This paper focuses on anisotropic elastic-plastic constitutive modeling of SAC (SnAgCu) solder grains because of their importance in modeling the behavior of oligocrystalline (few-grained) micron-scale solder joints that are increasingly common in heterogeneous integration. Such grain-scale anisotropic modeling approach provides more accurate assessment of the mechanical response of solder interconnects in terms of predicting different failure modes, failure sites, and variability in time-to-failure. Anisotropic plasticity is represented using Hill–Ramberg–Osgood (RO) continuum plasticity model, which utilizes Hill's anisotropic plastic potential along with a RO power-law plastic hardening flow rule. Mechanistically motivated empirical scaling factors are proposed to extrapolate the stress–strain response for different grain sizes/shapes and for different coarseness of microstructures within each grain (generated with different cooling rates). This scaling factor can therefore also capture the effects of microstructural coarsening due to isothermal aging. This goal is achieved by first conducting monotonic tensile and shear tests on monocrystalline and oligocrystalline SAC305 solder joints containing grains of various geometries and also intragranular microscale (dendritic and eutectic) structures of various coarseness. The grain structures are characterized for each tested specimen using electron backscattered diffraction (EBSD). The Hill–RO model constants and the empirical scaling factors are then estimated by matching grain-scale anisotropic elastic-plastic finite element models of each tested specimen to the measured stress–strain behavior, using an inverse-iteration process. Grain shape is seen to influence the sensitivity of the effective stress–strain curves to the applied stress state (i.e., to the orientation of the principal stress directions) relative to (i) the material principal directions and (ii) the geometric principal directions of grains with high aspect ratio. Limitations of the current results and opportunities for future improvements are discussed.
This section provides the background and motivation for the present work and presents a review of the relevant literature.
1.1 Background and Motivation.
The semiconductor industry has opted for heterogeneous integration of diverse individually designed and manufactured chiplets and components in a single package to maintain the pace of improvement of functionality, performance, and cost. A diverse collection of dies of different technologies are integrated into a package side-by-side and/or in three-dimensional stacks so that the resulting system-in-package (SiP) can perform enhanced functions in a small form factor [1,2]. Heterogeneous integration inevitably leads to hierarchical interconnection architectures, resulting in solder interconnections of a wide range of characteristic dimensions and pitch, of the order of 101–102μm. This miniaturization has resulted in SAC solder joints with one grain (monocrystalline) or, at most, a few grains (oligocrystalline). At this length scale, the mechanical response of each solder joint is highly sensitive to the grain structure and the anisotropic single-crystal properties, which are significantly different from the bulk isotropic polygranular solder mechanical properties commonly found in the literature. SAC305 consists of 96.5% β-Sn, which has a body-centered tetragonal (BCT) crystal lattice (a = 0.58 nm, c = 0.317 nm, and c/a = 0.545), and strong anisotropy of mechanical and thermo-mechanical properties, as shown in Fig. 1 [3,4]. The modulus of elasticity in the  direction at 20 °C (= 68 GPa) is considerably larger than that in the  direction (= 24 GPa). Furthermore, the coefficient of thermal expansion (CTE) in the  direction (=30 ppm/°C) is almost twice as large as that in the  direction. The anisotropic elastic stiffness constants for β-Sn have been characterized by several researchers in the past and used for modeling Tin-rich SAC solder joints in lieu of SAC properties [5,6]. Jiang et al.  quantified the elastic stiffness constants for SAC305 shown in Table 1 by considering the microstructural differences between SAC and Sn grains.
The grain-scale behavior of oligocrystalline SAC joints is therefore strongly dependent on the grain structure and can hence be significantly different (in structural coordinates) for each grain. In addition, another source of variation in the behavior of SAC305 joints is the heterogeneous microstructure within each grain. As mentioned above, this variability also evolves with age of the joint since both of these microstructural features (grain structure and the heterogeneous structure within each grain) continue to evolve as the joint ages during the life cycle.
The entire hierarchy of the heterogeneous structure of SAC joints, from macro to nanolength scale, has been classified into different tiers by Cuddalorepatta et al. , as shown in Fig. 2, and is summarized here from the highest to the lowest length scale. Tier 4 consists of a macroscale package assembly that contains multiple solder interconnects. Tier 3 consists of a single oligocrystalline solder joint. Tiers 0–2 consist of the microstructure within each SAC grain. Each SAC grain consists of pro-eutectic β-Sn dendrites surrounded by a eutectic Sn–Ag microconstituent. There are also embedded Cu6Sn5 intermetallic compound (IMC) microconstituents. This micromorphology within each grain is termed tier 2. The tier 1 eutectic Ag–Sn microconstituent consists of nanoscale Ag3Sn IMC particles distributed in β-Sn matrix. These IMC particles block the dislocation motion, leading to dispersion strengthening in the eutectic microconstituent. The crystalline microstructure (BCT) of the β-Sn matrix is labeled as tier 0.
Another challenge in characterizing the mechanical behavior of solder joints arises from the length-scale effect of the solder joint specimen used for testing. As an example, in uniaxial tensile tests, as the aspect ratio (length orthogonal to the loading direction, divided by height parallel to loading direction) of solder joints increases, the global triaxial stresses (due to Poisson mismatch with the Cu substrates and with the interfacial Cu–Sn IMC layers) significantly increase the extensional forces required to yield the solder joint [9–12]. In addition, traditionally used long bar dog bone specimens lack the interaction of grain boundaries with Cu substrates and interfacial IMC layers that lead to stress concentrations. This stress triaxiality is also seen in functional solder joints in leadless components (like quad flat no-lead package, leadless chip resistor/capacitor (LCR/LCC)) and in die attaches. The resulting load–displacement behavior is quite different from that seen in testing of uniaxial bars, where the dimension parallel to the loading direction is usually many times the cross-sectional dimension orthogonal to the loading direction. This is one of several reasons why it is informative to evaluate the mechanical properties of solder joints using specimens whose geometry is representative of functional solder joints.
Owing to the severe anisotropy and variability seen in the mechanical behavior of solder joints, many researchers have studied the impact of solder microstructure on solder mechanical behavior and failures.
Matin et al. investigated deformations in solder joints subjected to thermal cycling and correlated those microscopic observations with stress fields obtained from elastic anisotropic FE analysis . In addition, Park et al. [14,15] characterized strain fields in temperature-cycled SAC solder balls with single and multicrystal microstructures using digital image correlation (DIC). Cross-polarized microscopy was conducted to correlate the damage sites with the microstructure. The highest strain concentrations and cracking were along the grain boundaries and at the interfaces between the Cu pad and solder in oligocrystalline solder joints. It is important to note that these physical realities cannot be captured correctly in finite element analysis (FEA) simulations without grain-scale modeling. Modeling the joint as a homogeneous single entity will entirely miss the first of these two damage modes (intergranular damage) and misrepresent the correct stress-state driving the second of these two damage modes (interfacial damage).
Arfaei et al.  studied the effect of several grains and their orientation on cyclic mechanical fatigue of solder balls under shear mode. Fatigue lifetime for the multicrystal (two to three grains) solder ball was longer than single-grain solder balls. Early failures were observed in single-crystal solder joint with grains oriented along , , and  directions. Similarly, Bieler et al.  looked at the effect of Sn grain size and orientation on the thermomechanical reliability of solder joints. Orientation imaging microscopy (OIM) analysis indicated that the severity of damage correlated with the orientations that had very high CTE values ( direction). Maximum CTE mismatch is observed when the crystal  direction is oriented parallel to the substrate and is a worst-case scenario. Xu et al.  examined the grain structure in ball grid array solder balls and correlated that with fatigue failures. The grain structure of different solder balls within a ball grid array package was sensitive to the joint location and effective cooling rate (which varied from exterior to interior of the package). The fatigue resistance improved as the number of grains increased, possibly due to reduced mismatch at grain boundaries with decreasing grain size. Matilla et al.  investigated the microstructural changes caused by the damage during temperature cycling of a solder ball in a chip-scale package. As expected, localized recrystallization into smaller grains was observed in the regions with stress concentrations. Creep deformation and crack propagation were influenced by the formation of grain boundaries after recrystallization.
Cuddalorepatta et al. [20,21] proposed a mechanistic isotropic multiscale modeling methodology to capture the dominant creep deformation mechanisms such as dislocation climb and detachment and the influence of dispersion strengthening in the Sn–Ag eutectic phase on the secondary creep response of microscale as-fabricated SAC305 solder specimens. Building upon Cuddalorepatta's work, Mukherjee et al. [22,23] developed an anisotropic mechanistic multiscale crystal viscoplasticity (CV) modeling framework that describes the anisotropic transient and steady-state creep behavior of single-crystal SnAgCu alloy, based on its heterogeneous eutectic and dendritic microstructures, and dislocation mechanics along the dominant slip systems within the crystal. Jiang et al. [24,25] built upon Mukherjee's work to develop a methodology for determining continuum anisotropic creep model for each grain (using Hill's anisotropic potential and Garofalo creep flow model) using the multiscale CV modeling framework for SAC single crystals, in conjunction with single-crystal results available in the literature. Jiang et al.  also determined a grain boundary creep model, which was combined with the single-grain model to predict the behavior of multicrystal specimens. The grain boundary model constants were calibrated using experimental creep test results from oligocrystalline SAC305 solder joints.
In summary, a proper understanding and quantification of solder microstructure are needed to model the anisotropic grain-scale behavior of small oligocrystalline solder joints. However, the present literature on experimental characterization and continuum-modeling of anisotropic properties for SAC305 is minimal. In the absence of reliable anisotropic properties of SAC305 and complexities in modeling, engineers have traditionally relied on more approximate homogenous isotropic mechanical properties for estimating stresses and strains in the solder joints. Such simplification misses the important impact of grain-scale microstructural anisotropy and heterogeneity on solder joint behavior and cannot predict the variability in their mechanical response. A failure mode impossible to predict by homogenous isotropic model is intergranular cracking within oligocrystalline solder joint. Modeling solder interconnects as homogenous isotropic domains not only misrepresent the strong anisotropic behavior of solder grains but also miss the critical stress concentrations caused by mismatches between neighboring grains in solder joints. Another disadvantage of homogenous isotropic model is that it cannot predict the failure variability caused by the joint-to-joint variabilities of grain morphologies, using simulation-based finite element tools. So, cost of experimentation will be higher since there is no other way to assess the expected microstructure-induced variability in failure data. Hence, this type of anisotropic grain-scale modeling is important for predicting the joint-to-joint variability of the mechanical behavior of oligocrystalline solder joints. Furthermore, grain-scale information about local stress and strain concentrations within the grain and at grain boundaries can provide a more accurate assessment of different damage modes such as intergranular/transgranular damage within the bulk of the solder joint or damage near interfacial IMC layers. Several factors affect the microstructure of lead-free solder alloys, such as composition , time above liquidus during soldering and subsequent cooling rates [28,29], aging conditions [32–36], and surface plating materials . Most of these papers focus on the effect of evolving microstructure on isotropic creep properties, as discussed in Secs. 1.2 and 1.3.
1.2 Effect of Cooling Rate on Constitutive Properties of SAC Solder Alloy.
The cooling rate during the assembly process affects the solder microstructure and thermo-mechanical behavior of solder joints. However, the literature on the effect of cooling rate (and the resulting solidification rate and crystallization rate) on SAC solder behavior is somewhat limited. Wei et al.  studied the effect of cooling rates (0.14 K/s, 1.7 K/s, ∼100 K/s) on the microhardness of SAC305 solder joints. The β-Sn pro-eutectic dendrite size and spacing decreased as the cooling rate increased. In addition, faster cooling rates resulted in a finer eutectic microstructure, consisting of closely spaced and uniformly distributed small Ag3Sn particles, resulting in greater dispersion strengthening. Conversely, a slower cooling rate produces a significantly coarser eutectic structure with larger and fewer (with increased spacing) Ag3Sn IMC particles in the eutectic region. Consequently, Vickers microhardness increased with increasing solidification rates .
Similarly, Lee et al.  studied the SAC305 microstructure under different cooling rates: 2.5 °C/s, 9 °C/s, 21 °C/s, and 63 °C/s. The β-Sn dendrite size (620 μm2, 550 μm2, 380 μm2, and 30 μm2) and volume fraction (55.3%, 50.2%, 42.5%, and 27.8%) monotonically decreased as the cooling rate increased. The Ag3Sn IMC was shaped like a leaf at the lowest cooling rate and changed to ellipsoidal particles at the highest cooling rate. In addition, the tensile strength decreased by 34% between the highest cooling rate (60.8 MPa) to the lowest cooling rate (39.5 MPa). Mueller et al.  analyzed the effect of cooling rate, composition, and solder volume on the microstructure of lead-free solder joints. Cooling rates used in this study were 0.14 K/s, 1.1 K/s, and 10.9 K/s. An increase in cooling rate was seen to reduce the size of IMC particles and spacing between β-Sn dendrites, whereas the area fraction of the eutectic region in the entire joint increased. Mutuku et al.  investigated the effect of cooling rate, aging time on microstructure, and shear fatigue lifetimes of different SAC solder joints. The mechanical properties of lead-free solder joints were found to vary in direct correlation with the size and spacing of dispersoids in the joint. The solder joint strength and shear fatigue life increased as the cooling rate increased.
1.3 Effect of Isothermal Aging on Constitutive Properties of SAC Solder Alloy.
Numerous researchers have reported that postreflow isothermal aging temperature and duration influence the solder microstructure and mechanical properties, similar to the influences seen for different cooling rates. Solder microstructure continuously evolves throughout its life, starting from the assembly process to its end of life. This evolution of the stress–strain response of SAC305 is important to understand since microstructural degradation or enhancement can both influence the solder's fatigue durability. Chauhan and Mukherjee  quantified microstructural parameters such as size, volume fraction, and spacing of nanoscale IMC particles (Ag3Sn, Cu6Sn5), and Sn dendrites in SAC305 solder alloy during 24–1000 h of aging at 100 °C. These microstructural parameters were used in a microstructurally motivated, mechanistic, multiscale, isotropic, CV model to study the effect of aging on the creep behavior of SAC305 solder joints. The Ag3Sn IMC particle size and spacing monotonically increased with increasing aging duration. In contrast, the evolution of the microscale Cu6Sn5 phase wasn't sensitive to aging duration. The volume fraction of pure Sn dendrites increased monotonically from 59% in the as-soldered condition to 72% after 1000 h of aging. Secondary creep resistance of SAC305 was seen to decrease with aging since this coarsening of the eutectic phase (in particular the increased Ag3Sn particle spacing and concurrently decreased pure tin dendritic size) reduced the dispersion strengthening of the eutectic region of the alloy. This trend was measured experimentally and also predicted with the CV model.
Lall et al.  conducted uniaxial tensile tests at different test temperatures (25 °C, 75 °C, 100 °C, and 125 °C) and different strain rates (1 × 10−3, 1 × 10−4, and 1 × 10−5 s−1) on SAC305 bulk solder bars aged at 100 °C for different durations (0–12 months). The solder's yield strength and ultimate tensile strength were seen to decrease significantly with the duration of aging and test temperature for a given strain rate. Similar trends were also observed for reflow-cooled specimens as well. Fu et al.  studied the effect of long-term aging of SAC305 and SAC405 solder specimen on their cyclic stress–strain response. The tests were conducted at 25 °C and 1 × 10−3 strain rate. The test specimens were subjected to two aging temperatures (25 °C and 125 °C) for various aging periods (0, 5, 10, 20, 30, 45, 60, 80, 110, 150, 200, 260, and 360 days). A microscopic examination of a fixed region of a sample was conducted at the end of each aging period. The Ag3Sn particle size and number were measured at a fixed location in the specimen by analyzing the scanning electron microscope (SEM) images using image processing tools. The stress–strain hysteresis area ΔW was found to reduce significantly after the first few days of aging and then stabilize. This was correlated with the coarsening of Ag3Sn IMC particles and the weakening of the pure tin dendritic structure. Fu and Wu [35,36], in different studies, have quantified the evolution of microstructure at different intervals during the aging period.
1.4 Overall Approach.
This study characterized the anisotropic plastic behavior of SAC305 solder joints by conducting uniaxial tensile and shear monotonic tests on monocrystalline and oligocrystalline SAC305 specimens. Anisotropic continuum plasticity models, based on Hill's potential Ramberg–Osgood (RO) hardening rule, was developed, and model constants were calibrated by matching grain-scale FEA simulations of the tests with the measured stress–strain results. As the microstructure and mechanical behavior of solder joints vary based on the cooling rate and grain size, the microstructure of single and multicrystal specimens was quantified and its influence on stress–strain behavior was incorporated in Ramberg–Osgood model constants using the Orowan and Hall–Petch hardening relations.
2 Experimental Details and Results
Details of the test specimens, test setup, test method, test results, and OIM analysis conducted in this study are provided in this section.
2.1 Specimen Details.
The grain-scale solder specimen reported here for monotonic tensile and shear behavior of the solder joints has been traditionally used by this research group in the past [9,37,38]. Schematic drawings of specimens with solder joint geometries of low aspect ratio (LAR) and high aspect ratio (HAR) are shown in Fig. 3. Aspect ratio is defined here as the ratio of the length of the copper–solder interface (x-dimension of the solder joint in Fig. 3) to the spacing between these two interfaces (y-dimension of the solder joint in Fig. 3). Both HAR and LAR specimens were fabricated in two different geometric configurations suited to facilitating uniform stress distributions in tensile and shear tests. The soldering profiles used in this study for manual wire-soldering (sample provided in Fig. 4) were different from those used for reflow-soldering. As a result, extensive microstructural characterization and microstructure-sensitive modeling were combined in this study to permit the results to be extrapolated to different microstructural morphologies resulting from other soldering profiles. Literature demonstrated that in an equiaxed solder joint, sufficiently low cooling rate (LCR) during soldering resulted in a single crystal . Therefore, LAR specimens (shown in Fig. 3(a)) and low cooling rate (∼0.23 °C/s) (shown in Fig. 4) were selected for fabricating single-crystal solder joint specimens. In contrast, higher cooling rate (HCR) (∼7 °C/s) resulted in multicrystal LAR specimens, as was also seen in HAR specimens in a previous study . The multicrystal HAR specimens were fabricated with ∼200 μm tall solder joints, to make the length scale of the test specimen approximately comparable with the length scale of functional solder joints in electronics. This geometric matching is important because solder joint mechanical response was seen to be sensitive to the length scale. Table 2 shows the different types of specimens used in this study, grouped by the cooling rate, aspect ratio, and the grain morphology. These specimens consist of one or at most 4 grains (depending upon the cooling rate) and are referred to as grain-scale solder specimens.
A spacer was inserted between the two copper platens to maintain the joint height for each specimen (1000 μm for LAR specimens and 200 μm for HAR specimens). The specimens were ground to remove the excess pool of solder, and the specimens' thickness (z-dimension) was reduced to approximately 1000 μm. The platen geometry and solder geometry of each specimen were characterized in detail to permit subsequent estimation of average stress and strain levels generated during the testing. The bare copper interface of copper platen was roughened using 600 grit silicon carbide (SiC) paper and organic solderability preservative (OSP) surface finish was applied on the copper surface. Since thickness of Cu6Sn5 interfacial IMC layer is much smaller than bulk solder, the influence of interfacial IMC layer on load–deflection curve can be ignored. The specimens were isothermally aged for 63–72 h at 125 °C, to partially stabilize the eutectic and dendritic (tiers 1 and 2) microstructures within each grain and also to partially relieve the residual stresses generated during the fabrication steps.
Before testing, cross-polarized microscopy was conducted to determine the grain morphology of each solder joint. For example, cross-polarized microscope images in Fig. 5 show single-crystal and multicrystal solder joint specimens. The front and rear faces, as well as selected orthogonal cross section, were imaged, to verify whether this fabrication process was capable of yielding single-crystal specimens throughout the entire thickness of the specimen. The average grain area for single-crystal specimens was ∼1.4 mm2. The different cooling rates used to control the grain morphology (single versus multicrystal specimens) also resulted in different eutectic and dendritic (tiers 1 and 2) microstructures within each grain. SEM observations in Fig. 6 revealed examples of the resulting tier 1–2 microstructural differences between LCR and HCR specimens. The images show dark-gray pro-eutectic β-Sn dendritic lobes surrounded by a matrix of Ag–Sn eutectic mixture. The bright round and elliptical particles in the eutectic region are Ag3Sn IMCs. The morphological differences (average dendrite volume fraction, average size, and spacing of Ag3Sn IMC particles within the eutectic region) were quantified using a commercial image processing software image-pro and the results are provided in Table 3. Image processing steps are provided in detail in the Appendix. Results show that a low cooling rate during soldering resulted in approximately 270% increase in Ag3Sn particle size, 190% increase in Ag3Sn particle spacing, and 15% decrease in area fraction of dendritic phase. OIM analysis was conducted on the tested specimens using electron back-scatter diffraction (EBSD) to characterize the lattice orientation of each Sn grain in the specimen. Each specimen had a unique mechanical response owing to the uniqueness in its grain structure. EBSD images of tested specimens along with the Euler angles (Bunge convention ) and area of individual grains are presented in Figs. 7 and 8.
|Ag3Sn particle size||Area fraction|
|Cooling rate (°C/s)||Aging duration (h) (at 125 °C)||Area (μm2) (mean)||Diameter (D) (μm) (mean)||Ag3Sn particle spacing (L) (μm) (mean)||Dendrite||Eutectic||Difference between Ag3Sn particle spacing and diameter (L–D) (μm)|
|Ag3Sn particle size||Area fraction|
|Cooling rate (°C/s)||Aging duration (h) (at 125 °C)||Area (μm2) (mean)||Diameter (D) (μm) (mean)||Ag3Sn particle spacing (L) (μm) (mean)||Dendrite||Eutectic||Difference between Ag3Sn particle spacing and diameter (L–D) (μm)|
2.2 Monotonic Tensile and Shear Test Results.
Monotonic tensile and shear tests were conducted at room temperature on monocrystalline and oligocrystalline SAC305 solder joint specimens using a custom thermo-mechanical microscale test frame. Deformation within the solder was characterized using a digital image correlation (DIC) system. The test setup is described in detail in a prior study . Tensile or shear displacements were applied at the rate of 1 μm/s. The relative displacement , within the solder (between the Copper interfaces), was continuously measured during the test using DIC, and the shear or tensile engineering strain in the solder joint is estimated (). The load was measured by the load cell, and the nominal shear or tensile stress was estimated () based on the geometry of each specimen. Figure 9 presents axial stress versus engineering strain curves generated from uniaxial monotonic tensile tests. Figure 10 presents shear stress versus engineering shear strain curves obtained from uniaxial monotonic shear tests. The specimen ID number corresponds to the grain structure provided in Figs. 7 and 8 in Sec. 2.1.
3 Anisotropic Elastic-Plastic Constitutive Properties
Anisotropic elasticity was modeled using the elastic stiffness constants for SAC305 (Table 1) established in a prior study . Anisotropic continuum plasticity models, based on Hill's potential and Ramberg–Osgood hardening rule for SAC305 single crystal, are developed in this section. The model constants were calibrated by matching FEA simulations of the uniaxial monotonic tensile and shear tests on single-crystal specimens, with the experimentally measured stress–strain results. These model constants were then used to demonstrate the anisotropy of single-crystal plastic behavior.
3.1 Anisotropic Continuum Plasticity Model (Hill's Potential With Ramberg–Osgood Hardening Rule).
In this study, plastic hardening rule for deformation beyond the yield limit is modeled with Ramberg–Osgood model for deformation plasticity (shown in Eq. (8)), which relates Hill's stress to Hill's plastic strains . This model was used in this study to model the plastic flow behavior of SAC305. Here, is the strength coefficient, whereas is the strain hardening exponent. The anisotropic yield strength ratios and the plasticity model constants were calibrated using the FEA of the tested specimens (listed above in Sec. 2). The calibration method and results are presented next in Sec. 3.2.
3.2 Calibration of Hill and Ramberg–Osgood Model Constants.
Single-crystal stress–strain data were used to determine Hill's yield strength ratios using a parametric anisotropic elastic-plastic finite element approach. This is a manually conducted parametric study, for preliminary verification of the viability of using the Hill's potential and Ramberg–Osgood hardening rule for grain-scale modeling of solder joints. Future studies should use more systematic machine-learning optimization approaches to find the best-fit model constants. The tensile and shear single-crystal specimen geometries were modeled in FEA, with material principal axes defined in corresponding local material axes, based on the grain orientations revealed by OIM. The FEA model shown in Fig. 11 comprises the exposed portion of the specimen that lies between the grips of the test frame. Assigned average element size for copper region was 100 μm, whereas average element size for solder region was 50 μm. General purpose eight-node linear brick element with reduced integration (designated as C3D8R in abaqus) was selected for both copper and solder regions. The solder joint consists of an anisotropic monocrystalline or oligocrystalline solder domain sandwiched between two copper substrates.
Elastic stiffness constants shown in Table 1 were used to model the anisotropic elasticity of SAC305. The bottom surface of the specimen was constrained, and a monotonic tensile or shear displacement was applied on the top face. Eight-node linear brick elements were used to model the specimen, and the copper substrate was modeled as a linear elastic-isotropic material (elastic modulus: E = 110 GPa, Poisson's ratio: ν = 0.35). Cu–Sn interfacial IMC layers were not considered in the FEA model since the interfacial IMCs are assumed to have negligible impact on the overall average solder constitutive properties. Average applied stress was estimated by summing reaction forces at the fixed end (in the y-direction for axial loading and in the x-direction for shear loading) and dividing the total reaction force by the cross-sectional area of the solder joints at the narrowest section of the fillet. The average strain was calculated by dividing relative displacement (measured across the top and bottom Copper interfaces; in the y-direction for axial loading and in the x-direction for shear loading) by the height of the solder joint. Engineering stress–strain response curves were constructed using the average stress and average strain.
Engineering stress–strain responses along the loading direction for each single-crystal specimen were recorded and compared with average experimental stress–strain curves for multiple single-crystal specimens. The average input plasticity properties (plots of Hill's stress versus Hill's inelastic-strain) were simultaneously optimized (by a minimum fitting error by maximizing the R2 value) for each parametric case, to obtain the most reasonable possible fit of FEA predictions versus experimental results. Figure 12 shows different combinations of yield strength ratios used for the parametric study.
Based on these combinations, the effect of varying following parameters on stress–strain predictions was evaluated: (A) tensile yield strength ratios; (B) shear yield strength ratios; (C) both tensile and shear yield strength ratios; and (D) ratio between shear yield strength ratios. The following simplifications were made due to the BCT symmetry of the Sn crystal, where  and  crystallographic directions are similar: and . In this study, all yield strength values have been normalized by (which is the stiff  c-direction of the BCT crystal). The corresponding Ramberg–Osgood plot of Hill's stress and Hill's plastic strain was measured from the uniaxial tensile test in the  direction. The goodness of fit of the anisotropic elastic-plastic material constants was assessed by the dual criteria of (i) minimizing the mismatch (maximizing R2 value of the fit) between FEA and experimental engineering stress–strain curves (using metric M1); and (ii) capturing the differences observed due to different crystal orientations (using metric M2):
Mismatch between FEA and experimental stress–strain behavior (M1): the ability of anisotropic Hill FEA model to predict both average tensile and average shear stress–strain responses of all tested single-grained specimens (averaged across multiple grain orientations) is accomplished by minimizing the RMS difference between average FEA model and average experimental stress–strain responses across multiple grain orientations.
- Anisotropy metric (M2) is the ability of anisotropic Hill FEA to predict the severity of anisotropy caused by different grain orientations, in tensile and shear specimens. Anisotropy metric is defined as follows:The tested specimens with the most dissimilar grain orientations were selected for assessing metric M2. Thus, M2 = 1 represents the best case (when the FEA model predicts the same difference in stress–strain response for the different selected specimens, compared to experiments). M2 < 1 implies that FEA model lacks sufficient sensitivity to predict the measured anisotropy while M2 > 1 indicates that the selected anisotropic properties make the FEA model overly sensitive to different grain orientations.
Fitting accuracy (R2) is shown in Fig. 13, and M2 metrics for anisotropy sensitivity study are shown in Fig. 14. Based on both of these two considerations, for all the cases examined in this preliminary parametric study, constants presented in Tables 4 and 5 were found to generate the most reasonable FEA stress–strain response (shown in red box in Figs. 13 and 14). The corresponding Ramberg–Osgood power-law hardening rule relating Hill's stress to Hill's equivalent inelastic strain ( = 88 and = 0.225) is shown in Fig. 15. These Hill's constants and Ramberg–Osgood model constants are relevant for large SAC305 single crystals with coarse dendritic and eutectic microstructures that are common when the solder is either solidified and crystallized using slow cooling rate or is isothermally aged for extended periods after solidification. Variability in plastic properties due to anisotropy along different directions for tensile and shear loading is shown in Fig. 15. Methods to scale these elastic-plastic properties for finer microstructures and smaller grains are discussed in Sec. 4. The FEA stress–strain predictions generated using this set of constitutive properties were compared to experimental results shown in Figs. 16 and 17. The anisotropic Hill FEA predicted the anisotropy between specimens with different orientations reasonably well. Further improvement will possibly require a more complex function than the single-term power-law to represent the more complex plasticity behavior exhibited by the experimental stress–strain results. The limitation of using single-term power-law plasticity model was also observed by the fact that the Hill FEA consistently overpredicted the response when the specimens are loaded in tension and consistently underpredicted the shear stress–strain response.
4 Dependence of Elastic-Plastic Properties on Grain Size and Microstructure Within the Grain
As discussed in Sec. 3, the cooling rate used during fabrication of single-crystal solder joint specimens was about an order of magnitude lower than that typically used during the assembly process of functional electronics. The low cooling rate results in significantly coarser eutectic and dendritic microstructure, when compared to those with conventional cooling rates. Coarser microstructures are also common under isothermal aging. Another source of differences in mechanical elastic-plastic response can come from grain-size dependence. Therefore, the single-crystal elastic-plastic Hill–Ramberg–Osgood elastic-plastic properties presented in Table 4/Table 5 and Fig. 16 need to be scaled as a function of microstructural features and grain size to estimate the mechanical behavior across a wide range of fabrication and isothermal aging conditions.
The micromorphological influences discussed above were modeled as appropriate strengthening mechanisms due to the microstructural features (tiers 1 and 2) or the grain size (tier 3). The scaling factors were determined using a top–down empirical approach from tier 3 scaling factor to tier 1 scaling factor. The single-crystal Hill's stress versus inelastic Hill's strain data were scaled based on tiers 1–3 microstructural parameters. The procedure for estimating the scaling factors is discussed below in Secs. 4.1 and 4.2.
4.1 Scaling Factors for Average Grain Size (Tier 3).
where is the yield strength, is a model constant (representing the strength of very large grains), is the strengthening coefficient, and is the average grain diameter. The average grain diameter is approximated as . Figure 18 shows the relationship between the solder yield stress and square root of the average grain diameter. The grain size-dependent properties for SAC solder (composite) were iteratively determined by matching grain-scale anisotropic FEA with experimental stress–strain results of HCR multicrystal specimens #5–10. As shown in Figs. 7 and 8, specimens #5–10 have a wide range of grain sizes available for calibration of the Hall–Petch model constants. Details regarding the grain-scale anisotropic FE modeling procedure are provided in Sec. 4.4.
4.2 Scaling Factors for Dendritic (Tier 2) Volume Fraction.
At the next length scale (tier 2 of Fig. 2), the heterogeneous microstructure of the SAC solder composite was approximated as ellipsoidal β-Sn dendritic regions (inclusions) surrounded by the Ag–Sn eutectic region (matrix). The average stresses in the solder composite were approximated by considering the load-sharing between the Ag–Sn eutectic phase and β-Sn dendrite using a three-phase sphere model proposed by Christensen  (refer to Eqs. (4.12)–(4.16) from Christensen's book ) for isotropic linear elastic materials. This is clearly an approximation since the Sn-rich SAC305 solder material is not isotropic. As the cooling rate (or isothermal aging) changes the eutectic region's strength and the dendrites' volume fraction (and hence volume fraction of the eutectic region), the effective homogenized strength of the SAC composite alloy also changes. Examples of these changes are shown below in Sec. 4.3.
In addition, as the grain size strengthening mechanism is ideally attributed to Sn and not eutectic Ag–Sn phase of SAC solder, grain size-dependent yield strength for Sn was estimated using Christensen's model as shown in Fig. 19. β-Sn stress–strain curve from the prior work by Kariya et al. [43,44] was treated as a reference stress–strain curve for infinitely large grain, which was scaled based on the tier 1–3 microstructures. for Sn is greater than SAC (10.35) indicating that the Sn yield strength has greater sensitivity to grain size than SAC yield strength. The grain size dependence of Sn was diminished after homogenization with eutectic Ag–Sn phase which is insensitive to grain size.
4.3 Scaling Factors for Eutectic (Tier 1) Microstructure.
where is a model constant, is the shear modulus, is the Burger's vector, is the interparticle spacing, and is the radius of the dispersed particles. The yield strength of the eutectic region depends on the Orowan stress. Therefore, as the particle size (r) and spacing (L) increase, as in the case of slow-cooled (or isothermally aged) specimens, the yield strength of the eutectic region reduces in inverse proportion to (L − 2r). Figure 20 shows tier 1 scaling factors determined for scaling the eutectic yield strength as a function of cooling rate. Since this study only consisted of specimens fabricated using two (low and high) cooling rates, the tier 1 scaling model is approximated to be linear.
4.4 Comparison of Grain-Scale Finite Element Analysis Predictions With Experimental Results.
Grain-scale anisotropic FEA models of multicrystal HCR specimens (LAR: #5 and 6; HAR: #7–10) were used to: (i) calibrate tier 3 scaling factors and; (ii) compare the anisotropic Hill FEA predictions with experimental results. As mentioned before, multicrystal HCR specimens #5–10 consist of 16 different grain size variations that were used for calibration of Hall–Petch equation. For example, full FEA models (along with mesh) of tensile and shear specimens (LAR and HAR) are provided in Fig. 21. Different colors indicate different grains and their local material axis is determined using EBSD. Hill's yield strength ratios and Hill's equivalent stress–strain curves along with local material axes are used to model each grain. Cyclic displacements and boundary conditions were applied (depending upon tensile or shear specimen) in the same way as discussed in Sec. 3.2 and as shown in Fig. 11.
At the same displacement level, Hill's stress contour plots for specimens #5, 6, 7, and 10 are shown in Fig. 22. The grain-scale anisotropic FEA modeling methodology successfully predicts the severity and location of stress concentrations in solder joints. The grain-scale anisotropic FEA model predictions for different specimens were compared with the experimental results as shown in Figs. 23 and 24. As a result of this study, Fig. 25 shows Hill's stress versus Hill's inelastic strain curves for different grain sizes of HCR solder joints. Figure 26 shows Hill's stress versus inelastic strain curves of infinitely large grain for low and high cooling rates. These curves can be used to conduct anisotropic grain-scale FE analysis of oligocrystalline solder joints.
5 Summary and Conclusions
Monotonic tensile and shear tests were conducted on monocrystalline and oligocrystalline SAC305 solder joint specimens to characterize the anisotropic plastic behavior of SAC305 solder joints. The anisotropic plasticity of each grain was modeled using Hill's yield criterion and a Ramberg–Osgood hardening model. The single-grain solder stress–strain results were used to determine Hill's yield stress ratios and to characterize the plastic response of SAC305 solder joints. Each specimen was modeled in FEA, and the influence of each yield strength ratio on solder joint stress–strain response was evaluated by parametrically varying the ratios systematically and comparing the engineering stress–strain response with experimental results. The performance of different yield strength ratios was evaluated by assessing their ability to represent all the experimental data in an average sense, and their ability to capture the influence of grain orientation on anisotropic plastic behavior. An optimized set of yield strength ratios, that satisfies both considerations above, was obtained in this study. The anisotropic FEA, based on these fitted material properties, overpredicted the strength of tensile specimens and underpredicted the same in shear specimens. This limitation was probably because the single-term power-law plasticity assumption considered in this study was too simplistic to represent the complex plastic behavior of the solder materials.
The single-crystal specimen was attained using low cooling rate during soldering process, which resulted in significant differences in the tiers 1 and 2 microstructure compared to higher cooling rate specimens. In addition, higher cooling rate leads to formation of multiple grains in the solder joint, resulting in different tier 3 microstructures compared to low cooling rate specimens. The differences in microstructure lead to higher yield strength in high cooling rate specimens compared to low cooling rate specimens due to a combined contribution from Orowan hardening and Hall–Petch effect. Therefore, an empirical scaling methodology has been developed to scale the single-crystal stress–strain results (with low cooling rate) to multicrystal stress–strain curves (with higher cooling rate). The scaling factors are influenced by parameters such as the Ag3Sn particle size and spacing, volume fraction of dendritic and eutectic phases, and grain sizes. The FEA conducted in this paper successfully demonstrates that the microstructurally motivated grain-scale anisotropic model provides better prediction of the variability in the stress–strain response of tested specimens than the homogenous isotropic model.
For FEA simulation of oligocrystalline solder joint, material properties need to be scaled as per intermetallic (Ag3Sn) size and particle spacing and size of each grain (Figs. 18–20). Note that orientation of grains determined using EBSD is also another input in FEA. The grain-scale anisotropic properties determined in this study can be used by engineers to (i) assess the variability in mechanical behavior of solder joints resulting from different grain orientations; (ii) supplement the failure analysis of tested joints, by accurately evaluating the stress concentrations at failure sites. These applications will be explored at depth in a future paper.
This work is sponsored by the members of the CALCE Electronic Products and Systems Consortium (EPSC) at the University of Maryland, College Park.
CALCE EPSC at University of Maryland, College Park (Funder ID: 10.13039/100008510).
Conflict of Interest
There are no conflicts of interest.
Appendix: Image Processing for Quantification of Tiers 1 and 2 Microstructure
The differences in the solder microstructure resulting from variation in cooling rates during soldering and postsoldering aging duration were quantified using a commercial image processing software image-pro. The SEM images of different specimens presented in Fig. 6 clearly show variation in (a) Sn dendrite size, (b) Ag3Sn particle size and spacing, and (c) Area fractions of Sn-dendrite and eutectic phases. Therefore, each SEM image was processed to measure these microstructural parameters, which were used in the material models. The specimens were polished with 0.05 μm colloidal silica suspension for 30–45 min to polish and slightly etch the surface to reveal Ag3Sn IMC particles.
The SEM image was calibrated using the scale provided at the bottom right side of each image. “Smart Segmentation” tool was used to measure the area fractions of Sn-dendrites and Ag3Sn eutectic phase. At first, the tool was trained by manually locating the object region and the background region as a reference. The Smart Segmentation tool isolated the object regions from the background in the SEM image and measured the area of object regions in the image as shown in Fig. 27.
The low-pass and watershed filters were applied to threshold and segment the image. Thresholding reduced the images to two intensity levels. This operation converted the gray-scale image to the binary image such that the background was changed to black and the Ag3Sn particles were set to white as shown in Fig. 28. The Ag3Sn particles were then selected, and their size was recorded.
A Voronoi diagram was used to quantify the spatial characteristic of Ag3Sn particles such as particle spacing. As shown in Fig. 29, the Voronoi diagram divides the image into different regions (Voronoi cell) surrounding the reference object (Ag3Sn particles). Each cell consists of set of points that are closest to the object than to other object. The minimum center-to-center distance between each object was extracted using this tool. The circular regions represent the corresponding distance between the Ag3Sn particle and its closest neighbor.