Mg2Sn-induced whisker growth on the surfaces of Mg/Sn/Mg ultrasonic-assisted soldering joints

Cao Huijun ,Li shiqin ,Zhang Yinggan ,Zhu Yichen, ,Li Mingyu ,Zhang Zhihao

1.College of Materials (Xiamen 361000) &Shenzhen Research Institute (Shenzhen 518055),Xiamen University,China;2.School of Transportation Engineering,Xiamen City University,Xiamen 361005,China;3.Sauvage Laboratory for Smart Materials,School of Materials Science and Engineering,Harbin Institute of Technology (Shenzhen),Shenzhen 518055,China

Abstract In this paper,the phenomena of Mg2Sn-induced Sn whisker growth were explored on the surfaces of Mg/Sn/Mg ultrasonic-assisted soldering joints after aging treatment.The in-situ observation and thermal analysis confirmed that the formation and the corrosion of Mg2Sn nanoparticles were the dominant reason of Sn whisker growth.The Mg2Sn accumulation at the grain boundaries would pin the dislocation slip and affect the continuity of whisker growth,and the boundary angle would thus play a decisive role in the growth shape of Sn whiskers due to the pining effect of Mg2Sn.This study might be conducive to elucidating the growth behavior of Sn whiskers and provide the exploration strategy to further improve the bonding strength of Mg/Sn/Mg ultrasonic-assisted soldering joints.

Key words Mg alloys,ultrasonic wave,solder joint,Sn whisker,microstructure,growth mechanism

0 Introduction

Nowadays,the magnesium (Mg) alloys have attracted widespread public attention in the fields of aerospace,automotive and electronics owing to the low densities and excellent specific mechanical properties[1].However,the poor ductility of Mg alloys had made working difficult and invariably limited their practical applications[2–3].To achieve robust complex-assembled products from simple components,high-quality joining techniques of Mg alloys were desired but challenging.

Some approaches were developed to establish metallurgical joining between Mg alloys (and other metals),including brazing[4],transient liquid phase bonding[5],friction stir welding[6]and ultrasonic -assisted soldering (UAS)[7–9].Above all,the UAS method had multiple advantages that make it ideal for joining Mg alloys.For instance,it could obtain reliable interconnect joints at a comparatively low temperature (<300oC) within a very short time,which might effectively avoid the weakening of base materials near joining zones[8–9].Moreover,the surface oxide films of Mg alloys could be removed by ultrasonic cavitation effect;thereby,the UAS processes were always conducted in an air atmosphere without any flux agents[10].In addition,because of the homogenization effect of microstructure and composition by ultrasonic wave,the UAS joints are much more robust and reliable compared to traditional solder joints[11].

Despite its potential as an ideal joining method for Mg alloys,the application of UAS might suffer from a major challenge that the numerous tin (Sn) whiskers could be created on the surfaces of UAS joints after thermal aging at room temperature[9].Because the solder matrix would be continuously loosened up with increasing Sn whisker length,the mechanical properties of resultant joints might be degraded over time.Previous studies confirmed that the whisker growth was a creep process,and there were three necessary conditions for occurrence of Sn whisker growth,including the discontinuous protective layers of SnO2[12],the continuous source to produce compressive stresses[13,14] and the grain boundaries for Sn diffusion[15].The inhabitation of whisker growth required determining the inherent mechanism that enabled control of growth mode.Although certain growth behaviors have been revealed[16–20],the underlying mechanism for Sn whisker growth in Mg/Sn/Mg UAS joints are still missing,and it is unclear why the final shapes of certain Sn whiskers on such joints were curved with kinks and striations rather to be straight and long[21].In this paper,the growth characteristics of Sn whiskers in Mg/Sn/Mg UAS joints were explored and the intrinsic mechanism for whisker growth was systematically analyzed.

1 Experimental procedures

1.1 Structure preparation

Fig.1 showed a self-designed UAS equipment to fabricate Mg/Sn/Mg joints.An induction heating apparatus with a maximum frequency of 1.5 MHz and an operating power of 6 kW was used as a heating source.An iron (Fe) mould was fixed at 3 cm above the water-cooled copper (Cu) coil.When an alternating current at certain frequency was supplied to the coil,the mould would be heated rapidly.Two AZ31B Mg alloy sheets with the sizes of 25 mm × 15 mm ×3 mm and 20 mm × 10 mm × 3 mm,respectively,and one Sn-based sheet with a size of 20 mm × 10 mm × 0.3 mm were sandwiched and positioned in the mould.Subsequently,the Mg/Sn/Mg sandwiched structure was pressed by an ultrasonic horn with a pressure of 0.6 MPa.When the temperature of the structure was ramped to 250 ℃ and held for 10 s,the ultrasonic vibration was conducted for 5 s by the ultrasonic horn with ultrasonic power of 200 W and frequency of 28 kHz,respectively.Finally,the UAS equipment was switched off and the resultant joints were cooled down to 25 ℃ in water.

Fig.1 Schematic of fabrication of a Mg/Sn/Mg joint using a self-designed UAS equipment

1.2 Structure characterization

The standard grinding and polishing operations were carried out on aforesaid joints in the cross -sectional direction.The as-polished samples were placed in a drying oven,aged at 25 ℃ and detected by a scanning electron microscopy (SEM,Hitachi SU70) and an auger electron spectroscopy (AES,Ulvac PHI700) at each selected time point.To explore the intrinsic growth mechanism,certain Sn whiskers were cut by a focused-ion beam (FIB,FEI Helios NanoLab 600i) and detected by a transmission electron microscopy(TEM,Philips Tecnai G2 F20) with an energy dispersive spectroscopy (EDS).Also,a differential scanning calorimetry test (DSC,Netzsch STA 449F3) was conducted with four consecutive heating-cooling cycles in temperature ranges of 25–250 ℃ at a rate of 1 ℃/min in air,and the DSC sample (50 mg) was scraped from the solder of a UAS joint.

1.3 Simulation method

Based on the density functional theory,the theoretical calculation was performed by Vienna ab initio simulation package code in conjunction with the projector-augmented wave.The generalized gradient approximation of Perdew-Burke-Ernzerh approach of pseudopotentials was adopted to describe the exchange-correlation energy.The 4 × 4 × 2 superlattices ofβ-Sn phase (space group of I41/amd;a=5.830 8 ?,c= 3.181 0 ?) with certain Mg atoms inserted were modeled and analyzed by the VESTA.The energy cutoff was set to be 550 eV.The Gamma centered scheme was used with 1 × 1 × 1k-point.In addition,the formation energy of Mg2Sn (space group of Fmm;a= 6.763 0 ?)was also detected.

2 Results

2.1 Microstructure evolution of UAS joints after thermal aging

Fig.2 showed two Mg/Sn/Mg UAS joints after thermal aging at 25 ℃ for 1?8 d.On the 1std,the Mg alloy substrates and Sn matrix were bonded well with each other,and the wavy-like interfaces were distinctly observed.Because of the influence of ultrasonic vibration and pressure,the liquid Sn might be partially squeezed out from the soldering seam during UAS process.Hence,the width of the Sn matrix was detected to be ～250 μm,less than that of original solder sheet (300 μm).Moreover,since the ultrasonic vibration could promote the atomic dissolution and diffusion[8],plenty of Mg atoms would migrate from the substrates to the Sn matrix during UAS process.Therefore,numerous Mg2Sn grains with irregular block-like and fine seaweedlike features were emerged in UAS joints.In addition,plenty of Sn protuberances (colored by violet) were extruded from the Sn surface,and corresponding nucleation density per unit area was calculated to be ～3.37 × 103number/mm2,far larger than previous studies[14,22,23].

Fig.2 Microstructures of Mg/Sn/Mg UAS joints after thermal aging for (a) 1 d and (b) 8 d

However,on the 8thd (Fig.2 b),the nucleation density of extruded structures (i.z.,protuberances and whiskers) per unit area was calculated to be ～3.56 × 103number/mm2.This implied that only few Sn protuberances were newly generated on the Sn surface within 8 d.Compared with the sizes of protuberances in Fig.2 a,most of them had grown to a certain extent,but only a limited number of Sn whiskers were observed.This indicated that only partial protuberances grew to be long whiskers,and the rest had stopped growing at a certain time.Noteworthily,the closer to the Mg alloy substrates,the smaller number of whiskers but the larger number of protuberances.Based on EDS analysis,the concentration of the Mg atoms was relatively lower in the whisker-rich region than that in the protuberance-rich region.Hence,the distribution and concentration of Mg atoms in the Sn matrix might have an important effect on the nucleation and growth of Sn whiskers.In addition,most of these extruded structures exhibited curved shapes with kinks and striations,similar like in previous studies[17,21–23]

2.2 Formation reason of extruded Sn structures in UAS joints

The limited formation and selective growth of extruded Sn structures during aging were two important characteristics in aforesaid observation.The former implied that the formation probability of Sn protuberances might dramatically decrease,whereas the later indicated that the driving force and resistance for Sn whisker growth should be competitive with each other.Robertson et al.found that[24],the Mg2Sn phase would be corroded in humid conditions at room temperature owning to the galvanic corrosion mechanism,and the reaction can be described as

During water-assisted grinding and polishing,the Mg2Sn phase could transfer to Mg(OH)2andβ-Sn phases.Since the molar volumes of Mg2Sn,β-Sn and Mg(OH)2phases were 46.61,16.33 and 24.71 cm3/mol respectively,this reaction might lead to a 11.95% shrinkage in volume(Table 1).Meanwhile,the Mg(OH)2phase would dissolve in water,only leaving behind corrosion channels on the surface of Sn matrix.

Table 1 Physical properties of Mg2Sn,β-Sn and Mg(OH)2 phases

After polished,the surface of Sn matrix would be re-oxidized in air,and a continuous SnO2layer might be regenerated.When the sample was dried in an oven,the capillary force might compel the residual water contained in corrosion channels to travel back to the Sn surface.Further,the Mg2Sn grains closely linking with these channels might react with the water and collapse itself.In this case,the SnO2layer would be partially destroyed,being suitable sites for Sn protuberance formation.Hence,numerous protuberances were generated at the beginning.However,owing to the exhaustion of residual water,only few Sn protuberances could be formed in the following days.To verity aforesaid speculation,an in-situ observation of the local Sn matrix in a Mg/Sn/Mg UAS joint was conducted at 25 ℃ for 1 d?8 d.As shown in Fig.3,three Sn protuberances were extruded from Sn matrix on the 1std;however,as the residual water could not exist in a vacuum chamber,no newly generated Sn protuberances were emerged after the observation on the 1std.Moreover,some sunken channels were detected,directly demonstrating the occurrence of Mg2Sn corrosion after polished.Accordingly,the formation reason of extruded structures might be related to the galvanic corrosion of Mg2Sn phase.

Fig.3 In-situ SEM observation of a local Sn matrix in a Mg/Sn/Mg UAS joint during aging at 25 ℃ for 1 d?8 d

In addition,only the Protuberance-A exhibited a fast growth behavior in the first 7 d and stopped to grow after the 7thd,whereas the sizes of the Protuberance-B and Protuberance-C had never changed after the 1std.Through calculation,the Protuberance-A grew similar in length at each time interval between two time points of observation;because each small straight segment had rotated to certain angle relative to a previous segment,a curved whisker with striated surfaces was finally observed.Liu et al[22].proposed that Sn whiskers with longitudinal striations were produced in air due to the confining effect of SnO2layers,and the faceted segments would be observed in vacuum.Based on our observation,although the growth rate of Protuberance-A (3～5 μm/d) was slightly slower than that in Liu’s study(～20 μm/d),no faceted segments were detected.This indicated that the Protuberance-A stopped growth during observation.Interestingly,although the Protuberance-A rotated intermittently,its striated feature (e.g.,depth and width) was preserved intact.This meant that the driving force for whisker growth and the confining resistance of SnO2layers were stable throughout the period.Hence,the variation of growth direction of Protuberance-A should be largely induced by the growth resistance,rather to the driving force.

2.3 Driving force for Sn whisker growth in UAS joints

Undoubtedly a Sn protuberance sprouting from the Mg2Sn corrosion channel required a continuous driving force to be a long whisker.In previous study,the continuous formation of Cu6Sn5phase at grain boundaries of Sn matrix could provide the compressive stress for whisker growth[24].Hence,we speculated that Mg2Sn phase in our experiment may play a similar effect.Fig.4 a showed another in-situ SEM observation of a local Sn matrix during aging at 25 ℃ for 1 d?15 d.Besides the observation of fast growth of a Sn protuberance,some nanoscale black spots were concurrently generated on the surface of Sn matrix (as marked by the white dotted-line box).Based on the AES analysis (Fig.4 b),the Mg concentration in this region after aging for 15 d was relatively higher than that for 1 d.If we excluded the stoichiometry of SnO2,these black spots had a near stoichiometry of Mg2Sn,meaning that some Mg2Sn nanoparticles had precipitated from Sn matrix in the Mg/Sn/Mg UAS joint during aging treatment.

Fig.4 c depicted DSC heating flow signals for one sample subjected to four consecutive heating-cooling cycles.The sample was directly scraped from the solder of a UAS joint.Based on the phase diagram of the Mg-Sn binary system,the eutectic transformation ofL?Mg2Sn+β-Sn occurred at a temperature of 203.8 ℃ with respect to Mg concentration of 9.5at.%[25].Thus,the endothermic peaks around ～205 ℃ in the four heating curves should be related to Mg2Sn dissolution in liquid solder.Unexpectedly,the endothermic peaks for the 2nd,3rdand 4thheating curves (corresponding to red,blue and green lines,respectively) were maintained at 205.6 ℃,0.5 ℃ higher than that for the 1stheating curve (corresponding to black line).Through calculation,the Mg concentration in solder with endothermic peaks at 205.6 and 205.1 ℃ were 9.64 and 9.60 at.%,respectively.One explanation was that a few of Mg atoms were supplied in solder during the 1stheating process.Another possible cause was that some Mg2Sn grains might exist in solder before the 1stheating,producing a decrease of melting point of the solder to certain extent.

Fig.4 Experimental verification for Mg2Sn formation and growth (a) In-situ observation of local Sn matrix during aging at 25 ℃ for 1?15 d;(b) Two AES curves of the same region on the 1st and 15th d;DSC heating flow curves of the same sample in the temperature ranges of (c) 50?250 ℃ and (d) 75?150 and 150?250 ℃

Fig.4 d showed DSC heating flow curves in the temperature ranges of 75?150 and 150?250 ℃.A broad exothermic peak in the black line was detected in the temperature range of 75?120 ℃,but it did not reappear in the red,blue or green lines.According to previous studies[9,26,27],the solid solubility of Mg atoms inβ-Sn was estimated in the range of 0.1?0.2at.% at 200 ℃,and the actual Mg concentration in a UAS joint might reach a supersaturation level due to ultrasonic cavitation and fast cooling.Therefore,a part of Mg atoms existed in solid-solution states after UAS process.However,the supersaturated Mg atoms were metastable,and an exothermic reaction of 2Mg+β-Sn→Mg2Sn would occur at elevated temperature (e.g.,75?120 ℃).Because the solid solubility of Mg atoms was limited in a normal cooling process,the aforesaid reaction was merely detected in the 1stheating process but absent in other heating processes.In addition,the four exothermic peaks in the temperature range of 190??220 ℃ were almost in the same area.The small difference was that the endothermic peak with broader width and lower height was observed in the black line compared to those in other lines.It demonstrated that some Mg2Sn particles before dissolved in solder during the 1stheating process were in the nanoscale range,completely consistent with our SEM observation in Fig.4 a.

Based on Table 1,if Mg atoms were located at substitutional sites ofβ-Sn lattices,the formation of Mg2Sn phase would produce a 5.29% expansion in volume.However,if the Mg atoms were located at interstitial sites,the formation of Mg2Sn phase would have a 285.43% expansion in volume.It should be emphasized that the volume expansion over 2% would produce significant compressive stresses on the surface of the solid matrix[28].Thus,no matter the substitutional-site or interstitial-site occupation of Mg atoms inβ-Sn lattices,enough compressive stresses could be provided the driving force to drive Sn whisker growth in the Mg/Sn/Mg UAS joints.

2.4 Resistance of Sn whisker growth in UAS joints

Although large amounts of Sn protuberances were generated at the beginning,only a limited number of whiskers were formed after a period of aging,particularly closer to the Mg/Sn interfaces.To explore the inherent reason behind the difficulty of whisker growth,a Mg/Sn/Mg UAS joint after aging at 25 ℃ for 8 d was prepared,and two Sn whiskers located,respectively,near and far away from Mg/Sn interfaces were cut by the FIB and detected by the TEM.

Fig.5 a showed one Sn whisker located near the Mg/Sn interface (colored by violet).After FIB cutting (Fig.5 b),the whisker had a curved shape,and the matrix beneath it was loose with voids and cracks.Moreover,few Mg2Sn grains were detected at the whisker root,directly proofing that the Sn whisker was prone to extruding from corrosion channels of Mg2Sn phase.Based on TEM image in Fig.5 c,numerous black nanoparticles and several dislocations were dispersed.Especially in enlarged views of Fig.5 a?c,the black nanoparticles were distributed in all over the region-A,gradually clustered along dislocation lines in the region-B,and accumulated at the interface between the whisker and matrix in the region-C.Considering the phenomena of Sn whisker growth were time -dependent,the tendentious distribution of these black nanoparticles greatly reflected the change of growth behavior of Sn whiskers.

Fig.5 Sn whisker located near the Mg/Sn interface in a Mg/Sn/Mg UAS joint after aging at 25 ℃ for 8 d (a) SEM image in the top-view direction and (b) SEM image in the cross-sectional direction;(c) TEM image of cross-sectional microstructure.In addition,the enlarged images of Region-A,Region-B and Region-C were marked in c

Chiu et al.found that[29],a pile of dislocations would be created when a Sn whisker experienced an abrupt growth.Considering that the expansion reaction of 2Mg+β-Sn→Mg2Sn would provide enough driving force,we believed that there must be certain resistance to block whisker growth.To explore the resistance source,the microstructure of the region between whisker root and Sn matrix in Fig.5 c were re-studied in Fig.6.The selected-area electron diffraction analysis (SAED) showed that these black nanoparticles were the Mg2Sn phase.Moreover,based on the line-scan analysis in Fig.6 b,the Mg atoms were poor in the whisker root but were rich in the matrix,and an obvious segregation of Mg atoms was detected at the interface between them.In addition,the farther away from this interface,the higher Mg concentration in the whisker;however,the Mg concentration was stable in the matrix.Fig.6 c and d showed the high-resolution TEM images of the yellow and green box regions,respectively,in Fig.6 a.Based on the lattice fringe and SAED patterns,the former was viewed along the [11] direction ofβ-Sn phase,while the latter was viewed along the [001] direction ofβ-Sn phase.A huge orientation difference between the whisker and matrix meant that there was a high-angle grain boundary ofβ-Sn phase between them.Moreover,the regions in the white dotted -line boxes of Fig.6 c and d were processed by a standard Fourier space based band-pass filtering.Based on the resultant Fourier filtered images and inverted Fourier filtered images reconstructed by spatial frequencies of certain planes,the defects in this whisker root were composed by dislocations and layer faults,while the defects in the matrix were mainly composed by lattice distortions.

Fig.6 Sn whisker located near the Mg/Sn interface(a) Microstructures and (b) Mg concentration distributed in the region between the whisker root and Sn matrix.High-resolution TEM images with inserted SADE patterns in the (c) yellow and (d)green box regions in a.The Fourier filtered and inverted Fourier filtered image reconstructed using the spatial frequencies of(24) planes were inserted in c,while the Fourier filtered and inverted Fourier filtered image reconstructed using the spatial frequencies of (400) planes are inserted in d

Both Mg2Sn nanoparticles and solid-solution Mg atoms could cause the lattice distortions in the matrix.However,when the Sn atoms in the matrix went across the grain boundary to build a whisker,the Mg2Sn nanoparticles were more prone to be intercepted compared to solid-solution Mg atoms.Hence,the grain boundary would act in effect as a sieve to separate Mg2Sn nanoparticles in the matrix.Because solid-solution Mg atoms might transform continuously to the nanoparticles of Mg2Sn phase,Mg concentration of whisker root would decrease with increasing time.Meanwhile,the separated Mg2Sn nanoparticles would block the dislocation motion and pin grain boundaries[30–31];accordingly,the whisker growth would be intermittent,and the abrupt whisker growth could occur only when the accumulated stress overcome the pinning growth resistance.In addition,the whisker growth would finally stop owning to an excessive accumulation of Mg2Sn nanoparticles at the grain boundary.

Fig.7 showed a TEM image of a Sn whisker located far away from the Mg/Sn interface (colored by violet).Based on the inserted SEM image and the SAED pattern of Region-A,the whisker had a rod-like shape with the growth axis of [100] direction ofβ-Sn phase.Similar SAED patterns were also detected in the Region-B and Region-C,and the orientation angles between Region-A and Region-B and between Region-A and Region-C were calculated to be 3°and 14°,respectively,in the (001) plane ofβ-Sn phase.In other words,low -angle grain boundaries existed between whisker-A and matrix-B and between whisker-A and matrix-C.Moreover,based on local enlarged views of Fig.7 a?d,some black Mg2Sn nanoparticles were detected;however,the number of the nanoparticles,no matter in the whisker or in the matrix,was far fewer compared with that observed in Fig.5.Because the Mg concentration in the matrix would decrease with increasing distance from the Mg/Sn interfaces,the amount of solid-solution Mg atoms should have a similar trend;therefore,less Mg2Sn nanoparticles would be generated.In addition,two dislocation lines were observed in Fig.7 a.The Dislocation-a was probably a screw type,while the Dislocation-b was probably of mixed edge-screw type.Because few dislocation lines were detected in this whisker,its growth was continuous rather to be intermittent.

Fig.8 a showed the local scanning TEM image of the region between the whisker root and Sn matrix in Fig.7.The inserted SEM image implied that the whisker-A had also grown on the edges of Mg2Sn particles,but these particles had collapsed during FIB cutting,only leaving some voids in the matrix beneath whisker root.The linescan EDS profile across the whisker-A and matrix-B in Fig.8 b showed that no Mg segregation occurred at the grain boundary of 3°;however,an obvious Mg segregation was detected between the whisker-A and matrix-C at the grain boundary with 14° angle in Fig.8 c.Combined with Fig.6 b,the grain boundaries with higher angles seemed more beneficial to Mg adsorption than those with lower angles.Because the low-angle grain boundary had a limited effect on Mg segregation,the effects of grain boundary sliding or dislocation movement could not be realized;finally,the straight and long whisker might be generated.

Fig.7 TEM image of cross-sectional microstructure for a Sn whisker located far away from the Mg/Sn interface and the local enlarged views of the regions A,B,C,and D were highlighted by the white dotted-line boxes,while the SAED patterns of the regions A,B,C and E were highlighted by the white circles

Fig.8 Sn whisker located far away from the Mg/Sn interface (a) Local TEM image of the region between the whisker root and Sn matrix in Fig.7,and corresponding line-scan EDS profiles for (b) the red line-b and (c) the black line-c

3 Discussion

Spontaneous growth of Sn whiskers depended on atomic diffusion driven by stress gradient from surrounding matrix to oblique grain boundaries of shallow grains[25–29].Since the level of accumulated stresses could hardly exceed the yield strength of Sn matrix,the diffusional creep was considered to be dominant in whisker growth.In addition,owing to high homologous temperature of Sn (e.g.,0.58 at 20 ℃),both lattice diffusion and grain-boundary diffusion were comparable.

The Nabarro-Herring creep was a typical creep mode controlled by lattice diffusion,and the early classical models to describe whisker growth were mostly established based on this mechanism[17,30].However,with the development of FIB technique,oblique grain boundaries were detected beneath Sn whiskers.Accordingly,the Coble creep,controlled by grain boundary diffusion,was currently one of the most recognized mechanisms for Sn whisker growth[18,27,29].

Similar to the Sarobol’s finding[18],some grain boundaries beneath Sn whiskers were observed in Fig.5 c and Fig.8 a;however,because the impurities could pin the dislocations and restrain dislocation movement,the impurity drag creep,involving both diffusional flow and dislocations,might be more crucial in Mg/Sn/Mg UAS joints.Moreover,the reaction of 2Mg+β-Sn→Mg2Sn would make solder expansion in volume,producing the strain energy stored in the matrix.However,due to the kinetic reason,the precipitated size of Mg2Sn particles was varied with aging time.Thus,the driving force for whisker growth should not only be affected by the concentration of solid-solution Mg atoms but also be influenced by aging time.In addition,the sliding along the sliding plane was restricted by lattice friction stress (i.e.,Peierls-Nabarro stress);thus,the surface reconstruction of whisker should also be an important resistance for whisker growth.

3.1 Volumetric change of the solder

The solid solution Mg atoms might occupy two types of defect sites (i.e.,substitutional sites and interstitial sites) inβ-Sn lattices.If all Mg atoms in solder entirely transferred to Mg2Sn nanoparticles,the volumetric change rate (Δv/vSn)would be

for interstitial occupation of Mg atoms,where Δvis volumetric solder increase,vSnis original solder volume,VMg2sn,VSnandVMgare the molar volumes of Mg2Sn,β-Sn and Mg,respectively,andcsandciare the concentration of substitutional Mg atoms and interstitial Mg atoms,respectively,inβ-Sn.

No matter substitutional-site or interstitial-site occupation,the concentration of solid-solution Mg atoms would make the Gibbs free energy increase inβ-Sn lattices.The stability ofβ-Sn lattices was closely correlated to its cohesive energy,and the cohesive energy per atom (ECoh) was calculated by the difference between the total energy of superlattice per atom (ETot) and the sum of the isolated atomic energy of Mg (EMg) and Sn (ESn).Because Mg atoms were randomly dispersed inβ-Sn lattices after UAS,the cohesive energy per atom could be defined as

wheremandnrefer to the numbers of Mg and Sn atoms,respectively,in a unit superlattice.Based on DFT,a 4 × 4 × 2 superlattice with 128 atoms was applied to model the solidsolution Mg atoms inβ-Sn lattices (Fig.9).Apparently,the cohesive energy was influenced by both solid-solution type and the number of solid-solution atoms.As the formation energy (EFor) of Mg2Sn phase was ?0.220 eV/atom,the superlattice would split only whenECoh≥EFor.Hence,the solid-solution Mg atoms would transfer to Mg2Sn if the number of substitutional Mg atoms exceeded 59 atoms (ECoh≥?0.220 eV/atom) or the number of interstitial Mg atoms exceeded 10 atoms (ECoh≥ ?0.230 eV/atom).Combining Eqs.1 and 2,Δv/vSnwas 6.13% for substitutional occupation and was 7.24% for interstitial occupation.Hence,both types of solid-solution Mg atoms could induce serious distortion inβ-Sn lattices,further being the driving force for whisker growth.In addition,from the perspective of actual Mg concentration,both types of solid-solution Mg atoms should coexist in the Mg/Sn/Mg UAS joints.

Fig.9 Formation energy of solid-solution Mg atoms in superlattice of β-Sn phase

3.2 Growth size of Mg2Sn nanoparticles

Based on our TEM observation,the spherical Mg2Sn nanoparticles would be precipitated from supersaturatedβ-Sn matrix during aging at 25 ℃,and the concentration of Mg atoms distributed around a nanoparticle in a spherical coordinate can be described in Fig.10.If the nucleation of one nanoparticle occurred at the beginning of aging,the radius of nanoparticle (r) with a time (t) would ber0= 0 att=0.Moreover,when this nanoparticle grew at a rate of dr/dt,the Mg flux absorbed by the nanoparticle would be equal to Mg flux supplied by the diffusion fromβ-Sn matrix (we could ignore the molar volume change of Sn atoms in this reaction for simplicity).That was

Fig.10 Derivation of precipitation equation for a Mg2Sn nanoparticle in the β-Sn matrix (a) Diffusion in spherical coordinates (b) Concentration curve of Mg atoms around the nanoparticle,and (c) Phase diagram of Mg-Sn

whereDis the diffusional coefficient,cis the Mg concentration around the nanoparticle in theβ-Sn matrix,andRis the diffusion distance from certain point of the matrix to the center of the nanoparticle.Based on the equilibrium of Mg atoms on the sides at Mg2Sn/β-Sn interface,the concentration of Mg atoms distributed around this nanoparticle inβ-Sn matrix was constant and would satisfy

whereu=c·R.The solution of Eq.6 wasu=k1·R+k2orc=k1+k2/R.The boundary conditions were

and the concentration gradient at the Mg2Sn/β-Sn interface could be expressed by

Combining Eqs.(5)?(8),the growth rate of a Mg2Sn nanoparticle could be

Because the very low solubility of Mg inβ-Sn at room temperature,the corresponding diffusivity had not been reported.We could assume that the diffusivity of Mg inβ-Sn was equal to the self-diffusivity of Sn,and its value was nearly 10?21?10?22m2/s[31].Usingcs= 9.6%,cβ= 0,cMg2sn=66.7%,we expected a Mg2Sn nanoparticle of 3.15? 9.97 nm after aging for 4 d,which greatly agreed with our observation in Fig.4 a.

3.3 Grain boundary pinning

When a Sn whisker grew,the substances (e.g,Sn atoms,solid-solution Mg atoms and Mg2Sn nanoparticles) surrounding the whisker root would be sucked up by the grain boundary between the whisker and the matrix.Undoubtedly,the Sn atoms and solid-solution Mg atoms were more prone to penetrate the grain boundary,sustaining whisker growth;however,the Mg2Sn nanoparticles were too large to pass through the grain boundary and were more likely intercepted due to the dragging effect.The pinning force was generated to block boundary movement and accumulated to be one of main resistances for Sn whisker growth.Once the driving force could not offset the resistances,the growth would stagnate.Notably,when a Mg2Sn nanoparticle with a radius ofrpinned at the grain boundary,a conical influence zone with a vertical angle of 2θwould be created (as shown in Fig.11),and corresponding pinning force (FPin) could be

Fig.11 Sketch map of Mg2Sn nanoparticles pinning at the grain boundary

whereδwas the energy of grain boundary.The maximum value wasFM=π·r·δatθ= 45°.If we assumed that the Mg2Sn nanoparticles in theβ-Sn matrix were precipitated at a constant nucleation rate per unit volume (I),the nanoparticle number generated between the timeτandτ+dτwould beVβ·I·dτ,whereVβis the molar volume ofβ-Sn matrix.The radius of nanoparticle at a timetwasWhen a Sn whisker grew,the Mg2Sn nanoparticles would be captured by grain boundary;thus,the maximum pinning force per unit volume at a timetwould be

Because the extended volume of a nanoparticle was Δvt=4π/3[2f·D·(t?τ)]3/2,the maximum extended volume of nanoparticles per unit volume at a timetwas

whereKis the bulk modulus.Undoubtably,based on Eqs.(11) and (13),theFwincreased faster than theFMwith increasing time;thus,the whiskers would grow.However,the whisker growth would result in stress release,and theFwwould inevitably decease with increasing time.WhenFw

4 Conclusion

The Sn protuberances and whiskers would be extruded from the Sn surfaces of Mg/Sn/Mg UAS joints because of the precipitation of the Mg2Sn nanoparticles.The solidsolution Mg atoms were existed in Sn matrix after UAS and would further transform to Mg2Sn nanoparticles during aging at room temperature.No matter the substitutional-site occupation or interstitial-site occupation,these solid-solution Mg atoms would react with Sn atoms and the expansion reaction of 2Mg +β-Sn→Mg2Sn would become the dominant source of the driving force for whisker growth.Meanwhile,due to the Mg2Sn corrosion,the continuous surface oxide layers of Sn matrix would be broken,further being the second necessary condition for whisker growth.In addition,Mg2Sn nanoparticles would be sucked up by grain boundaries beneath Sn whisker roots and pin the dislocation slip.The pinning effect of Mg2Sn nanoparticles would be varied with boundary angle,and the large-angle grain boundary would be more prone to form the curved whiskers.The competition between the driving force and resistance would affect the continuity of whisker growth,and the abrupt growth would lead to the formation of numerous dislocations in Sn whiskers.The loose Sn matrix would be produced with whisker growth,significantly worsening the bonding strength of the Mg/Sn/Mg UAS joints;thus,the further work should be how to decrease the amount of solid-solution Mg and inhibit the precipitation of Mg2Sn nanoparticles during UAS process.