Eur. Phys. J. AP 5, 19–31 (1999) THE EUROPEANPHYSICAL JOURNALAPPLIED PHYSICS

c© EDP Sciences 1999

Liquid metal embrittlement: A state-of-the-art appraisal

B. Joseph, M. Picat, and F. Barbiera

CEA-CEREM/SCECF, BP 6, 92265 Fontenay-aux-Roses Cedex, France

Received: 18 June 1998 / Revised: 29 September 1998 / Accepted: 22 October 1998

Abstract. Although the risk of embrittlement of materials exposed to liquid metals has been recognized formany years, its prediction remains problematical insofar as the knowledge of the mechanisms involved inthe phenomenon is limited. However, Liquid Metal Embrittlement (LME) is of prime interest because therisk of damage exists wherever the handling of liquid metals is required in various industrial or scientificalfields (chemical plants, power-producing systems, soldering process, ...). The interest for this phenomenonneeds thus to be pursued. The present paper reviews experimental results about the occurrence of LMEand the influence of different parameters, and presents a number of mechanisms which have been proposedto explain LME.

PACS. 81.90.+c Other topics in materials science

1 Introduction

The phenomenon of Liquid Metal Embrittlement (LME)may be defined as the brittle fracture, or loss in ductility,of a usually ductile material in presence of liquid metal.Some authors add to this definition the necessary influ-ence of an external load or the presence of internal resid-ual stresses: actually, it has long been known that LME,during a tension test, results in a significant reduction inelongation to failure and fracture strength [1]. However,it is also possible for some ductile materials to becomebrittle without any stress, the best example being theAlsolid – Galiquid system [2].

Embrittlement in liquid metal environments has beenthe subject of many reviews describing the phenomenolog-ical features of the LME, the prerequisites for its occur-rence, the effects of variables including grain size, strainrate, temperature, metallurgical state... (Rostoker et al. in1960 [3], Stoloff in 1968 [4], Nicholas et al. in 1979 [5], Oldin 1980 [6], Kamdar in 1973, 1983 and 1987 [7–9]). Actu-ally, a qualitative explanation of LME has not yet emergedand its prediction is still missing, despite decades of re-search. Empirical predictions of LME susceptibility havebeen proposed, such as embrittled systems usually havea low mutual solubility and a lack of intermetallic com-pounds, but this tentative correlation is not completelyreliable. It is therefore appropriate to review LME in thelight of more recent work, to provide a current assessmentof this phenomenon.

In the second section of this review, we will make asummary of the experimental results reported in the liter-ature in order to compare the basic characteristics of LME

a e-mail: barbier@cyborg.cea.fr

and the influence of parameters, to the predictions of mod-els. In the third section, we will focus on the most impor-tant models which have been advanced so far to describethe LME. We will underline the elementary mechanismsinvolved in each model, and propose a diagram that reca-pitulates the links between those mechanisms. Then, wewill do an analysis of each model based on experimentalresults and on comments found in the literature.

2 Experimental results on LME

The main purpose of this section is to present experimen-tal facts about LME, in order to compare these with theresults of the models that will be detailed further. Af-ter a characterization of the phenomenon, we will anal-yse the influence of the different parameters involved.Experimental data depend a lot on the systems stud-ied, and for a given system, differs sometimes from oneauthor to another.

2.1 Description of the occurrence of LME

Under certain experimental conditions, LME can be quitedramatic and one of the peculiarities of this phenomenonis the crack propagation rate which is reported to bevery high. Other striking results are the drastic reductionof strain to rupture and the modification of stress-straincurves in presence of liquid metal. These first experimentalresults have allowed the characterization of the LME phe-nomenon by simply studying some parameters that havebeen modified from an inert medium to a liquid metalenvironment.

20 The European Physical Journal Applied Physics

Fig. 1. Dependence of zinc monocrystals fracture strain ontemperature and environment : (a) air, (b) mercury (afterRozhanskii et al. [12]).

2.1.1 Rate of crack propagation

The most striking characteristic of LME is a very highpropagation rate, in comparison with the rate in air orvacuum. Different results can be found in the literature,but they are all in the range of one centimeter to severalmeters per second [7,10]. This difference in propagationrate is mainly due to important differences in the char-acteristics of LME that exist from one solid metal-liquidmetal system to another.

According to some authors, the propagation is not thecontrolling stage: by instance, Glickman [11] asserts theexistence of a slow precritical crack propagation stage,which is the controlling stage of failure. During this stage,the propagation rate is 10−7 to 10−3cm s−1. In fact, thecontrolling stage, whose rate is quite low, induces a de-layed failure that is not observed in every system.

2.1.2 Strain to rupture

In almost every tension test in liquid metal reported inthe literature, strain to rupture is drastically reduced incomparison to the same test in vacuum or in air. For ex-ample, Rozhanskii et al. [12] showed (Fig. 1) that in a de-fined range of temperature the strain to rupture is larglyreduced by the presence of liquid metal.

This limited range of temperature is called “ductilitytrough”, because it corresponds to the range of tempera-ture where the solid metal shows a loss of ductility. Forexample, in Figure 1, under 150 C fracture strain is dras-tically reduced, and above 150 C, it is equal in air and inliquid metal. The lower limit of this trough is a tempera-ture close to the melting temperature of the liquid metal.At a higher temperature, we can observe a recovery ofductility. We will study the influence of the temperature(cf. Sect. 2.4.1).

2.1.3 Stress-strain curves

When we compare the stress-strain curves obtained byseveral authors, two main trends are emerging. The firstone, sustained by Westwood [13] and Stoloff [14], claims

Fig. 2. Embrittlement of polycrystalline aluminium: compari-son between the behavior of aluminium in vacuum and in var-ious mercury solutions (after Westwood et al. [13]).

Fig. 3. Stress-strain curves of Armco iron specimens testedin bismuth (continuous lines) and in vacuum (dashed lines) at350 C, 400 C and 550 C (after Popovich [15]).

that until fracture occurs, the stress-strain behaviour ofthe solid metal is the same as that in the unwetted condi-tions (Fig. 2). The second one, sustained by Popovich [15],asserts that there is an initial promotion of plastic flow,and consequently, a different behaviour of the metal inpresence of liquid metal before rupture (Fig. 3).

In the two cases, the elastic domain is unchanged inpresence of a liquid metal: the yield stress is not modifiedby an embrittling environment. However we can point outthat in Popovich experiments, a part of the plastic zoneis the same in vacuum and liquid metal.

2.2 Analysis of the influence of the liquid and solidmetals compositions

Compositions of solid and liquid metals are reportedto influence LME, that is to modify propagation rate,strain to rupture... In this subsection, experimental resultsabout this influence on the severity and the domain ofapplication of LME, are presented.

B. Joseph et al.: Liquid metal embrittlement: A state-of-the-art appraisal 21

2.2.1 Composition of the liquid metal

The effects of the liquid metal composition are often sig-nificant. If we consider the effects of additions to the em-brittling metal, the result can be either a decrease of theembrittlement or an increase of it. Sometimes, it may alsohave no influence on the phenomenon.

Besides this influence on the degree of the embrittle-ment, the composition of the liquid metal can even modifythe range of manifestation of LME. By instance, Preeceand Westwood [16] have shown that the brittle to ductiletransition temperature of aluminium (which correspondsto the upper limit of the “ductility trough”) is increasedby gradually adding gallium to a mercury environment.

Composition of the liquid metal also modifies sev-eral physico-chemical parameters of the solid metal-liquidmetal system (interfacial energy, mutual solubilities, ...)which are supposed, by many authors, to influence themechanism of LME, as we will examine in the next sub-section.

2.2.2 Composition of the solid metal

The effect of alloying (additions to initial compositionof the solid metal) has been widely studied in the liter-ature, and the results are different from one author toanother. Some additions can increase the effect of LME,some others can prevent the solid metal from it. However,the fact that the main influence of an alloying elementtakes place at the level of grain-boundaries is commonlyaccepted. A study of the deformation at fracture, as afunction of the composition of the solid metal has shownthat the maximum effect on LME corresponds to a concen-tration of alloying additions in solid solution which ensuresadsorption saturation at grain boundary [11].

2.2.3 Microstructure of the solid metal

The hardness and deformation behaviour of the stressedsolid can affect its susceptibility to LME, the hardest ma-terials normally being more severely embrittled. Thus, themicrostructure of the solid metal influences its suscepti-bility to LME. However, this increase in the severity maybe offset if it is achieved by cold-working, since this candecrease the grain size [17].

Actually, the average size of the grains is pointed outby many authors to have two distinct effects on LME: inthe first place, LME is enhanced by an increase in grainsize and there is a linear relation between the true tensilestrength and d−1/2, where d is the average grain size, i.e.LME follows the Cottrell-Petch relationship of grain-sizedependence on fracture stress [18]; in the second place,the brittle to ductile transition temperature is increasedby an increase in grain size and there is a linear relationbetween this temperature and log(d) [19].

Fig. 4. Interfacial energy between copper and bismuth-leadmelts at 350 C as a function of lead concentration in melt [20],and grain-boundary energy at 950 C as a function of antimonyconcentration in copper [21] (after Glickman [11]).

2.3 Analysis of the influence of the physico-chemicalproperties of solid and liquid metals

In this section, we examine the physico-chemical interac-tions between the two metals (formation of intermetallics,wettability, mutual solubilities and penetration of liq-uid metal atoms in the solid metal), which are directlyinfluenced by their composition and microstructure.

Many authors have tried to classify all the systems byconsidering their physico-chemical properties. The num-ber of properties that are relevant for several systems isimportant, but very few of them seems to be appropri-ate for all the systems. Here is the analysis of the mostimportant of these properties; others, as electronegativ-ity, could have been presented, but they do not lead to abetter classification nor explanation of the phenomenon.

2.3.1 Interfacial and grain-boundary energies

Among the physico-chemical parameters of the solidmetal-liquid metal system, the interfacial energy γSL (forthe “crystal-melt” interface) and the grain-boundary en-ergy γb, are of first importance, and they are considered inalmost every LME model (Sect. 3), and thus, their varia-tion is very likely to influence the characteristics of LME.As an example, we can see in Figure 4 that on the oneside, the composition of liquid metal modifies the interfa-cial energy [20], and on the other side, the composition ofthe solid metal induces a variation of the grain-boundaryenergy [21].

2.3.2 Formation of intermetallics

Some authors (Kamdar [7], Shunk and Warke [22], ...)have introduced the notion of “specificity” of the LMEphenomenon, that is the fact that only certain liquidmetals embrittle certain solid metals.

22 The European Physical Journal Applied Physics

For this purpose, they have proposed an empirical lawwhich claims that a system of a solid metal and a liquidmetal that forms intermetallics can not undergo LME.However, Chaevski and Popovich [23] noticed that thisempirical law is followed by some systems (gallium reactswith Armco iron to form intermetallics and no LME is ob-served) but not for all (tin forms intermetallics with ironand it produces LME).

2.3.3 Mutual solubilities

Another aspect of the “specificity” of LME is the require-ment of low mutual solubilities. According to Kamdar [7],it could be related to difficulty of propaging a brittle crackin a solvent environment because dissolution processes willtend to blunt the crack. However, it is sometimes statedthat a very limited amount of solubility of a liquid metalin a solid metal is required to facilitate wetting [3]. Oncemore this criterion of “low mutual solubilities” knows ex-ception such as the copper-bismuth or the aluminium-galium systems and thus cannot be used as permanentrule.

2.3.4 Wetting

Wetting is often considered as a necessary condition forLME to occur. If the contact between the solid metal andthe liquid metal is interrupted by the presence of an oxidefilm, or by removal of the liquid metal after preliminarycontact, LME does not occur.

It has to be noticed that the conditions for this “speci-ficity”, that is low mutual solubilities and no formation ofintermetallics, are in conflict with the condition of goodwetting since many studies associated good wetting withsome degree of mutual solubility [3]. This discrepancy ledStoloff [4] to regard parameters derived from wetting dataas being irrelevant to the interpretation of LME. How-ever, Old [5,6] reports that this view must be revised inthe light of advances made in surface study techniques.In fact, wetting studies can now be conducted on cleansurfaces and excellent wetting can be achieved even withsystems having negligible solubilities. Consequently, thesenotions have no longer to be opposed. Anyway, the contacthas to be good to produce LME, and the growth of crackscan be stopped if the supply in liquid metal is limited orinterrupted. The issue of the supply of liquid metal hasbeen extensively studied by Gordon [24] (Sect. 3.7.2).

2.4 Analysis of the influence of the externalparameters

Many improvements have been done in the field of theinfluence of external parameters. Lots of experimental re-sults have given a strong basis to compare to LME models.This list of parameters is not complete, but it correspondsto the most commonly studied.

2.4.1 Temperature

As it has been pointed out in Section 2.1.2, LME occursin a certain temperature range, called “ductility trough”.This domain is generally considered as depending on. Thecompositions of both solid and liquid metals, the structureof the solid metal and the external conditions. Inside thisdomain, the susceptibility of embrittlement may, in mostcases, remain unchanged with temperature.

The lower temperature of the “ductility trough” cor-responds in many systems, to the melting point of theembrittling metal, but some embrittling metals have beenreported to induce embrittlement in a solid state. Thisphenomenon has been called “Solid Metal Embrittlement”(SME). Anyway, this lower limit of the temperature do-main of occurrence of LME is supposed to depend onlyon the nature of the embrittling metal. The higher tem-perature seems to depend mainly on the strain rate, asexplained in the next subsection.

Beyond this influence on the domain of existenceof LME, temperature has been reported to influencethe kinetics of LME. According to Glickman, the rate-controlling stage is a slow subcritical growth, whichis thermally activated. For the solid copper-equiatomicbismuth-lead bath system, the effective activation energyis Q = 0, 38 eV [11].

2.4.2 Strain-rate

According to Glickman, the relation between the strainrate and the crack propagation rate is nearly linear forthe bismuth-copper system [11]. For Popovich, at slowstrain rates, thus in conditions favorable for intense stressrelaxation, the only effect of adsorption-active media isto ease plastic flow, i.e. to inhibit LME. With a de-crease in strain-rate, the upper temperature of the “duc-tility trough” drops, which leads to narrowing of thetemperature range of brittleness or to its disappearance.

2.4.3 Stress

In some definitions of LME [7], an applied or residualstress (σ) is a necessary prerequesite. In this context,the embrittlement of aluminium by liquid gallium, whichcan occur without any stress, is defined as another phe-nomenon than LME. In fact, the main problem about thepresence of stress is to know whether the metal studied isfree of stress or is containing high stress concentrations.Actually, grain-boundaries and other obstacle may serveas a stress concentrator.

LME would occur only with stresses exceeding a“threshold” stress. The latter is not reported by all theauthors, because some of them may have worked on a sys-tem for which the threshold stress is very low. Moreover,there are experimental results about stress dependenceduring creep tests: according to Glickman, the time to rup-ture τc is a linear function of σ−2 [11], and for Popovich,the stress dependence of creep rate, plotted in logarithmiccoordinates, is linear [25].

B. Joseph et al.: Liquid metal embrittlement: A state-of-the-art appraisal 23

Fig. 5. Diagram illustrating all the elementary steps involved in the different LME models.

2.4.4 Time of exposure before testing

In many studies, it was proved that preliminary contact ofthe solid metal in the embrittling medium did not enhancebrittleness. By example, some systems such as aluminium-mercury and (zinc-copper) alloy-mercury seem not to besensitive to the time of exposure before testing, since thefracture stress is not influenced by it [26].

3 Mechanisms of LME

Here are presented different models that have been pro-posed so far to explain the LME phenomenon. This listis not complete, but the main purpose is to present themost outstanding models. For each of them, we recall theelementary mechanisms which are involved and we giveits description. Then we report an analysis showing itslimits or/and its possible improvements compared to othermodels.

The models are summarized in Figure 5. This diagramdescribes, for each model, the sequence of the elementarymechanisms leading to the propagation of cracks in pres-ence of stress and liquid metal. They are represented byarrows, which are numbered in order to link the follow-ing description of the models to the diagram. It allows

to make a quick comparison between the different modelsand to see whether they use common elementary mech-anisms or not. Furthermore, if we make this hypothesisthat several mechanisms can occur at the same time toproduce LME, as it has been proposed, this diagram givesa general overview of all the mechanisms that could beinvolved.

3.1 Summary of the different approaches of LME

There are six different models of LME mechanism:

• the dissolution-diffusion based models of Robertson(Sect. 3.2) and Glickman (Sect. 3.3);• the brittle fracture theory based model, denominated

SJWK (Sect. 3.4), proposed by Stoloff, Johnson,Westwood and Kamdar;• the ductile failure based models of Lynch (Sect. 3.5) and

Popovich (Sect. 3.6);• the liquid metal atoms penetration based model of

Gordon (Sect. 3.7).

The concept of the adsorption-induced surface energylowering is the central point of all these models, exceptedfor Robertson who does not mention the occurrence of

24 The European Physical Journal Applied Physics

Fig. 6. Crack of length 2L, thickness 2r, and tip radius r,subject to an applied stress σa [30].

adsorption. This concept has first been introduced by Re-binder and it is known as the Rebinder effect (RE). TheRE is one of the effects exerted by a liquid or a gaseousmedium on the mechanical properties of solids [27–29]. Itappears as a plastic flow increase and a strength reduction.The origin of the RE is the surface free energy reductioncaused by adsorption which is a localized physicochem-ical interaction between a solid and the medium at theinterphase boundary.

The main problem in using the Rebinder effect to de-scribe LME is that it has a pure thermodynamic character.That is the reason why many authors have improved hiswork by adding molecular mechanisms and/or by addingkinetic modelization of the phenomenon.

3.2 Robertson model

Robertson [30] has presented his dissolution-diffusionbased model in 1966. The main purpose of his theory is togive an expression for the crack propagation rate. There isno reference to a crack initiation stage, consequently thefinal expression concerns only the propagation rate. Thismodel uses only macroscopic, thermodynamic and elasticconcepts.

3.2.1 Elementary Mechanisms

The presence of liquid metal at the crack tip results ina dissolution of the solid metal (Fig. 5, arrow 1) which isenhanced by stress and capillarity effects (Fig. 5, arrow 2).The dissolved atoms diffuse then through the liquid metaland the crack propagates (Fig. 5, arrow 3).

3.2.2 Description

Let us consider a crack (Fig. 6) of length 2L, of thickness2r and of tip radius r, subject to an applied stress σa.The stress distribution around the crack tip establishes agradient in the chemical potential at the tip, which causesa diffusion flux through the liquid away from the tip.

The magnitude of the stress at the crack tip, due tothe applied stress σa is given, for a long narrow crack, bythe relation:

σ = 2σa

√L

r· (1)

The concentration of the solute in the liquid, C, is equalto the equilibrium concentration in the presence of an un-stressed flat surface, C0, plus the excess concentration dueto the stress σ at the crack tip, ∆C(σ), plus the excessconcentration due to capillarity, ∆C(γ).

After calculations, the excess concentrations are givenby:

∆C(σ) =C02ΩLσ2

a

ErkT(2)

∆C(γ) = −C0Ωγ

kTr(3)

with Ω, the atomic volume of the solid, E the Young’smodulus of the solid metal, k the Boltzmann constant, Tthe temperature and γ the solid-liquid interfacial energy.

Consequently, the concentration in the vicinity of thestressed crack tip is:

C = C0 +C0Ω

kTr

(2Lσ2

a

E− γ

)· (4)

The flux, J , of solute atoms away from the crack tip isgiven by Fick’s first law:

J = −D

(∂C

∂x

)x=0

(5)

where D is the diffusion coefficient of the solute in theliquid.The velocity of the crack is defined by:

v = −ΩJ . (6)

Finally, the expression for the tip velocity is:

v =

(C0DΩ

2γ

kT

)1

r2

(2Lσ2

a

Eγ− 1

)· (7)

The effect of capillarity, as expressed by the (−1) in equa-tion (7), provides a threshold stress for crack propagation.Setting v equal to zero gives the classical Griffith criterionfor the critical stress to obtain propagation:

σc =

√Eγ

2L· (8)

We can deduce the maximum velocity from equation (7):the tip radius cannot be less than about d, the atomic di-ameter of the lattice atoms, and σ cannot be greater thanE. If γ/r is small compared to E/2, then the maximumvelocity is given by the product of the three terms:

vmax =1

2k

Ω2E

d

C0D

T(9)

where 12k is a numerical constant, Ω

2Ed

is determined solely

by the properties of the solid, and C0DT depends on the

interaction of the solid and the liquid and contains mostof the temperature dependence of the crack velocity.

B. Joseph et al.: Liquid metal embrittlement: A state-of-the-art appraisal 25

3.2.3 Analysis

Equation (9) gives vmax ≈ 15 cm s−1 for copper in mercurywhich is of the same order as those observed by Rhineset al. [10] that were in the range of 1 to 10 cm s−1. Con-sequently, the present dissolution-diffusion model predictsmeasurably high crack velocities.

Popovich [15,25,31] notices that, in agreement withthe model, the degree of embrittlement must increase withan increase in the solubility of the solid metal in the liquidmetal, which does not agree with practical observation.Moreover, according to him, when the solution process be-comes important, the crack tip is blunted and brittlenessis suppressed.

Futhermore, Glickman [32] notes that this model is cer-tainly oversimplified: it does not take into account the pos-sibility of relaxation of those elastic stresses during blunt-ing of the crack, giving here the relation ∂L/∂t ∼ σ2

aL.This equation is inconsistent with test data reported byGlickman: ∂L/∂t = α(σa −∆).

Concerning temperature, it affects the expression forthe crack velocity directly through the factor C0D/T , asnoted earlier. This product increases as the temperatureincreases, predicting that the crack velocity will increasewith increasing temperature. But Kamdar [7] underlinesthat the increased velocity with increasing temperature isopposed to the observed behavior of most embrittlementcouples. Moreover, Gordon [24,33] says that this theorypredicts an abrupt and substantial change in temperaturedependence of crack propagation rate at the embrittlermelting point, since its main temperature dependence de-rives from the activation energy for the diffusion of thebase metal atoms in the embrittler. He has shown thatit was not true for indium embrittlement of steel. Any-way, contradictory results exist in litterature: for Rhineset al. [10], crack propagation rate increases with increas-ing temperature, but for Westwood [34], embrittlement isindependent of temperature over a considerable tempera-ture range.

Finally, Robertson remarks that in all of the discus-sion, the crack-tip radius r has been left as an unspecifiedparameter. The tip-radius will depend on the differentialrates of dissolution at the tip and on the dynamic yieldingbehavior around the tip. This yielding behavior can varymarkedly with composition, heat treatment and tempera-ture. But it appears difficult to predict the dependence ofr on T , and hence, it is not all clear how the crack velocityvaries with temperature.

3.3 Glickman model

According to Glickman [32], LME is a clear manifesta-tion of the Rebinder effect (cf. Sect. 3.1). He adds to thisapproach, a mechanism based on dissolution and diffu-sion of solid metal atoms. Glickman worked mainly withsolid copper and lead-bismuth melt. His model of thephenomenon is mainly kinetic, but also thermodynamicbecause it is based on Rebinder effect.

3.3.1 Elementary mechanisms

For Glickman, the nucleation and subcritical growth ofcracks which take place along grain boundaries are asso-ciated with selective dissolution in the melt of solid metalatoms at the crack tip (Fig. 5, arrow 1). This dissolu-tion is enhanced by the adsorption of liquid metal atoms(Fig. 5, arrow 4), which favours the nucleation of dis-locations (Fig. 5, arrow 5). These dislocations are cre-ating an atomic roughness at the solid-liquid interface(Fig. 5, arrow 6) which enhance dissolution of solid metalatoms (Fig. 5, arrow 7). This is followed by rapid diffu-sion of solid metal atoms from the crack tip and crackpropagation (Fig. 5, arrow 3).

3.3.2 Description

Glickman asserts that failure kinetics need to be known tounderstand the mechanism of LME. However these failurekinetics have to coincide with the rate at which the inter-atomic bonds at the crack tip are broken and reorganized“boundary kinetics” and not with the the rate of melt flowalong the crack “transport kinetics”. Since the rate v ofmelt flow along a capillary crack of length l is v ∼ 1/l, itis difficult to obtain such data for macroscopic cracks (inthis case, it is only possible to study “transport kinetics”),and consequently the rate of development of microscopiccracks has to be studied.

According to Glickman, there are three kinetic stagesfor microcracks failure:

• Crack initiation, which results from selective dissolu-tion of the grain boundaries in the melt. The melt-filledgrooves are formed where boundaries exit to the sur-face. The basic outlines of the mechanism of this processare described by the well-known theory of the growth ofthermal-etching grooves by Mullins [35]. For the copper-bismuth system, the cracks switch from initiation topropagation at a length of approximately 10 µm.• Crack precritical propagation, which is the controlling

stage of failure. It is a slow crack propagation whoseduration, which determines the time-to-rupture τc, de-pends on the crack rate during this stage and on thecritical length lc at the moment of transition fromsubcritical to hypercritical growth.The rate of subcritical growth of cracks does not dependon their length, and its variation with the applied stressis approximately described by the linear function:

v =∂L

∂τ≈ α(σ −∆) (10)

where L the crack length, α the crack “mobility”, σthe applied stress, ∆ the threshold stress of subcriticalgrowth and τ the time.• Crack supercritical propagation, which is due to the

achievement of a certain critical deformation levelin the plastic deformed region near the crack tip,according to data obtained by Glickman.

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Measurements of lc and stress σc at the moment of frac-ture showed that regardless of the conditions of defor-mation (air or any liquid metal, creep or active tension),the transition from the subcritical to the hypercriticalfracture stage occurs at a constant value of the productσclc. Glickman has developed a possible mechanism toexplain the kinetics of precritical stage: the mechanismof dissolution-precipitation.

According to this author, atoms of the solid at thecrack tip dissolve in the melt, diffuse in it, and become pre-cipitated on the crack walls at a distance approximatelyequal to the crack opening δ. Both the dissolution and thereverse process of crystallization out of the melt take placeat discontinuities on elemental steps formed on solid sur-face, the concentration of discontinuities Cj correspondingto its thermal equilibrium level:

Cj ∝ exp

(−UjkT

)· (11)

The higher the value Cj , the faster the rate of dissolutionand crack growth v:

v ≈ Cjvj(w1/3/h). (12)

Here, h denotes the average distance between dislocationsteps at the crack front, ω is the atomic volume and vjis the rate of displacement of the discontinuities alongthe step, which is determined by the density of the diffu-sional flow of solid atoms from the discontinuity into themelt and which increases with increasing stress σ, diffusioncoeffcient DL, and solubility of solid metal in the melt C0.

The role of ωSL in the framework of this model isassociated with the variation in energy of formation ofdiscontinuities Uj :

Uj ≈ SSLγSL − SGBγb (13)

where SSL is the increase in the area of the metal-meltinterface due to the formation of discontinuities, and SGBis the accompanying decrease in the grain-boundary area.The decrease in γSL and the increase in γb produce anexponential increase in the crack growth rate as a resultof the increase in the concentration of nuclei of dissolution,i.e. discontinuities at the crack tip.

In the framework of this model, it was possible to de-rive a formula for crack growth rate v in a form similar toequation (10) and to express the crack mobility α(T ) andthe threshold stress ∆ in terms of the parameters of themetal-melt system:

v ≈

C0ω4/3 exp

(−SSLγSL − Sbγb

kT

)DL(σ −∆)

RhkT(14)

with

∆ =(2γSL − γb)E

πσl· (15)

Thus, in the case of LME, the accelerated fracture underthe influence of surface-active melts is associated with thesimultaneous influence of two factors:

– the exceptionally high diffusional permeability of themelt filling the crack cavity, which ensures fast diffu-sional removal of dissolved atoms from the crack tip;

– the high degree of atomic roughness of the metal-meltinterfaces which is related to the low specific surfaceenergy at these interfaces and which ensures a highdissolution rate.

3.3.3 Analysis

In the equation (10), α and ∆ are independent of σ andτ , but α is much more sensitive to the physicochemicalconditions of deformation than ∆. Glickman has com-pared these theorical results of the variations of α to itsexperiments for the copper-bismuth and copper-mercurysystems and has found a good accordance.

In comparison with the Robertson model, the analysisproposed by Glickman presents some improvements (dif-ference between nucleation and propagation, better inter-pretation of the influence of temperature). However, hismodel still faces the issue of the solubility of the solidmetal which is supposed, contrary to experimental data,to enhance the embrittlement. He denies this allegationand claims that his mechanism still operates when thesolubility tends to zero: such systems often contain impu-rities which react with the solid metal to form compoundssoluble in the melt. He also proposes another possible ex-planation, with the mass transfer from crack tip to themetal bulk by volume or grain boundary diffusion or evenalong the melt-metal interface: in the absence of solubil-ity, the mechanisms of crack growth are analogous to thoseoperating in vacuum, and the part played by the melt islimited to increasing the effectiveness of the metal surfacein its role of the source of point defects taking part in thediffusional mass transfer from the crack tip.

At last, the contradiction with Gordon [24,33] aboutthe abrupt transition in temperature dependence at theembrittler melting point is still under consideration.

3.4 SJWK model

This model described in [7], is based on the “weaken-ing of interatomic bond” mechanism (Fig. 7), which isa particular case of the brittle fracture theory. It managesto rationalize the effects of many experimental variables:temperature, strain rate, grain size, slip character, butit is not capable of predicting susceptibility of a newsubstrate-environment system.

3.4.1 Elementary mechanisms

A liquid metal atom B at the crack tip A − A0 reducesthe surface energy (Fig. 5, arrow 4), and consequently thestrength of interatomic bonds (Fig. 5, arrow 8). As thestress s is applied, the stress acting on interatomic bondA−A0 eventually exceeds its now reduced breaking stress,so that the bond breaks, the crack propagates to the next

B. Joseph et al.: Liquid metal embrittlement: A state-of-the-art appraisal 27

Fig. 7. Displacement of atoms at the tip of a crack. Thebond A−A0 constitutes the crack tip, and B is a liquid metalatom [7].

solid atoms A1 (Fig. 5, arrow 9) and the liquid metal atombecomes stably chemisorbed on the freshly created sur-face. This procedure is then repeated until the specimenfails. The cracking process is limited by the arrival of liq-uid metal atoms at the crack tip and it is assumed thatliquid is able to keep up with the propagating crack tip.

3.4.2 Description

Stoloff and Johnston [36] and Westwood and Kam-dar [37], working independently, postulated that a per-fectly straightforward proportional relationship must ex-ist between surface energy and interatomic bonds energy.Their model, denoted SJWK, utilized a relation first de-rived from Gilman [38], to show that cracking facilitatedby the lowering of elastic surface energy must inevitablylead to a lowering of plastic work.

This relationship between surface energy and inter-atomic bonds energy is as follows: for a crack radius ρ, sub-mited to a normal stress σ and a shear stress τ , the workassociated with fracture is proportional to γρ/a0 whereγ is the elastic surface energy and a0 is the radius of anelastic crack. Any factor that facilitates crack blunting atthe tip of a growing crack increases ρ, resulting in highenergy absorption or even crack arrest.

If the work done in breaking interatomic bonds is thenequated to the surface-free energy of the subsequently cre-ated fracture surface γ, we obtain the maximum stress σmto break A−A0 bonds:

σm =

(Eγ

a0

) 12

· (16)

Consequently, the adsorption of liquid metal atoms, whichreduces the surface energy, decreases the strength of inter-atomic bonds σ. This adsorption can also cause reductionsin the shear strength τ of atomic bonds at the crack tip,but these effects will be localized at the surfaces of thecrack tip and will not be felt at distances greater than sev-eral atomic spacings due to electronic screening effects. As

a result, the ratio σ/τ will decrease and according to thebrittle fracture theory, the crack will develop by cleavage.

Concerning crack initiation at the surface, it will alsobe facilitated by the adsorption of liquid metal atoms,and if chemisorption is strain activated, it will occur pref-erentially at sites of stress concentration, such as in thevicinity of piled-up groups of dislocations at high anglegrain-boundaries.

3.4.3 Analysis

As the ratio σ/τ represents the tendancy to cleavage,Kamdar explains the LME phenomenon by a reduc-tion in σ, with τ unaffected. Nevertheless, according toPopovich [31], the liquid metal increases the activity ofdislocations, which should result in a change in the valueof τ . Moreover, he asserts that LME is a ductile phe-nomenon at a microlevel: the use of brittle fracture theorymay not consequently be used here. Consequently, Kam-dar acknowledged in recent year [9], that some modelsbased on plastic flow, like Lynch’s one were relevant, butonly in few cases.

Nevertheless, Dimelfi [39], taking for granted the pres-ence of plastic flow in LME, asserts that the the “weak-ening of interatomic bond” model is not contradictorywith plastic flow. According to him, in usual ductile mate-rial, local elastic strains are not sufficient to break atomicbonds until the bulk material undergoes significant plas-tic flow. The work-hardening aspect of plastic strain allowsan accompanying increment of elastic strain and hence al-lows sufficient elastic strains for bond breaking and crackpropagation. The presence of plastic flow is then con-sistent with SJWK model, and such an explanation al-lows to explain the influence of cold work, strain rateand temperature.

According to Glickman, all indicates that surface-active liquids role in LME consists only in accelerating thesubcritical growth and crack nucleation and not in takingpart in the decrease of the fracture thoughness, disputinghere SJWK model. Moreover, he also contests the straight-forward proportional relationship between surface energyand interatomic bonds energy.

3.5 Lynch model

Lynch’s model [40–43], which is also based on “the de-crease of the strength of the interatomic bonds”, differsfrom SJWK’s model, because it assumes that crack prop-agation does not occur by atom rupture of the bonds. Thismodel gives an explanation for the presence of ductility ata microlevel. It is mainly based on metallographic inves-tigations on many different systems.

3.5.1 Elementary mechanisms

Adsorption of liquid metal atoms (Fig. 5, arrow 4) is low-ering the strength of interatomic bonds at the crack tip

28 The European Physical Journal Applied Physics

Fig. 8. Diagram illustrating mechanisms of crack growth inD6ac steel in inert and liquid mercury environment [40].

(Fig. 5, arrow 8), which eases the nucleation of disloca-tions (Fig. 5, arrow 10) and promotes intensive slip ofthem, which lead to the occurence of voids (Fig. 5, ar-row 11) and coalescence of them with the propagatingcrack (Fig. 5, arrows 12 and 13).

3.5.2 Description

The consequence of the adsorption of liquid metal atomsat the crack tip is the weakening of interatomic bonds,thereby facilitating the nucleation of dislocations (Fig. 8).These dislocations, injected from crack tips on suitablyinclined slip planes, produce crack advance (as well asopening) and, when extensive adsorption-induced dislo-cation injection occurs from crack tips, only small strainsare required to link up crack tips with voids formed in aplastic zone ahead of cracks.

These voids are nucleated at small particles, disloca-tion cell boundaries, or slip-band intersections ahead ofcracks, and serve to resharpen cracks so that crack-tip-opening angles are smaller than the angle between the twoslip planes. This process produces fracture surfaces thatare microscopically dimpled but macroscopically parallelto low-index crystallographic planes bisecting the two ac-tive slip planes, with crack fronts parallel to the line ofintersection of crack planes with slip planes.

In inert environments, dislocations are not injectedfrom crack tips to any extents, so that coalescence ofcracks and voids involves predominantly egress of disloca-tions nucleated from near crack-tip sources. Only a smallportion of such dislocations emerges exactly at crack tip toproduce an increment of crack advance: most dislocationseither egress behind crack tips producing only blunting or

do not intersect crack tips. Thus, relatively large strainsare required to produce coalescence cracks with voids, sothat larger and deeper dimples are observed on fracturesurfaces, when dislocations egress rather than dislocationinjection predominates.

3.5.3 Analysis

The most important aspect of this model is that the de-layed failure is not explained, as Gordon [24,33] under-lined it in his study about delayed failure. However, Lynchis totally aware of that fact and explains it as follows:crack initiation sometimes involves the slow developmentof a notch of critical depth and acuity, by injection ofdislocations facilitated by adsorption.

Moreover, Kamdar [7] notices that adsorption effectson dislocations will be limited to several atomic spacingsdue to electronic screening effects, and will not be felt atlarger distance in the bulk of the solid ahead or in thevicinity of the crack tip. According to him, these effectswill be secondary in importance when compared to break-ing of bonds where liquid is continuously adsorbed at thetip of a propaging crack.

3.6 Popovich model

Popovich [15,25,31] presents a mechanism for LME, basedon the Rebinder effect (cf. Sect. 3.1) and on enhancedplastic flow. At the thermodynamic aspect developed byRebinder, he adds an atomistic approach. His model isonly qualitative and give no quantitative expression forthe time to rupture or for the crack propagation rate.

3.6.1 Elementary mechanisms

This mechanism is based on the fact that, adsorbed liq-uid metal atoms (Fig. 5, arrow 4) enhance the outlet ofdislocations to the surface (Fig. 5, arrow 5). These dislo-cations could lead to pile-ups at grain boundaries (Fig. 5,arrow 14) which result in rapid work hardening (Fig. 5,arrow 15) and crack propagation (Fig. 5, arrow 16).

3.6.2 Description

Popovich proposes that adsorbed liquid metal atoms pro-mote plastic flow in specific solid metals, which results in amore important dislocations activity (Fig. 9). He assumesthat this easing of plastic flow is caused by the reductionof the shear stress τ , under the action of the liquid metal.Then, the increased dislocation activity can rapidly lead toenhanced work hardening in a localized region and, conse-quently, premature failure. He also proposes that adsorbedspecies can activate new slip planes, which results in en-hanced plastic flow and work hardening. Alternatively, ifthe large number of dislocations is able to transfer thestrain to neighboring grains by crossing grain boundaries,

B. Joseph et al.: Liquid metal embrittlement: A state-of-the-art appraisal 29

Fig. 9. Diagram explaining the Popovich model.

then overall ductility could actually be increased by theadsorption.

Adsorption-LME manifests itself at lower tempera-tures where relaxation processes are limited, because atelevated temperature, relaxation is easier and thereforeductility is restored. Furthermore, at rapid strain rates,relaxation processes are slow so that rapid work hardeningwould also be expected to cause embrittlement. Concern-ing grain size, Popovich explains the experimental results(Sect. 2.2.3) by the fact that coarsed-grain structures aremore strain hardened and less relaxed when the averagegrain size is increased. Actually, decreased strain hard-ening in fine-grain structures is explained by the easiermovement of dislocations in fine-grain structure as a re-sult of the presence of a multitude of boundaries with ahigh energy of disorientation.

Thus, adsorption-LME would be expected to be moresevere at high strain rates, at elevated temperatures andwith high average grain size.

3.6.3 Analysis

This model by Popovich has a number of similarities withLynch’s one. Most of the comments that have been doneabout Lynch’s model are still relevant here. Consequently,we will concentrate our analysis on the main differencesbetween the two models based on ductile failure.

Actually, Popovich proposes that the adsorbed liquidmetal, besides the nucleation of dislocations, promotesnew slip planes and work hardening, the latter being notdeveloped in Lynch’s model. Moreover, the influence ofthe dislocations, for Popovich, can be felt at a great dis-tance, which is impossible according to Kamdar, becauseof electronic screening effects.

3.7 Gordon model

This model is based on the penetration of liquid metalatoms along the grain boundaries. The concept proposedby Gordon and An [24,33], is that the actual crack nu-cleation event is not the rate-controlling step in the crackinitiation, but rather than during an incubation periodthere is a preparation process which is rate-controlling. For

Gordon, the present approach must allow to explain thedelayed fracture, often observed in experimental studies.

3.7.1 Elementary mechanisms

During this incubation period, embrittler atoms are ab-sorbed (Fig. 5, arrow 4) and then penetrate (Fig. 5, ar-row 17) by stress-aided diffusion (Fig. 5, arrow 18) a shortdistance into base metal grain boundaries. In the penetra-tion zones the presence of the embrittler atoms lowers thecrack resistance and increase the difficulty of slip. Whena sufficient concentration of embrittler atoms has beenbuilt up to some critical depth in one penetration zone,cracks nucleation takes place (Fig. 5, arrow 19), proba-bly at the head of already existing dislocation pile-upswhere the stress has become supercritical for the loweredcrack resistance, and then the crack propagates (Fig. 5,arrow 20).

3.7.2 Description

Initiation stage

The embrittler atoms penetration consists of two steps,namely:

1. change of the embrittler atoms from the adsorbed tothe dissolved state on the surface;

2. subsequent diffusion and penetration along preferentialpaths-usually grain boundaries.

The rate of both steps is accelerated by increased stress.The corresponding nucleation time depends on the twosteps as follows:

tn ∼ exp

(∆GS

RT

)exp

(∆Gd

RT

)(17)

where ∆GS and ∆Gd are the activation free energies forsteps 1 and 2, respectively.

Gordon does not give atomic details of the nucleationprocess itself, but reports that once the crack is nucleated,it grows extremely rapidly.

Moreover, he proposes a schematic diagram (Fig. 10)that represents the effect of stress and temperature on thetotal time to produce penetration zones during the crackinitiation stage. Superimposed on the diagram is a dottedline marked threshold stress, σth, which gives σth versustemperature (not a function of time).

If a sample preheated to T4 is tested at any fixed σlevel between σi4 (for which the loading time equals thepenetration zone formation time) and σth (which is thestress below which no LME failure takes place), such as,for example, σ1 (Fig. 10), delayed crack initiation will oc-cur. The initiation time will be equal to t1 − tσ1 , wheretσ1 is the time to load to σ1 and t1 is the time to formpenetration zones at T4 and stress level σ1.

30 The European Physical Journal Applied Physics

Fig. 10. Schematic diagram showing time to develop penetra-tion zones as a function of stress and temperature of test, andrelationship to crack initiation time. T9 > T8 > . . . > T1 [33].

Propagation stage

Concerning the propagation stage, it is well establishedthat the propagation of LME cracks cannot proceed with-out the continual supply of embrittler to the crack tip. It isnevertheless possible that the embrittler transport processmay not be the rate-determining step in the propagation.Repeated renucleation along the crack path could some-times be necessary, and this process could be slower thanthe embrittler transport.

The posssible embrittler transport mechanisms thathave been suggested for LME are:

– vapor transport;– bulk liquid flow;– grain-boudary diffusion;– heterogeneous diffusion of a couple of layers of the em-

brittler over the crack surface;– surface self-diffusion of the embrittler.

In Gordon’s analysis, it appears clearly that the em-brittler transport mechanism for SME consists of a sur-face diffusion of embrittler atoms over a layer of embrittlerthick enough so that the moving of the embrittler atoms inthe top atom layer is essentially self-diffusing. The trans-port mechanism in LME is bulk liquid flow; liquid metalswhich wet the base metal can penetrate to the tips ofeven very sharp cracks under the impetus of surface ten-sion, contrary to some statements in the literature. For afew high-vapor pressure embrittlers, such as Zn, Cd andpossibly Hg, vapor transport could play a role. For all theother embrittlers, the rate of evaporation from the solidor liquid embrittler makes vapor transport too slow.

3.7.3 Analysis

Gordon manages to explain many phenomenological char-acteristics of LME and SME with this model, such asdelayed failure, strain rate effects, solute effects, grainsize effects and effects of cold work. However, accordingto Popovich [31], this model does not explain the differ-ence in the incubation period in SME and LME, since therate of penetration will not depend on the solid or liquidnature of the coating. At last, according to Gordon, themain weak point of this model is that the existence of athreshold stress is not explained.

4 Conclusion

The survey has been concerned with the most outstand-ing models that have been proposed so far in the literatureto describe liquid metal embrittlement. Unfortunately, thereview does not allow an easy comparison between them.In fact, we cannot conclude that one model is more ap-propriate than another one for a specific solid metal-liquidmetal couple. Attempts to apply these models in a systemhave often been carried out taking into account experi-mental results obtained in specific conditions. Therefore,we think it is difficult to draw general conclusions via suchlimited data.

Some models have been developed to be in agreementwith the behavior of large number of systems. In spite ofsome advances, the difficulty still remains to understandand to predict why a particular embrittling species at-tacks a particular metal and how the embrittled metal willcrack. As a consequence, the approach which consists incomparing the results obtained in two different systems isnot relevant, since the elementary mechanisms which si-multaneously work in one system may not be the sameas those involved in the other. For example, diffusion-penetration mechanism should not be eliminated becauseit is not involved in some systems. In fact, it takes place inthe embrittlement of copper-bismuth system [44,45], likeother elementary mechanisms such as adsorption. For thisreason, studying the elementary processes that occur inthe embrittlement of a specific solid-liquid couple may leadto improved understanding. The chemisorption process,which is assumed in most of the models, should be studied.Liquid penetration at sites of high stress concentrations(e.g., grain boundaries) and its effect on embrittlementshould be also more investigated in the future. For exam-ple, in a recent paper, a mechanism of grain boundary pen-etration via the development of a macroscopic amorphouslayer has been proposed to explain the loss of ductilityin the aluminum-gallium system [46]. Such a mechanismcould be expanded to other systems. It is necessary there-fore to investigate new experimental work in that fieldincluding microstructural and mechanical studies.

This work has been supported by the European Union in theframe of the fusion technology programme (task UT-SM&C-LME).

B. Joseph et al.: Liquid metal embrittlement: A state-of-the-art appraisal 31

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