Modelling of Ag3Sn coarsening and its effect on creep of Sn–Ag eutectics

Materials Science and Engineering A 427 (2006) 60–68

Modelling of Ag3Sn coarsening and its effect on creep of Sn–Ag eutectics Jicheng Gong ∗ , Changqing Liu, Paul P. Conway, Vadim V. Silberschmidt Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, UK Received 9 August 2005; received in revised form 7 April 2006; accepted 11 April 2006

Abstract A new constitutive model, which can account for the solder’s microstructure and its evolution, is proposed to describe the creep behaviour of the Sn–Ag eutectic phase. In this model, the threshold stress, being a function of the particle size, volume fraction and distribution of Ag3 Sn intermetallic compound (IMC), is introduced to build the relationship between the creep behaviour of the Sn–Ag solder and its microstructure. Evolution of the eutectic phase’s microstructure is accounted for in terms of the coarsening model. Both the creep strain rate and hydrostatic stress’s influence are taken into account in the IMC coarsening model. The proposed model is implemented into the commercial finite element code ABAQUS. The creep deformation due to the applied stress and IMC coarsening are discussed in the case of a flip chip solder joint. The obtained results show that the shape of the solder joint influences the particle distribution caused by heterogeneous coarsening. The solder joint is softened due to microstructure evolution over a long range of time. © 2006 Elsevier B.V. All rights reserved. Keywords: Pb-free solder; Constitutive model; Threshold stress; Coarsening; Finite elements

1. Introduction In the past five decades, the Sn–Pb solder has been the major material to assemble electronic components in the electronic industry. This has been mainly due to its low cost, good solderability, a low melting temperature and proper interfacial bonding reliability. However, the Pb and Pb-containing compounds are identified as the most toxic chemicals, and the Sn–Pb solder in the electronics is the main source of Pb contamination in the environment [1]. Under the new waste electrical and electronic equipment (WEEE) legislation, aimed to prevent environmental contamination with lead, the use of Pb in consumer electronics is banned after July 1, 2006 in the European Union [2]. Among the Pb-free alternatives, the Ag–Sn solder is one of the most promising candidates because it can provide compatible properties with the Sn–Pb solder [3]. While the Sn–Ag solder shows better creep resistance and consequently higher fatigue performance than the Sn–Pb solder in the bulk specimen, it does not suggest that this material can provide better reliability in microscopic joints, because the characteristic size of its microstructure (e.g. grain size) can be comparable to the solder joint size, when it becomes smaller than 100 ␮m in diameter. In this case, the creep and dam-

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0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.04.034

age behaviour shifts from polycrystals-based mechanisms to the intra-granular based ones. Few numerical studies have been done in this particular area. Hence a model that correlates mechanical integrity to the microstructure of micro-solder joints is of particular interest. The general numerical approach to reliability study of solder joints can be implemented in two steps: (i) finite element (FE) analysis and (ii) prediction of the fatigue life. In the first step, the material’s description, including a constitutive model for the solder material, are introduced into the geometry model. At the same time, the initial and boundary conditions are applied. Then the FE analysis is performed to calculate response of stresses and strains in the solder joints as well as a respective hysteresis loop. In the second step, a suitable fatigue model is chosen. The FE results are substituted into the fatigue model to predict the number of cycles to failure. The major drawback of this method lies in its assumption that the microstructure and corresponding properties of the solders remain constant during the entire loading process. So the analysed results of the FE analysis are usually for the forth loading cycle because they will reach a stable state at that stage. In fact, the stress response of solder materials to constant strain decreases under cycling load [4,5]. Thus, this approach underestimates the fatigue life of solder joints. The recently developed cohesive zone model [6,7] combines these two steps together and successfully predicts the damage accumulation in the solder joints. But it attributes degradation of

J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68

the solder joint only to the material deterioration, neglecting the influence of microstructure coarsening. In fact, the coarsening behaviour of both Sn–Pb and Sn–Ag solders is rather apparent [8] and it is well-known that the microstructure has a significant influence on mechanical properties of materials. To account for this effect, Jung and Conrad [9–11] and Vianco et al. [12] quantitatively studied the strain effect on the grain growth in the Sn–Pb solder under isothermal conditions. Mei et al. [13] took advantage of the Vianco’s results and coupled the effect of microstructure evolution with his continuum damage model based on Dorn’s constitutive equation:
 ADGb b p  σ n ε˙ c = , (1) RT d G where ε˙ is the steady-state creep rate, A the dimensionless constant, D the appropriate diffusion coefficient, G the shear modulus, b the magnitude of the Burgers vector, R the gas constant, d the grain size, σ the applied stress, n the stress exponent and p is the inverse exponent for the grain size. Later, Dutta [14] used a similar constitutive model to account for the coarsening process of Ag3 Sn particles in the Sn–Ag solder. Dorn’s equation is suitable to describe the creep behaviour associated with the grain size, while the recent results of Orientation Imaging and Electron Backscatter Diffraction [15] showed that a large number of Sn dendrites and the surrounding eutectic phase oriented in the same crystal direction, indicating that Sn–Ag solder joints are single- or multi-crystals rather than polycrystals. In this case, the grain size is not accurate enough to represent the solder’s microstructure and the substructure within grains should be taken into account. Our previous studies showed that the size and distribution of Ag3 Sn particles are non-uniform in the eutectics of the Sn–Ag solder, which can result in potential reliability issues for smallscale joints. The aim of this paper is to explore a model that can study this effect on reliability of joints at a microscopic scale. To describe details of the microstructure of eutectics while avoiding a large amount of additional elements in the model, three state variables are used to describe the microstructure of the eutectic phase. Besides being able to deal with complicated reaction between particles and dislocations, this method is capable of describing the microstructure evolution. In the constitutive model, the concept of athermal detachment is adopted to describe the reaction between Ag3 Sn particles and dislocations, and the creep threshold is quantified based on the size and distribution of these particles. At the same time, the growth of Ag3 Sn particles in ␤-Sn matrix is introduced into the constitutive model to capture the effect of microstructure evolution on mechanical properties of the solder joint. The proposed model is implemented for the case of a solder joint in flip chip assembly.

rate during solidification [16,17]. Under the cooling rate close to the equilibrium condition, large needle- or rod-like Ag3 Sn IMCs are formed randomly in the ␤-Sn matrix, and the solder is mainly composed of eutectics. However, in the reflow process, the cooling speed is significantly higher, and the alloy system deviates from equilibrium condition. Large Sn dendrites yielded and they are separated by the Sn–Ag eutectic phase, which is composed of ␤-Sn and small Ag3 Sn particles in the ␤-Sn matrix. Since the eutectic phase shows much better creep resistance than Sn dendrites and it is a continuous body within the whole solder, it is reasonable to consider that this phase controls the deformation of solder joints. Thus, this paper is focused on the creep behaviour of the Sn–Ag eutectics. To describe the microstructure of the Sn–Ag eutectic phase, three state variables are introduced: the volume fraction of Ag3 Sn particles, their diameter and inter-particle spacing. Note that all the three state variables are mean values, averaged over a small area to capture the local microstructure; at the same time, the area of averaging should be large enough to avoid the sudden change in properties at the interface between Ag3 Sn and the Sn matrix. Since the boundary between Sn dendrites and eutectic phase is pined by IMC particles, limiting the growth of these two phases [18], there is almost no Ag atoms consumed in or supplied to the eutectic phase. At the same time, solubility of Ag in the Sn solvent is near zero in the whole temperature range at a solid state. The local volume fraction of Ag3 Sn does not change with temperature or microstructure coarsening if excess Ag in Sn solvent is released after aging. In other words, the volume fraction can be considered to be an unchanged field under a thermal cycling load. Assuming Ag3 Sn particles to be ideal spheres, the size of local IMCs can be quantified by their mean diameter. Based on the assumption that the Ag3 Sn particles are arranged orderly in a small area, their interparticle spacing λ can be introduced as a function of the local diameter d and volume fraction f:


π λ=d √ 3 2f

1/3 .

(2)

So there are two independent parameters for the microstructure of the eutectic phase. Fig. 1 illustrates the microstructure of the eutectic phase.

2. Constitutive model 2.1. Microstructure of Sn–Ag eutectic phase The microstructure of the eutectic Sn3.5Ag solder is not uniform. The most important reason for this is the cooling

61

Fig. 1. Schematic substructure of eutectic phase in Sn–Ag solder.

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J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68 Table 1 Parameters of the athermal detachment model for Sn–Ag solder

2.2. Effect of Ag3 Sn IMCs on creep behaviour of Sn–Ag eutectics

kr

It has been known that the Sn–Ag solder exhibits a high stress exponent and a much better creep-resistant performance than pure Sn. One explanation is that Ag3 Sn particles at the phase boundaries pin the slide between Sn dendrites and the eutectic phase. A more acceptable explanation is that Ag3 Sn particles in the Sn matrix prevent the movement of dislocations when pipe diffusion dominates under high stresses. Since there is almost no phase-boundary sliding during creep deformation [19], matrix creep plays a dominant role in this process. Thus the reaction between IMC particles and dislocations, the existence of which is verified by Kerr’s TEM results [20], controls the creep behaviour of the Ag–Sn solder under high stresses. The effect of dispersion particles on the time-independent deformation of alloys is generally explained by the Orowan bowing process. In this mechanism, the dislocation line cannot cut through a particle but bows between particles under deformation. So an additional stress is required to overcome the energy increase arising from elongation of the dislocation line. This additional stress is the well-known Orowan stress: σOr =

0.84 MGb , λ − 2r

(3)

where M is the Taylor factor and r is the mean radius of particles. For alloys at high temperatures, dispersed particles also increase the stress needed to keep the creep strain rate, and the power law model is revised as:

  σeq − σth n Q ε˙ c = A exp − , (4) G RT where A is a material constant, σ eq the equivalent stress and σ th is the creep threshold stress. But the creep threshold stress cannot reach the full value of Orowan stress, and the ratio between them seems to be unchanged with temperature [21]. Both the local climb and general climb mechanisms were used to explain this phenomenon. In the former, the dislocation line can be released from the particle surface freely. But the value of the creep threshold stress in this model is too small to have any effect on creep. In the latter, the dislocation line moves tightly alone the interface between a particle and matrix during climbing. But the void flux in the matrix unravels dislocations from the particle/matrix interface and relaxes the dislocation line. R¨osler and Arzt [22] questioned the mechanisms of the local and general climb. They pointed out that dislocation detachment from the particle/matrix interface controls the bypassing process. In their opinion, dislocations are attracted by particles since the dislocation energy at the particle/matrix interface is lower than that in the matrix. So the threshold stress arises from the energy increase when dislocations leave the particle rather due to the elongation of the dislocation line at the beginning of the climb. The intensity of attraction between the particles and dislocations can be characterized by the relaxation factor kr . When kr = 1, the particle has no effect on the dislocation movement; when kr = 0, the threshold stress is equal to the Orowan stress. The creep threshold

0.87

M

b (m)

v

n

Q (kJ/mole)

3

0.21 × 10−9

0.36

6.5

65

stress is expressed as:  σth = σOr 1 − kr2 .

(5)

This model was supported by the in situ TEM investigation of the oxide-dispersion strengthened NiAl, Ni3 Al and FeAl [23], which demonstrated that the climb takes a very short time and detachment dominates the entire bypassing process. Recently, Kerr and Chawla [20] and Dutta et al. [24] adopted the creep threshold stress to study the strengthening effect of Sn3 Ag particles on creep behaviour of Sn–Ag solder, and successfully explained creep data of this solder. In this paper, the contribution of the size and distribution of Ag3 Sn particles to the creep threshold stress are quantified by Eqs. (3) and (5) for the constitutive Eq. (4). Hence, the local mechanical properties of Sn–Ag eutectics are correlated to its microstructure. The necessary parameters for these equations are based on Kerr and Dutta’s results and listed in Table 1. 2.3. Coarsening model During service, the microstructure of solder joints is unstable because the cooling rate during solidification is too high to get an equilibrium phase and the service temperature is relatively high due to its low melting temperature. Microstructure coarsening takes place under these conditions, which results in deterioration of the mechanical properties of the solder material. As mentioned, the growth of Sn dendrites and eutectics is not obvious. So the microstructure coarsening of the Sn–Ag solder mainly refers to the growth behaviour of Ag3 Sn particles. Since the solubility of Ag in the Sn solvent is very small, and, consequently, it is easy to reach the saturation state, the growth of Ag3 Sn particles can be described by the model for the second phase growth in a saturated solution [25]:
 −Qgr m m D − D0 = Kt exp , (6) RT where D and D0 are the current (at time t) and initial mean dimensions of the second phase, respectively, Qgr the activation energy for the rate-controlling process, m the phase size exponent and K is a constant. The value of m is determined by the rate-controlling mechanism. Table 2 lists the growth mechanisms with the corresponding to values of m. Gibson et al. [26] quantitatively studied the Ag3 Sn growth rate in the Ag–Sn solder and found that the particle growth rate fits Eq. (6) quite well when m = 3. This result indicates that the Ag3 Sn particle growth in Sn matrix is controlled by the volume diffusion. According to the Lifshitz and Slyozov’s model [27], the growth rate of the

J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68 Table 2 Mechanisms of the second phase growth and exponents of the phase size Phase size exponent

Growth mechanism

2 3 4 5

Solute atoms transfer across the particle/matrix interface Volume diffusion Diffusion along grain boundaries or interfaces Diffusion along triple junctions or dislocations

second phase based on volume diffusion is written as: r˙ =

B1 γs Vm C0 Dsol , 3r 2 RT

(7)

where r˙ is the second phase growth rate, B1 a material constant, γ s the interfacial energy between the particle and matrix, Vm the molar volume of the second phase and C0 is the equilibrium solute concentration in the matrix. Dsol is the effective solute diffusivity in the matrix, which is proportional to the total void density ρ: Dsol = D0 ρ,

(8)

where D0 is a material parameter. Under isothermal conditions, Dsol can be presented as:
 −Qth Dsol = D0 ρth = D0 ρ0 exp , (9) RT where ρth is the thermally-induced void density, ρ0 a material constant and Qth is enthalpy required to form one mole of voids. But Eq. (9) is not able to accurately describe the Sn–Ag solder coarsening during thermo-mechanical deformation, because the external stress and respective strain produce additional voids, enhancing the diffusion rate and accelerating Ag3 Sn coarsening. It is found that these excess voids are mainly due to the combination effect of the local hydrostatic stress and inelastic strain [14]. Actually, formation of one void under the hydrostatic stress σ h needs additional energy σ h Ω, where Ω is the atomic volume. So the void density due to the hydrostatic pressure would be: 
−Qth − N0 σh Ω + ρh = ρ0 exp , (10) RT where N0 is the Avogadro number. The decrease in the density of voids due to dissipation should be:
 −Qth + N0 σh Ω ρh− = ρ0 exp . (11) RT The total void density due to the hydrostatic stress is the difference between ρh+ and ρh− :

 −Qth −N0 σh Ω sinh ρh = ρ0 exp . (12) RT RT

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Under the constant strain rate, the excess in saturated voids due to inelastic strain is proportional to the strain rate [28]. Using the method similar to [14], the saturated void density due to the hydrostatic stress and constant strain rate is combined into the flowing form:

 −Qth −N0 σh Ω ∗ sinh (1 + N ε˙ ), (13) ρ = ρ0 exp RT RT where N is a material constant. Assuming the change rate of the void density to be proportional to the creep rate and considering dispersion between the void density at current state and saturated state, the evolution equation for void density can be obtained in the form:
 ρ p q ρ˙ = h0 1 − ∗ ε˙ , (14) ρ where h0 , p and q are positive material’s parameters. Substituting Eqs. (13) and (14) into Eqs. (7) and (8), we get the microstructure coarsening model for the Sn–Ag eutectic phase. As the deformation rate adopted in the implemented model remains constant and is very slow, the void density is ready to reach the saturated state. Hence the coarsening model in the implemented model is simplified:

 B1 γs Vm C0 Dsol −Qth −N0 σh Ω r˙ = exp sinh (1 + N ε˙ ) 3r 2 RT RT RT (15) Table 3 lists the parameters for Eq. (15). 3. FE model As the service temperature of solder joints exceeds half of its melting temperature, creep begins dominating the inelastic deformation. Under this condition the plastic strain is negligible, and the total strain is composed of elastic, thermal and creep components: εtotal = εe + εth + εc ,

(16)

where εe is the elastic strain, εth the thermal stain and εc is the creep strain. In the implemented model, the solder joint is assumed to be under isothermal conditions. Thus, the thermal strain vanishes. Following the Hooke’s law, the elastic strain can be written as: εe = D−1 e σ,

(17)

where De is the elastic stiffness matrix. The creep strain is described by the constitutive Eq. (4). The equivalent stress in

Table 3 Parameters for the coarsening model of the Sn–Ag solder B1 (m3 /mole) 5 × 10−3

γ s (J/m2 )

Vm (m3 )

0.5

1 × 10−5

C0 (mole/m3 )

D0sol (m2 /s)

20

7 × 10−7

Qsol (kJ/mole)

Ω (m3 )

N (s)

51.5

0.027 × 10−9

3000

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J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68

Table 4 Procedure of introduction of microstructure of Sn–Ag solder and its evolution into constitutive model Step

Description

1

Define the initial value of state variables (λ0 , d0 and f0 ) for the solder microstructure before calculations Substitute the state variables into Eqs. (2), (3) and (5) to obtain the creep threshold stress for the constitutive Eq. (4) at the beginning of the time increment Carry out FE analysis using constitutive Eq. (4) Update the state variables according to Eqs. (15) and (2) at the end of the time increment Repeat steps 2, 3 and 4 till the expected number of cycles is finished Finish the FE analysis

2

3 4 5 6

this equation can be expressed as:  3 σeq = S : S, 2

(18)

where S is the deviatoric stress. The hydrostatic stress in Eq. (15) is expressed as: σh =

σ1 + σ2 + σ3 3

(19)

where σ 1 , σ 2 and σ 3 are the principle stresses. The procedure to introduce the microstructure into constitutive model during FE analysis is clarified in Table 4. Considering calculation efficiency and the further work on the model, the initial model is built in two dimensions using 4nodes quadrilateral finite elements. The model consists of two Cu pads and a solder joint between them. The distance between the two pads is fixed as 60 ␮m and the interfacial area between the pad and joint is 80 ␮m in diameter. Three solder joints of different shapes are adopted. Their diameters at the middle section are 100, 80 and 60 ␮m, respectively. In this paper, they are referred to Types a, b and c, respectively. The dependence of the Young’s modulus E of the SnAg solder on temperature T

Fig. 2. Radial displacement of the top pad with time.

is: E(MPa) = −184.62(T − 273) + 52415. The shear modulus is: G = E/2(1 + υ). 4. Results and discussion 4.1. Ag3 Sn particle coarsening during mechanical cycling In our numerical simulations, the cyclic load is applied in the following way: the bottom pad is fixed in all directions and a cycling radial displacement boundary condition, shown in Fig. 2, is applied to the top pad. The joint’s temperature is chosen as 353 K to avoid the low-stress state while acquiring a proper creep rate. Based on the microstructure data for the Sn–3.5Ag solder from [24], the original size (radius) and volume fraction of Ag3 Sn particles are assumed as 3 × 10−7 m and 0.07, respectively. Fig. 3 shows the field of Ag3 Sn particle’s size at 1/4 cycle. It can be seen that the IMC particles at the sharp-angle corner between the solder and the pad grow quicker as a result of stress

Fig. 3. Distribution of size of Ag3 Sn particles in joint (Type a) at 1/4 cycle.

J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68

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Fig. 4. Distribution of size of Ag3 Sn particle in joint (Type a) at 3/4 cycle.

concentration there, which may cause faster inelastic deformations and, consequently, more voids. Because the right-top and left-bottom corners of the solder joint are at the tensile stress state during the first half of the cycle, the particle-coarsening process is more apparent than in two other corners, which are in compression. It can also be seen that Ag3 Sn particles at the area near the interface grow more quickly than in other areas. Fig. 4 shows the distribution of the particle size at 3/4 cycle. It demonstrates an opposite trend to 1/4 cycle but not as obvious as in Fig. 3 because it is also affected by the accumulation process during the previous half cycle. Fig. 5 reflects the growth of particles at the left-top and right-top corners. It clearly shows that the growth rate of particles at the right corner is higher at the beginning but decreases after half cycle. The particle size at the two corners reaches approximately the same value at the end of the first cycle.

When the size of the solder joint is reduced as shown in Fig. 6, stress concentration at the corners decreases with the decrease in angle value. So particle coarsening there is not as pronounced as in the previous case (Figs. 3 and 4). Fig. 6 also shows a trend of the particle-coarsening rate to increase at the centre of the solder joint (point D in Fig. 6). This trend is more pronounced in Type c as shown in Fig. 7. The coarsening data for point D in Fig. 8 demonstrates that this increase is not a relative one due to the drop in coarsening at the corners but an absolute one. Since the creep rate in this area is approximately the same in all three types of joints, this result is mainly due to the decrease in the hydrostatic stress (scalar variable) when the volume of solder joints decreases. An interesting phenomenon is that the most rapid coarsening does not occur at the corners but at the surface of joints of about 1/4 of its height away from the pads in Type c. Actually, the stress concentration position at the surface shifts from point B to the area near point C with the decrease in the joint’s volume. At the same time, point C in Type c undergoes tensile stress during the entire cycling process. Thus, this phenomenon is a result of the combination of these two effects. The coarsening rate for Point A, the site near the centre of the interface, is the smallest, on average, and does not depend on the shape of solder joints type as shown in Fig. 8. Fig. 8 also shows that the particle growth behaviour changes from Types b to c is more apparent than from Types a to b. Distributions of the size of Ag3 Sn particles are not given for other cycles since they display the similar trends as in the first cycle. 4.2. Effect of microstructure coarsening on the solder mechanical properties

Fig. 5. Growth of Ag3 Sn particles at the top corners in joint (Type a) during the first cycle.

In order to study the influence of coarsening on the mechanical response of the solder joint, the cycling displacement boundary condition is applied to the right end of the top pad. The point that is used to plot data is situated on the left side of loading

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J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68

Fig. 6. Distribution of size of Ag3 Sn particles in joint (Type b) at the end of first cycle.

Fig. 7. Distribution of size of Ag3 Sn particles in joint (Type c) at the end of first cycle.

Fig. 8. Ag3 Sn particle size at points A, B, C and D at the end of first cycle for various types of joints. The positions of points A, B, C, and D in Type a correspond to that in Types b and c.

position in the top pad (point E). Fig. 9 presents the radial stress component against time during the first 20 cycles in the Type b joint. While the movement rate of the top pad and its maximum displacement remain the same in each cycle, the maximum reacting stress decreases gradually due to microstructure coarsening. Hysteretic loops in Fig. 10 demonstrate the softening behaviour of solder bump under long-term loading. It does not begin with the first cycle because the solder joint hysteretic loop usually reaches a stable shape after four initiate cycles. The results show that the softening behaviour becomes rather obvious after 80 cycles. Although larger Ag3 Sn particles hinder the movement of dislocation to a greater degree, the growth of IMCs reduces the interparticle spacing. Actually, the decline rate of interparticle spacing is higher than that of Ag3 Sn particles. So the creep threshold stress decreases during the microstructure evolution process, resulting in the softening behaviour. The softening behaviour of SnPb and SnAg solders under the cycling load can be divided into three stages: the rapid increasing

J. Gong et al. / Materials Science and Engineering A 427 (2006) 60–68

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Fig. 11. Radial softening in joints of Types a, b and c. Fig. 9. Evolution of radial stress component in point E in Type b under cycling loading.

stage, steady-state stage and accelerated stage. In most cases, these stages of the softening behaviour are attributed to damage formation during deformation. While the damage model can predict softening in the tertiary stage quite well because the damage evolution is accelerated by the existing damage, this character contradicts to the decrease of the softening rate at the primary stage. In our model, the amplitude of the reaction stresses decreases for all the three joints drops with the number of cycles as shown in Fig. 11. In fact, the driving force of the particle growth, namely reduction in the interfacial energy between the matrix and particles, decreases with the growth of particles. So the particle growth rate decreases with time, resulting in the decrease in the softening rate of SnAg solder due to coarsening. If we combine the softening contribution due to microstructure coarsening and damage evolution (curves 1 and 2 in Fig. 12) together, the shape of the obtained curve 3 in Fig. 12 fits quite well with the experimental results throughout the three soften-

Fig. 10. Hysteretic loops for the 4th, 80th and 160th cycle in point E in Type b.

ing stages [4,5]. Therefore, the deterioration of the Sn–Ag solder may be the result of the combination of microstructure coarsening and damage formation, with microstructure coarsening dominating at the primary stage. From Fig. 11, it can be concluded that Type c is the best one because its softening rate is the lowest. However, the initial maximum stresses in joints of Types a and b, 92.1 and 90.5 MPa, respectively, are much higher than that in Type c, 78.8 MPa. Thus, the use of Types a and b is preferable. Since mechanical properties and the softening rate of Types a and b are similar, but the volume of Type b is much less than that for Type a, Type b is therefore recommended. This conclusion is based on only the analysis of material deterioration due to microstructure coarsening or the softening at the primary stage. A more comprehensive understanding that is able to account for the softening behaviour at the steady and tertiary stages can be obtained after the damage evolution is introduced into the model. This part of work is under investigation and will be published later.

Fig. 12. Softening due to microstructure coarsening (curve 1), damage accumulation (curve 2) and due to their combination (curve 3).

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5. Conclusion

References

A constitutive model for microstructure evolution is proposed for SnAg solder to simulate coarsening of Ag3 Sn IMCs and to study its effect on mechanical properties. This constitutive model is used to model the mechanical behaviour of a flip chip solder joint under cycling load. The obtained results demonstrate:

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1. The volume and/or shape of the solder joint influence the extent of IMC coarsening and distribution of Ag3 Sn particles. In the large joints, particles grow fast at the sharp angle corners. When the volume of the solder joint decreases, the fast growth sites at the surface of the joints shifts away for the pads to 1/4 of its height. At the same time, the particle growth rate at the centre of the joints increases. 2. Microstructure coarsening makes a contribution to the solder softening under cycling. This softening mechanism can account for the solder deterioration at the primary stage when softening rate decreases with time. If combined with the damage mechanism, the proposed model can be generalized to all stages of the softening process. 3. Considering the volume, initial mechanical properties and softening behaviour, the joint having a diameter of 80 ␮m seems to have an optimised result among three types of joint geometries. However, this conclusion is based on the analysis of the primary stage, when softening due to microstructure coarsening dominates. Acknowledgements Financial Support by the Engineering and Physical Sciences Research Council’s Innovative Manufacturing and Construction Research Centre at Loughborough University under GR/R64483/01P are gratefully acknowledged.