Introduction

The particle mobility \({M}_{i}\) and diffusivity \({D}_{i}\) of mobile species \(i\) such as atoms, molecules, ions, and electrons are two of the most familiar and important transport parameters in materials science and engineering. They are related by the well-known Einstein relation,

$$\frac{{D}_{i}}{{k}_{\text{B}}T}={M}_{i},$$
(1)

where \({k}_{\text{B}}\) is the Boltzmann constant, and \(T\) is temperature in Kelvin. We would like to point out that the Einstein relation is only valid for the transport of idealized particle systems, for example, noninteracting particles, ideal solutions, or chemical species that obey Raoult’s or Henry’s Law in a solution.

Chemical capacitance1 is a much less known thermodynamic quantity. It will be shown that the factor (1/\({k}_{\text{B}}T\)) in the Einstein relation is actually a chemical capacitance. The main objectives of this article are to:

  • provide a general thermodynamic definition of capacitance, including chemical capacitance;

  • use chemical capacitance to establish a simple but general relation between the chemical mobility and diffusivity of mobile species in a material; this relation can be considered as a generalized version of the Einstein relation between mobility and diffusivity and is applicable to any mobile species, regardless of whether they are noninteracting or mutually interacting; and

  • introduce a new concept of chemical stiffness. Its relation to chemical capacitance is analogous to the relation between mechanical stiffness (or bulk modulus) and mechanical compliance (or compressibility).

Capacitance or capacity

Capacitance or capacity can be generally defined as the change in the amount of a transportable substance/energy per unit change of its corresponding potential.2 Several examples of transportable substances/energies and their corresponding potentials are provided in Table I.

Table I Examples of mobile species/properties and their corresponding potentials and capacitances,2–3 where V is volume and p is pressure.

The most straightforward of these is the transport of internal energy or enthalpy, where the corresponding potential is the temperature of the system. In this case, the capacitance is the well-known heat capacity, defined as simply the change of either internal energy (for isochoric systems) or enthalpy (for isobaric systems) per unit change in temperature. A corresponding entropy capacity can also be defined, as shown in Table I.

Another common example from Table I is the transport of electrical charge governed by the corresponding change in electrical potential, that is, the voltage. The electrical capacitance is simply the change in electrical charge per unit change in electrical potential.

Chemical capacitance

Total chemical capacitance

Let us now turn our attention to chemical capacitance. Following the nomenclature in Table I, we consider a system where \({N}_{i}\) represents the number (or moles) of particles of a chemical species \(i\) and \({\upmu }_{i}\) represents its corresponding chemical potential. The total chemical capacitance, \({C}_{{N}_{i}}\), of the species \(i\) in a system is defined as the change in the amount of that chemical species, \({dN}_{i}\), per unit change in chemical potential, \(d{\upmu }_{i}\), of the same species:1

$$\mathrm{Chemical \,capacitance}=\frac{\mathrm{Change\, in\, amount \,of \,chemical\, species } \, i}{\mathrm{Change \,in\, chemical \,potential \,of \,species}\,i}.$$
(2)

Mathematically, this can be expressed as:

$${C}_{{N}_{i}}={\left(\frac{\partial {N}_{i}}{\partial {\upmu }_{i}}\right)}_{T, p\,\mathrm{ or\, V},{N}_{j\ne i}}=\frac{1}{{\left(\frac{{\partial }^{2}F}{\partial {N}_{i}^{2}}\right)}_{T, {\text{V}},{N}_{j\ne i}}}=\frac{1}{{\left(\frac{{\partial }^{2}G}{\partial {N}_{i}^{2}}\right)}_{T, p,{N}_{j\ne i}}},$$
(3)

where \(F\) is the Helmholtz free energy and \(G\) is the Gibbs free energy of the system. The subscripts in Equation 3 represent the thermodynamic quantities to be kept constant during the changes, and \({N}_{j\ne i}\) indicates that the number of moles of all components \(j\) except component \(i\) are kept constant.

It is often more convenient to employ a constant pressure condition for the transport of atoms, ions, and molecules, whereas for electrons it is often more convenient to consider constant volume, since the electron energy states are typically computed under isochoric conditions. It should be noted that the chemical capacitance for a single-component system is undefined under the condition of constant temperature and pressure due to the restriction imposed by the Gibbs–Duhem relation.2

Chemical capacitance density or specific chemical capacitance

To introduce chemical capacitance density, we use \({n}_{i}\) to represent the concentration or number density of species \(i\), that is, the number of particles of chemical species \(i\) per unit volume. The chemical capacitance density, \({c}_{{n}_{i}}\), of a chemical species \(i\) in a material is then defined as the change in the concentration of chemical species \(i\), \({dn}_{i}\), per unit of change in chemical potential, \(d{\upmu }_{i}\), of that species:

$${c}_{{n}_{i}}={\left(\frac{\partial {n}_{i}}{\partial {\upmu }_{i}}\right)}_{T,{n}_{j\ne i} }.$$
(4)

Relative chemical capacitance

The relative chemical capacitance, \({c}_{{n}_{i}}^{r}\), of a chemical species \(i\) in a material is defined as the relative change in the concentration of chemical species \(i\), \({dn}_{i}/{n}_{i}\), per unit of change in chemical potential, \(d{\upmu }_{i}\), of that species:

$${c}_{{n}_{i}}^{r}=\frac{1}{{n}_{i}}{\left(\frac{\partial {n}_{i}}{\partial {\upmu }_{i}}\right)}_{T,{n}_{j\ne i}}={\left[\frac{\partial {\text{ln}}\left({n}_{i}\right)}{\partial {\upmu }_{i}}\right]}_{T, {n}_{j\ne i}}.$$
(5)

Chemical stiffness

The chemical stiffness \({s}_{{n}_{i}}\) is defined as simply the inverse of the specific chemical capacitance. For example,

$${s}_{{n}_{i}}=\frac{1}{{c}_{{n}_{i}}}={\left(\frac{\partial {\upmu }_{i}}{\partial {n}_{i}}\right)}_{T,{n}_{j\ne i}}.$$
(6)

Likewise, the relative chemical stiffness \({s}_{{n}_{i}}^{r}\) is defined as the inverse of the relative chemical capacitance:

$${s}_{{n}_{i}}^{r}=\frac{1}{{c}_{{n}_{i}}^{r}}={\left(\frac{\partial {\upmu }_{i}}{\partial {\text{ln}}\left({n}_{i}\right)}\right)}_{T, {n}_{j\ne i}}={n}_{i}{\left(\frac{{\partial }^{2}f}{\partial {n}_{i}^{2}}\right)}_{T,{n}_{j\ne i}}={n}_{i}{\left(\frac{{\partial }^{2}g}{\partial {n}_{i}^{2}}\right)}_{T, p,{n}_{j\ne i}},$$
(7)

where \(f\) is the Helmholtz free energy density (i.e., the Helmholtz free energy per unit volume) and \(g\) is the Gibbs free energy density (i.e., the Gibbs free energy per unit volume).

Chemical diffusivity

Following Fick’s First Law, chemical diffusivity \({D}_{i}\) is the proportionality constant relating the negative of the chemical concentration gradient \(\nabla {n}_{i}\) of species \(i\) and the flux density \({\overrightarrow{J}}_{i}\) of the corresponding chemical species \(i\),

$${\overrightarrow{J}}_{i}=-{D}_{i}\nabla {n}_{i}.$$
(8)

Chemical mobility

Chemical mobility \({M}_{i}\) is the proportionality constant linearly relating the particle velocity \({v}_{i}\) of species \(i\) and the force that the particle is subjected to \({F}_{i}=-\nabla {\upmu }_{i}\):

$${v}_{i}={M}_{i}{F}_{i}=-{M}_{i}\nabla {\upmu }_{i},$$
(9)

where \(\nabla {\upmu }_{i}\) is the chemical potential gradient of species \(i\).

Relationship among chemical mobility, diffusivity, and capacitance

To relate the chemical mobility to the chemical diffusivity, let us rewrite the chemical concentration gradient \(\nabla {n}_{i}\) in terms of chemical potential gradient \(\nabla {\upmu }_{i}\),

$${\overrightarrow{J}}_{i}=-{D}_{i}\nabla {n}_{i}=-{D}_{i}\frac{\partial {n}_{i}}{\partial {\upmu }_{i}}\nabla {\upmu }_{i}.$$
(10)

We can then relate the particle flux density to the drift velocity \({v}_{i}\) and then to the chemical potential gradient as

$${\overrightarrow{J}}_{i}={n}_{i}{v}_{i}={n}_{i}{M}_{i}{F}_{i}=-{n}_{i}{M}_{i}\nabla {\upmu }_{i}.$$
(11)

Comparing Equations 10 and 11, we arrive at a simple, but completely general expression relating the chemical diffusivity (\({D}_{i}\)) to the chemical mobility (\({M}_{i}\)) through the relative chemical capacitance (\({c}_{{n}_{i}}^{r}\)),

$${D}_{i}\frac{\partial {n}_{i}}{\partial {\upmu }_{i}}={n}_{i}{M}_{i}$$
(12)

or

$$\mathrm{Diffusivity }\,{D}_{i}=\frac{{M}_{i}}{\frac{\partial {\text{ln}}\left({n}_{i}\right)}{\partial {\upmu }_{i}}}=\frac{\mathrm{Mobility }\,{M}_{i}}{\mathrm{Chemical \,Capacitance }\,{c}_{{n}_{i}}^{r}}.$$
(13)

Similarly, we can relate the chemical diffusivity and mobility through the relative chemical stiffness,

$$\mathrm{Diffusivity }\,{D}_{i}={M}_{i}\frac{\partial {\upmu }_{i}}{\partial {\text{ln}}\left({n}_{i}\right)}=\mathrm{Mobility }\,{M}_{i} \times \mathrm{Stiffness }\,{s}_{{n}_{i}}^{r}.$$
(14)

Thus, the purely thermodynamic quantities of either chemical capacitance or chemical stiffness provide a completely general link between the purely kinetic properties of diffusivity and mobility.

Examples

Chemical diffusion of species \(\mathbf{i}\)

The chemical potential of species \(i\) in a multicomponent solution can be expressed in terms of its activity coefficient \({\upgamma }_{i}\) and mole fraction \({x}_{i}\),

$${\upmu }_{i}={\upmu }_{i}^{o}\left({n}_{i}^{o}\right)+{k}_{\text{B}}T{\text{ln}}\left({{\upgamma }_{i}x}_{i}\right),$$
(15)

where \({n}_{i}^{o}\) is the number density of pure chemical species \(i\).

The chemical capacitance and stiffness are given by

$${c}_{{n}_{i}}^{r}=\frac{1}{{\left[\frac{\partial {\upmu }_{i}}{\partial {\text{ln}}\left({n}_{i}\right)}\right]}_{T,p,{n}_{j\ne i}}}=\frac{1}{{{k}_{\text{B}}T\left[\frac{\partial {\text{ln}}\left({\upgamma }_{i}\right)}{\partial {\text{ln}}\left({n}_{i}\right)}+1\right]}}=\frac{1}{{k}_{\text{B}}T\left[\frac{\partial {\text{ln}}\left({\upgamma }_{i}\right)}{\partial {\text{ln}}\left({x}_{i}\right)}+1\right]},$$
(16)
$${s}_{{n}_{i}}^{r}=\frac{1}{{c}_{{n}_{i}}^{r}}={\left[\frac{\partial {\upmu }_{i}}{\partial {\text{ln}}\left({n}_{i}\right)}\right]}_{T,p,{n}_{j\ne i}}={k}_{\text{B}}T\left[\frac{\partial {\text{ln}}\left({\upgamma }_{i}\right)}{\partial {\text{ln}}\left({x}_{i}\right)}+1\right],$$
(17)

which is directly related to the familiar thermodynamic factor \(\uppsi\) in chemical diffusion,

$$\uppsi =\left[\frac{\partial {\text{ln}}\left({\upgamma }_{i}\right)}{\partial {\text{ln}}\left({x}_{i}\right)}+1\right],$$
(18)
$${c}_{{n}_{i}}^{r}=\frac{1}{{k}_{\text{B}}T\uppsi },$$
(19)
$${s}_{{n}_{i}}^{r}={k}_{\text{B}}T\uppsi .$$
(20)

Therefore, the chemical diffusion coefficient is related to the chemical mobility via

$${D}_{i}=\frac{{M}_{i}}{{c}_{{n}_{i}}^{r}}={M}_{i}{s}_{{n}_{i}}^{r}={M}_{i}{k}_{\text{B}}T\uppsi ={M}_{i}{k}_{\text{B}}T\left[\frac{\partial {\text{ln}}\left({\upgamma }_{i}\right)}{\partial {\text{ln}}\left({x}_{i}\right)}+1\right].$$
(21)

The simplest example involves a chemical species \(i\), which obeys Henry’s Law or Raoult’s Law,

$$\left[\frac{\partial {\text{ln}}\left({\upgamma }_{i}\right)}{\partial {\text{ln}}\left({x}_{i}\right)}+1\right]=1,$$
(22)

and we arrive at the familiar Einstein relation between chemical diffusivity and mobility,

$${D}_{i}={M}_{i}{k}_{\text{B}}T,$$
(23)

where \({k}_{\text{B}}\) is Boltzmann’s constant.

Let us consider another relatively simple example. For a binary A–B regular solution with a constant regular solution parameter \(\upalpha\),

$${k}_{\text{B}}T{\text{ln}}\left({\upgamma }_{A}\right)=\upalpha {x}_{B}^{2},$$
(24)

where \({\upgamma }_{A}\) is the activity coefficient of component \(A\), and \({x}_{B}\) is the mole fraction of component \(B\). For a binary regular solution, the chemical capacitance of the two components is the same and given by

$${c}_{{n}_{A}}^{r}=\frac{1}{{k}_{\text{B}}T\left[\frac{\partial {\text{ln}}\left({\upgamma }_{A}\right)}{\partial {\text{ln}}\left({x}_{A}\right)}+1\right]}=\frac{1}{\left[-2\upalpha {x}_{A}{x}_{B}+{k}_{\text{B}}T\right]}={c}_{{n}_{B}}^{r}.$$
(25)

The corresponding chemical stiffness is

$${s}_{{n}_{A}}^{r}={s}_{{n}_{B}}^{r}=\frac{1}{{c}_{{n}_{A}}^{r}}=-2\upalpha {x}_{A}{x}_{B}+{k}_{\text{B}}T.$$
(26)

The relation between intrinsic diffusivity and mobility for a binary regular solution is then

$${D}_{A}=\frac{{M}_{A}}{{c}_{{n}_{A}}^{r}}={M}_{A}{s}_{{n}_{A}}^{r}={M}_{A}{k}_{\text{B}}T\uppsi ={M}_{A}\left[-2\upalpha {x}_{A}{x}_{B}+{k}_{\text{B}}T\right],$$
(27)
$${D}_{B}=\frac{{M}_{B}}{{c}_{{n}_{B}}^{r}}={M}_{B}{s}_{{n}_{B}}^{r}={M}_{B}{k}_{\text{B}}T\uppsi ={M}_{B}\left[-2\upalpha {x}_{A}{x}_{B}+{k}_{\text{B}}T\right].$$
(28)

One can easily see that if \(\upalpha>0\), \(-2\upalpha {x}_{A}{x}_{B}+{k}_{\text{B}}T=0\) defines the spinodal of a binary system, at which the chemical capacitance is zero, and the chemical stiffness is infinite or undefined. When \(-2\upalpha {x}_{A}{x}_{B}+{k}_{\text{B}}T<0\), a homogeneous binary solution is unstable, and the chemical capacitance, chemical stiffness, and the intrinsic diffusivity of each component, as well as the chemical diffusivity, all become negative.

Electron transport in a nondegenerate semiconductor

$${\upmu }_{n}={E}_{c}+{k}_{\text{B}}T{\text{ln}}\left(\frac{n}{{E}_{c}}\right).$$
(29)

The chemical stiffness is given by

$${s}_{n}^{r}=\frac{1}{{c}_{n}^{r}}={\left[\frac{\partial {\upmu }_{n}}{\partial {\text{ln}}\left(n\right)}\right]}_{T,p,{n}_{j\ne i}}={k}_{\text{B}}T.$$
(30)

Therefore,

$${D}_{n}={M}_{n}{s}_{n}^{r}={M}_{n}{k}_{\text{B}}T.$$
(31)

Note that in most semiconductor literature, the symbol \({\upmu }_{e}\) is used to represent the electrical mobility. Here, we reserve \(\upmu\) for chemical potential and therefore represent electrical mobility by \({M}_{e}\). Electrical mobility is related to particle mobility \({M}_{n}\) by the equation \({M}_{n} = {M}_{e}/e\) (where \(e\) is the amount of elementary charge):

$${D}_{n}={M}_{n}{k}_{\text{B}}T=\frac{{M}_{e}{k}_{\text{B}}T}{e}.$$
(32)

Analogies of chemical transport to thermal and electric conduction

To briefly discuss the generality of relating the mobility and diffusivity through a capacitance, we examine the familiar heat conduction and electric conduction.

Heat conduction

Based on linear kinetics, the heat flux density \({\overrightarrow{J}}_{Q}\), which, under the constant volume condition, is the same as the internal energy flux density \({\overrightarrow{J}}_{u}\), can be written as

$${\overrightarrow{J}}_{Q}={\overrightarrow{J}}_{u}=-{D}_{u}\nabla u=-{D}_{u}\frac{\partial u}{\partial T}\nabla T=-{D}_{u}{c}_{v}\nabla T,$$
(33)

where \({D}_{u}\) is thermal diffusivity, \(u\) is internal energy density, \({c}_{v}\) is constant volume heat capacity per unit volume, and \(\nabla T\) is temperature gradient.

The heat flux density can also be written in terms of heat conductivity,

$${\overrightarrow{J}}_{Q}=-k\nabla T,$$
(34)

where \(k\) is the thermal conductivity.

By comparing Equations 33 and 34, one can easily see that the thermal diffusivity \({D}_{u}\) is related to the thermal conductivity \(k\) through the heat capacity \({c}_{v}\), similar to the chemical capacitance connecting the chemical diffusivity and the chemical mobility,

$${D}_{u}=\frac{k}{{c}_{v}}.$$
(35)

Similarly, under constant pressure, the heat or enthalpy diffusivity is related to the heat conductivity through the constant pressure heat capacity,

$${D}_{h}=\frac{k}{{c}_{p}}.$$
(36)

Electric conduction

Similar to heat conduction and chemical diffusion, the electric current density \(\overrightarrow{I}\) can be written as

$$\overrightarrow{I}=-{D}_{q}\nabla \uprho =-{D}_{q}\frac{\partial \uprho }{\partial \upphi }\nabla \upphi =-{D}_{q}{c}_{q}\nabla \upphi ,$$
(37)

where \({D}_{q}\) is charge diffusivity, \(\uprho\) is charge density, \(\nabla \upphi\) is the electric potential gradient, and \({c}_{q}\) is the electric capacitance per unit volume given by

$${c}_{q}=\frac{\partial \uprho }{\partial \upphi }.$$
(38)

The current density flux can also be written in terms of charge mobility \({M}_{e}\) or electric conductivity \(\upsigma\),

$$\overrightarrow{I}=-\uprho {M}_{e}\nabla \upphi =-\upsigma \nabla \upphi .$$
(39)

By comparing Equations 37 and 39, we arrive at a similar expression relating charge diffusivity \({D}_{q}\) to electric conductivity \(\upsigma\) through the electric capacitance \({c}_{q}\),

$${D}_{q}=\frac{\upsigma }{{c}_{q}}.$$
(40)

Summary

  • Mobility is a purely kinetic quantity.

  • Chemical capacitance and its inverse, chemical stiffness, are purely thermodynamic quantities.

  • The chemical diffusivity contains information about both the kinetics and thermodynamics of a material. It is related to mobility and chemical capacitance or stiffness by the following generalized Einstein relation,

    $$\mathrm{Diffusivity }\,{D}_{i}=\frac{\mathrm{Mobility }\,{M}_{i}}{\mathrm{Relative\, Chemical \,Capacitance }\,{c}_{{n}_{i}}^{r}}=\mathrm{Mobility }\,{M}_{i}\times \mathrm{Relative \,Stiffness }\,{s}_{{n}_{i}}^{r}.$$
  • The relation between mobility and diffusivity through capacitance is general, for example, in heat conduction, the thermal diffusivity and thermal conductivity are related by heat capacity, and in electric conduction, the electrical charge diffusivity and electrical conductivity are related by the electrical capacitance.

Acknowledgments

We thank Fei Yang at Penn State for his comments. L.-Q.C.’s effort is supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020145 as part of the Computational Materials Sciences Program. L.-Q.C. also appreciates the generous support from the Hamer Foundation through a Hamer Professorship at Penn State. J.C.M. acknowledges the Dorothy Pate Enright Professorship.

Endnotes

  1. 1.

    J. Maier, Chemical resistance and chemical capacitance. Z. Naturforsch. B J. Chem. Sci75 (1–2), 15 (2020)

  2. 2.

    L.-Q. Chen, Thermodynamic Equilibrium and Stability of Materials, 1st edn. (Springer, Singapore, 2022)

  3. 3.

    L.-Q. Chen, Chemical potential and Gibbs free energy. MRS Bull. 44(7), 520 (2019)