Hyun-Suk Kim
Lecture 2:
Diffusion & Electrical Conductivity 51570-00, Spring 2018
Ceramics Materials: Science & Engineering
(세라믹스 특론)
1. Introduction
▪ The solid state is far from static!
Thermal energy keeps the atoms vibrating vigorously about their lattice
positions and continuously bumping into each other and exchanging energy with their neighbors and surroundings. Every now and then, an atom will gain sufficient energy to leave its mooring and migrate. This motion is
termed diffusion, without which the sintering of ceramics, oxidation of metals, tempering of steels, precipitation hardening of alloys, and doping of
semiconductors, just to name a few phenomena, would not be possible.
▪ A prerequisite for diffusion and electrical conductivity is the presence of point and electronic defects. In many ceramics, diffusion and electrical conductivity are inextricably (불가분하게) linked for two reasons.
The first is that ionic species can be induced to migrate under the influence of a chemical potential gradient (diffusion) or an electric potential gradient (electrical conductivity). In either case, the basic atomic mechanism is same, and one of the major conclusions of this chapter is that the diffusivity of a
given species is directly related to its conductivity.
The second important link is that the defects required for diffusion and electrical conductivity are often created in tandem (동시에).
For example, as discussed in Chap. 6, the reduction of an oxide can result in the formation of oxygen vacancies and free electrons in the conduction band.
This not only renders the oxide more electrically conductive but also increases the diffusivity of oxygen in that oxide.
𝑂 ⇒ 𝑂 𝑔 𝑉 2𝑒′
2. Diffusion
▪ There are essentially three mechanisms by which atoms will diffuse.
(1) Vacancy mechanism: (Fig 7.1a)
-involves the jump of an atom or ion from a regular site into an adjacent vacant site
(2) Interstitial diffusion (Fig. 7.1b)
-requires the presence of interstitial atoms or ions
(3) Interstitialcy mechanism (Fig. 7.1c) -less common mechanism
-an interstitial atom pushes an atom from a regular site into an
interstitial site
▪ In all cases, to make the jump, the atom has to squeeze through a narrow passage, which implies it has overcome an activation energy barrier.
This barrier is known as the energy of migration (Fig. 7.1d) for the diffusing interstitial ion.
▪ Phenomenological Equations
▪ For many physical phenomena that entail transport whether it is a charge, mass, or momentum the assumption is usually made that the flux 𝐽 is
linearly proportional to the driving force 𝐹, or 𝐽 𝐾𝐹
where 𝐾 is a material property (see Chap. 5).
In the case of diffusion, the relationship between the flux 𝐽 and the concentration gradient 𝑑𝑐/𝑑𝑥 is given by Fick’s first law, namely,
𝐽 𝑚𝑜𝑙
𝑚 𝑠 𝐷 𝜕𝑐
𝜕𝑥
𝑚 𝑠
𝑚𝑜𝑙 𝑚 · 𝑚
where 𝐷 is the chemical diffusion coefficient of species A in matrix B.
The self-diffusivity D of an atom or ion is a measure of the ease and frequency with which that atom or ion jumps around in a crystal lattice
(hopping) in the absence of external force, i.e., in a totally random fashion.
Experimentally, it has long been appreciated that 𝐷 is thermally activated and could be expressed as
𝐷 𝐷 exp (1)
where 𝑄 is the activation energy for diffusion which is not a function of temperature, whereas 𝐷0 was a weak function of temperature.
It also has been long appreciated that diffusivity depends critically on the stoichiometry and purity level of a ceramic. To understand how these
variables affect 𝐷, the phenomenon of diffusion has to be considered at the atomic level.
▪ Atomistics of Solid State Diffusion
▪ The fundamental relationship relating the self-diffusion coefficient 𝐷 of an atom or ion to the atomistic processes occurring in a solid is
𝐷 𝛼Ω (2)
where Ω is the frequency of successful jumps, i.e., number of successful
jumps per second; is the elementary jump distance which is on the order of the atomic spacing; and is a geometric constant that depends on the crystal structure.
Eq. (2) can be derived from random walk theory considerations. A particle after 𝑛 random jumps will, on average, have traveled a distance proportional to 𝑛 times the elementary jump distance .
𝑥 ∝ 𝑛
It can be easily shown that, in general, the characteristic diffusion length is related to the diffusion coefficient 𝐷 and time 𝑡 through the equation
𝑥 𝐷𝑡
▪ Frequency Ω is the product of the probability of an atom’s having the requisite energy to make a jump and the probability that the site adjacent to the diffusing entity is available for the jump, or
Ω 𝜃 (3)
From this relationship, it follows that to understand diffusion and its
dependence on temperature, stoichiometry, and atmosphere requires an understanding of how and 𝜃 vary under the same conditions.
Each is dealt with separately in the following subsections.
From which it follows that:
𝑛 ∝ 𝐷𝑡 Rearranging yields
𝐷 ∝ 𝑛
𝑡 ∝ Ω
where Ω is defined as 𝑛/𝑡, or the number of successful jumps per second.
Jump frequency
▪ For an atom to jump from one site to another, it has to be able to break the bonds attaching it to its original site and to squeeze between adjacent atoms, as shown schematically in Fig. 7.1b and d.
This process requires an energy ∆𝐻∗ , which is usually much higher than the average thermal energy available to the atoms. Hence at any instant only a fraction of the atoms will have sufficient energy to make the jump.
Therefore, to understand diffusion, one must answer the question:
At any given temperature, what fraction of the atoms have an energy ∆𝐻∗ and thus capable of making the jump?
To answer this question, the Boltzmann distribution law is invoked, which states that the probability 𝑃 of a particle having an energy ∆𝐻∗ or greater is given by:
𝑃 𝐸 ∆𝐻∗ 𝑐𝑜𝑛𝑠𝑡. exp ∆𝐻∗ 𝑘𝑇
where 𝑘 is Boltzmann’s constant and 𝑇 is the temperature in Kelvin.
It follows that the frequency with which a particle can jump, provided that an adjacent site is vacant, is equal to the probability that it is found in a state of sufficient energy to cross the barrier multiplied by the frequency 0 at
which that barrier is being approached.
ex𝑝 ∆ ∗ (4)
where 0 is the vibration of the atoms and is the order of 1013 s-1.
For low temperatures of high values of ∆𝐻∗ , the frequency of successful
jumps becomes vanishingly small, which is why, for the most part, solid-state diffusion occurs readily only at high temperatures.
Conversely, at sufficient high temperatures, that is, 𝑘𝑇 ≫ ∆𝐻∗ , the barrier ceases to be one and every vibration could, in principle, result in a jump.
Probability 𝜽 of site adjacent to diffusing species being vacant
▪ The probability of a site being available for the diffusing species to make the jump will depend on whether one considers the motion of the defects or of the ions themselves. Consider each separately.
Defect diffusivity
▪ As noted before, the two major defects responsible for the mobility of atoms are vacancies and interstitials (Fig. 7.1).
For both, at low concentrations (which is true for a vast majority of solids) the site adjacent to the defect will almost always be available for it to make the jump and 𝜃 1.
(1) For an interstitial, by combining Eqs. (2), (3) and (4), with 𝜃 1, the interstitial diffusivity 𝐷 is given by
𝐷 𝛼 exp ∆𝐻∗ , 𝑘𝑇
where ∆𝐻∗ , is the activation E needed by the interstitial to make the jump.
(2) For a vacancy, the probability of a successful jump is increased -fold, where is the number of atoms adjacent to that vacancy (i.e., the
coordination number of the atoms), since if any of the neighboring
atoms attains the requisite energy to make a jump, the vacancy will make a jump. Thus, for vacancy diffusion
exp ∆𝐻∗ 𝑘𝑇
Combining this equation with Eqs. (2) and (3), again assuming 𝜃 1, yields
𝐷 exp ∆ ∗ (4)
Atomic or ionic diffusivity
▪ In contrast to the defects, for an atom or ion in a regular site, 𝜃 ≪ 1, because most of these are surrounded by other atoms.
The probability of a site being vacant is simply equal to the mole or site fraction, denoted by (lambda), of vacancies in that solid. Thus the frequency of successful jumps for diffusion of atoms by a vacancy mechanism is given by
Ω exp ∆𝐻∗ 𝑘𝑇
where the factor appears because the probability of a site next to a
diffusing atom being vacant is increased -fold. The diffusion coefficient is given by
𝐷 exp ∆ ∗ (5)
Comparing Eqs. (4) and (5) reveals an important relationship between vacancy and ion diffusivity, namely,
𝐷 𝐷
Given that usually 1, it follows that 𝐷𝑖𝑜𝑛 𝐷𝑣𝑎𝑐 , a result that at first sight appears paradoxical after all one is dealing with the same species.
By noting that , where 𝑐 and 𝑐 are the concentrations of vacancies and ions, respectively, we see that
𝐷 𝑐 𝐷 𝑐
The defects move often (high D) but are not that numerous the atoms move less frequently, but there are a lot of them.
(6)
▪ In deriving Eq. (5), for the sake of simplicity, the effect of the jump on the vibration entropy was ignored. This is taken into account by postulating the existence of an excited equilibrium state, albeit of very short duration, that affects the frequency of vibration of its neighbors and is associated with an entropy change given by ∆𝑆∗ 𝑘𝑙𝑛 ν′/ν , where ν and ν’ the frequencies of vibration of the ions in their ground and activated states, respectively.
A more accurate expression for 𝐷𝑖𝑜𝑛 thus reads 𝐷 exp ∆ ∗
where ∆𝐺∗ is defined as
∆𝐺∗ ∆𝐻∗ 𝑇∆𝑆∗
Fig. 7.5 (a) Schematic of activated state during diffusion
(7)
Putting all the pieces together, one obtains a final expression that most resembles Eq. (1), namely,
𝐷 exp∆𝑆∗
𝑘 exp ∆𝐻∗
𝑘𝑇 𝐷 exp 𝑄 𝑘𝑇
Now the physics of why diffusivity takes that form should be clearer.
The temperature dependence of diffusivity of some common ceramics is shown in Fig. 7.2.
Pure ceramics
At a given temperature, the diffusion coefficient in ceramics can vary by over 10 orders of magnitude.
The temperature dependence, or activation energy, can also vary by over an order of magnitude, from about 0.5 eV to more than 8 eV.
The values of the activation energies 𝑄 and their variation with temperature are quite useful in deciphering the nature of the diffusion process.
For instance, if is thermally activated, as in the case of intrinsic point defects, then the energy needed for the defect formation will appear in the final expression for 𝐷. If, however, the vacancy or defect concentration is fixed by impurities, then is no longer thermally activated but is proportional to the concentration of the dopant.
H.W. Worked Example 7.1, 7.2
Impure ceramics (doping) Schottky formation
energy (intrinsic)
Energy of migration
Worked Example 7.1
For Na+ ion migration in NaCl, ∆𝐻∗ is 77 kJ/mol, while the enthalpy and entropy associated with the formation of a Schottky defect are, respectively, 240 kJ/mol and 10 𝑘 (see Table 6.2).
(a) At approximately what temperature does the diffusion change from extrinsic (i.e., impurity-controlled) to intrinsic in an NaCl-CaCl2 solid solution containing 0.01 percent CaCl2? You can ignore ∆𝑆∗ .
(b) At 800 K, what mole percent of CaCl2 must be dissolved in pure NaCl to increase DNa+ by an order of magnitude?
𝐶𝑎𝐶𝑙 ⟹ 𝑉 𝐶𝑎· 2𝐶𝑙
2𝑁𝑎𝐶𝑙
▪ Diffusion in a Chemical Potential Gradient
▪ In the foregoing discussion, the implicit assumption was that diffusion was totally random, a randomness that was assumed in defining D by Eq. (2).
This self-diffusion, however, is of no practical use and cannot be measured.
▪ Diffusion is important inasmuch as it can be used to effect compositional and microstructural changes.
In such situations, atoms diffuse from areas of higher free energy, or
chemical potential, to areas of lower free energy, in which case the process is no longer random but is now biased in the direction of decreasing free energy.
Fig. 7.5b: Diffusion of an ion down a chemical potential gradient.
▪ Consider Fig. 7.5b, where an ion is diffusing in the presence of a chemical potential gradient (𝑑𝜇/𝑑𝑥).
If the chemical potential is given per mole, then the gradient or force per atom, 𝑓, is given by
𝑓
1 𝑁
𝑑𝜇 𝑑𝑥 where 𝑁𝐴𝑣 is Avogadro’s number.
Consequently, the difference between the energy barrier in the forward and backward direction is
𝑓 /
Chemical potential per atom
(8)
The forward rate for the atom to jump is thus proportional to
𝛼0exp ∆𝐺∗ 𝑘𝑇 while the backward jump rate is
𝛼0exp ∆𝐺∗ 𝑘𝑇
and appears here because, as noted earlier, for a jump to be successful, the site into which it is jumping must be vacant.
Also, 𝛼 is the same constant that appears in Eq. (2)it is a factor that takes into account that only a fraction of the total hops are hops in the 𝑥
direction. For instance, in cubic lattices for which 6, only one-sixth of successful jumps are in the forward x direction, that is, 𝛼 1/6 and thus 𝛼 1 (𝛼 1/𝜁).
The existence of a chemical potential gradient will bias the jumps in the forward direction, and the net rate will be proportional to
In general, chemical potential gradients are small compared to the thermal energy, i.e., / 𝑘𝑇 ≪ 1, and Eq. (9) reduces to (𝑒 ≅ 1 𝑥 for small 𝑥)
The average drift velocity 𝑣 is given by , which, when combined with Eqs. (10) and (8), gives
𝛼0 exp ∆𝐺∗
𝑘𝑇 exp ∆𝐺∗ 𝑘𝑇
𝛼0exp ∆𝐺∗
𝑘𝑇 1 exp
𝑘𝑇 (9)
𝛼0
𝑘𝑇 exp ∆𝐺∗
𝑘𝑇 (10)
𝑣 𝛼 0
𝑘𝑇 exp ∆𝐺∗ 𝑘𝑇 𝑓
And the resulting flux
where 𝑐𝑖 is the total concentration (atoms per cubic meter) of atoms or ions diffusing through the solid. Since the term in brackets is nothing but 𝐷𝑖𝑜𝑛
[Eq. (6)] it follows that
𝐽 𝑐 𝐷
𝑘𝑇 𝑓
This equation is of fundamental importance because
1. It relates the flux to the product 𝑐 𝐷 . The full implication of Eq. (6), namely, that 𝐷 𝑐 𝐷 𝑐 , should now be obvious when one is
considering the diffusion of a given species, it is not important whether one considers the ions themselves or the defects responsible for their motion:
the two fluxes have to be, and are, equal.
𝐽 𝑐 𝑣 𝑐
𝑘𝑇 𝛼 0exp ∆𝐺∗
𝑘𝑇 𝑓 (11)
(12)
2. It relates the flux to the driving force, 𝑓. Given that 𝑓 has the dimensions of force, Eq. (11) can be considered a true flux equation in that it is identical in form to 𝐽 𝐾𝐹.
Note also that this relationship has general validity and is not restricted to chemical potential gradients. For instance, as discussed in the next
sections, 𝑓 can be related to gradients in electrical or electrochemical potentials as well.
3. It can be shown (see App. 7A) that for ideal and dilute solutions Eq. (12) is identical to Fick’s first law (the following equation).
H.W. Appendix 7A
𝐽 𝑚𝑜𝑙
𝑚 𝑠 𝐷 𝜕𝑐
𝜕𝑥
𝑚 𝑠
𝑚𝑜𝑙 𝑚 · 𝑚
▪ Diffusion in an Electric Potential Gradient
▪ The situation where the driving force is a chemical potential gradient has just been addressed. If, however, the driving force is an electric potential gradient, then the force on the ions is given by
𝑓 𝑧 𝑒𝑑∅
𝑑𝑥
where ∅ is the electric potential in volts and 𝑧𝑖 is the net charge on the moving ions. The current density 𝐼
· is related to the ionic flux 𝐽
· ,by 𝐼 𝑧 𝑒𝐽
Substituting Eqs. (13) and (14) in Eq. (12) shows that
𝐼 𝑧 𝑒𝐽 𝑧 𝑒𝑐 𝐷
𝑘𝑇 𝑓 𝑧 𝑒𝑐 𝐷
𝑘𝑇 𝑧 𝑒 𝑑∅
𝑑𝑥
which, when compared to Ohm’s law 𝐼 𝜎 ∅ [this will be derived in the next section!], yields
(13)
(14)
𝜎 𝑧 𝑒 𝑐 𝐷 𝑘𝑇
𝑧 𝑒 𝑐 𝐷 𝑘𝑇
where 𝜎 is the ionic conductivity.
This relationship is known as the Nernst-Einstein relationship, and it relates the self-diffusion coefficient to the ionic conductivity. The reason for the connection is obvious: in both cases, one is dealing with the jump of an ion or a defect from one site to an adjacent site. The driving forces may vary, but the basic atomic mechanism remains the same.
In applying Eq. (15), the following points should be kept in mind:
1. The conductivity 𝜎 refers to only the ionic component of the total conductivity (next section for more details).
2. This relationship is valid only as long as 𝜃 for the defects 1 (i.e., at high dilution).
(15)
3. The variable 𝑐 introduced in Eq. (11) and now appearing in Eq. (15) represents the total concentration of the diffusing ions in the crystal.
For example, in calcia-stabilized zirconia which is an oxygen ion
conductor, 𝑐 is the total number of oxygen ions in the crystal and not the total number of defects.
On the other hand, in a solid in which the diffusion or conductivity occurs by an interstitial mechanism, 𝑐 represents the total number of interstitial ions in the crystal, which is identical to the number of defects.
