Microstructure evolution of tin-lead solder

Transcription

1 Mirostruture evolution of tin-lead solder R.L.J.M Ubahs, P.J.G. Shreurs, and M.G.D. Geers Eindhoven University of Tehnology, Department of Mehanial Engineering, Setion of Materials Tehnology, PO Box 53, 56 MB Eindhoven, the Netherlands. Abstrat Mirostrutural length sales are relatively large in typial soldered onnetions. The mirostruture whih is ontinuously evolving is known to have a strong influene on damage initiation and propagation. In order to make aurate life-time preditions by numerial simulations, it is therefore neessary to take the mirostrutural evolution into aount. In this work this is aomplished by using a diffuse interfae model based on a strongly nonloal variable. It an be seen as an extension of the Cahn-Hilliard model, whih is weakly nonloal sine it depends on higher order gradients whih are by definition onfined to the infinitesimal neighbourhood of the onsidered material point. Next to introduing a truly nonloal measure in the free energy, this nonloal formulation has the advantage that it is numerially more effiient. Additionally the model is extended to inlude elastially stored energy as a driving fore for diffusion after whih the entire system is solved using the finite element approah. The model results in a omputational effiient algorithm whih is apable of simulating the phase separation and oarsening of a solder material aused by ombined thermal and mehanial loading. Introdution Solder joints serve two important purposes. Firstly, they form the eletrial onnetion between the omponent, e.g. a hip or resistor, and a substrate, e.g. a motherboard, and seondly, they onstitute the mehanial bond that holds the omponent to the substrate. The most ommonly used solder alloy up till now is the lead-tin system. When an eletroni devie is in use, solder onnetions will be subjeted to mehanial stresses and strains. These arise beause of the mismath between the oeffiients of thermal expansion (CTE) of the eletroni omponent and the board onto whih it is onneted. Beause solder materials are multiphase systems there are also differenes between CTEs of the phases within the solder leading to internal stresses. Note that these stresses an arise even in a single phase material if there are different CTEs for different lattie diretions. The on and off swithing of a system thus leads to yli thermomehanial loads, resulting in stresses and strains in the solder joint. The soldered onnetion an also be subjeted to external loads aused by e.g. bending of the substrate. Sine solder alloys are required to operate at high homologous temperatures their deformation is governed by high temperature mehanisms suh as reep, rerystallisation, oarsening, and reovery. An important mehanial property of solder joints is undoubtedly their resistane to fatigue under thermal yling. Sine failure of a solder joint will render a omplete devie useless, it is important to predit the lifetime of suh a onnetion. Beause of time limitations solder joints are often tested for their thermomehanial fatigue properties by subjeting them to aelerated tests methods. However, during these aelerated experiments damage mehanisms might be invoked whih do not our during normal usage. Preditions based on aelerated yling tests are therefore unreliable. Thermomehanial modelling an be used during the design proess to simulate the real life onditions of a solder joint, and thus give an aurate predition of its lifetime, with a signifiant redution in ost and time when ompared to prototype testing. The properties of a solder joint are highly dependent on its mirostruture, whih hanges onsiderably during its lifetime. The high homologous temperature at whih solder materials operate provokes mirostrutural oarsening, resulting in a markedly different mirostruture after only a short time after solidifiation, as an be seen in figure. During thermomehanial yling of a soldered onnetion the oarsening proess will not be homogeneous. Frear et al. [8] found that regions of high shear strain within an euteti tin-lead onnetion oarsened more. These regions of inhomogeneous oarsening are know to be rak nuleation sites during thermal yling [6; 7; 8]. It is also well-known that thermal yling leads to faster oarsening rates than stati thermal aging [], possibly due to the mehanial stresses whih arise within the material due to differenes in thermal expansion oeffiients. Another fator whih influenes the diffusion proess is the surfae tension between different phases. It ontrols the number, steepness, orientation, and the final shape of the interfaes within the oarsened struture. The magnitude of the surfae tension an vary in different diretions of the rystal lattie. 2 A truly nonloal diffuse interfae theory Beause the mirostruture has a large influene on the material properties and oarsening readily ours during the lifetime of a soldered onnetion, it is important to take the mirostrutural evolution into aount during numerial simulation of fatigue damage propagation. Speifially in tin-lead solder oarsening will our beause of the high solubility of tin in lead and vie versa. Various attempts have been made to inlude a mirostutural variable like the average grain size in the stress-strain relation [; 2; 2]. Viano et al. [3] and Frear et al. [7] also inlude grain size evolution in their models. In the literature

2 Free energy To omplete the diffusion equation the free energy funtion is needed. At any time there will be different kinds of energy available in a system. Firstly, eah phase will have different energy values for different ompositions, whih onstitutes the onfigurational energy F. Seondly, interfaes will ontribute to the surfae energy, denoted F γ, and thirdly, stresses and strains will ause elasti energy, indiated as Fε, to be stored in the system. So F = F + F γ + Fε J/mol (3) The onfigurational free energy denotes the ontribution of the pure phase. For a system where only binary interations take plae the funtion will have the following form: F = i g i + R T i ln i + i j χ ij (4) i i i j>i where g i and χ ij are oeffiients dependent on temperature. Spinodal points Figure : Optial mirographs illustrating the oarsening of the Pb-rih phase partiles in 63Sn-37Pb samples under stati aging onditions. Left: Quenhed in liquid nitrogen. Right: aged at 5 C for 5 hours. Free energy binodal points Composition several equations an be found that desribe the evolution of the grain size [4; 3]. No information about the distribution or shape of the grains is inorporated however. This information an be found by using of a diffuse interfae theory [3]. Diffuse interfae theories are based on the notion that the free energy is not only a funtion of a loal order parameter, but also depends on the value of this parameter in neighbouring points. In this setion first the diffusion equation will be introdued, after whih a diffuse interfae theory will be presented based on a truly nonloal order parameter. Diffusion equation The diffusion equation reads: ρ d i dt = [ˆρ i ( v i v)] = j i () where the j i is the diffusion mass flow per unit area per unit time. Using the phenomenologial approah of lassial irreversible thermodynamis [9], the following expression for the diffusion mass flow is found: ( ) j i = ρ M F (2) i where F is the speifi Helmholtz free energy and M the mobility tensor. This means that for a system to be in equilibrium the derivative of the Helmholtz free energy with respet to the omposition, also alled the hemial potential, has to be homogeneous throughout the system. Figure 2: Free energy urves for the α and β phase, and binodal and spinodal points. For a two-phase material (phases α and β), the onfigurational energy will be a double-well potential, see figure 2. It an be onstruted from the onfigurational energy urves of the individual phases by assuming that at a ertain omposition only the phase with the lowest energy will exist. Equilibrium is then found when the hemial potential is homogeneous throughout the system. Note that this does not mean that the mass fration has to be homogeneous. The onfiguration with the lowest possible free energy is found when, due to phase separation, all of the material has either one of two ompositions defined by the so-alled binodal points. These binodal points an be found by onstruting a double tangent to the free energy urve. If the binodal points are plotted at different temperatures the phase diagram will be obtained. The elastially stored energy is due to stresses and strains whih our in the material. Thus in order to alulate it, a onstitutive model needs to be hosen and the momentum equation, has to be solved. As a first approximation linear elastiity is assumed and inertia effets and body fores are left out of onsideration, yielding the momentum equation σ = C : ε = (5) where C is the fourth-order material tensor, σ denotes the stress tensor, and ε is the strain tensor defined as 2 { u +

3 ( u) } The elasti energy is then defined as Fε = 2 ε : C : ε (6) Note that the elastially stored energy term introdues a fourth-order gradient of the displaements into the diffusion equation. Beause the system will be solved by using a finite element method it is onvenient to eliminate these fourth-order gradient terms. This is aomplished by introduing a new independent variable, ψ, defined as the derivative of the elasti energy with respet to the loal mass fration. Thus an additional equation is added to the system: ψ = 2 ε : C : ε (7) The surfae free energy aounts for energy due to interfaes between phase regions. This means that it annot be only a funtion of the loal mass fration,. The omposition of the surroundings of a material point has to be taken into aount as well. In diffuse interfae theories this is done by taking the free energy to be dependent on both the loal omposition and its derivatives. These models an be onsidered to be weakly nonloal, sine the derivatives of the omposition are by definition onfined to the infinitesimal neighbourhood of the onsidered material point. In this work the free energy is taken to be a funtion of the loal mass fration and a weighted averaged value of this variable, denoted. This variable is alulated by solving an impliit Helmholtz equation: l 2 2 = (8) where l is an internal length parameter defined as l 2 = ξ 2 g(ξ) dω (9) 2 Ω Ω with g a normalised weighting funtion and ξ a loal oordinate vetor. The surfae energy is next related to the square of the differene between the nonloal and loal mass fration: F γ = 2 κ ( )2 () where κ is the surfae tension oeffiient tensor. Total system of equations The system of equations whih has to be solved, onsists of the diffusion equation for the loal mass fration, an impliit Helmholtz equation for the nonloal mass fration, the momentum equation for stresses and strains, and the equation for the derivative of the elastially stored energy with respet to the loal mass fration: [ ρ M ρ d dt = l 2 2 = σ = ψ = 2 ε : C : ε ( F )] κ ( ) + ψ () 3 Finite element implementation The system of equations is solved using the finite element method. The first step towards a finite element sheme is time disretisation of the diffusion equation. This is done by using the impliit or bakward Euler sheme, whih is unonditionally stable and first-order aurate. Next a linearised weighted residual form of the equations is onstruted in whih essential boundary onditions are taken are off. Finally all variables are disretised and interpolation funtions are hosen, linear for all variables. The total system is then divided into two systems of two equations whih are solved deoupled. The first set ontains the diffusion equation and the nonloality equation. The seond set momentum balane and the derivative of the elastially stored energy. In matrix form the systems then are: K int K δ = f (2) int K K δ f ext int K uu δũ u u = f f (3) int K ψu K ψψ ψ δψ f These systems are then solved with a onjugant gradient solver in order to redue the required amount of omputer memory ompared to using a diret solver. 4 Model parameters The euteti tin-lead system has been hosen fore these simulations, beause it has been most extensively studied in the literature. For this system the α phase is a lead-rih BCC rystal struture, whih for the euteti forms islands in a tin-rih matrix, the β phase. The neessary material data were obtained from literature, and are summarised in table. The internal length parameter l is taken to be in the order of magnitude of the real interfae width, and κ is hosen suh that surfae energy has the same order of magnitude as the onfigurational energy. The mobility of the α phase is supposed to be signifiantly lower than that of the β phase, hene it is assumed to be negligible. The mobility of the β is fitted using the omparison between a twodimensional simulation and an ageing experiment, whih will be disussed later. All the parameters are assumed to vary linearly with the mass fration. Configurational free energy Figure 3 shows the onfigurational part of the Helmholtz free energy urve for the tin-lead system at 5 C. Where it is assumed that at a ertain omposition the material will assume the phase whih has the lowest free energy at that omposition. The data used to alulate this urve is taken from MTDATA R [5; ]. Next the urve is fitted to the following funtion to failitate the finite element routine. F =g + g 2 ( ) + g 3 ln + +g 4 ( ) ln( ) + g 5 ( ) (4)

4 Table : Material data for the tin-lead solder system. mass density of tin ρ Sn [kg/m 3 ] 726 mass density of lead ρ P b [kg/m 3 ] 3 Young s modulus of tin E Sn [N/m 2 ] Young s modulus of lead E P b [N/m 2 ] 6. 9 Poisson s ratio of tin ν Sn [ ].36 Poisson s ratio of lead ν P b [ ].44 CTE of tin α Sn [/K].67 5 CTE of lead α P b [/K] surfae tension oeff. κ [J].6 mobility of α-phase M α [m 5 /(J s)] mobility of β-phase M β [m 5 /(J s)].2 2 internal length sale l [m] 25 9 onfigurational energy [J/m 3 ].35 x mass fration Figure 3: Configurational free energy F for the tin-lead system at 5 C. Resulting parameter values for T=5 C are: g =.36 9, g 2 =.52 9, g 3 =.36 9, g 4 =.6 9, and g 5 = One-dimensional simulations One-dimensional alulations have been performed to investigate the behaviour of the model. Two different initial onfigurations are used, see figure 4. First the initial ondition was taken to be a step funtion in the mass fration in the middle of the domain in order to investigate the influene of the internal length sale parameter l and the surfae tension oeffiient κ on the equilibrium interfae whih develops. The effet of external loading has been studied as well. Seondly, spinodal deomposition was investigated starting from a homogeneous, euteti, mass fration distribution with random imperfetions imposed. mass fration [ ] Figure 4: Initial onditions for one-dimensional analyses. Left: step funtion; Right: homogeneous distribution with random imperfetions imposed. Disontinuous interfae Calulations were performed to investigate the influene of the internal length sale parameter, l, and the surfae tension oeffiient, κ. When using the expliit formulation for alulating the nonloal field, whih is idential to the Cahn-Hilliard model, inreasing κ will have the same effet as inreasing l 2. Both these parameters serve as a multipliation fator for the Laplaian of the loal mass fration. However for the impliit formulation this is no longer the ase. For this ase, substituting the nonloality equation in the diffusion equation yields ρ d dt = [ ρ M ( F + κ l2 2 + ψ )] (5) However, even if the produt κ l 2 is kept onstant, interfae width still dereases with an inreasing value of l. see figure 5. This is due to the fat that hanging this parameter hanges the nonloal field whih in turn affets the loal mass fration. The internal length sale an be onsidered a measure for the distane at whih an interfae an affet the bulk material. The surfae tension remains a multipliation fator whih determines the amount of influene an interfae an exert. Figure 6 shows the influene of the internal length sale while the surfae tension oeffiient is kept onstant. The parameter influenes the interfae width linearly. The interfae width is defined as the distane over whih the mass fration deviates.5 or more from the bulk mass fration, either of the α or β phase. If another value for the deviation is hosen the results are affeted quantitatively but not qualitatively. In figure 7 the interfae width as a funtion of the square root of κ is shown. Similarly to the influene of l, here also a linear relationship an be seen. It should be noted that the model is more sensitive to variations of κ than of l. This effet an also be seen in figure 5. Under the influene of straining the shape of the interfae between the two phases also hanges. Interfaial width dereases under the influene of either ompression or tension, see figure 8. This an be explained by the fat that the interfae is determined by the ratio between the surfae energy and the total free energy. If this ratio dereases, as in the ase of external loading, interfaes beome steeper and onsequently thinner. The reason for the loss of symmetry of these results at larger deformations ould be aused by the small strain assumption used in the model. Homogeneous distribution One-dimensional simulations of the phase separation and onsequent oarsening proess have been performed. The initial ondition is a homogeneous distribution of the tin mass fration at the euteti omposition of the tin-lead system, =.63. To trigger spinodal deomposition random imperfetions on the mass fration are imposed (normal distribution with mean and variane.), see figure 4. The imperfetions are the same for all alulations. First a alulation is performed without external loading

5 interfaial width [µm] internal length sale [µm] Figure 5: Interfae width as a funtion of the internal length sale, keeping the produt κ l 2 onstant interfaial width [µm] internal length sale [µm] Figure 6: Interfae width as a funtion of the internal length sale l, keeping κ onstant. interfae width [µm] square root of surfae tension oeffiient x 5 Figure 7: Interfae width as a funtion of the square root of the surfae tension oeffiient κ, keeping l onstant. interfaial width [µm] Figure 8: strain strain [ ] Interfae width versus applied marosopi Figure 9: Development of one-dimensional tin-lead system starting from a homogeneous distribution with random imperfetions imposed. l = 25 nm and κ = 2 GJ (κ l 2 = 2.5 mj m 2 ). From top to bottom: t=.4,.9,.4, 4.8, and 5 hours. and l = 25 nm and κ = 2 GJ, see figure 9. The figure shows the apability of the model to desribe phase separation and oarsening. Next a alulation with external loading of % global tensile strain is also performed to demonstrate the effet of elastially stored energy on the system. The results are shown in figure. In this ase deomposition is faster and the final struture shows slightly more oarsening. The results show a shift of the binodal points, downward for the α phase and upward for the β phase, and steeper interfaes ompared to the referene alulation. 6 Two-dimensional simulations To validate the model an experiment has been performed of the ageing of euteti tin-lead solder. The speimen was rapidly ooled in liquid nitrogen and onsequently aged at 5 C. During the ageing proess photographs where made of the same area using an Sanning Eletron Mirosope. Some of the results are shown in figure. Next a simulation was performed taking the initial ondition from the SEM photograph made immediately after ooling. The results of the simulation are shown in figure next to SEM photographs of approximately the same area at the same time. The simulated mirostruture exhibits break-up and oalesene of phase regions, and phase regions growing and dissolving, similar to what an be observed experimentally. In order to make a omparison between the numerial simulations and the experimental data the results need to be

6 .5 t = hours t = 2 hours Figure : Development of one-dimensional tin-lead system starting from a homogeneous distribution, with an imposed global strain of %. l = 25 nm and κ = 2 GJ (κ l 2 = 2.5 mj m 2 ). From top to bottom: t=.4,.9,.4, 4.8, and 5 hours. t = 6 hours quantified. Sine minimization of surfae energy, and thus minimization of surfae area, is a driving fore for the diffusion proess, the total surfae area per unit volume is taken as a quantifiable number to be ompared. The results of this quantifiation is next used to fit the unknown mobility of the β phase. This parameter is found to be.2 2 m 5 /(J s). Figure 2 shows the surfae area development over time after fitting the mobility of the β phase. The agreement between experimental and numerial data is very good. 7 Conlusions In this paper an extension of the Cahn-Hilliard model is presented to desribe the mirostruture evolution of tinlead solder. The free energy density is taken to be a funtion of the loal and a strongly nonloal mass fration whih an be alulated by solving a Helmholtz equation. The differene between the nonloal and the loal mass fration is assoiated with the surfae energy. The derivative with respet to loal mass fration of this energy ats as a driving fore for the oarsening proess. Inorporation of additional driving fores into the free energy is straightforward. In this way the effet of stresses on diffusion is also aounted for through the elastially stored energy. The model inludes an internal length sale l and a surfae tension oeffiient κ as parameters. The produt κ l 2 appears as a multipliation fator for the derivative of the surfae energy with respet to the mass fration in the diffusion equation. One-dimensional simulations are used to investigate the influene of these parameters on interfae t = 9 hours t = 2 hours t = 5 hours Figure : Experimental and numerial thermal ageing of tinlead solder at 5 C, (size is µm).

7 fatigue of solder joints, Journal of Eletroni Materials 28(9), Interfae area per unit volume [ / µm ].2.8 experimental numerial, M=.2e time [hours] Figure 2: Interfae area per unit volume for experimental and numerial results, see figure. development and oarsening. It is found that inreasing either the internal length sale or the surfae tension oeffiient inreases the interfae. Inreasing l while hanging κ appropriately to keep the produt κ l 2 onstant leads to a steeper interfaes. The influene of external loading on the phase separation and oarsening proess is also investigated. It is found that elastially stored energy, either due to tensile or ompressive loading, auses a shift of the binodal points and dereases the interfae width. Furthermore, it inreases the deomposition rate and oarsening. The model is validated by omparing a two-dimensional simulation with experimental data. The model is able to predit break-up, oalesene, growth and dissolution of phase regions, similar to what an be observed experimentally. The total surfae area per unit volume is used to quantify the mirostruture. The results are in good agreement with experimental data. Aknowledgements This researh is supported by the Tehnology Foundation STW, applied siene division of NWO and the tehnology programme of the Ministry of Eonomi Affairs. We would like to thank M.A. Matin for providing the experimental data. Referenes [] Basaran, C. and Chanderoy, R. [998], Mehanis of Pb4/Sn6 near-euteti solder alloys subjeted to vibration, Applied Mathematial Modelling 22, [5] Dreyer, W. and Müller, W. [2], A study of the oarsening in tin/lead solders, International Journal of Solids and Strutures 37, [6] Frear, D. [989], Thermomehanial fatigue of solder joints: A new omprehensive test method, IEEE Transations on Composites, Hybrids, and Manufaturing Tehnology 2(4), [7] Frear, D., Burhet, S., Neilsen, M. and Stephens, J. [997], Mirostruturally based finite element simulation on solder joint behaviour, Soldering and Surfae Mount Tehnology 9(), [8] Frear, D., Grivas, D. and Morris, Jr., J. [988], A mirostrutural study of the thermal fatigue failures of 6Sn-4Pb solder joints, Journal of Eletroni Materials 7, 7 8. [9] Groot, S. d. and Mazur, P. [984], Non-equilibrium thermodynamis, Dover Publiations, In., New York. [] Hake, P., Spreher, A. and Conrad, H. [998], Mirostruture oarsening during thermo-mehanial fatigue of Pb-Sn solder joints, Journal of Eletroni Materials 26(7), [] MTData [998], Npl Databank for Materials Thermohemistry, Tehnial report, National Physial Laboratory, Queens Road, Tedington, Middlesex, TW LW. [2] Pan, T.-Y. [99], Thermal yling indued plasti deformation in solder joints - part ii: Aumulated deformation in through hole joints, IEEE Transations on Components, Hybrids, and Manufaturing Tehnology 4(4), [3] Viano, P., Burhet, S., Neilsen, M., Rejent, J. and Frear, D. [999], Coarsening of the sn-pb solder mirostruture in onstitutive model-based preditions of solder joint thermal mehanial fatigue, Journal of Eletroni Materials 28(), [2] Basaran, C. and Chanderoy, R. [2], Using finite element analysis for simulation of reliability tests on solder joints in miroeletroni pakaging, Computers and strutures 74, [3] Cahn, J. and Hilliard, J. [958], Free energy of a nonuniform system, I: Interfaial energy, The Journal of Chemial Physis 28(2), [4] Conrad, H., Guo, Z., Fahmy, Y. and Yang, D. [999], Influene of mirostruture on the plasti deformation kinetis, fatigue rak growth rate, and low-yle