eqnarray*}{\rm{d}}S={\left(\frac{\partial S}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}T+\displaystyle \sum _{i=1}^{{M}_{1}}{\left(\frac{\partial S}{\partial {X}_{i}}\right)}_{T,{{\mathscr{A}}}_{1}{\rm{\backslash }}{X}_{i}}{\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray}+\,\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\left(\frac{\partial S}{\partial {Y}_{i}}\right)}_{T,{{\mathscr{A}}}_{1}{\rm{\backslash }}{Y}_{i}}{\rm{d}}{Y}_{i},\end{eqnarray}$
where $\begin{eqnarray}{{\mathscr{A}}}_{1}\triangleq \left\{{X}_{1},{X}_{2},\cdots ,{X}_{{M}_{1}}{\rm{;}}{Y}_{{M}_{1}+1},{Y}_{{M}_{1}+2},\cdots ,{Y}_{N-1}\right\}.\end{eqnarray}$
${{\mathscr{A}}}_{1}$ is shorthand for a system with N degrees of freedom, representing a set of selected free-state independent variables (equation (15 )), in addition to the conjugate quantities of T and S ($T\notin {{\mathscr{A}}}_{1},$ $S\notin {{\mathscr{A}}}_{1}$). Either way, it contains the elements involving generalized coordinates Xi (i = 1, 2, … , M1) and generalized forces Yi (i = M1 + 1, M1 + 2, … , N − 1), where 0 ≤ M1 ≤ N − 1. Calculating equation (14 ), its first term becomes zero during an isothermal process10 ), the second line), the second term of equation (14 ) turns out to be14 ) is16 ) and (17 ) to be substituted into equation (14 ) for integral operation, finally it will be 18 ) is the expression of the micro-change in entropy given by (ΔS)T under the condition of an isothermal process.
$\begin{eqnarray}{\rm{d}}T=0.\end{eqnarray}$
Referring to Maxwell's relations (equation ( $\begin{eqnarray}{\left(\frac{\partial S}{\partial {X}_{i}}\right)}_{T,{{\mathscr{A}}}_{1}{\rm{\backslash }}{X}_{i}}=-{\left(\frac{\partial {Y}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}.\end{eqnarray}$
Similarly, the third term of equation ( $\begin{eqnarray}{\left(\frac{\partial S}{\partial {Y}_{i}}\right)}_{T,{{\mathscr{A}}}_{1}{\rm{\backslash }}{Y}_{i}}={\left(\frac{\partial {X}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}.\end{eqnarray}$
Applying equations ( $\begin{eqnarray}{\left({\rm{\Delta }}S\right)}_{T}=-\displaystyle \sum _{i=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {Y}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{X}_{i}+\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\int }_{{y}_{i1}}^{{y}_{i2}}{\left(\frac{\partial {X}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{Y}_{i}.\end{eqnarray}$
Equation (5. Adiabatic temperature change
Next, let us further study temperature T as function of entropy S:20 ) becomes17a ) and (17b ), the latter two terms of equation (20 ) respectively are21 ) and (22 ) into equation (20 ), it will be28 ) is the expression of the micro-change in temperature given by (ΔT)S under the condition of an adiabatic process. It can be seen that, for the first term of which, the generalized coordinates Xi (i = 1, 2, … , M1) have a positive contribution to the total temperature change, while for the second term, the generalized forces Yi (i = M1 + 1, M1 + 2, … , N − 1) contribute opposite effects to it.
$\begin{eqnarray}T=T\left(S{\rm{;}}{X}_{1},{X}_{2},\cdots ,{X}_{{M}_{1}}{\rm{;}}{Y}_{{M}_{1}+1},{Y}_{{M}_{1}+2},\cdots ,{Y}_{N-1}\right).\end{eqnarray}$
dT can be expanded by the following, along with ${{\mathscr{A}}}_{1}$ as a shorthand $\begin{eqnarray*}{\rm{d}}T={\left(\frac{\partial T}{\partial S}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}S+\displaystyle \sum _{i=1}^{{M}_{1}}{\left(\frac{\partial T}{\partial {X}_{i}}\right)}_{S,{{\mathscr{A}}}_{1}{\rm{\backslash }}{X}_{i}}{\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray}+\,\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\left(\frac{\partial T}{\partial {Y}_{i}}\right)}_{S,{{\mathscr{A}}}_{1}{\rm{\backslash }}{Y}_{i}}{\rm{d}}{Y}_{i}.\end{eqnarray}$
During an adiabatic process, the first term of equation ( $\begin{eqnarray}{\rm{d}}S=0.\end{eqnarray}$
Using equations ( $\begin{eqnarray}{\left(\frac{\partial T}{\partial {X}_{i}}\right)}_{S,{{\mathscr{A}}}_{1}{\rm{\backslash }}{X}_{i}}=-{\left(\frac{\partial T}{\partial S}\right)}_{{{\mathscr{A}}}_{1}}{\left(\frac{\partial S}{\partial {X}_{i}}\right)}_{T,{{\mathscr{A}}}_{1}{\rm{\backslash }}{X}_{i}}=\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}{\left(\frac{\partial {Y}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}},\end{eqnarray}$
$\begin{eqnarray}{\left(\frac{\partial T}{\partial {Y}_{i}}\right)}_{S,{{\mathscr{A}}}_{1}{\rm{\backslash }}{Y}_{i}}=-{\left(\frac{\partial T}{\partial S}\right)}_{{{\mathscr{A}}}_{1}}{\left(\frac{\partial S}{\partial {Y}_{i}}\right)}_{T,{{\mathscr{A}}}_{1}{\rm{\backslash }}{Y}_{i}}=-\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}{\left(\frac{\partial {X}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}},\end{eqnarray}$
where including the properties of the reciprocal theorem and reciprocity theorem $\begin{eqnarray}{\left(\frac{\partial x}{\partial y}\right)}_{z}={\left[{\left(\frac{\partial y}{\partial x}\right)}_{z}\right]}^{-1},\end{eqnarray}$
$\begin{eqnarray}{\left(\frac{\partial x}{\partial y}\right)}_{z}{\left(\frac{\partial y}{\partial z}\right)}_{x}{\left(\frac{\partial z}{\partial x}\right)}_{y}=-1.\end{eqnarray}$
Also, the heat capacity ${C}_{{{\mathscr{A}}}_{1}},$ has been defined, which shows how much heat is required to change the temperature of the system by per unit through an endothermic reaction or an exothermic reaction $\begin{eqnarray}{C}_{{{\mathscr{A}}}_{1}}={\left(\frac{\partial Q}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}=T{\left(\frac{\partial S}{\partial T}\right)}_{{{\mathscr{A}}}_{1}},\end{eqnarray}$
in the case where all the variables of ${{\mathscr{A}}}_{1}$ are given at constants $\begin{eqnarray}{{\mathscr{A}}}_{1}=\left\{{X}_{1},{X}_{2},\cdots ,{X}_{{M}_{1}}{\rm{;}}{Y}_{{M}_{1}+1},{Y}_{{M}_{1}+2},\cdots ,{Y}_{N-1}\right\}={\rm{Const}}.\end{eqnarray}$
Generally, heat capacity ${C}_{{{\mathscr{A}}}_{1}}$ is a function of all the variables that are represented in ${{\mathscr{A}}}_{1}$ $\begin{eqnarray}{C}_{{{\mathscr{A}}}_{1}}={C}_{{{\mathscr{A}}}_{1}}\left({X}_{1},{X}_{2},\cdots ,{X}_{{M}_{1}}{\rm{;}}{Y}_{{M}_{1}+1},{Y}_{{M}_{1}+2},\cdots ,{X}_{N-1}\right).\end{eqnarray}$
More specifically, ${C}_{{{\mathscr{A}}}_{1}}$ has its simplest form if there is a constant value $\begin{eqnarray}{C}_{{{\mathscr{A}}}_{1}}={\rm{Const}}.\end{eqnarray}$
Substituting equations ( $\begin{eqnarray*}{\left({\rm{\Delta }}T\right)}_{S}=\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\left[\displaystyle \sum _{i=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {Y}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{X}_{i}\right.\end{eqnarray*}$
$\begin{eqnarray}\left.-\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\int }_{{y}_{i1}}^{{y}_{i2}}{\left(\frac{\partial {X}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{Y}_{i}\right].\end{eqnarray}$
Equation (6. Application: cooling to low temperatures
Equation (28 ) is an important result, describing the micro-change in temperature of a specific system, which occurs through different pathways (i.e., endothermic or exothermic) but also an adiabatic process. Below are a few examples of the results in different systems under the application of different force-field interactions.
6.1. Simple system
For an adiabatic process being subjected to a single force field, for example, the change is in the volume or pressure of the p−V system. In this case, there is exactly one of all elements in the set of ${{\mathscr{A}}}_{1},$ that is28 ) to calculate (ΔT)S at i = 1, a pair of conjugate quantities Xi and Yi are corresponding to31 ) is preceded by a negative sign, while equation (32 ) is a positive sign. As a result, it is through ‘adiabatic expansion' (V1 < V2) or ‘adiabatic decompression' (p1 > p2) that creates the cooling effects in the p−V system. Conversely, when V1 > V2 or p1 < p2, it heats up.
$\begin{eqnarray}{{\mathscr{A}}}_{1}=\left\{V\right\}{\rm{or}}{{\mathscr{A}}}_{1}=\left\{p\right\}.\end{eqnarray}$
Utilizing equation ( $\begin{eqnarray}{X}_{1}\to V,\,{Y}_{1}\to -p.\end{eqnarray}$
The results are obtained $\begin{eqnarray}{\left({\rm{\Delta }}T\right)}_{S}^{{\rm{expansion}}}={\int }_{{V}_{1}}^{{V}_{2}}{\left(\frac{\partial T}{\partial V}\right)}_{S}{\rm{d}}V=-\frac{T}{{C}_{V}}{\int }_{{V}_{1}}^{{V}_{2}}{\left(\frac{\partial p}{\partial T}\right)}_{V}{\rm{d}}V,\end{eqnarray}$
$\begin{eqnarray}{\left({\rm{\Delta }}T\right)}_{S}^{{\rm{decompression}}}={\int }_{{p}_{1}}^{{p}_{2}}{\left(\frac{\partial T}{\partial p}\right)}_{S}{\rm{d}}p=\frac{T}{{C}_{p}}{\int }_{{p}_{1}}^{{p}_{2}}{\left(\frac{\partial V}{\partial T}\right)}_{p}{\rm{d}}p,\end{eqnarray}$
where CV is the heat capacity at a constant volume and Cp is the heat capacity at a constant pressure. Particularly, note that equation (6.2. Electromagnetic system
Instead of mechanical interactions, the electric field and magnetic field are exploited to regulate the temperature of the system, while ignoring the deformation in the geometry of dielectric or magnetic materials. According to its correspondence relationship, specifically for the magnetic field parameters28 ), there is the effect of adiabatic demagnetization28 ), showing the effect of adiabatic demagnetization34 ) and (36 ), a picture of the temperature change based on experimental results can be further provided, by quantifying the confirmation of the correlation between the electromagnetic parameters (i.e., M and H, P and E). As a matter of fact, it is more applicable to equations (34b ) and (36b ) since the magnetic field intensity H and the electric field intensity E are readily quantifiable and controllable in practice.
$\begin{eqnarray}{X}_{i}\to M,\,{Y}_{i}\to {\mu }_{0}H,\end{eqnarray}$
where M is magnetization intensity and H is the magnetic field intensity. Applied to a single magnetic field by equation ( $\begin{eqnarray}\begin{array}{c}{\left({\rm{\Delta }}T\right)}_{S}^{{\rm{demagnetization}}}\\ =\,{\int }_{{M}_{1}}^{{M}_{2}}{\left(\frac{\partial T}{\partial M}\right)}_{S}{\rm{d}}M=\frac{{\mu }_{0}T}{{C}_{M}}{\int }_{{M}_{1}}^{{M}_{2}}{\left(\frac{\partial H}{\partial T}\right)}_{M}{\rm{d}}M,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{\left({\rm{\Delta }}T\right)}_{S}^{{\rm{demagnetization}}}\\ =\,{\int }_{{H}_{1}}^{{H}_{2}}{\left(\frac{\partial T}{\partial H}\right)}_{S}{\rm{d}}H=-\frac{{\mu }_{0}T}{{C}_{H}}{\int }_{{H}_{1}}^{{H}_{2}}{\left(\frac{\partial M}{\partial T}\right)}_{H}{\rm{d}}H.\end{array}\end{eqnarray}$
Similarly, the correspondence of the electric field parameters is $\begin{eqnarray}{X}_{i}\to P,\,{Y}_{i}\to E,\end{eqnarray}$
where P is polarization intensity, E is electric field intensity. Also by equation ( $\begin{eqnarray}\begin{array}{c}{\left({\rm{\Delta }}T\right)}_{S}^{{\rm{depolarization}}}\\ =\,{\int }_{{P}_{1}}^{{P}_{2}}{\left(\frac{\partial T}{\partial P}\right)}_{S}{\rm{d}}P=\frac{T}{{C}_{P}}{\int }_{{P}_{1}}^{{P}_{2}}{\left(\frac{\partial E}{\partial T}\right)}_{P}{\rm{d}}P,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{\left({\rm{\Delta }}T\right)}_{S}^{{\rm{depolarization}}}\\ =\,{\int }_{{E}_{1}}^{{E}_{2}}{\left(\frac{\partial T}{\partial E}\right)}_{S}{\rm{d}}E=-\frac{T}{{C}_{E}}{\int }_{{E}_{1}}^{{E}_{2}}{\left(\frac{\partial P}{\partial T}\right)}_{E}{\rm{d}}E.\end{array}\end{eqnarray}$
According to equations (7. Application: multi-caloric effects
Looking again at equation (28 ), the summation terms of which are separately written as38a ) (M1 = 1) and (38b) (M1 = 0), refering to examples in section 6 . For temperature effects resulting from multiple force fields without inter-field coupling (i > 1), as exemplified in scenarios involving single-phase materials, the changes in temperature are calculated independently, and it is crucial to discern whether each force field makes a positive or negative contribution to the comprehensive temperature effects.
$\begin{eqnarray}{\left({\rm{\Delta }}T\right)}_{S}=I+{II},\end{eqnarray}$
where $\begin{eqnarray}I=\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\displaystyle \sum _{i=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {Y}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{X}_{i},\end{eqnarray}$
$\begin{eqnarray}{II}=-\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\int }_{{y}_{i1}}^{{y}_{i2}}{\left(\frac{\partial {X}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{Y}_{i}.\end{eqnarray}$
For temperature effects of a single force field (i = 1), we employ equation (Advancing the investigation into the realm of multi-caloric effects necessitates consideration of the correlation among physical parameters responding to multiple force fields operating on multi-phase materials. Herein, we assume a linear dependence between a pair of non-conjugate quantities, specifically, that generalized coordinates and generalized force fields conforming to
$\begin{eqnarray}{Y}_{i}={\alpha }_{{ij}}{X}_{j}\,\mathrm{and}\,{X}_{i}={\beta }_{{ij}}{Y}_{j},\end{eqnarray}$
wherein αij and βij are symmetric tensors that, as functions of temperature, satisfy αij (T) = αji (T) and βij (T) = βji (T); they represent the generalized inverse matrix of each other: [αik] [βkj] = [Δij].According to the assumption of equation (39 ), concerning equation (38 ), it yields40 ) and (41 )
$\begin{eqnarray*}I=\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\displaystyle \sum _{i=1}^{{M}_{1}}\displaystyle \sum _{j=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {\alpha }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{X}_{j}{\rm{d}}{X}_{i}\end{eqnarray*}$
$\begin{eqnarray*}=\,\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\left[\Space{0ex}{5.5ex}{0ex}\displaystyle \sum _{i=j=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {\alpha }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{X}_{j}{\rm{d}}{X}_{i}\right.\end{eqnarray*}$
$\begin{eqnarray*}\left.+\,\displaystyle \sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {\alpha }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{X}_{j}{\rm{d}}{X}_{i}\right]\end{eqnarray*}$
$\begin{eqnarray*}=\,\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\left[\Space{0ex}{5.5ex}{0ex}\displaystyle \sum _{i=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {\alpha }_{{ii}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{X}_{i}{\rm{d}}{X}_{i}\right.\end{eqnarray*}$
$\begin{eqnarray*}\left.+\,\displaystyle \sum _{\begin{array}{c}i,j=1\\ i\lt j\end{array}}^{{M}_{1}}{\int }_{\left({x}_{i1},{x}_{j1}\right)}^{\left({x}_{i2},{x}_{j2}\right)}{\left(\frac{\partial {\alpha }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}\left({X}_{j}{\rm{d}}{X}_{i}+{X}_{i}{\rm{d}}{X}_{j}\right)\right]\end{eqnarray*}$
$\begin{eqnarray*}=\,\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\left[\Space{0ex}{5.5ex}{0ex}\displaystyle \sum _{i=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {\alpha }_{{ii}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left(\frac{{X}_{i}^{2}}{2}\right)\right.\end{eqnarray*}$
$\begin{eqnarray}\left.+\,\displaystyle \sum _{\begin{array}{c}i,j=1\\ i\lt j\end{array}}^{{M}_{1}}{\int }_{\left({x}_{i1},{x}_{j1}\right)}^{\left({x}_{i2},{x}_{j2}\right)}{\left(\frac{\partial {\alpha }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left({X}_{i}{X}_{j}\right)\right].\end{eqnarray}$
Similarly, it yields $\begin{eqnarray*}{II}=-\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\int }_{{y}_{i1}}^{{y}_{i2}}{\left(\frac{\partial {X}_{i}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}{Y}_{i}\end{eqnarray*}$
$\begin{eqnarray*}=\,-\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}\left[\Space{0ex}{5.5ex}{0ex}\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\int }_{{y}_{i1}}^{{y}_{i2}}{\left(\frac{\partial {\beta }_{{ii}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left(\frac{{Y}_{i}^{2}}{2}\right)\right.\end{eqnarray*}$
$\begin{eqnarray}\left.+\,\displaystyle \sum _{\begin{array}{c}i,j={M}_{1}+1\\ i\lt j\end{array}}^{N-1}{\int }_{\left({y}_{i1},{y}_{j1}\right)}^{\left({y}_{i2},{y}_{j2}\right)}{\left(\frac{\partial {\beta }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left({Y}_{i}{Y}_{j}\right)\right].\end{eqnarray}$
Finally, by combining equations ( $\begin{eqnarray*}{\left({\rm{\Delta }}T\right)}_{S}=I+{II}=\frac{T}{{C}_{{{\mathscr{A}}}_{1}}}{\rm{\cdot }}\left[\Space{0ex}{5.5ex}{0ex}\displaystyle \sum _{i=1}^{{M}_{1}}{\int }_{{x}_{i1}}^{{x}_{i2}}{\left(\frac{\partial {\alpha }_{{ii}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left(\frac{{X}_{i}^{2}}{2}\right)\right.\end{eqnarray*}$
$\begin{eqnarray*}-\,\displaystyle \sum _{i={M}_{1}+1}^{N-1}{\int }_{{y}_{i1}}^{{y}_{i2}}{\left(\frac{\partial {\beta }_{{ii}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left(\frac{{Y}_{i}^{2}}{2}\right)\end{eqnarray*}$
$\begin{eqnarray*}+\displaystyle \sum _{\begin{array}{c}i,j=1\\ i\lt j\end{array}}^{{M}_{1}}{\int }_{\left({x}_{i1},{x}_{j1}\right)}^{\left({x}_{i2},{x}_{j2}\right)}{\left(\frac{\partial {\alpha }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left({X}_{i}{X}_{j}\right)\end{eqnarray*}$
$\begin{eqnarray}\left.-\,\displaystyle \sum _{\begin{array}{c}i,j={M}_{1}+1\\ i\lt j\end{array}}^{N-1}{\int }_{\left({y}_{i1},{y}_{j1}\right)}^{\left({y}_{i2},{y}_{j2}\right)}{\left(\frac{\partial {\beta }_{{ij}}}{\partial T}\right)}_{{{\mathscr{A}}}_{1}}{\rm{d}}\left({Y}_{i}{Y}_{j}\right)\right].\end{eqnarray}$
Equation (42 ) shows a result of temperature effects, in response to a linear dependence between the composite materials and multiple force fields. It can be seen that, the initial two terms of equation (42 ) represent an independent contribution from each one of force fields, while the latter two terms of equation (42 ) represent a coupled consequence from a variety of force fields. What's more, the differential parameters denoted by generalized coordinates ${X}_{{\rm{i}}}$ have a positive contribution to temperature effects, whereas those of generalized force fields ${Y}_{{\rm{i}}}$ contribute oppositely, incorporating coupling effects. On the basis of equation (42 ), it is possible to know the diverse sources of multi-caloric effects and predict the temperature variations in multi-phase materials. This insight is aimed at guiding the design and development of thermodynamic systems with versatile responsive capabilities and optimized functional performances.
8. Conclusions
In this paper, we leverage the method and theory of exterior derivative forms to formulate Maxwell relations being characterized by generality and universality, a contribution not reported in previous literature. Upon this methodology, we present the formulations detailing isothermal entropy changes and adiabatic temperature changes for arbitrary combinations of conjugate physical quantities. As a matter of fact, this theoretical framework covers all aspects of thermodynamic systems comprehensively, demonstrating its applicability to a wide range of materials, including both single-phase and multi-phase materials, and its adaptability to different scenarios, whether in single or multiple force fields (such as electric fields and magnetic fields). Our further study allows us to identify the contributions from multi-caloric effects, facilitating a clear differentiation between independent and interdependent roles of force fields, ultimately predicting their performances of entropy changes and temperature changes within thermodynamic systems. The foregoing content of Maxwell relations provides insight into the fundamental thermodynamic behaviors, paving a way for creating specialized thermodynamic materials, and further investigation of novel thermodynamic mechanisms. The comprehensive endeavor contributes to the advancement of traditional understanding and holds significant implications for both theoretical and practical research.