Supervised by: A. De Vita, J.R. Kermode, O.A. Von Lilienfeld

Studying stress-corrosion in silica: taking multiscale simulations to the petascale

Marco Caccin

Open questionsH2O dissociation - crack advance in SiO2

J. Phys. D: Appl. Phys. 42 (2009) 214006 M Ciccotti

Figure 6. Effect of temperature on the crack propagation insoda-lime glass (from Wiederhorn and Bolz (1970)).

For the same reasons, the region I data can also be fitted bya different exponential equation proposed by Pollet and Burns(1977) and applied to glasses by Lawn (1993):

vI = v0 exp(αG) = A exp!

−"Ea

kBT

"exp

!−α(G − G0)

kBT

",

(3)

where the energetic balance has a more central role, as will bediscussed later concerning the threshold at G0.

The Wiederhorn equation (1) has been the most used in thedescription of region I in glasses since it provides a more directinterpretation of the chemical reactions between a glass of agiven composition and the environment at the crack tip. Inthis framework, the basic mechanism of the stress-corrosionreaction was associated with the stress-enhanced thermalactivation of a dissociative hydrolysis reaction representedin figure 7 for the case of silica glass (Michalske andFreiman 1983). Michalske and Bunker (1984) supported thepositive effect of stress on the reaction rate by a molecular-orbital simulation of the reaction on distorted siloxane bonds.They later provided experimental support for their theory bymeasuring the rate of hydrolysis of a series of strained silicatering structures (Michalske and Bunker 1993). For other glasscompositions (figure 8), several other reactions are possible,but they share the similar feature of being stress-enhanced andthermally activated, thus explaining the exponential (Arrheniuslike) term in equation (1), where "Ea can be interpretedas an activation enthalpy in the absence of stress and b canbe expressed in terms of an activation volume (of moleculardimensions) by relating the stress intensity factor to the cracktip stress (Wiederhorn and Bolz 1970). According to thechemical reaction rate theory (Glasstone et al 1941), thepreexponential term represents the activity of the reactants inthe moist atmosphere and the exponent m (generally close to

Figure 7. Basic mechanism of the stress-corrosion reaction (fromMichalske and Freiman (1983)).

1) is related to the nearly first order of the stress-corrosionreaction (Wiederhorn et al 1974).

For propagation in liquid environment, the term p/p0

is substituted by the activity of water molecules whichis strongly dependent on the composition of the liquidenvironment. Notably, it is strongly affected by the pHand ionic concentration of some species related to the glasscomposition (Wiederhorn and Johnson 1973). Apart fromwater, other molecules that are effective for stress-corrosionsuch as ammonia, hydrazine and formamide have in common(a) the capability of donating both a proton and an electron atthe two ends of a molecule and (b) a molecular diameter inferiorto 0.5 nm (Michalske and Bunker 1987). The first property isnecessary to present a concerted reaction of adsorption andscission of the siloxane bridges of the glass network (figure 7,bottom). The small diameter is necessary for the reactivemolecules to be able to reach the strained bonds at the crack

6

✦ Ciccotti, M. (2009). Stress-corrosion mechanisms in silicate glasses. Journal of Physics D: Applied Physics, 42(21), 214006.✦ Wiederhorn, S. M. (1967). Influence of Water Vapor on Crack Propagation in Soda-Lime Glass. Journal of the American Ceramic Society, 50(8), 407-414.

J. Phys. D: Appl. Phys. 42 (2009) 214006 M Ciccotti

When describing the macroscopic fracture properties of amaterial, it is the relation of the external variables G, K whichis studied as a function of v. Even if assuming a uniquelydefined kinetic law v(G∗, K∗), these local variables remainhidden, and the presence of a process zone will cause the non-uniqueness of the macroscopic fracture laws v(G, K). Thisinduces a time- or crack-length dependent behaviour (R-curve)that may lead to controversial interpretation (Lawn 1993).These types of modelling will be shown later to be particularlyeffective in rationalizing the propagation threshold behaviour(cf section 3.4).

As a concluding remark to this section, it is generallyuseful to identify the different space, time and energy scales ofeach symbolic block (cf figure 2) in order to estimate whetherthe different dissipation mechanisms will have time to activateand how they will influence the energetic balance or the fracturepropagation laws. Several spatial scales are present in theproblem, such as the sample lateral dimensions and thickness,the loading displacement, the crack length, the basic grainsize of the material (molecular rings for glass), the size of theprocess zone (or of each shell of a multiple process zone), theenclave size, the length of the one or several cohesive zones inthe crack cavity. Typical time scales are given by the loadingor displacement rate, the crack velocity, the stimulation timeof each equivalent block and its characteristic relaxation time,the rate of thermal activation of different mechanisms, the rateof transport of different relevant chemical species in the bulkor in the crack cavity. We will try to keep these space and timesscales in mind while proceeding in the description of the crackpropagation kinetics in glasses and especially when trying toget a deeper insight into the stress-corrosion mechanisms.

3. Crack propagation kinetics

A huge variety of possible oxide glasses may be created bymelting together variable amounts of oxides, within specificcompositional ranges that are strongly influenced by thequench speed (cf Zarzycki (1991)). The details of the chemicalreactions and the competition between different propagationmechanisms will be altered, but many typical features of crackpropagation remain substantially similar. The most influentialdifference is between glasses essentially made by network-former oxides (SiO2, GeO2, B2O3, P2O5, etc) and glassescontaining significant amounts of modifier oxides (Na2O, K2O,CaO, etc). For that reason in the following discussion silicaglass will mainly be described as a representative of the firstclass and a typical soda-lime glass will be used as a paradigmto represent the second class of glasses.

The subcritical fracture propagation properties of glassesare generally studied using samples that allow the stable slowpropagation of a single fracture such as the double torsion(DT) (Evans 1972), the dual cantilever beam (DCB) (Srawleyand Gross 1967) or the double cleavage drilled compression(DCDC) (Janssen 1974). The measurement of the crackpropagation velocity as a function of the applied stress intensityfactor generally presents three (or four) characteristic regions,figure 4.

Figure 5. Effect of humidity on the crack propagation in soda-limeglass (from Wiederhorn (1967)).

3.1. Region I: stress-corrosion regime

In region I, corresponding to the stress-corrosion regime, thecrack propagation velocity is a strongly increasing function ofthe stress intensity factor K; it has an almost linear dependenceon humidity in moist air (figure 5) and it increases withtemperature (figure 6). All these dependences can generallybe fitted by the Wiederhorn (1967) equation:

vI = v0 exp(αK) = A

!pH2O

p0

"m

exp!

−"Ea − bK

RT

", (1)

where pH2O is the partial pressure of the vapour phase in theatmosphere, p0 is the total atmospheric pressure and R thegas constant; A, m, "Ea and b are four adjustable parametersthat take into account the dependence on the glass composition(cf interpretation below).

However, due to the extremely strong dependence ofvelocity on the stress intensity factor K and to the limitedrange of variation of K , the data in region I may also be fittedby a power law expression as in the Maugis (1985) approach:

vI = v0(K/K0)n (2)

the exponent n being generally between 12 and 50 for silicateglasses (Evans and Wiederhorn 1974). This form is, by theway, particularly useful (and indeed used) for developingsimple analytical predictions of the life times in static anddynamic fatigue tests (Davidge et al 1973). The difficultyin determining the most adequate relation results in a stronguncertainty in the predictions over very long periods.

5

• Bond breaking mechanism • H2O diffusion to reaction site

• Reaction kinetics

Open questions• Near - crack tip stress field and

plasticity• Role of H2O ahead of crack

front

crack tip and the critical applied stress intensity factorfor crack closure. Because the condensate only fills thecrack tip to about 100 nm, a high-resolution instru-ment such as an AFM has to be used to determine thecondensate length accurately. This measurement isachieved using the so called tapping mode of measure-ment. When the probe tip comes in contact with thecondensate, a sharp change in retardation is observedand the condensate length can be measured very accu-rately, Fig. 7. Measured in this way, a Laplace pressureof !36 " 5 MPa was determined for a crack tip in sil-ica glass at 40% relative humidity. Also, from the clos-ing forces, an adhesive energy, Go = 180 " 20 mJ/m2

was obtained, which is reasonably close to the energyrequired to create two water surfaces, 144 mJ/m2.

This technique allows important information to beobtained on the nature of the crack tip condensate,which constitutes the local environment for stress cor-rosion crack propagation in a moist atmosphere. In sil-ica glass, the condensate composition was shown to bevery close to pure water and to be in stationary equilib-rium with the moist atmosphere.42 By contrast, thecrack tip condensate size in soda lime silicate glass canevolve in time due to alkali ion diffusion and exchangewith hydronium ions at the crack tip.43

The coupled modeling of crack tip shielding,induced by water penetration into glass and the evolu-tions of the local crack tip environment, can be veryuseful to predict the shape of the v–K curves in thestress corrosion cracking of silicate glasses as a functionof their composition and external environmental.49

Water Penetration at Crack Tips During CrackGrowth

The first authors to demonstrate that water pene-trated into material surrounding a moving crack in sil-ica glass were Tomozawa et al.,50 who argued thatwater penetrates because of the dilatation of the silicaglass due to the high tensile stresses at the head of thecrack tip. As the crack propagates through the water-penetrated glass, the water is left behind on the fracturesurface. This water could be detected using the tech-nique of nuclear reaction analysis,50 which measuresthe concentration of hydrogen atoms within the glass.The details of the experimental technique are given inreference (50). The resolution of the technique was suf-ficient to show that the concentration of hydrogen inthe water-penetrated surface exceeded that of the oil/etched specimen (cf. caption in Fig. 8) for whichadsorbed water was only at the surface.

To obtain a theoretical estimate of the experimen-tal curve in Fig. 8a, the rate of diffusion of water intothe tip of a crack from the crack opening has to be cal-culated. This was performed recently by Wiederhornet al.47 The diffusion equation for water into glass wassolved in cylindrical coordinates and included the effectof the mechanical swelling stresses on the diffusivity. Adata curve was estimated from the theoretical penetra-tion curve (the curve calculated in reference 47), andthe instrument curve given in the paper by Tomozawaet al. (the curve obtained from the oil/etched specimen,Fig. 8). The predicted experimental curve from

Fig. 7. 15(a) Typical atomic force microscopy (AFM) height image of a crack tip. Note the depression in the vicinity of the crack tip. (b)Typical AFM phase image of a crack tip. The zone of the phase shift at the front of the crack marks the region of water condensation.The size of the image is 400 nm2. The scale to the right of the images is 5 nm for the height image and 5° for the phase image. Takenfrom reference (49).

Colouronline,B&W

inprint

POOR

QUALITY

FIG

8 International Journal of Applied Glass Science—Wiederhorn, et al.

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✦ Wiederhorn, S. M., Fett, T., Guin, J.-P., & Ciccotti, M. (2013). Griffith Cracks at the Nanoscale. International Journal of Applied Glass Science, 4(2), 76–86.

ApproachJ. Phys. D: Appl. Phys. 42 (2009) 214006 M Ciccotti

The brittleness of glass can thus be read as the elevatedefficiency of conversion of elastic energy into surfaceenergy, suggesting a relatively weak contribution of plasticdeformation and other bulk dissipation mechanisms, whichwould involve at most a region of a few nanometres aroundthe crack tip. The values of Kc for most glasses are of theorder of 1 MPa m1/2, which is rather low, especially whencompared with the 50 fold larger values in metals where theenergy dissipation by bulk ductility is dominant in the fractureenergy.

In the fracture mechanics framework the subcriticalpropagation by stress-corrosion can be well rationalized andrepresented as a v(K) or G(v) relation for K < Kc

(Wiederhorn and Bolz 1970). This was shown to generallyinvolve three propagation regions (cf figure 4). Region Icorresponds to the cited stress-corrosion regime, where thecrack propagates due to a stress-enhanced corrosion reaction,which is strongly dependent on the environment. In region IIthe propagation velocity is limited by the transport kineticsof environmental corrosive molecules to the crack tip. Inregion III, where the stresses are strong enough to inducethe bond breaking in the absence of corrosion reactions, thevelocity rises very sharply with K , eventually reaching thecritical propagation at Kc. For some glass compositions, athreshold behaviour appears for K = K0 leading to what isgenerally called region 0 (Michalske 1977).

Phenomenological equations such as Wiederhorn’s(cf section 3.1) can describe the dependence of crackvelocity on stress and on environmental parameters for mosttypical glasses and are compatible with a thorough consistentmodelling based on the sharp-crack atomic-bonding paradigm(Lawn 1993). Yet the detailed nature of the stress-corrosionmechanisms that occur at the crack tip has been debated fordecades (Marsh 1964b, Maugis 1985, Gehrke et al 1991,Tomozawa 1996), and a general disagreement can be foundon the relevance of several accessory phenomena that mayparticipate in the stress-corrosion mechanisms at differentstages of the process. Stress-corrosion can involve a complexinterplay between the diffusion of reactive molecules (mainlywater) into the crack cavity and into the glass network,the corrosion (or dissolution) of the network itself and themigration of weakly bonded alkali ions under chemical orstress gradient (Gehrke et al 1991, Bunker 1994). Allthese phenomena are typically very slow under ambientconditions in the unstressed material, but they can significantlyaccelerate in the highly stressed neighbourhood of the cracktip, depending especially on the nature of the environment andof its confinement. Their time scales will be progressivelyaccelerated in a series of smaller and smaller shells surroundingthe crack tip (cf figure 1), defined by the stress being largerthan some critical value. However, since the crack tip ispropagating, the different phenomena will be able or not tospread in each shell depending on a competition between theirtime scale in each shell and the crack propagation velocity.Since the size of the activated shells can vary from micrometresto nanometres depending on the interplay between spaceand time scales, the modelling of these phenomena (whichwill be reviewed in more detail in the following sections)

Figure 1. Graphic representation of the mechanical elements forfracture mechanics.

requires nanometre scale resolved investigation techniques andquestioning about the relevant physical laws at the nanoscale.Such studies should be very promising to solve the debatesby direct observation and to relate the phenomenologicalparameters to the specific composition and structure of glasses.

The development of several nanoscale investigationtechniques in the 1980s has led to significant insights into thecomprehension of the different mechanisms (cf sections 4.3–4.6). In particular SEM, TEM and AFM measurementsof both post-mortem crack surfaces and direct in situobservations of the crack tip neighbourhood have permittedthe investigation of the space and time scales of occurrence ofthese phenomena under specific conditions. The progressiveincrease in resolution down to the micrometre scale of severalstructural and compositional analysis techniques (such asRaman, IR, Brillouin spectroscopies, x-ray, electron andneutron scattering, NMR, XPS, SIMS) have also permittedinvestigations of the alterations of the bulk glass near thecrack tip or at freshly fractured surfaces. These techniques,combined with the increasing power of molecular andquantum dynamics simulations, permit great insights into theunderstanding of the combination between mechanical andchemical processes acting at the crack tip.

Section 2 first focuses on some relevant concepts offracture mechanics. In section 3 the most solidly establishedfeatures of the slow crack propagation kinetics in glassesare described and interpreted in the framework of the classicsharp-crack atomic-bonding paradigm proposed in the 1970s.Section 4 of the review presents a deeper discussion of therelations between the crack propagation and the chemicalmechanisms of stress-corrosion, along with a critical analysisof the points which are still controversial and of the recentexperimental evidence and efforts which have been madeto make them clearer. A concluding section discussesthe perspectives and promises of the development of thisinteresting research field.

2

QM region K-way partition

Multiscale LOTF

(QM/MM)

Atomistic model

Approach: why partitioning?

• n QM core atoms, m � n2 ‘bu�er’ atoms

• N + M valence electrons (N � n, M � m)

• DFT code time to solution and cost � (N + M)p

• K-way partitioning of QM core region t.t.s. � ((N/K + M �))p

• Targets: reduce t.t.s. (better weak scaling),reduce CPU time per calculation (better strong scaling)

Approach: distributed multi-scale MD

QUIP proxy

QUIPN clusters

QM/MM MD with N QM atoms

N forces {F(1)

i}, {F(2)i}, …, {F(M)

i}

QM Box 1Cluster 1

Forces {F(1)i} QM calculation on Cluster 1

QM Box 2Cluster 2

Forces {F(2)i} QM calculation on Cluster 2

64 nodes

64 nodes

64 nodes

QM Box MCluster M

QM calculation on Cluster M

64 nodes..

.

BG/Q Execute NodesFront End Node

Forces {F(M)i}

Partitioning scheme• Input: atoms connectivity map

(bonds) ⟺ graph

• Target: optimal load balance. Split system in K parts• Convex (minimal edgecut)

• Equal size

✦ Fiedler, M. (1975). A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czech. Math. J., 25(4), 619-633.

• Proposed solution: combined method• Iterative spectral bisection

• Merging of 2 smallest neighbouring regions

• Subgraph digestive ripening

Results: partition quality

Results: DFT scalingStrong scaling - crack front in Silicon Weak scaling - crack front in a-SiO2

The Machine Learning front

• Big calculations generate big data.

• How to draw relation atomic structure - observable? Machine learning

• Which properties can be learnt?

• Any scalar function of atomic coordinates (potentially)

• Vectorial property of local environment: forces

• How accurately?

• The bigger the database, the better

• Importance of accurate estimation of prediction error

✦ Rupp, M., Tkatchenko, A., Müller, K. R., & von Lilienfeld, O. A. (2012). Fast and accurate modeling of molecular atomization energies with machine learning. PRL, 108(5), 058301.✦ Bartók, A. P., Payne, M. C., Kondor, R., & Csányi, G. (2010). Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons. PRL, 104(13), 136403.✦ Li, Z. (2014). Submitted.

Predicting error on predictions…• Kernel ridge regression

• α: regression coefficients

• Strong evidence of correlation error - regression coefficient

• …can we build a 2nd layer ML to predict α of new configuration?

Kij = k(Di, Dj)

↵ = (K + �I)�1t

tnew =X

i

↵i k(Di, Dnew)

Ongoing and perspective work

Powerful computational framework applicable to:

• Dry vs. wet dynamical fracture in silica (crystalline, amorphous)

• Bond breaking reaction mechanism

• Influence of penetrated water on elastic stress field

• Study of 3D crack fronts: are crack fronts curved?

• Flexoelectricity in crack cavities

• Database generation of DFT calculations:

enabling ML atomistic configuration ⟷ physical observable

Thank you for your attention!

• Substitute ‘automate’ with ‘use bigger computer’. Argument still holds.