General fluxes and forces

4.7 General fluxes and forces

In the last section we discussed the 3D version of a flux of particles. Now we will generalize this for the flux of any species/property \(\Gamma\). The current density of such flux is

\begin{equation*} \vec{j}_\Gamma = \frac{1}{A} \frac{d \Gamma}{d t} = - D \vec{\nabla} \Gamma = L \vec{X} \quad , \label{j_Gamma} \end{equation*}

(4.60)

\(L\) can be taken as a phenomenological coefficient.
\(\vec{X}\) is a thermodynamic force driving the current density \(\vec{j}_\Gamma\).

Due to the ”principle of correspondence” each of these forces corresponds to its flux so that the product \(\vec{j}_\Gamma \vec{X}\) gives the entropy production (in [J / Ks])

\begin{equation*} \frac{dS}{dt} = \vec{j}_\Gamma \vec{X}\geq 0 \quad , \label{dSdT} \end{equation*}

(4.61)

The most relevant and typical pairs of fluxes and forces are summarized in the following table


	Heat transfer	Expansion	Diffusion	Chemical reaction	Electrical current

\(X\)	\(\Delta(1/T)\)	\(\Delta(p/T)\)	\(-\Delta(\mu_k/T)\)	\(A_r/T\)	\(U/T\)

\(j\)	\(\frac{\partial Q}{\partial t}\)	\(\frac{\partial V}{\partial t}\)	\(\frac{\partial n_k}{\partial t}\)	\(\frac{\partial \xi}{\partial t} = v\)	\(\frac{\partial q}{\partial t} = I\)

For the diffusion of many (real) systems not \(\Delta c\) (as for ideal systems) but \(\Delta \mu\) represents the driving force.
\(A_r = \frac{dG}{d\xi}\) is the chemical affinity

Eq. (4.60) is the starting point for the linear irreversible thermodynamics. Due to the linear approximation it is only a good approximation for irreversible changes close to equilibrium.
Extension 1: \(\vec{j}\) and \(\vec{X}\) do not need to be parallel, e.g. for the transport in anisotropic crystals; so in general \(L\) is a tensor.

Extension 2: Coupling of different forces and fluxes, e.g. diffusion driven by concentration and temperature gradient; so in general \(L\) is a high dimensional tensor coupling all forces to all fluxes:

\begin{equation*} \begin{split} j_1 & = L_{11} X_1 + L_{12} X_2 + \ldots + L_{1n} X_n\\ j_2 & = L_{21} X_1 + L_{22} X_2 + \ldots + L_{2n} X_n\\ \vdots \; & = \quad \vdots \qquad \qquad \vdots \qquad \qquad \qquad \vdots\\ j_n & = L_{n1} X_1 + L_{n2} X_2 + \ldots + L_{nn} X_n \end{split} \end{equation*}

(4.62)

Due to the Onsager theorem the tensor \(L\) of the phenomenological coefficients is always symmetric, i.e.

\begin{equation*} L_{ij} = L_{ji} \end{equation*}

(4.63)