Ionic and Electronic Conduction in Nonmetallic Phases - ACS

4 Ionic and Electronic Conduction in Nonmetallic Phases J O H N W.

PATTERSON

Engineering Research Institute and Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 20, 2016 | http://pubs.acs.org Publication Date: January 31, 1979 | doi: 10.1021/bk-1979-0089.ch004

PART I

Open Circuit Theory and Parabolic Tarnishing Kinetics

Historical Background About the turn of the century and shortly thereafter, certain developments in mathematical physics and in physical chemistry were realized which were to prove important in the theory of mass and charge transport in solids, later. Einstein(1) and Smoluchowski(2) i n i t i a t e d the modern theory of Brownian motion by idealizing it as a problem in random f l i g h t s . Then some seventeen years or so later, Joffee(3) proposed that i n t e r s t i t i a l defects could form inside the l a t t i c e of ionic crystals and play a role in electrical conductivity. The f i r s t tenable model for ionic conductivity was proposed by Frenkel(4), who recognized that vacancies and i n t e r s t i t i a l s could form internally to account for ion movement. Figure 1 is a schematic representation of Frenkel's notion: an atom or ion can get dislodged from i t s normal site to form an interstitial-vacancy pair. He further proposed that they do not a l ways recombine but instead may dissociate and thus contribute to diffusional transport and e l e c t r i c a l conduction. They were free to wander about in a "random walk" manner essentially equivalent to that of Brownian motion... this meant they should exhibit a net d r i f t in an applied f i e l d . In the fluids considered by Einstein and Smoluchowski, a l l species large and small are capable of substantial migration at a l l times. In solid crystals, however, only the i n t e r s t i t i a l atoms and those next to vacant sites can enjoy any significant amount of motion. Thus, i t was realized that the concentrations of mobile defects are the important things, at least in connection with the ionic conductivity of crystals. An elaborate analysis of thermodynamic equilibria of point defects was then developed by Wagner and Schottky (5z§_) in which the laws relating defect concentrations to impurities, ambient p a r t i a l pressures and temperature were worked out in detail. Wagner followed this in 1933 by combining v i r t u a l l y a l l the foregoing concepts to explain the phenomenon of parabolic tarnishing of metal in aggressive environments ^— . He assumed that transport of neutral species was negligible compared to that of ions and 0-8412-0471-3/79/47-089-096$07.50/0 © 1979 American Chemical Society

Brubaker and Phipps; Corrosion Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1979.


4.

PATTERSON

Conduction

in

Nonmetallic

Phases

ο ο ο ο ο ο ο ο ο ο οίο ο ν

ο ο -ο ο ο ο ο ο Figure 1.

Frenkel defect in a crystal lattice


97


98

CORROSION

CHEMISTRY

electronic carriers, which no doubt must have seemed a rather bold assumption to many others at that time. But beyond that he also assumed that a l l the charged carriers migrated with Einstein-type mobilities under the combined influences of free energy (i.e., chemical potential) gradients and electric f i e l d s . In this way, he was able to transform the problem of parabolic scaling of metals to one of diffusional transport in a electrochemical medium. The implication of Wagner's theory was that the parabolic scaling rate constants, many of which were known for certain gasmetal combinations, should be quantitatively related to two basic types of information. A) Thermodynamic information, such as the formation free energy for the scale and p a r t i a l pressure or a c t i v i t y data for the environment... and B) E l e c t r i c a l property information for the scaling compound, such as e l e c t r i c a l conductivity, ionic and electronic transference numbers or the open c i r c u i t emf established over the growing scale. In addition, the theory predicted that a measurable emf would be established over the tarnishing layer, and moreover, that this voltage could be used to infer the average ionic and electronic transference numbers of the scale. Prior to Wagner's treatment, of course, there had been no reason whatever to think that the electrical properties of the scale compound should bear any relation to oxidation rates, nor was there any reason to think that voltages should appear over these coatings. And so the metal scaling systems known to exhibit parabolic growth kinetics were soon selected for the c r i t i c a l testing of Wagner' s theory. The conductivity-emf-transference number data required by the theory were measured and the scaling rate constant predicted. Then independent measurements of the same scaling constant were obtained, as for example from weight gain measurements and the results compared.(10-12) . Astounding agreement was found and much excitement began to spread in corrosion engineering c i r c l e s because Wagner's theory suggested many ways to possibly control or stop scaling. But this optimism was somewhat overdone because so few practical situations conform to the parabolic scaling prerequisites. Nevertheless, Wagner's theory has been applied and reviewed extensively and this continues up to the present time(13-19) Presently, the major interest in Wagner's theory derives mainly from the open c i r c u i t emf relationships rather than the scaling rate predictions. Only a limited number of the metal scaling systems important to modern technology s t r i c t l y conform to the conditions required by Wagner's theory, because of various complicating factors (abrasion, thermal gradients, contaminations, and so on). However, a variety of new solid electrolyte materials has been developed for remote sensor applications(20) , fuel c e l l battery applications(20) f laboratory investigations of thermodynamic data(21-23)^ -Q f these applications the solid electrolyte element can be expected to perform according to o r

I n

a

Q


4.

PATTERSON

Conduction

in

Nonmetallic

99

Phases

Wagner's tarnishing theory but significant modifications are required for the non-open c i r c u i t c a s e s ) . in short, we can say that the present understanding of solid-state electrochemistry is largely due to Wagner's electrochemical theory of tarnishing and moreover that the concepts as he i n i t i a l l y elucidated them have remained in tact to the present time. In part II of this paper, less simple mixed conducting phases and non open c i r c u i t conditions w i l l be considered


2

4

Physical Processes Occurring During Parabolic Scaling Patterson (19) has given a detailed discussion of Wagner's theory complete with flow-diagram summaries for the derivations of the quantitative formulas. But these derivations are much too elaborate to develop here. We w i l l invoke a simplified model o r i ginally put forward by Hoar and Price (—^ , but this comes later. F i r s t l e t us outline the physical processes that can and do occur in parabolic scaling. Figure 2 f a c i l i t a t e s the physical description by showing the compound M X^ (oxide, fluoride, chloride, etc.) attached as a scaling layer to the substrate of metal M on the l e f t . The layer is exposed to an atmosphere containing X 2 gas molecules ( O 2 , F , C l 2 f N / etc.) on the right where the X molecules become adsorbed on the surface and eventually get incorporated as ions. Ultimately, however, the scaling rate—whether measured by weight gains or scale thicknesses — is just the total assimilation rate of X atoms by the scale. This can happen in either of two ways: by combining with cations which emerge at L or by d i f fusing in as anions. For simplicity, assume for the moment that only cations are mobile in the scale. Even though no X anions can move, adsorbed X atoms can s t i l l be assimilated because M cations can migrate from the other side to combine with the adsorbed gas atoms. Note however, that the scale would grow only at the gas side in this case. On the other hand, i f anions are mobile, but cations are not, the adsorption-dissociation process X 2 (g) 2 X (ads) on the right is followed by anion migration through the scale and growth occurs at the M, M Xb interface (left side). These two situations can be distinguished experimentally by identifying markers inside the scale. The apparent change of depth of the marker with time identifies the mobile ionic species. If the scale is an ionic compound, X 2 molecules, which are neutral, can be incorporated only as anions, which are negatively charged X ions. Conversely, each incorporated M atom must become a positively charged cation. This has important electrochemical consequences because the region at L is forced to become increasingly electron-deficient or positively charged as the negative anions are "formed" and then wander off toward the metal-rich side. Similarly, the metal at the l e f t accrues a surplus of negative charge due to the electrons liberated when M atoms enter the scale as cations. a

2

2

2

a



100

CORROSION C H E M I S T R Y

Eventually these oppositely charged outside layers cause a very strong coulombic f i e l d to build up, but i t must be positive on the gas side and negative at the metal side. But of course this polarity necessarily opposes the migration of both anions and ca tions and increasingly so as the charging continues to build. That is, the incorporation of X atoms by the scale is self defeat ing i f eletrochemical forces are at play . In fact, i f there were no electronic leakage paths connecting the two surfaces, a l l cat ion and anion migration would soon grind to a halt. At this point, no further weight gain would be observed but an emf on the order of volts would be observed. However, some degree of internal shorting within scales always occurs because scaling layers always exhibit at least a trace of internal conduction by electronic carriers. For this reason, i t is best to think of scales as mixed (ionic and electronic) conduc tors. The effect of the internal short is to bleed off some of the charge,on the surfaces and thus weaken the coulombic f i e l d accord ingly. This would allow ionic migration to continue indefinitely but at an ever-decreasing rate as the thickness of the scale in creases. In principle then, real scaling layers never achieve the completely arrested state alluded to above as a limiting case, in stead, they thicken indefinitely but according to the parabolic law. And since a l l the mobile species are presumed to be charged, applied electric fields or currents can be used to significantly alter the thickening kinetics and morphologies of mixed conducting scaling systems. Now i f the scale exhibits a large conductivity for electrons (large compared to the ionic conductivity), massive internal short ing occurs and no coulombic retardation of ion movement is possible. In this situation, scaling proceeds as fast as the unhindered ions can diffuse through the scale. And although the growth kinetics w i l l s t i l l be parabolic, no perceptible open c i r c u i t emf w i l l be found over the scaling layer. This is the limiting case in which electronic conduction dominates in the scale. This limiting case is at the opposite extreme to that in which ionic conduction dominates, as discussed f i r s t . Equivalent Circuit Description of Mixed Conduction in Solids In essence chemical free energy differences cause the ions to migrate during scaling. This is similar to the way batteries induce current flow in resistors. In parabolic scaling the cations mi grate outward to the gas surface while the anions move inwards to ward the metal. The molar free energy μ (often called the chemical potential) for the metal is highest at the metal side but very low at the gas surface, and vice versa for the non metal species Χ or X. Thus, every time a mole of metal migrates out from the inside/ the total free energy of the entire system drops by the amount 2

II

I

μ - u Μ Μ μ

= Δμ M M


1

4.

PATTERSON

Conduction

in

Nonmetallic

Phases

101

Here as elsewhere below, a double prime indicates the gas side at L whereas a single prime quantity refers to the metal side at zero. Thus, the free energy change of equation 1 is a drop because y is very low compared to \x which prevails at the metal interface. By the same token every mole of X passing in the opposite direction changes the free energy inventory by m

m

ι

ii

h Μχ - h y 2

X 2

= -h Δ μ

Χ 2

2

and is also a decrease as μ refers to the gas side. These free energy drops act as the chemical driving forces which cause the ionic migrations to take place in the f i r s t place. They continue to persist but become less and less effective as the coulombic fields build up at the opposite sides of the scale. In any case, we begin to see how an e l e c t r i c a l formulation of the chemical process of scaling might become possible. The actual ion ic migrations amount to internal e l e c t r i c a l currents which are being driven at the expense of the chemical free energy inventory, just as occurs when batteries discharge through a c i r c u i t of resis tors. A successful equivalent c i r c u i t approach to Wagner's theory was worked out by Hoar and Price They developed a simple volt age divider c i r c u i t which gives quantitative formulas for emf and scaling rate that are very similar to those derived more rigorously by Wagner. A linear lumped version of their proposed c i r c u i t is shown in Fig. 3. The subscripts 1, 2 and 3 refer to M cations, X anions and electrons respectively. The voltages and V are given by the expressions


Χ 2

2

v

A

l = - V

V

2

Z

1

= - h Ap

P

X 2

3 /Z F

4

2

Since the valences ζ of the cations and anions are always opposite in sign and the two chemical potential differences are always opposed in scaling, i t follows that both voltages w i l l have the same sign. As a matter of fact, i t can be shown that they are also of precisely the same value for two component scales. This equality, which de rives from the so called Gibbs-Duhem relation between μ-^ ^ ^2' merely means that the two batteries and V are equivalent to, and hence may be replaced by, the single one shown dotted—in on the figure. If voltages and V simulate the chemical driving forces ac ting on the ions during scaling, the resistors and R simulate the scale's resistance to cation and anion migration. Its imped ance to electron flow is represented by the single resistor R 3 which is shown connected a l l the way across the scale thickness. The resistor R 3 simulates the electronic leakage path which weakens the coulombic fields and allows ion migration to continue indefi nitely rather than halting. The usual definitional formulas hold anc

2

2

2


CORROSION C H E M I S T R Y

102

SUBSTRATE METAL M

AMBIENT ATMOSPHERE (ACTUAL X GAS, PARTIAL PRESSURE = p"

SCALING LAYER

2

ab EFFECTIVE

CATIONS

X GAS £

PARTIAL PRESSURE = Pi

•I ANIONS J -


2

I

2

ELECTRONS J -

3

3

χ = L Industrial and Engineering Chemistry, Product Research and Development Figure 2.

Physical arrangement of a metal (M) undergoing scaling (39)

Yl

R

l

h

iCATIONS ν

ιI ! -ι 1

CIRCUIT ANALOG OF SCALE

*ion

^2jANIONS R

2

2

l

ELECTRONS

EXTERNAL LEADS

-ο Ε oSCALE EMF Industrial and Engineering Chemistry, Product Research and Development Figure 3.

DC circuit analog to simulate scaling according to electrochemical theory (39)


PATTERSON

4.

Conduction

in

Nonmetallic

103

Phases

for these resistors R

i -

1 / G

i =

i Ï

1

Ionic and Electronic Conduction in Nonmetallic Phases - ACS

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