The grain boundary groove (GBG) developing at the ceramic substrate under the liquid metal is evident, yet not fully explained influencing appearance in describing the wetting phenomena at liquid metal/ceramics interface. The focus here is on modeling of the phenomena at/around a groove between grains depending on grooves’ geometry. Based on atomic force microscopy results, the groove efficiency assessment is provided as a function of the transferred mass quantity and related to grooves geometry. The transferred mass quantity and, according to it, the groove efficiency at parabolic GBG is about 10 % higher comparing to the triangular GBG

A. Patarić

K. Raić

M. Mihailović

Z. Gulišija

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Wetting phenomena of grooves at liquid metal/ceramics interface

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321METALURGIJA 55 (2016) 3, 321-324M. MIHAILOVIĆ, A. PATARIĆ, K. RAIĆ, Z. GULIŠIJA WETTING PHENOMENA OF GROOVES AT LIQUID METAL/CERAMICS INTERFACEReceived – Primljeno: 2015-10-16Accepted – Prihvaćeno: 2016-02-25Original Scientific Paper – Izvorni znanstveni radM. Mihailović, A. Patarić, Z. Gulišija, ITNMS, Belgrade, K. Raić, Fac-ulty of Technology and Metallurgy, Belgrade, SerbiaThe grain boundary groove (GBG) developing at the ceramic substrate under the liquid metal is evident, yet not fully explained influencing appearance in describing the wetting phenomena at liquid metal/ceramics interface. The focus here is on modelling of the phenomena at/around a groove between grains depending on grooves’ ge-ometry. Based on atomic force microscopy results, the groove efficiency assessment is provided as a function of the transferred mass quantity and related to grooves geometry. The transferred mass quantity and, according to it, the groove efficiency at parabolic GBG is about 10 % higher comparing to the triangular GBG. Key words: liquid metal, ceramics, interface, grain boundary grooves, wettingINTRODUCTIONThe wetting phenomena are important for under-standing the metal/ceramics joining process and further process development. The liquid metal/ceramic sub-strate systems can be encountered from metals process-ing, tools manufacturing, in high temperature and cor-rosion protection applications, in automotive industry and especially in microelectronics [1-7]. Despite devel-oped practical applications, the interface bonding of these two juxtaposed components is not fully explained yet. ‘Reaction product control’ theory claimed that the interface reactions take control over the wetting mecha-nism at interface [8,9]. The other theory stated that cap-illary effects and adsorption of metals onto the ceramic substrate with triple line ridging are crucial for control-ling the wetting phenomenon, not chemical reactions [1,2]. Properties of the resulting metal/ceramic interface extremely depend on the thickness, uniformity and po-rosity of the bonding layer, which may be the result of non-homogeneous chemical transformation that starts at the ceramic surface and propagates into the liquid metal [5-7,10,11]. Lattice matching of the two adjacent materials is an-other significant factor governing characteristics of the re-action layer formation at metal-ceramic interface [11-13]. So-called surface topography, such as roughness and other irregularities on the macroscale, or grain bounda-ry grooves (GBG) and lattice pits at microscale, signifi-cantly modifies properties of the liquid metal/solid ce-ramic interface. Finally, aside of the reactive or non-re-active wetting, it is influenced by the mechanisms oc-curring at nanoscale [14-18]. ISSN 0543-5846METABK 55(3) 321-324 (2016)UDC – UDK 669.7.0:532.5:666.3/.7.04:621.49=111Experimental studies of wetting phenomena on sub-strates patterned by pits, grooves and wedges with finite depths show the strong influence of nanostructure on the adsorption behaviour, comparing to that on flat sub-strates [14]. It is stated that the classical contact angle considerations were invalid for the tip of a spreading film. The shape, size and layout of the grain boundary grooves (GBG) grooves should be taken into account.MODEL OF GBG WETTINGThe phenomena at grain boundary grooves (GBG) on metal/ceramic interface are presented here as an original theory, encompassing the geometry of grooves influence on mass transport. In modelling GBG pro-cesses here, the analogies between mass and heat trans-fer, according to the transport phenomena standpoint, are applied [19]. Due to the capillary forces, the leading edge of the liquid droplet develops a ridge at the solid–liquid–va-pour triple junction, recorded in several metal-ceramic systems using AFM [1]. Grain boundary grooving is enhanced in the area under the liquid, and the groove cross-sections of interest are parabolic or triangular, ac-cording to recorded AFM profiles [4].Having in mind the analogies between mass and heat transfer, a mathematical analysis similar to the ex-tended surfaces modelling is applied here [20]. The generalized differential equation for the groove concen-tration profile is expressed based on consideration of the steady-state mass balance over the differential ele-ment of the groove depth dy, with planes parallel to the groove base at y and y+dy, (Figure 1). Here, y is a dis-tance from the groove tip to an observed element of the groove depth. The limiting profile curves are expressed as x = ±f2(y). 322M. MIHAILOVIĆ et al.: WETTING PHENOMENA OF GROOVES AT LIQUID METAL/CERAMICS INTERFACE METALURGIJA 55 (2016) 3, 321-324Where:–  CA(l) – is the concentration in liquid metal– CA(s) – is the concentration in solid ceramics– Deff  –  the effective diffusion coefficient from the liq-uid metal into the porous solids–  k’’’ – is the surface rate constant – km – is the mass transfer coefficient– u – the groove perimeter– Lg – the groove length – ωg – the groove half-width– ϑ –  is excess of concentration between a point on the groove and the surroundings.To meet the terms of a steady state, the difference in mass transfer into and out of the element dy, must be balanced by mass dissipation from the lateral surface of the groove. This also postulates that the element dy on a random surface described by function f2(y) is equal in height to the element dy on the y-axis.The mass difference (dm) caused by the mass enter-ing the observing element by diffusion at y + dy and the mass leaving the GBG element by diffusion at y is:  (1) For ϑ= C − CA(S) and because of C A(S) is assumed to be constant, it follows dϑ = dC.When k’’’<<< km, than the k’’’ becomes rate limiting constant, and since both have the same dimension, k’’’ should substitute km, giving the following: dm = k’’’u (C − CA(S) ) dy  (2)or dm = 2 k’’’[Lg + f2(y)](C − C A(S)) (3)Here is to be applied the Murray–Gardner assump-tion, which states that the groove depth must be small in comparison to its length, Lg >> 2f2(y). Hence:  (4)Equating the equations (1) and (4), the general dif-ferential equation is obtained:  (5)The profile function f2(y), for any longitudinal groove is in the following form:  (6)For parabolic groove profile, the exponent here sat-isfies the groove geometry for n = 1/3, while for trian-gular GBG cross-section, the exponent satisfies geom-etry when n = 0.Introducing z, the groove performing parameter:  z = (k’’’ /Deff ωg)1/2 (7)and substituting it in eqn. (5), together with adequate profile function f2(y), and the groove depth l , the gov-erning differential equation for parabolic GBG profile for concentration excess ϑ(y) = C(y) – CA(S), turns into:  (8)while for triangular GBG profile it is as follows:  (9)RESULTS AND DISCUSSIONCalculation of the transferred mass quantity for each GBG starts by differentiating the equation for the par-ticular solution and evaluating the derivative at y = l. Designating the groove cross-section with S=ωgLg and applying expanded Bessel function for according to [20], the mass quantity transferred through the groove of a parabolic profile is:  (10)Accordingly, the mass quantity transferred through the groove of a triangular profile is:  (11)Where:l – is the groove depthI0 – modified Bessel function of the first kindIn –  modified Bessel function for different profile function exponent n, eqn. (6)The groove efficiency is the ratio of the actual and the ideal mass transfer, mideal=2k’’’lLgϑl , so:  (12)The zl parameter, which figures in both equations, for the quantity of mass transferred through the groove and for the groove efficiency, encompasses the groove dimension and mass transfer influence.Figure 1 Parabolic type of a grain boundary groove 323M. MIHAILOVIĆ et al.: WETTING PHENOMENA OF GROOVES AT LIQUID METAL/CERAMICS INTERFACEMETALURGIJA 55 (2016) 3, 321-324The groove mass transfer as well as the groove effi-ciency are related to groove perform parameter z, di-rectly depending on the grooves geometry, eqn. (7). When k’’’ and Deff are constant in a particular system, the groove efficiency is influenced by GBG geometry. Ob-serving GBG profiles obtained using atomic force mi-croscopy (AFM) and measurements of grooves by high resolution scan in the Ni/Al2O3 system [4], the calcula-tions can be performed according to the here proposed model of GBG wetting. For the surface rate constant estimated to be k’’’~10-5 m/s in this system [4], it is calculated that ωgDeff = 4,4 10-19 m3/s. Since the measured groove depth is l = 0,2 μm = 2 10-7 m [4], the groove performing parameter from eqn. (7) can be determined. Upon calculation, zl ≈ 1 is ob-tained.According to the reported AFM profiles, the cross-section of the GBG can vary in shape. The groove effi-ciency for both characteristic types of groove profiles, triangular and parabolic, using the proposed calculation is presented (Figure 2). It can be concluded that the maximum difference in groove efficiency between these characteristic groove types is at/around the value 1 for zl parameter, which is previously calculated based on the value of groove per-forming parameter z.Observing the dependence of groove efficiency for both characteristic types of grain boundary groove pro-files on zl parameter, it can be concluded that the maxi-mum difference in groove efficiency between these characteristic groove types is at/around the value 1 for zl parameter, which is calculated based on the value of groove perform parameter z. It is evident that groove efficiency, η, strongly de-pends on the groove profile. The efficiency of triangular GBG is below the parabolic value. The quantity of mass transferred through parabolic GBG is about 10 % high-er comparing to the mass transferred through triangular GBG.Since the groove perform parameter z, directly de-pends on the groove geometry, eqn. (7), it can be noted that with groove width decrease or depth increase, the zl parameter increases, reaching its maximum influence in the interval between 1 and 1,5 (Figure 2). The differ-ence in groove efficiency for different grooves geome-try, when zl parameter reaching value 0, is becoming less influential, meaning that with very wide grooves or small groove depth, the surface is becoming almost flat.With zl parameter increasing, the difference between characteristic groove profiles increases, and for zl > 0,2 the groove geometry influence is evident. The calculation has proven the prediction of the GBG wetting model, i.e. that grooves efficiency, η, strongly depends on the grooves profile. The efficiency of triangular GBG is below the parabolic value. The quantity of mass transferred through parabolic GBG higher comparing to the mass transferred through trian-gular GBG, implying that the groove efficiency, as well as liquid metal/ceramic joining can be influenced by GBG geometry.CONCLUSIONSThe wetting phenomena at liquid metal / ceramics interface are shown to be the GBG geometry dependent. Grain boundary grooves are, according to reported AFM profiles, classified as triangular or parabolic and completely evaluated from the governing differential equation, to groove efficiency, η. The calculated values, according to the analytical model, reveal evident differ-ences between profiles, giving the higher groove effi-ciency for parabolic profile and its maximum influence for parameter zl ≈ 1.Based on AFM measurements, the calculated groove efficiency for parabolic GBG is about 10 % higher com-paring to the triangular GBG. The quantity of mass transferred through parabolic concave GBG is, accord-ing to proposed equations, proportional to the grove ef-ficiency.This implies that the metal/ceramic joining process can be influenced by GBG geometry.AcknowledgementsThe authors wish to acknowledge the financial sup-port from the Ministry of Education, Science and Tech-nological Development of the Republic of Serbia through the projects TR 34002 and P-172005.REFERENCES[1]  E. Saiz, R.M. Cannon, A. P. Tomsia, Reactive spreading: adsorption, ridging and compound formation, Acta Mate-rialia 48 (2000) 4449–4462. doi:10.1016/S1359-6454(00)00231-7[2]  E. Saiz, A. P. Tomsia, R. M. Cannon, Triple line ridging and attachment in high-temperature wetting, Scripta Materialia 44 (2001) 159–164. doi:10.1016/S1359-6454(00)00231-7Figure 2  Groove efficiency η vs. zl parameter, for both groove profiles324M. MIHAILOVIĆ et al.: WETTING PHENOMENA OF GROOVES AT LIQUID METAL/CERAMICS INTERFACE METALURGIJA 55 (2016) 3, 321-324[3]  C. 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Wetting phenomena of grooves at liquid metal/ceramics interface

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