Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 1, 1978
19
Fundamental Concepts in the Scaling of Metals John W. Patterson Engineering Research Institute, Department of Materials Science and Engineering, Iowa State University, Arnes, lowa 500 1 1
Fundamental aspects of scale formation and growth on metals exposed to hostile gases at elevated temperature are presented. Quantitative formulas for voltages and thickening rates are derived in some detail by appealing to a simplified voltage divider circuit analog. This model simulates the electrochemical transport phenemona which operate during the parabolic scaling process and is useful for predicting the effects of various other external influences. Comparisons between theoretical predictions and experimental observations are discussed and the comments concerning the value of the theory in the perspective of corrosion protection engineering are offered. The possibility of using scaling theory to develop a better understanding of protection by organic coatings is also left as an implicit suggestion for the interested reader.
Introduction The primary purpose here is to present in simplified form the fundamental aspects of parabolic scaling of metals according to Wagner (1933,1951). An attempt will also be made to demonstrate how fundamental theory can be used to obtain valuable insight into certain other fields of science and technology. Thus, while the growth of a crack-free, adherent scale receives emphasis here, implications of the theory are cited for other applications as well. The effects of variables such as temperature, chemical activity or partial pressure, impurity additions, applied voltages, other metals in contact, etc. will be discussed in conjunction with the theoretical formulas that result and supporting experimental observations will be presented. First, however, a qualitative description of the scaling process will be given along with a simple dc analog (Choudhury and Patterson, 1971; Hoar and Price, 1938; Patterson 1974) to supplement the physicochemical aspects of Wagner’s (1952,1953) theory. Numerous in-depth reviews related to Wagner’s theory are available which provide much more discussion and supporting experimental evidence than can be provided here (Alcock, 1968; Hauffe, 1965; Jost, 1962; Kofstat, 1966; Kroger, 1964; Kubaschewski and Hopkins, 1962; Rapp and Shores, 1970; Wachtman and Franklin, 1968). General Description of Parabolic Scaling When metal is heated in the presence of oxygen, sulfur, or a halogen, a reaction-product compound quickly coats the surface. If this coating is sufficiently dense, and remains adherent, it effectively separates the metal and gas reactants and further reaction-scale thickening-requires the diffusion of substance through the entire thickness of the scale. Therefore, thicker scales grow slower than thin ones, and from this fact alone we can already make two rather profound predictions about the scaling process! The first and most obvious prediction is that oxidation layers on freshly exposed metals will grow a t a rapid rate compared to those which have been exposed for a very long time. A corollary is that spalling and cracking will result in accelerated attack. The second prediction, which deals with morphology, is that compact, adherent scales generally evolve into layers of remarkably uniform thickness, and visual examination studies confirm this prediction. Why is it that variations in thickness are not sustained? Clearly, no local region in the scale can either remain thinner or thicker than any of the surrounding regions simply because the thin regions grow faster and always “catch up” to the slower growing, thicker zones. If this were not the case, thick regions could grow as fast or faster than 0019-7890/78/1217-0019$01.00/0
thinner ones, and grossly irregular morphologies would be the norm. While any number of mathematical relations can be proposed to describe “decreasing rate” growth kinetics, the most important one leads to the so-called parabolic kinetics. In terms of the instantaneous scale thickness L and time t , this reaction rate relation may be written as dLldt = k / L where the parabolic rate constant k is a constant for a given metal, atmosphere, and temperature. Integration of eq 1 leads to the following so-called “parabolic” relationship between scale thickness and time L2 = 2kt
+ Lo2
(2)
where Lo2 is an integration constant. Both k and Lo in eq 2 depend on thermodynamic variables and transport properties (e.g., temperature, partial pressure, scale’compositions, diffusivities in the scale, and so on), but not on t or L explicitly. In the section entitled “Scaling Rate” we will arrive at a theoretical formula for the parabolic rate constant k in terms of transport properties (conductivities) of the scaling layer and will glean from it some rather interesting inferences concerning the scaling process. Figure 1 is a schematic representation of a uniformly thick Ma&, scale adhering to the surface of the substrate metal M. The nonmetal reactant is depicted as a diatomic Xz gas molecule to represent any of the species 0 2 , Nz, Sp, Brp, C12, etc. The chemical potential or activity of the nonmetal species X is highest at the scale/gas interface x = L where the ambient partial pressure P”x2 prevails. It is lowest a t the metal/scale contact at x = 0 where M, Ma& coexistence implies a partial pressure of P f x 2which , is much lower than Pf‘x2.This chemical potential gradient is attended by an X concentration gradient in the same direction. The corresponding concentration and chemical potential gradients for component M in the scale are necessarily directed in the opposite sense. These gradients come about automatically during scaling because incorporation of Xp molecules a t x = L causes the stoichiometry in this region of the scale to be slightly anionrich (cation-poor), but the reverse is true at x = 0 where incorporation of M atoms takes place. A moment’s reflection reveals that the prevailing gradients are directed so as to promote further reaction via diffusional transport through the scale. Thus, the metal atom concentration (or chemical potential) gradient induces cation migration toward the gas atmosphere side while the nonmetal gradient slopes downward as one moves toward the fresh metal substrate. Scale 0 1978 American Chemical Society
20
Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 1, 1978 SUBSTRATE METAL M
EFFECTIVE
1
X p GAS
I
’aXb
-1,
CATIONS J,-
-Iz
ANIONS J 2 c
cI ) ELECTRONS J3--
1-
x i 0
AMBIENT ATMOSPHERE (ACTUAL X z GAS, PARTIAL PRESSURE = P ” )
SCALING LAYER
X
xP
I I
x = 1
Figure 1.
growth, therefore, tends to continue indefinitely but a t a reduced rate as the scale gets thicker and thicker. Strictly speaking, the kinetics of scaling (parabolic or otherwise) cannot be used to determine whether scale growth proceeds by neutral transport or by simultaneous ion and electron transport (electrochemical processes). In any case, it is interesting that Wagner (1933) proposed his purely electrochemical theory a t a time when neutral transport was believed by many to be the dominant mechanism (middle 1930’s).However, in contrast to neutral transport models, the electrochemical theory leads quite naturally to the existence of a measurable emf over a growing scale and explains alterations in scaling kinetics brought about by externally imposed voltages, and contamination or dopant effects based on the charge compensation in the scale. Subsequently reported findings concerning these phenomena have convinced almost all present day workers that they should think in terms of the electrochemical theory for metal scaling. Presumably the protective qualities of organic coatings derive from considerations very similar to those which apply to parabolic scaling layers. In the case of paints, however, it is not certain whether neutral transport or electrochemical transport dominates in general. I t would seem that a clear answer to this question-if one exists-could provide important keys for improving the protective qualities of such coatings. This possibility motivates one to look for hints from Wagner’s theory because it successfully predicts a number of ways to enhance the protective nature of “X-ide” scales, and scales are also interposed between the metal and its environment, just as painted coatings are.
Electrochemical Description of Scaling Virtually all known scales are considered to be ionic a t least in the sense that all their constituent atoms are thought to be ions. Thus, Xp molecules acquired from the atmosphere can be incorporated only as anions, which are negatively charged X ions. Similarly, each M atom incorporated from the substrate becomes a positively charged cation. This has important electrochemical-type consequences because the region a t x = L is forced to become electron-deficient or positively charged. Negative charge is consumed there by the formation of X anions from neutral X2 molecules, but then the anions migrate “down gradient” toward the metal-rich side, carrying off their recently acquired negative charges. Similarly, the metal interface a t x = 0 acquires a surplus of negative charge because electrons are liberated there when anions arrive and M atoms become cations. As a special limiting case, assume that no short or leakage path connects these two zones. Then continued reaction causes the positive charge a t x = L to continue increasing while the negative polarity increases a t x = 0. The resulting coulombic field would only serve t o retard the leftward migration of anions and rightward drift of cations in Figure l, but eventually it would become strong enough to hold all the migrant ions a t bay and all ionic migration would cease. Careful experimental studies a t this point would reveal that
scale thickening has been halted and that a measurable emf-the order of volts-has developed over the scale. However, in practical cases some degree of shorting always occurs because actual scaling layers are found to exhibit a t least a trace of internal conduction by electronic carriers. For this reason, it is perhaps better to think of real scales as mixed (ionic and electronic) conductors. The internal electronic short serves to bleed off some of the stored charge, thereby weakening the coulombic field established under open circuit conditions. Thus, ionic migration is not arrested but proceeds a t a perceptible rate as evidenced by scale thickening. Observations involving parabolic scaling layers confirm this point because they do not achieve the completely arrested state discussed above as a limiting case. Instead, they thicken indefinitely according to the parabolic law of eq 2.
Dc Analog A convenient method for conceptualizing the electrochemical aspects of scaling is afforded by the dc “voltage divider” circuit shown in Figure 2. During open circuit scaling, the cations and anions migrate in opposite directions but they also have opposite charge; thus both transport positive current in the same direction. These two current modes are simulated by the parallel resistors R1 and R2, respectively. R3 simulates the internal leakage path for electronic carriers through the scale. Often we think of diffusion as being driven by concentration gradients. In the present situation, however, it is more expedient to consider the equivalent alternative viewpoint in which chemical potential (Gibbs free energy) differences serve as the effective driving forces for ion migration. These driving forces are simulated by the batteries labeled V1 and V2. Their values in volts are given in terms of the appropriate chemical potentials as follows: for cations, the impetus for migration is provided by the effective voltage VI (3) and for anions the “driving” voltage is given by V2 V2 = - [ p o x 2
- p ’ x 2 ] / 2 Z 2F
(4)
Here pi is the chemical potential (molar Gibbs free energy) of species i in J/mol, Zi is the valence of the corresponding ionic species (dimensionless), the Faraday, F = 96 500 ( C I equiv or JIV equiv), and the primes and double primes refer to conditions a t the x = 0 and x = L sides of the scale, re, > p ‘ ~and ~ , 2 2 is negative spectively. Because p ” 01 close to zero. It is in this sense that tion is a quantitative measure of the ionic conduction capability of the scale and as mentioned earlier, its value can be determined solely from open circuit voltage determinations without any conductivity measurements whatever! This striking fact is evident from eq 18 below which will now be derived with the help of equivalent circuit notions. From formulas 6 through 9, the following expressions for the ionic and total resistances of the scale are readily derived. For Rion,one finds
+
EXTERNAL LEADS
I
E
0
SCALE
A EMF
Figure 2.
A number of interesting features may be noted at this point. First of all, one may simulate the behavior of a virtually exclusive ionic conducting scale by simply letting the value of R3 approach infinity, in which case all current stops flowing as soon as charging through the ionic resistors is complete. In this case, the steady-state voltage over the scale will be identically given by eq 5 . The behavior of a predominantly electronic scale is simulated when R3 is made very small compared to either R1 or Rz, and in this situation the steady-state voltage will be close to zero because of the shorting effect. In both cases, the total corrosion rate or scale growth rate is directly proportional to the sum of the ion currents-namely I1 Zz-but because the scale is under open circuit conditions, this is also equal to -13. We will make use of this fact later. The left and right terminals of the equivalent circuit correspond to the substrate metal and the scalelgas interfaces, respectively, in the actual arrangement. Hence, they simulate the access points where an open circuit emf is to be measured or where an impressed voltage could be applied. It is unfortunate that a current meter cannot be inserted to measure the current inside the actual scale, because this internal current could be translated directly into growth rate as will be shown later.
+
Scale Voltages and Conductivities The partial conductivities of the scale are related to the equivalent circuit resistances as
R
=--
(i = 1,cations; 2, anions; 3, electrons) (6) uiA where the length of L of Ri and its cross sectional area A correspond to the thickness and area of the scale, respectively. These conductivities are often grouped into the ionic, electronic, and total conductivities as
’
Uion
UT
=
U1
= u1
+
a, =
U3
+ +
U3
Uz
(7)
U2
(8) =
Uion
+
4,
(9)
and into the following dimensionless ratios-known as the ionic and electronic transference numbers-which turn up frequently in the electrochemical theory of scaling (10)
and
t=e= e
UT
Ul
+ U0 23 + 6 3
(11)
Clearly, these ratios conform to the “unity” identity tion
+ t , = [tl + t z ] + t3 = 1
(12)
The ionic transference number tion = 1 - t , is a somewhat fundamental property of a given scale and its value can only
and for RT we have
Note that Rion and Re = R3 are arranged in series with the thermodynamic voltage V and that the equivalent circuit of Figure 2 amounts to a simple dc voltage divider. Under open circuit conditions no current can leave or enter the terminals in Figure 2. Hence, we have I1
+ I2 = -13
(15)
and, by inspection of Figure 2, we deduce that
(16) where eq 14 has been used to replace RT. The open circuit emf E over the terminals can now be written either as minus the ZR product for R3, or as V minus the ZR drop over the ionic resistor bank-both approaches will yield the same result. We proceed here with
E = -Z3R3= [$-VI
[2--]
(17)
or, after various cancellations and in view of eq 10, we find
This is the equation alluded to earlier which can be used to determine the ionic or electronic character of the scale’s electrical conductivity. As suggested earlier, direct measurements of qonand U, are not necessary; only measured values of E and independent information on the ambient P x P= P”x2 and on P‘x2 or AGf” are needed to get V from eq 5 . Interestingly, the values of tiondetermined in this way are free from errors in electrode contact areas which often complicate electrical conductivity measurements. However, when independent determinations of tion are made from conductivity measurements, they are generally found to be in excellent agreement with open circuit emf determinations of tion. In addition to contact area problems, it is quite difficult in practice to separate the ionic and electronic conductivities for independent measurement (Wagner, 1957); therefore, values of qonand ue are often inferred from an emf measurement of tion and one total conductivity measurement at each temperature of interest. Specifically, one measures the total conductivity UT of the Ma&, material, as for example with a moderately high frequency ac bridge, and combines the result with open circuit emf measurements. Then the values of uion and U, can be estimated from relation 18 as
22
Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 1, 1978
and
where the quantity bluZ1 has been eliminated in favor of 1/Z2 by invoking the combined charge neutrality-virtual stoichiometry relationship
+
Z ~ QZ2b = 0
because EIV = tion and [V - E ] / V = 1 - tion = t,. Because the ionic and electronic conductivities or transference numbers inferred from the emf method are in excellent agreement with those determined from conductivity measurements directly, this technique is frequently used to screen newly developed candidate materials in the search for solid electrolytes, i.e., for compounds in which tion is virtually unity. Most of these candidate materials consist of exotically doped, “man made” compounds which have never been observed to form a scaling layer on metals. I t may be restated here that these very valuable emf relations would not be available if transport through the compound in question was exclusively due to neutral carriers rather than ions and electrons. The success of Wagner’s electrochemical scaling theory in this regard has played a major role in the presently accepted ionic nature of nonmetallic compounds in general. Moreover, Wagner’s fundamental ideas about parabolic scaling have had impact in other areas of technology which would otherwise seem to be totally unrelated. As but one example, we mention here that eq 18 also serves as the basis for interpreting solid electrolyte emf sensor data. In this important new technological development, a solid electrolyte, for which tion is virtually unity, replaces the Ma& scale and P‘xZis fixed a t a known value on the so-called reference electrode side x = 0 (Rapp and Shores, 1970). Then, in view of eq 5, it follows that an unknown P”x2 can be followed continuously by knowing T (thermocouple) and monitoring E. Remote sensing of oxygen partial pressures in high-temperature hostile environments are now routinely carried out with oxide solid electrolytes and important improvements in high-temperature technology are now under widespread development (Voinov, 1976).
Scaling Rate Returning to the matter of parabolic scaling, we note that scale growth can proceed by either or both of two processes, a t least in principle. Certainly, metal atoms from the substrate could permeate through the scale until they emerge to combine with X 2 gas molecules, thus causing scale growth a t x = L. Alternatively, X 2 gas molecules assimilated as anions a t x = L may diffuse in the opposite direction, thus causing scale advancement into the metal substrate a t x = 0. For reasons to do with the energetics of diffusional transport, usually only one of these will predominate in practice. Hence, practical scaling systems can be distinguished as to whether the ion or the cation is the mobile species. However, to preserve generality in the treatment here, we will proceed by not ruling out either possibility. Since the current through R1 corresponds to cation migration through the scale, 11 is clearly a measure of the rate of growth taking place a t n = L, the scaleIgas interface. In particular, Z1/Z1is the cation arrival rate (in g-mol/s) a t x = L. However, according to the stoichiometry of the M,Xb compound, “a” cations must arrive a t x = L, to cause the assimilation of the “b” atoms or b / 2 molecules of X 2 . Similarly, -12/Z2F is the “molar” arrival rate for anions a t the metall scale interface ( x = 0) where scale growth can also occur. Thus, the total assimilation rate rix in moles of X per second is given by
(22) Also, +13 replaces - [ I 1 I21 in eq 21 because open circuit conditions are presumed. But rix is also related to the thickening rate dLldt, cross-sectional area A, and molar volume V of the scale as follows.
+
A dL/dt = Vrix/b
(23)
Replacing lix with 13IZ2F according to eq 21 and then invoking eq 16 and 5 allows one to solve eq 23 for dLldt. Carrying out the algebra and comparing the result with eq 1 yields the following theoretical expressions for the parabolic scaling rate constant k L dLldt = k = - [V/2bZ2F]V
* ‘JT
or, equivalently
The first bracketed term on the right side of eq 24 is virtually constant for a given scaling situation and cannot be exploited to control or alter the kinetics of scaling. The second term in brackets reflects the effect of the prevailing chemical potential or activity of X 2 gas. Certainly, the driving force for scaling could be eliminated by controlling P”xZ a t the value of P‘x2,and this would indeed make k = 0 and hence stop corrosion. In practice, however, one has to live with the prevailing value of Pr‘x2and so the third term, the conductivity ratio or “conductivity factor”, is the one of central importance here. Several interesting features come to light when one considers the so-called conductivity factor ueuion/u~.For example, if either of the quantities qonor u, is zero, k is likewise zero. Thus, steady-state scaling can proceed a t a finite rate only if neither vanishes even if the other conductivity is very high. This is consistent with the equivalent circuit model because replacement of either Rion or R3 (or both) with an :‘open” in the internal circuit “kills” the internal current and hence rix and dLIdt. Physically, this means that either a totally arresting coulombic field sets itself up over the scale (very high ionic diffusivities in the scale but no internal electronic shorting as discussed earlier) or that the diffusiuities of both ionic species are vanishing small. In real compounds a t elevated temperatures, of course, neither uion nor re is expected to vanish, but they may differ by many orders of magnitude depending on the temperature and/or prevailing Px2. Thus, we are led to consider some limiting cases. For example, if u, >> qon,both u, and UT leave the expression for k , and k then depends only on giantwhich in turn can be written in terms of the cationic or anionic diffusivity. Thus, the thickening kinetics of electronically conducting scales follow the kinetics of ionic diffusion and the slope of a log k vs. 1/T plot should be interpreted in terms of the activation energy for ionic diffusion. This may seem plausible enough, but now consider the other extreme where ionic conductivity dominates. In this case, uion >> u,, and fJion/(rT is sufficiently close to unity that this ratio can be replaced by unity in eq 24 and only g e remains. Thus, we have the interesting notion that electron transport determines the growth of solid electrolyte-type scales, even though ions can move about many orders of magnitude more rapidly in such scales than can the electronic carriers. Imagine how easy it would be to assume that the thickening of ionically conducting scales reflects the diffu-
Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 1, 1978
sivity of some ionic species. That is, one might erroneously try to identify the slope of a log k vs. 1/T plot with the activation energy for cation or anion diffusion in the scale. In reality, however, that slope corresponds to the activation energy for either n- or p-type electronic carriers and has nothing whatever to do with ionic conductivities! Thus, studies of the growth kinetics of ionic scales are studies of electronic transport! Thankfully, a quick experimental determination of E (for comparison with V ) can be used to signal the predominance of ionic conductivity which gives rise to this case. Weight gain often is measured instead of scale thickness when carrying out studies of parabolic scaling kinetics. The results of these studies can nevertheless be used t o obtain k values and these in turn may be compared with values calculated independently from eq 24. The values of T, P”xp,and P’x2 (or AGfO) must be known in addition to aionand getor the latter may be inferred from the scale’s total conductivity UT and emf ,E. These kinds of tests were carried out many years ago for numerous metal. metal “X-ide”) scaling systems which were known to conform to the conditions of parabolic scaling. The comparison between predicted and observed values for some selected scaling systems are shown in Table I. Again, the favorable agreement serves to substantiate the validity of the basic assumptions of the electrochemical theory of scaling. The primary effects of temperature and dopant additions (contamination of the scale by foreign species) have to do with their role in altering the partial conductivities of ionic and electronic resistances of the scale. All the partial conductivities of nonmetallic compounds are extremely sensitive to temperature according to an Arrhenius-type relation
v,
U,
= A, exp(-Qi/RT)
(i = 1 , 2 , 3 )
(25)
Here the activation energy Q L ,like A,, is usually found to be substaptially constant over wide ranges of T and both parameters ordinarily differ significantly for the different modes of conduction. The effects of dopant additions are generally explained in terms of how they alter the various carrier concentrations away from those of the uncontaminated material. Thus, the nature of conduction in any given scale can usually be completely revamped either by incorporation of foreign ions or by changing the temperature regime. To a much lesser extent, changes can also be brought about by altering the ambient P x 2 . In a rather broad sense, temperature may be considered the most important variable because for most all M,Xb compounds, the so-called extrinsic conductivity behavior due to dopant additions can often be overpowered in favor of intrinsic electronic-type conduction by merely increasing temperature. Of course, a change of structure can occur if a high solubility for the foreign species exists but temperature can also induce phase changes. Thus, dopant additions and temperature are thought to be the most important with P x p generally having the least significant effect on the conductivity properties.
23
At one time, it was thought that Wagner’s theory might find widespread use among corrosion engineers interested in lowering the scaling rate of metals. For example, if the scale exhibits very high levels of ionic transport, suppressing its electronic conduction (as for example, by suitable dopant additions) could provide the key to lowering its growth rate. However, if it is a semiconductor or exhibits metallic-type conduction, the same end is achieved by suppressing ion migration. The idea would be to alloy the substrate metal with the dopant species of interest in hopes that it will get incorporated into the scale as attack proceeds. Isolated successes have been reported in this regard and the results have been extensively reviewed by Kubaschewski and Evans (1962), Hauffe (1965), and Kofstad (1966) among others. But Wagner’s theory is of more academic than practical interest for corrosion engineers, primarily because the number of practical systems which also conform to the prerequisites for long term parabolic scaling are rather few in number. Moreover, the alloying that is necessary can have other major effects on the metal’s properties. However, the rather spectacular successes of the theory not only confirm the validity of the basic ideas, but also help to expand the base of fundamental information concerning the defect nature of solids in general and how to control it. In the latter regard, it is interesting to reflect on the influence these early successes might have had in leading to the present day understanding of solid-state electronic devices, such as metal oxide semiconductor-field effect transistors (MOSFET’s) among others.
Some Additional Implications of the Electrochemical Theory of Scaling A number of rather broad avenues of inference emerge from the electrochemical theory of scale growth. Inferences for open circuit conditions can be discussed either in terms of the dc analog circuit or in terms of the foregoing equations for E and for the kinetic rate constant k , eq 18 and 24, respectively. On the other hand, other important effects such as those due to applied voltages or contacting metals can also be discussed. Many other effects could also be rationalized on the basis of electrochemical scaling ideas, but the scope and intent of the present treatment demands that attention be restricted to only a few cases. Let us now consider the kinds of morphology effects that can result from external shorting (Ilschner-Gensch and Wagner, 1958).Figure 3 shows a cutaway view of a prototype M(M,Xb scaling system in contact with graphite (which acts as an inert metal short across the scale) and with a hostile atmosphere containing X2 gas. We will skip case (a) for the moment until after (b) and (c) are discussed. In case (b) the scale has a very high cationic conductivity compared to those of the anions and electronic carriers and scale growth occurs a t the X2 gas side of the scale. In the regions removed from the graphite short, the scale is very thin because its growth is limited by the trace electronic conduc-
Table I. Calculated and Measured Rate Constants for the Formation of Metal-Nonmetal Compounds a Rational rate constant equiv x cm-1 s1 Metal Nonmetal Compound 0 “C kcalcd kobsd Ag S (liquid) 2.4 X 1.6 X Ag2S 220 CUI 195 3.8 x 10-lo 3.4 x 10-10 cu 12 (gas) AgBr 200 2.7 x 10-l’ 3.8 x lo-” Ag Brz (gas) cu 0 2 ; p = 8.3 X atm cuzo 1000 6.6 X 6.2 x 10-9 cu 0 2 ; p = 1.6 X atm cuzo 1000 4.8 x 10-9 4.5 x 10-9 0 2 ; p = 2.3 X atm cu cu20 1000 3.4 x 10-9 3.1 x 10-9 cu 0 2 ; p = 3.0 X atm cuzo 1000 2.1 x 10-9 2.2 x 10-9 Kubaschewski and Hopkins (1962), p 77.
24
(0)
Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 1, 1978 X, GAS
X, GAS
MX2 CAL
MX2 CALE
Stole i s on electronic conductor
b) ~
c a ~i s e ~1 cationic
conductor
(5) scale is on anionic conductor
Figure 3.
tivity. Along the graphitehcale interface, however, the scale grows much more rapidly. The advancement rate along this interface is not limited by electron transport through the scale because electrons are shunted out to the advancing front via the graphite short. The graphite also prevents any coulombic field buildup along this interface so that the cations migrating along this path are free from the retarding fields set up in the other regions of the scale. Because the electrons cannot migrate through the gas atmosphere, the locally accelerated growth takes the form of a very thin sheet advancing rapidly outward. In case (c) the scale is a predominant anion conductor so that growth takes place a t the substrate interface. Again electrons limit the kinetics in the regions far removed from the graphite because Xz molecules cannot get incorporated into the scale until electrons arrive. Because of the graphite, however, an abundant supply of electrons always exists at the line contact shared by the scale, the gas phase, and the graphite. Incorporation along that three-phase contact occurs at a very rapid rate as the anions migrate swiftly inward along the shorted graphitehcale interface. Because the electronic short path extends to all points on the underside of the scale, advancement into the metal substrate does not produce the thin "pancake" morphology of case (b). Instead, a more toroidal-like geometry is expected. The explanation of case (a) is now very clear. Because the scale is already dominated by electronic conduction, the scaling rate is limited only by ion migration. The presence of an additional electronic shunt can do nothing to increase the drift velocities of the slow moving ions and so no advantage is gained. Thus, scale growth is not accelerated along the scale/graphite interface and no special morphological anomalies occur in its proximity. Considerations similar to those just outlined can in some cases be used to analyze reaction-product morphologies at the electrodes of solid state battery and fuel cell systems. In contrast to the foregoing external short effects, one may also consider the effect of applying a voltage to the terminals by means of an external supply. An important question has to do with the possibility of being able to halt scale thickening by applying such a voltage. By inspection of the dc analog one realizes that I 1 I2 can be made to vanish by applying a voltage which is equal to V in magnitude but opposite in sign. In other words, bucking off the internal thermodynamic voltage V arrests the currents I I and I 2 and this amounts to halting corrosion! The price one pays for this is the cost due to current drain through RB.The power drain in watts is given by V2/R3 which would probably be trivial for most solid electrolyte scales but not for those in which electronic conduction predominates because R3 would then be a very small value.
+
Final Comments The foregoing is an attempt to draw some of the interesting implications of parabolic scaling theory to the attention of those interested in other forms of protective coatings. The author claims no expertise whatever in the area of organic coatings and paint technology but feels that some of the basic ideas from scaling theory might be relevant. Perhaps some of the experimental methods used in scaling studies could prove useful in studying the nature of paint films. For example, open circuit emf studies or transport under applied voltage conditions might provide valuable information for understanding and controlling the protective qualities of organic films. The morphological evolution of scales has also been discussed, even though no direct counterpart is expected for paint films. In contrast to the case of scaling layers, mass transport through paint films should in no way alter their thickness. However, it is doubtful if any prepared metal surfaces can be regarded as free of oxide and very often scale formation is deliberately induced to improve paint adherence. Thus, it was assumed that the factors which govern the morphology or roughness of oxide scales would be of some interest to the organic coatings community. Note Added in Proof. The author has been informed of the recent death in Goettingen, West Germany, of Carl Wagner, upon whose work the entire subject of this paper is based. Recognition is hereby given to the remarkably broad spectrum of his contributions in the areas of physical chemistry, corrosion, and metallurgy. Literature Cited Alcock, C. B., Ed., "Electromotive Force Measurements in High Temperature Systems: Proceedings of a 1967 Symposium", American Elsevier, 1968. Choudhury, N. S.,Patterson, J. W., J. Electrochem. SOC.,118, 1380 (1971). Hauffe, K., "Oxidation of Metals", Plenum Press, New York, N.Y., 1965. Hoar, T. P., Price, L. E., Trans. Faraday SOC., 34, 867 (1938). Ilschner-Gensch. C., Wagner, C., J. Electrochem. SOC., 105, 198 (1958). Jost, W., "Diffusion in Solids, Liquids and Gases", Academic Press, New York, N.Y., 1962. Kofstat, P.. "High Temperature Oxidation of Metals", Wiley, New York, N.Y., 1966. Kroger, F. A,, "The Chemistry of Imperfect Crystals", North-Holland Publishing Co., Amsterdam, 1964. Kubaschewski, O., Hopkins, 6. E., "Oxidation of Metals and Alloys, Butterworth, London, 1962. Patterson, J. W., in N. M. Tallan, "Electrical Conductivity in Ceramics and Glass", Chapter 8, Marcel Dekker, New York, N.Y., 1974. Rapp, R. A., Shores, D. R., "Techniques in Metals Research", Part 2, Vol. IV, Runshak, Ed., Interscience, New York, N.Y., 1970. Voinov, M., in "Electrode Processes in Solid State lonics", M. Kleitz and J. Dupsey, Ed., p 431, NATO Advanced Study Institute Series, D. Reidel Company, 1976. Wachtman, J. B., Franklin, A. D.,Ed., "Mass Transport in Oxides, Proceedings of a 1967 Symposium", National Bureau of Standards Special Publication 296, US. Government Printing Office, Washington, D.C., 1968. Wagner, C., 2.Phys. Chem., 321,25 (1933). Wagner, C., in "Atom Movements", ASM, Cleveland, Ohio, 1951. Wagner, C., J. Electrochem. SOC.,99,346C (1952). Wagner, C., J. Phys. Chem., 57, 738 (1953). Wagner, C., "Proceedings of the International Committee of ElectroChemical Thermodynamics and Kinetics (CITCE), 7th Meeting, Lindau, 1955", p 361, Butterworths, London, 1957. T h e author gratefully acknowledges t h e financial and technical support provided b y the Iowa State University Engineering Research I n s t i t u t e d u r i n g t h e preparation of this manuscript and for similar support of previous research related t o parabolic scaling theory and solid electrochemistry. T h i s paper i s based o n a talk presented a t the 173rd N a t i o n a l Meeting o f the American Chemical Society, N e w Orleans, La., M a r 20, 1977.
