# Impact of Grain Boundary Density on Oxide Scaling Revisited

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## Abstract

A straightforward conceptual tool for discriminating between different oxide scaling processes deviating from the parabolic standard model is formulated. Grain boundary diffusion-controlled oxide scaling is generalized to include lateral grains coarsening. Building on traditional Wagner theory, attenuation of rates of inwards growing oxides owing to the gradual loss of grain boundary density is revisited. Two viable cases are identified. One has the rate of grain boundary density loss to be independent of the rate of oxide growth, while the second case takes the two instantaneous rates to be equal. Simple parabolic–logarithmic and superparabolic–cubic expressions are arrived at for the two cases, respectively. Usefulness is demonstrated by applying the models to published experimental data from 1990 to date. Upon arrival at the superparabolic–cubic behaviour, a generic mathematical form analogous to a ‘spring force’ attenuating the scale growth was identified. ‘Parabolic’, ‘cubic’ and ‘logarithmic’ scaling emerges as limiting cases.

## Keywords

Oxidation kinetics Model Scaling Grain boundary transport## Introduction

For several decades, the outstanding properties of oxide dispersion strengthened alloys have served as a vehicle for fundamental research at the forefront of industrial materials development of relevance to chemical engineering and energy conversion processes [1, 2]. The efficient generic high-temperature oxide scaling of alumina-forming ODS alloys—i.e. slow growing and well adhering—apparently contradicts their complex and diverse compositions and textures. However, it should be kept in mind that the complexities of said class of alloys—they comprise a base metal, e.g. Co, Fe(Ni) or Ni(Co), a scale-forming component (Al) and commonly chromium providing the third element effect [3, 4, 5, 6]—are actually *tuned* to achieve the sought near-ideal long-term scaling property. Additional fine-tuning, mainly aiming at improved scale/alloy adhesion, is achieved by the so-called reactive elements—e.g. Y, La, Zr, Hf—often appearing as submicron oxide particles in the ODS alloy [7, 8]. Beneficial impact has been correlated with change in electronic structure of oxide grain boundaries [9], and avoidance of scale spallation due to sulphur decoration at alloy–oxide interface by REs acting sulphur getters in the alloy [10]. Currently, the origins of often near-ideal oxidation properties at high temperatures are increasingly being unravelled and revisited by employing state-of-the-art analysis on the nanoscale in conjunction with atomistic first-principles modelling [11].

Inevitably, such emerging novel perspectives attract renewed interest in generic models for the transformation from early transient oxide formation to late ideal oxide growth kinetics in order to offer a semi-quantitative conceptual framework for alloys development in general. Building on Fickian diffusion, today still, the contemporary overall understanding of the oxidation of metals is based on the groundbreaking works of Tammann [12] and Wagner [13]. Also the decisive impact of grain boundaries, constituting easy paths for ion diffusion has long been recognized [14, 15, 16]. A recent attempt at connecting oxide grains coarsening to the oxide scaling as measured by inspecting oxide scale cross sections [17, 18] has served inspiration to the present effort. Thus, the objective of this study is to expose generic origins of near-ideal oxidation properties, so as to distinguish between beneficial oxide scaling and corrosion. To this end, two models are derived representing two different physical realities in that a particular alloy is taken to obey *either* ‘superparabolic—cubic’ *or* ‘parabolic–logarithmic’ short-term–long-term oxide scaling kinetics. Experimental data, from 1990 till present, form basis for the following discussion. As a consequence, no statistical analyses are provided here. It is, however, greatly encouraged, in conjunction with modelling, in view of the current limited predictive capacity of alloy development beyond empirical knowledge of the oxidation behaviour [19].

## Modelling Oxide Grain Coarsening During Scale Growth

The present effort relies on a clear distinction between corrosion and oxide scale formation. First and foremost, the formation of an adherent oxide scale acts to prevent breakaway corrosion. Also, sacrificing metal to form a barrier oxide often offers the alloy protection against a hostile environment. And while corrosion is often specific to the particular application, we understand the build-up of a protective oxide scale to be more generic in nature. In this spirit, below we explore to what extent grain boundary transport alone—as attenuated by change of grain boundary density exclusively taking place at the mobile reaction zone—may suffice to describe experimentally observed oxide scale growth kinetics. Thus, it is implied below that first-order kinetics in loss of grain boundary density *alone* is indeed insufficient in reproducing experiment, owing to unphysical limiting properties. This is in contrast to the two expressions at focus in the present study, emerging from second-order kinetics.

*δ*

_{GB}is the width of a grain boundary and \(r_{E} \left( t \right)\) is the lateral length of a growing grain. For the diffusion flux, it follows that

*exclusively*at the moving oxide front. It is discretized according to the so-called low-gradient approximation

*X*is the instantaneous scale thickness. Equation 2b may subsequently be converted into a scale growth velocity by writing

*V*

_{O}is the specific volume of the mobile species, such that

*C*

_{O}

*V*

_{O}= 1. The differential equation to solve becomes (see, for example, [11])

*cf*. Equation 1)

*C*

_{GB}on scale growth to exhibit first-order kinetics,

*C*

_{GB}(

*t*) =

*C*

_{GB}(0)e

^{−kt}, and hence, \(f\left( t \right) = f_{0} {\text{e}}^{ - kt} = \frac{{2\delta_{\text{GB}} }}{{r_{\text{o}} }}{\text{e}}^{ - kt}\). In this case, Eq. 4 becomes

*D*

_{GB}

*f*≫

*D*

_{L}(1 −

*f*). Indeed, the second fundamental assumption of the present study is that oxide scaling owing to lattice diffusion is vanishingly small. This is in order to unravel the impact of grains coarsening on subparabolic limiting scale growth, ubiquitous to oxidation of metals. Inasmuch as Eq. 5b displays parabolic limiting scale growth owing to the lattice diffusion channel, in its absence, the first-order kinetics model

*t*→ ∞. Here, we proceed by considering second-order kinetics \(\frac{{{\text{d}}C_{\text{GB}} }}{{{\text{d}}t}} = - kC_{\text{GB}}^{2}\) for the decay of grain boundary density, as was originally proposed by Fehlner and Mott [16]. They obtained \(C_{\text{GB}} = \frac{{C_{\text{GB}}^{0} }}{{1 + C_{\text{GB}}^{0} k \cdot t}} = \frac{1}{{r_{0} + k \cdot t}}\), i.e. \(f = \frac{{2\delta_{\text{GB}} }}{{r_{0} + k \cdot t}}\) for

*r*

_{G}(

*t*) =

*r*

_{0}+

*k*·

*t*. Furthermore, as a consequence of

*D*

_{L}→ 0, Eq. 4 can be modified to read

At this point, our deduction bifurcates as we may have the grains coarsening to be either implicitly or explicitly connected to the rate of scale growth (see the “Parabolic–Logarithmic Oxide Growth Mode” and “Superparabolic–Cubic Oxide Growth Mode” sections, respectively).

### Parabolic–Logarithmic Oxide Growth Mode

*k*= 0 constitutes the parabolic oxide growth limit. Hence, for

*k*≠ 0 we write

By taking this term to represent the potential energy of a classical spring, the parabolic–logarithmic scaling Eq. 8a may intuitively be understood to result from a ‘spring force’, which resists parabolic scale growth. Indeed, rapidly decelerating scale growth may imply a significant long-term remnant (electro-)chemical potential gradient across the scale, which may in turn leave the scale vulnerable to scavenging local (electro-)chemical diffusion processes (see, for example, Geers et al. [20]).

The latter tells of oxide scale growth by grain boundary diffusion in the absence of grains coarsening.

### Superparabolic–Cubic Oxide Growth Mode

*t*

_{0}, i.e. the initial curing time \(t_{0}^{c}\) for the grains coarsening that attenuates the scale growth. Also, we take

*α*= 1 to say that its influence is absorbed by the material specific

*D*

_{GB}2

*δ*

_{GB}. By inserting Eq. 10 into Eq. 3, we arrive at the corresponding time evolution of scale thickness

Linking the grains coarsening rate to that of the scale growth puts Eq. (11) on par with the parabolic \(t^{1/2}\) growth mode, in that besides the curing time *t* _{0} ^{ c} no additional parameters are introduced.

*t*≪

*t*

_{0}

^{ c}we have the superparabolic \(X \propto t^{{\frac{2}{3}}}\), i.e.

*t*

_{0}

^{ c}.

*t*). Thus, by differentiating Eq. 11 with respect to time, and for

*t*> 0 we have

*t*

_{0}

^{ c}≪

*t*, we obtain \(r_{\text{G}} \propto t^{{\frac{1}{3}}}\) independent of

*t*

_{0}

^{ c}

*t*<

*t*

_{0}

^{ c}the grain boundary “curing” is controlled by the rapid initial oxide growth rate. Thus, from Eq. 16 it is inferred that \(\frac{{{\text{d}}X}}{{{\text{d}}t}}t_{0}^{c} ( \equiv r_{0}\), see Eq. 10) → ∞ for

*t*→ 0. Here, contact can be made with lattice diffusion driven single-crystal growth (see Eq. 1) in that this suggests the existence of an initial grain boundary free oxide film, which initially at

*t*<

*t*

_{0}

^{ c}becomes increasingly fine grained as the oxide scale starts growing [21]. Thus, by allowing only grain boundary transport, the grain boundary density is forced to initially

*increase*owing to the retardation term in

In particular, it is noted that the top angle starts at 180° at *t* = 0 and also that the limiting ratio \(\frac{{r_{G} }}{X}\left( {t \to \infty } \right) = \frac{1}{3}\) reflects a limiting top angle of ~ 18.9°. At \(= \frac{{5t_{0}^{c} }}{4}\), i.e. where *r*_{G} = *r* _{G} ^{ c} , the ratio \(r_{\text{G}} /X\) reaches unity, corresponding to top angle of ~ 53.13°.

It is gratifying to note (1) how the two central asymptotes of the superparabolic–cubic model, Eqs. (12) and (13), come out clearly in Eq. 19f, as well as (2) the crucial role of *t* _{0} ^{ c} as it controls the transition between said two asymptotes.

## Results and Discussion

By comparing to the experimental literature, here we demonstrate the possible validity of the two fundamental assumptions—the predominance of grain boundary transport and the location of grains coarsening processes to the mobile reaction zone, vide supra—as well as the potential usefulness of the emerging conceptual understanding.

*X*and specific mass gain Δ

*m*is made according to

*effective grains coarsening*is introduced in the “‘Effective’ Grain Boundary Density in Oxide Scaling” section in order to demonstrate the versatility of the approach.

### Application of Superparabolic–Cubic Model

*t*

_{0}

^{ c}on the time evolution of oxide scaling. Excellent agreement with the mass gain curve by Quadakkers et al. [22] is obtained for the alloy PM2000 at 1200 °C in air (Fig. 3), by setting the prefactor to be 0.245 mg/cm

^{2}in Eq. 21 for

*t*

_{0}

^{ c}= 1 h, i.e. the scaling behaviour for

*t*

_{0}

^{ c}= 0.1 h; 0.5 h; 1 h; 10 h is shown in Fig. 3.

While initial scaling depends on the initial curing time (see Eq. 12), it is noted that the late asymptotic form is independent of *t* _{0} ^{ c} (see Eq. 13). This says that the prefactor in Eq. 21 is independent of the initial curing time, exhibiting only the temperature dependence of the grain boundary diffusivity and width, and the chemical potential difference, determined by gas and alloy compositions. Extracting this prefactor from finite time experiments, however, requires knowledge of initial conditions related to sample preparation for which there is limited control. By employing \(t_{0}^{c}\), the initial conditions may be compensated for, allowing the prefactor to be extracted from finite time exposure experiments. Moreover, there are cases where long-term exposures may not even be feasible. One example is the oxide scaling on zirconium alloys by water, for which the barrier oxide becomes reset by repeated breakdowns, empirically found to follow intermittent ‘cube-root kinetics’ in line with the present superparabolic–cubic model [23].

_{2}in Ar (oxidizing conditions), as well as 4% H

_{2}, 7%H

_{2}O in Ar (reducing conditions). Good agreements with the superparabolic–cubic scaling model are obtained in both cases (see Fig. 6a, b). Employing the equilibrium data of Young et al. [18] concerning oxygen activities at metal–oxide interface and at the oxide–gas interface, slightly modified values of \(\delta_{\text{GB}} D_{\text{GB}}\) as compared to that study are obtained for both the oxidizing and the reducing conditions (see Fig. 6c).

*t*

_{0}

^{ c}= 10 h (see inset in Fig. 8). For reference,

*t*

_{0}

^{ c}= 1 h; 20 h are also included in Fig. 8a. The large curing time (

*t*

_{0}

^{ c}= 10 h) compared to that at 1200–1300 °C (

*t*

_{0}

^{ c}= 1 h) may be owing to the lower temperature. We note that the parabolic–logarithmic model, according to Eq. (22), also provides excellent agreement for the duration of the 250 h exposure experiment (see Fig. 8e). Arbitrarily, for this model the rate of grains coarsening is taken to be 1.0 nm/h throughout the presentation. At long times, however, the logarithmic function detaches from the superparabolic–cubic model as expected (see Fig. 8d). In as much as longer exposure times are required in order to discriminate between the two oxide scaling models, here, we envisage that possible transitions between grains coarsening modes may be observed as well, being either coupled to or decoupled from the rate of oxide scale growth. In view of lack of knowledge regarding the true scaling kinetics in this case, the models may be employed to estimate best-case and worst-case mass gain scenarios.

### Application of Parabolic–Logarithmic Model

*t*

_{0}

^{ c}for the failing superparabolic–cubic model is 0.001 h. And in spite of early access to its cubic asymptote, it is insufficient in describing the rapidly flattening out of the experimental curve (see Fig. 10). Clearly, this behaviour is easily reproduced by the parabolic–logarithmic model.

We observe that the initial oxide grain boundary densities of the two specimens come out different, i.e. smaller initial grain width for the coarse-grained alloy *r*_{0} = 63 nm as compared to the *r*_{0} = 150 nm for the fine-grained alloy. This is in turn reflected in higher initial mass gain rate in case of the coarse-grained alloy (see Fig. 11b). It is tempting to suggest preference for the parabolic–logarithmic model, particularly for the coarse-grained PM2000, to reflect the fact that initially, the oxide grains coarsening rate cannot keep up with the rate of scale growth. It is conceivable that the ability to alleviate stress by allowing for formation of voids and pores at the moving alloy–oxide interface (cf., for example, Ref. [25]) is indeed a common feature when parabolic–logarithmic scaling is observed. Such scavenging processes would be symptomatic for the residual (electro-)chemical potential gradients owing to the rapid slowdown of oxide growth as represented by the effective ‘spring force’ attenuated oxide scaling (cf. Eqs. 3 and 8c).

### ‘Effective’ Grain Boundary Density in Oxide Scaling

_{2}–1%O

_{2}[29] is particularly instructive. TG reports dramatic suppression of oxidation rate under humid conditions, while microscopy reports no change in grain size distribution. Fundamentally, transport paths may be sealed either by explicit grains coarsening or by the ‘clogging’ of grain boundaries. Indeed, hydrogen residing in interstitial sites in the form of hydroxide ions was suggested in [29] to restrict transport in the crystalline grain boundaries. And yet both grains coarsening models developed here are able to describe the early subparabolic signatures of the TG data provided in [29] (see Fig. 12a, compare Fig. 8c–e). We arrive at the understanding that the continuous loss of ‘effective’ grain boundary density as reflected in the transport properties disagrees with a quasi-static poly-crystalline oxide microstructure. The apparent grain growth, which emerges from the TG, is taken to imply that easy ion diffusion paths continuously become sealed off by the hydrolysis process, e.g. by ‘glassy’ subnanointerfaces between grains being formed. This straightforward interpretation for wet conditions is contrasted by the dry exposures. Thus, the parabolic–logarithmic model cannot describe the observed ‘superparabolic’ scaling (see Fig. 12b). For this, only the superparabolic–cubic model is applicable (compare Eq. 12 (superparabolic) and Eq. 9b (parabolic)). Thus, in case of 1-mm specimen, 10 h curing time is employed (see Fig. 12b). Hence, the ‘waist’ in the grains coarsening evolution as function of scaling ends up at 12.5 h (compare Fig. 2c and Eq. 18b). The impact of this comparatively large curing time in relation to the 25-h duration of the experiment is not insignificant. This is further accentuated in the modelling of dry oxidation of the 0.25-mm specimen where the \(t_{0}^{c} \gg t\) limit becomes even more relevant (see Fig. 12b again). Here the ‘waist’ is at 125 h, i.e.

*t*= 1.25

*t*

_{0}

^{ c}for

*t*

_{0}

^{ c}= 100 h. While such scaling phenomenology at early stages is commonly hidden or deemed inaccessible, the superparabolic–cubic model offers a semi-quantitative overall contextual understanding. Failing to describe even qualitatively the scaling process of the 0.5-mm specimen under dry conditions may point at critical phenomena also present at 1000 °C, and not captured by the simple model. Having said this, it is gratifying to note how well the phenomenological second-order kinetics approach of Fehlner and Mott [16] as realized in the superparabolic–cubic model is able to capture essential aspects of the underlying complexities of the chromia formers, mainly under wet conditions but to some extent also in dry.

## Concluding Remarks

A simple, yet comprehensive approach to oxide scaling was described with emphasis on the impact of oxide grains coarsening on the oxidation of high-temperature alloys. It is based on two fundamental assumptions: Firstly, the scaling process is governed by grain boundary diffusion alone, and secondly, grains coarsening occurs exclusively at the moving oxide front. First- and second-order kinetics in loss of grains boundary density were considered, but in the light of said assumptions, only the latter was deemed physical. Expressions for oxide scaling were formulated, and essential aspects emanating from the lateral grains coarsening rate being connected to—or disconnected from—the rate of scale growth were articulated. Asymptotic forms of the two mutually excluding models were deduced and employed to discuss experimental scaling studies in the literature. The potential usefulness was illustrated by providing new estimated values of \(\delta_{\text{GB}} D_{\text{GB}}\) from experimental scaling experiments. The term ‘effective’ grains coarsening was introduced for chromia scaling in wet and dry N_{2}–1%O_{2} in order to explain subparabolic suppression of oxide growth in case of the former in spite of little difference in grain size distribution between the two. The phenomenology inherent in the superparabolic–cubic model was employed to articulate early scaling of chromia in dry N_{2}–1%O_{2}.

Arguably, the degree of synchronization between rates of scale growth and oxide grains coarsening may change during scaling, early between superparabolic \(X \propto t^{{\frac{2}{3}}}\) or parabolic \(X \propto t^{{\frac{1}{2}}}\), and late between cubic \(X \propto t^{{\frac{1}{3}}}\) or logarithmic \(X \propto \sqrt {\ln \left( t \right)}\). Yet, it should be born in mind that, inasmuch as both grains coarsening models display late subparabolic scale growth, in the absence of scale breakdown the parabolic lattice diffusion channel is expected to eventually prevail. Having said this, scavenging processes driven by residual (electro-)chemical potential gradients might disallow this limit to be reached (compare, for example, the cyclic barrier oxide breakdown prevalent for zirconia formers [30, 31]).

In summary, a conceptually straightforward and practical modelling approach is provided that allows discrimination between different oxidation mechanisms and the extraction of scaling kinetics information from mass gain experiments, benefiting future materials development, not least by identifying deviations from the ideal. Moreover, the ability to connect oxide scaling kinetics with activation energies for diffusion offers a bridge between experiment and first-principles atomistic model studies, to allow the pinpointing of the decisive transport channels for future tailored control.

## Notes

### Acknowledgements

This work was carried out within the Swedish High Temperature Corrosion Center (HTC).

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