Types of Diffusivities

Types of Diffusivities

When describing the diffusion coefficient of a material, it is important to precisely indicate the type of the diffusion coefficient (or, in short, diffusivity). Any lecture or textbook covering diffusion will most likely have at least some discussion about tracer diffusivities. But if you are not careful about the subtle nuances, you can easily confuse between different types that may differ by orders of magnitude. Unfortunately, diffusivity numbers in the literature are often reported without accurately specifying the type, so it is up to the readers to understand exactly what those numbers represent. Let’s take a closer look at this problem and not be confused again!

Tracer diffusivity

Tracer diffusivity is the diffusion coefficient of an isotope in the host, and most closely resembles the case in which Fick’s law is derived. Take an oxide crystal \mathrm{M}_2\mathrm{O}, consisting of \mathrm{M}^{+} and \mathrm{O}^{2-} ions, as an example. If we take the crystal into an oxygen chamber containing ^{18}\mathrm{O} isotopes, the isotopes will start to exchange with the surface oxygen in the crystal (most of which is ^{16}\mathrm{O}), and they will diffuse into the crystal interior. By measuring the depth profile of the isotopes, one can extract the diffusivity of ^{18}\mathrm{O}^{2-} ions, which is the tracer diffusivity. An asterisk is conventionally used to indicate tracer diffusivity, such as D^*_{\mathrm{O}^{2-}} (or simply D^*_{\mathrm{O}}) in this example.

Tracer diffusion is driven by configurational entropy, without involving stoichiometry changes or external fields, so it closely resembles the description of Fick’s law. Why would one be interested in tracer diffusivity? One reason is that it’s a proxy for the self-diffusivity of \mathrm{O}^{2-} ions. However, tracer diffusivity is not necessarily identical to the self-diffusivity of the corresponding ion. To understand the difference, let’s take a closer look at self-diffusivity.

Self-diffusivity (conductivity-derived diffusivity)

Self-diffusivity is the diffusion coefficient of a component of the host, and it is related to the ionic conduction of that component. The component can either be the regular or nominal ionic species consisting of the host (e.g. regular \mathrm{O}^{2-} ions) or a defect species responsible for the transport of that regular species (e.g. oxygen interstitials or vacancies). Both are illustrated in Figure 1. When the term self-diffusivity is used without specifying, it usually refers to the regular components. It is extremely important to distinguish between these two different self-diffusivities since their values can differ by many orders and their activation energies represent different physical phenomena.

Figure 1: Regular components and defects in a crystal lattice.
Figure 1: Regular components and defects in a crystal lattice. Self-diffusivity of regular components is small (much smaller than defect diffusivity) because regular species are only mobile when adjacent to defects.

Since self-diffusion is the exchange between its own species, it is not driven by concentration gradients. Then how is the diffusivity defined when Fick’s first law is not applicable? In another post, we examined how Fick’s law defines diffusivity but that the general diffusion equation of a single species (denoted with subscript k) is in the following form:

(1)   \begin{equation*} J_k = -\frac{D_k c_k}{RT}\nabla \tilde{\mu}_k. \end{equation*}

To describe charged species, here we replaced chemical potential \mu with electrochemical potential \tilde{\mu} = \mu + zF\phi, where z is the charge number of the species, F is the Faraday constant, and \phi is the electrical potential. D is diffusivity, c is concentration, J is particle flux, R is the gas constant, and T is temperature. According to this equation, we can drive particle flux with an electrical field instead of a concentration gradient, which is nothing but an ionic conductivity measurement! For example, self-diffusivity of \mathrm{O}^{2-} ions corresponds to D_{\mathrm{O}^{2-}}, meaning that it can be evaluated by measuring J_{\mathrm{O}^{2-}}, c_{\mathrm{O}^{2-}}, and \nabla\tilde{\mu}_{\mathrm{O}^{2-}} at a fixed temperature.

Conductivity and self-diffusivity are equivalent descriptions. We can write J_k in terms of ionic conductivity (derived in Footnote 1):

(2)   \begin{equation*}  J_k = - \frac{\sigma_k}{z_k^2F^2}\nabla \tilde{\mu}_k, \end{equation*}

where \sigma_k is the ionic conductivity of species k. The comparison between Eqs. (1) and (2) gives the relation between \sigma_k and D_k:

(3)   \begin{equation*}  \frac{D_k c_k}{RT} = \frac{\sigma_k}{z_k^2F^2}, \end{equation*}

which is the Nernst-Einstein relation. It is seen that measuring \sigma_{\mathrm{O}^{2-}} is equivalent to measuring D_{\mathrm{O}^{2-}}, which is why self-diffusivity may be referred to as conductivity-derived diffusivity.

Defect diffusivity (self-diffusivity of defects)

Defect diffusivity is the self-diffusivity of a particular defect. Because lattices are filled with regular species as shown in Figure 1, the transport of nominal species such as regular \mathrm{O}^{2-} ions requires defects. Let’s assume that the transport is enabled by oxygen interstitials, which we denote as \mathrm{O}^{''}_\mathrm{i} (Kröger-Vink notation). Then, J_{\mathrm{O}^{2-}} = J_{\mathrm{O}^{''}_\mathrm{i}} and also \sigma_{\mathrm{O}^{2-}} = \sigma_{\mathrm{O}^{''}_\mathrm{i}}. But recall that concentration is required to convert conductivity into self-diffusivity using the Nernst-Einstein relation, Eq. (3). Therefore, the identity \sigma_{\mathrm{O}^{2-}} = \sigma_{\mathrm{O}^{''}_\mathrm{i}} turns into:

(4)   \begin{equation*} D_{\mathrm{O}^{2-}} c_{\mathrm{O}^{2-}} = D_{\mathrm{O}^{''}_\mathrm{i}} c_{\mathrm{O}^{''}_\mathrm{i}}, \end{equation*}

or

(5)   \begin{equation*}  D_{\mathrm{O}^{2-}} = X_{\mathrm{O}^{''}_\mathrm{i}} D_{\mathrm{O}^{''}_\mathrm{i}}. \end{equation*}

Here X is the fraction of defects, typically much smaller than 1. If we assume that 0.1% of the \mathrm{O}^{2-} ions are interstitials, then D_{\mathrm{O}^{2-}} would be 1000 times smaller than the defect diffusivity D_{\mathrm{O}^{''}_\mathrm{i}}. Self-diffusion of \mathrm{O}^{''}_\mathrm{i} and \mathrm{O}^{2-} both refer to the same physical process in this example. However, one has a choice to either ascribe that physical process to a nominal concentration (the regular \mathrm{O}^{2-} ions) or an actual concentration of mobile species (the \mathrm{O}^{''}_\mathrm{i} defects). Consequently, the factor difference (X) is compensated by the self-diffusivity value, which is why the two diffusivity numbers differ dramatically.

Why would one be interested in defect diffusivity? It is because it represents the actual transport mechanism. For example, if one were to obtain the activation energy of both D_{\mathrm{O}^{2-}} and D_{\mathrm{O}^{''}_\mathrm{i}} (through a \ln{D} vs. 1/T investigation), the latter would reflect activation related to the motion of the oxygen interstitial. However, the former would also include the activation energy for interstitial formation (i.e. activation energy for X_{\mathrm{O}^{''}_\mathrm{i}}). Besides, for the important chemical diffusion process (which we will cover below), the relevant diffusivity information is that of the defects. Unfortunately, it is not always known what the dominant mobile defect is, and measuring the defect concentration could also be challenging.

Tracer vs. self-diffusivity

Based on the explanation so far, it might be tempting to declare that tracer diffusivity D^* is identical to self-diffusivity D (of a regular species k), and I’ve indeed seen this statement in a few places. However, the two are generally different by a factor defined as the Haven ratio H:

(6)   \begin{equation*} D^*_k = H D_k. \end{equation*}

Because, H is on the order of 0.1, and sometimes is close to 1, it could be hard to distinguish experimentally. Perhaps that is the reason why it is sometimes dismissed.

Many mechanisms are lumped into H, but the most important one is the correlation factor (you can approximately replace H with the correlation factor for the discussion here). This factor accounts for any correlation between subsequent jumps such as having a higher probability for a backward jump right after a forward jump. The correlation factor is bound between 0 and 1, and 1 means that each jump is independent. For defects like vacancies or interstitials, which are typically dilute relative to the total sites, the environment seen from the defect will be, most of the time, identical before and after the jump. Thus, for self-diffusion, which is mechanistically the diffusion of the mobile defect itself, jump correlation effects are typically minimal. However, tracer isotopes could be different, depending on how its diffusion is mediated by defects. When the mobile defect is a vacancy and the tracer isotope makes a jump to a vacancy site, then the tracer isotope has a higher probability going backward because that is where the vacancy is. In this case, the correlation factor will be less than 1 (for a diamond crystal structure it is 0.5). If the mobile defect is an interstitial, then the tracer isotope mobilizes by becoming an interstitial itself, to which there would be minimal correlation effects.

Figure 2: Chemical Diffusion.
Figure 2: An example of chemical diffusion in battery applications. For the net transport of the neutral species Li, ambipolar diffusion of charged species is required. The same physical process can be described with either regular species (left) or defect species (right).

Chemical diffusivity

So far we have discussed diffusion processes that don’t change the chemical stoichiometry of the host material. But for many practical applications, changing the stoichiometry through a diffusion process is the primary objective. This diffusion process is called chemical diffusion. The most distinguishing feature is that it involves the diffusion of at least one positive and one negative species, hence also called ambipolar diffusion.

For example, in Li-ion batteries, Li species are stored in a host material, and the insertion and extraction of Li require chemical diffusion. This process is illustrated in Figure 2: the net change is the diffusion of neutral species Li, but microscopically, a pair of positive and negative components (a \mathrm{Li}^+ & \mathrm{e}^- pair) must diffuse. Because of the requirement of local charge neutrality (or, more generally, conservation of local charge density), the particle fluxes of these two species cannot change independently.

Rather than going through detailed derivations here (which I will post someday), let’s gain some understanding by examining the final equations. When there is no net current (additional requirements are needed with non-zero net current), the particle flux of the neutral species M is:

(7)   \begin{equation*}J_\textrm{M} = \tilde{D}_\textrm{M} \nabla c_\textrm{M}.\end{equation*}

Here, \tilde{D} is the chemical diffusion coefficient, conventionally denoted with a tilde accent. Notice the similarity to Fick’s first law. However, here \tilde{D} describes the combined property of more than one species, so one should be cautious to not erroneously mix-match different types of diffusivities.

Let’s consider a generic case where \mathrm{M} \leftrightarrow \mathrm{M}^{z+} + z\mathrm{e}^-. In this case, chemical diffusivity can be expressed as:

(8)   \begin{equation*}  \tilde{D}_\mathrm{M} = \left(\frac{1}{z^2F^2}\right) \frac{\sigma_i \sigma_e}{\sigma_i + \sigma_e} \frac{\partial \mu_\mathrm{M}}{\partial c_\mathrm{M}}. \end{equation*}

Here, \sigma is conductivity, and the subscript denotes ionic and electronic species. Chemical potential of the neutral species is defined as \mu_\mathrm{M} = \tilde{\mu}_{\mathrm{M}^{z+}} + z\tilde{\mu}_{\mathrm{e}^-}.

The factor {\partial \mu_\mathrm{M}}/{\partial c_\mathrm{M}} in Eq. (8) is responsible for the the so-called “enhancement effect”. This factor accounts for the discrepancy where the real driving force for diffusion is \nabla \mu (explained here) whereas the actual change of interest is \delta c. If a given change in c results in a larger change in \mu, then the actual driving force is much bigger and it would appear as if a given \nabla c is driving faster diffusion.

The factor {\sigma_i \sigma_e}/({\sigma_i + \sigma_e}) in Eq. (8) is the total conductivity for connecting two conductors with \sigma_i and \sigma_e in series. Note that the factor has a value smaller than whichever conductivity is lower between the two. Therefore, chemical diffusion could be rate-limited by either the positive or negative species. Why does the conduction of the two species act as a series process rather than a parallel one? It is because of the charge neutrality requirement. For example, even if electron conduction is fast, the neutral species transport cannot happen until the ionic part catches up and maintains charge neutrality. Therefore, any statement about chemical diffusivity not considering at least two species is inherently incorrect.

Because of the series-process nature of chemical diffusion, it might be tempting to conclude that, if one process is rate-limiting then the counter species for ambipolar diffusion has no impact. However, this suspicion is not necessarily true because of the enhancement factor. From \mu_\mathrm{M} = \tilde{\mu}_{\mathrm{M}^{z+}} + z\tilde{\mu}_{\mathrm{e}^-}, and \mathrm{M} \leftrightarrow \mathrm{M}^{z+} + z\mathrm{e}^-, we can decompose the enhancement factor into contributions from each species:

(9)   \begin{equation*} \frac{\partial \mu_\mathrm{M}}{\partial c_\mathrm{M}} = \frac{\partial\tilde{\mu}_{\mathrm{M}^{z+}}}{\partial c_{\mathrm{M}^{z+}}} + \frac{z\partial\tilde{\mu}_{\mathrm{e}^-}}{\partial c_{\mathrm{e}^-}/z}. \end{equation*}

In the ideal-solution limit, \delta \tilde{\mu}_{\mathrm{M}^{z+}} would be RT\delta\ln c_{\mathrm{M}^{z+}}. If we account for non-ideal behavior with a thermodynamic factor \Gamma, Eq. (9) turns into:

(10)   \begin{equation*} \frac{\partial \mu_\mathrm{M}}{\partial c_\mathrm{M}} =RT\left( \frac{\Gamma_i}{c_{\mathrm{M}^{z+}}} + \frac{z^2 \Gamma_e}{c_{\mathrm{e}^-}}\right). \end{equation*}

It is seen that even a less conducting species could possibly have a big contribution in the enhancement factor because the terms of each species are added, and they involve inverse concentration rather than conductivity or diffusivity. To see the combinations of contributions more clearly, we can plug in Eq. (10) into Eq. (8) and also convert conductivities into diffusivities using the Nernst-Einstein equation (when using Eq. (3), note that z_{\mathrm{M}^{z+}}= z and z_{\mathrm{e}^-} = -1). Then chemical diffusivity simplifies to:

(11)   \begin{equation*}  \tilde{D}_\mathrm{M} = \Gamma_i D_i t_e + \Gamma_e D_e t_i. \end{equation*}

Here, transference numbers indicate the contribution of a partial conductivity to total conductivity, such as t_e = \sigma_e / (\sigma_e + \sigma_i). It is seen that, even for electron-conducting (ion-limiting) materials (i.e. t_e \gg t_i), electronic diffusivity might still be important if D_e \gg D_i overwhelms the difference in transference number.

Some authors refer to the quantities \Gamma_i D_i and \Gamma_e D_e in Eq. (11) also as chemical diffusivity, which is an unfortunate source of confusion. In this case, one should distinguish the two types of “chemical diffusivity” by finding whether it is describing a charged species or a neutral species.

Which type of diffusivity matters for chemical diffusion?

Chemical diffusivity is the ultimate property relevant for many applications, but what type of diffusivity is relevant for determining \tilde{D}? From intuition, one could suspect that it is the defect diffusivity (self-diffusivity of defects) since they describe the actual species that are mobile (as in the right column of Figure 2). If we try to check out this guess from the discussion in the previous section, you will notice that we did not specify any defects. We discussed everything with regular components, so does that mean the relevant diffusivity number is the self-diffusivity of regular components? The answer is no, and the intuition was correct. But this point is not trivial to understand until you breakdown the phenomenological parameters with defect model cases. This difficulty is perhaps why so many literature reports erroneously use self-diffusivity or tracer diffusivity to estimate chemical diffusivity. Fortunately, if you followed the discussion up to here, the hard work is now about to pay off.

The secret is hidden in the thermodynamic factor \Gamma in Eq. (11). If you plug in self-diffusivity numbers of regular components into Eq. (11), then the corresponding thermodynamic factors will turn out to be enormously large. By contrast, if defect diffusivity numbers are plugged in, \Gamma factors will turn out to be “normal” and in many cases simply 1 (for dilute defect concentrations). The difference in \Gamma corresponds to the fraction of defects X, exactly the factor that related self-diffusivity and defect diffusivity in Eq. (5). In other words, chemical diffusivity will be generally in between the two mobile defect diffusivities, and these will be larger than the self-diffusivity of regular components by a factor of 1/X.

For those interested, here is how the math works out. Let’s say the regular component is \mathrm{O}^{2-} and the dominant mobile defect is an interstitial \mathrm{O}^{''}_\mathrm{i}. The thermodynamic factors are:

(12)   \begin{equation*} \Gamma_{\mathrm{O}^{2-}}= \frac{c_{\mathrm{O}^{2-}}}{RT}\frac{\partial \mu_{\mathrm{O}^{2-}}}{\partial c_{\mathrm{O}^{2-}}}; \quad \Gamma_{\mathrm{O}^{''}_\mathrm{i}} = \frac{c_{\mathrm{O}^{''}_\mathrm{i}}}{RT}\frac{\partial \mu_{\mathrm{O}^{''}_\mathrm{i}} }{\partial c_{\mathrm{O}^{''}_\mathrm{i}}}. \end{equation*}

Note that the \partial\mu / \partial c factors are identical because they represent the same physical quantity. However, c_{\mathrm{O}^{2-}} X = c_{\mathrm{O}^{''}_\mathrm{i}}, where the fraction X is typically a very small number. Therefore,

(13)   \begin{equation*} \Gamma_{\mathrm{O}^{2-}} D_{\mathrm{O}^{2-}} = \Gamma_{\mathrm{O}^{''}_\mathrm{i}}  D_{\mathrm{O}^{2-}} / X = \Gamma_{\mathrm{O}^{''}_\mathrm{i}} D_{\mathrm{O}^{''}_\mathrm{i}}, \end{equation*}

where the second equality is from Eq. (5).

Recap

Self-diffusivity usually refers to the diffusion coefficient of regular ions. Defect diffusivity is the self-diffusivity of defect species, and it is much larger than the self-diffusivity of regular ions. Tracer diffusivity is similar in orders-of-magnitude as the self-diffusivity of regular ions. Chemical diffusion always involves at least one positive and one negative species. Chemical diffusivity is a weighted average between the diffusivities of mobile defects.

Footnote 1

Conductivity is defined as the current driven by the electrochemical potential gradient of electrons \nabla \tilde{\mu}_{\mathrm{e}^{-}} under no temperature or compositional gradients. Taking oxygen as an example, since \mu_{\mathrm{O}} = \tilde{\mu}_{\mathrm{O}^{2-}} + 2\tilde{\mu}_{\mathrm{e}^{-}}, the condition of no compositional gradient (\nabla\mu_{\mathrm{O}}=0) is equivalent to -2\nabla \tilde{\mu}_{\mathrm{e}^{-}} = \nabla\tilde{\mu}_{\mathrm{O}^{2-}}. Ionic current density is zFJ. Therefore, ionic conductivity is:

(14)   \begin{equation*} \sigma_{\mathrm{O}^{2-}} = \frac{2FJ_{\mathrm{O}^{2-}}}{-\nabla \tilde{\mu}_{\mathrm{e}^{-}}/F} =  \frac{4F^2J_{\mathrm{O}^{2-}}}{-\nabla \tilde{\mu}_{\mathrm{O}^{2-}}}. \end{equation*}

The general expression for this ionic conductivity equation is Eq. (2).

Last modified: Sep. 05, 2020

Photo Credit: Arthur Mazi
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