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Review Article

# Creep Constitutive Models Suitable for Solder Alloys in Electronic AssembliesPUBLIC ACCESS

[+] Author and Article Information
Subhasis Mukherjee

Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: smukher1@umd.edu

Mohammed Nuhi

Department of Reliability Engineering,
University of Maryland,
College Park, MD 20742

Abhijit Dasgupta

Professor
Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: dasgupta@umd.edu

Professor
Department of Reliability Engineering,
University of Maryland,
College Park, MD 20742
e-mail: modarres@umd.edu

Contributed by the Electronic and Photonic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received December 29, 2015; final manuscript received April 4, 2016; published online June 8, 2016. Assoc. Editor: Eric Wong.

J. Electron. Packag 138(3), 030801 (Jun 08, 2016) (13 pages) Paper No: EP-15-1143; doi: 10.1115/1.4033375 History: Received December 29, 2015; Revised April 04, 2016

## Abstract

Most solders used in electronic systems have low-melting temperature and hence experience significant amount of creep deformation throughout their life-cycle because typical operational and test conditions represent high homologous temperature. Phenomenological and mechanistic models used in the literature for predicting creep response of both bulk and grain scale specimens are reviewed in this paper. The phenomenological models reviewed in this paper are based on purely empirical observations of the creep deformation behavior or derived from qualitative interpretation of the underlying microscale mechanisms. These models have some intrinsic disadvantages since they do not have explicit mechanistic dependence on microstructural features. Therefore, the constitutive relations derived using the above models are difficult to extrapolate beyond the test conditions. This paper also reviews how some of the above limitations can be mitigated by using mechanistic or microstructurally motivated models. Mechanistic models are capable of estimating the material creep response based on the detailed physics of the underlying mechanisms and microstructure. The microstructure and constitutive response of the most popular family of lead-free solders, namely, SnAgCu (SAC) solders, are significantly different from those of previously used eutectic Sn37Pb solder. The creep deformation in Sn37Pb solder occurs primarily through diffusion-assisted grain-boundary sliding. In SAC solder joints, dislocation-based creep deformation mechanisms such as glide, climb, detachment, and cross-slip appear to be the dominant mechanisms in coarse-grained joints. Mechanistic creep models are therefore based on the deformation mechanisms listed above.

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## Introduction

Eutectic and near-eutectic Sn–Pb solder joints have either lamellar structure, in which alternate platelets of Pb-rich and Sn-rich phases are formed in coarse colonies or in which a near-equiaxed structure of Pb islands is embedded in a polycrystalline Sn matrix. The Sn grains are randomly oriented and are relatively small compared to the typical characteristic length scale of the interconnect, thus making the alloy statistically isotropic. Rapid cooling produces the lamellar structure, while slower cooling tends to produce the equiaxed structure. The creep deformation in Sn–Pb solder occurs primarily through grain-boundary sliding (and phase-boundary sliding) assisted by grain-boundary diffusion. Sustained cyclic mechanical deformation at elevated temperature can result in significant coarsening of the phases.

On the other hand, lead-free SAC solder joints have a coarse-grained microstructure with only a few large anisotropic Sn grains throughout the joint after reflow. Within each grain, the pro-eutectic Sn is mostly in dendritic form, with the space between the dendritic lobes filled in with a eutectic Sn–Ag phase that consists of nanoscale Ag3Sn intermetallic compounds (IMCs) present in a Sn matrix. In addition, there are micron-scale Cu6Sn5 IMC precipitates with either straight or branched rodlike structures of hexagonal cross section, mostly present at Sn dendrite boundaries and grain boundaries, which obstruct grain-boundary sliding [1]. Furthermore, SAC melt temperatures are significantly higher than those of Sn–Pb solders. Hence, in SAC solder joints, dislocation glide, climb, detachment, and cross-slip appear to be the dominant creep deformation mechanisms, resulting in significantly higher creep resistance than SnPb. The dislocation movement results in the formation of complex dislocation networks, dislocation cells, and subgrains in each dendritic lobe. Subsequent recrystallization tends to be preferentially along these dendrite boundaries and/or along the dislocation cell boundaries.

Sn phase in SAC alloys can solidify in different morphologies depending on the solidification conditions (e.g., cooling rate) and the metallurgical conditions that are governed by the nominal composition of the solder and the metallization on the printed wiring board and components [2]. In order to rationalize the formation of the observed microstructure, it is informative to examine the solidification of solder interconnects with the help of the equilibrium phase diagram. It should be kept in mind, however, that the equilibrium diagrams do not contain information about the nonequilibrium effects of cooling rate or the morphology of the phases formed. Varying the Ag content in SAC alloys contributes to varying volume fraction of pro-eutectic Sn dendrites versus the eutectic phase and also affects the size and distribution of nanoscale Ag3Sn intermetallic precipitates within the eutectic phase [3]. The wide range of IMC morphologies (obtained through various compositions and processing techniques) and the presence of a few large anisotropic Sn grains in coarse-grained SAC solders contribute to heterogeneity and joint-to-joint variation in the mechanical behavior of SAC alloys [4]. In contrast, eutectic SnPb material exhibits smaller grains, more homogeneous mechanical properties, and hence less joint-to-joint variability in the mechanical behavior.

Typical creep deformation consists of three stages as shown schematically in Fig. 1. Primary creep regime is defined by the region where the strain rate (slope of the above curve) gradually decreases with time. It is followed by secondary or steady-state creep regime where the strain rate becomes constant and then finally the strain rate increases over time in the tertiary creep regime, finally leading to creep rupture of the material.

The creep equivalent strain rate (/dt) depends on a multitude of variables. These include:

• Stress parameters such as temperature T, von Mises' stress σ, cumulative equivalent strain (ε), and time (t).

• Microstructural state variables such as grain size (d), intermetallic precipitate density (average size r and spacing λ), dislocation density (ρi), and mobility (Mi).

• Material parameters such as diffusion constants (Dv), atomic volume (Ω), and Burger's vector (b).

Thus, phenomenological creep models usually use empirically motivated functional forms to express the creep rate as a function of the driver variables Display Formula

(1a)$ε˙=f(T, σ, ε, t, d, b, ......, Ω)$

Most phenomenological modeling approaches simplify this functional dependence into a product of separable terms, as exemplified in the following equation: Display Formula

(1b)$ε˙=f1(T) f2(σ) f3(ε) f4(t) f5(d) f6(b) .....fn(Ω)$

In the case of lead-free solders, these phenomenological models have been shown to have limitations in simulating the influence of complex interaction effects between different creep drivers. Several such models are discussed below. Additional research is recommended to aid in selecting material parameter values, recommending improved testing methods, and/or developing new phenomenological models that better address these complexities.

There are three dominant mechanisms that play a significant role in any creep deformation process:

• Dislocation creep (glide, climb, cross-slip, and detachment),

• diffusion matrix creep (Nabarro Herring creep—volume diffusion of interstitials and vacancies inside the grain), and

• diffusion grain-boundary creep (Coble creep—diffusion of interstitials and vacancies along grain boundary).

###### Dislocation Creep.

Dislocation creep is more dominant at high stress and moderate homologous temperature. The motion of dislocations during time-dependent creep deformation is hindered by the crystal structure itself (i.e., grain boundaries). Moreover, discrete obstacles such as solute atoms, precipitates, and other dislocations block the motion of gliding dislocations. At high homologous temperature, blocked dislocations can free themselves by mechanisms of climb, detachment, cross-slip, and formation of partial dislocations. The diffusion of vacancies through the lattice or along the dislocation core drives the dislocation to change its slip plane and to pass beyond the obstacle. Atoms diffuse into or out of the dislocation core, leading to dislocation climb [59]. Equivalent creep rate for such mechanisms is given by [10] Display Formula

(2)$ε.d=ADGbkT(σG)n$

where subscript d refers to dislocation creep, A is a material constant, D is the diffusion coefficient, G is the shear modulus, b is the Burgers vector, σ is the applied von Mises' stress, n is the power-law exponent, k is the Boltzmann constant, and T is the absolute temperature.

###### Diffusion Creep.

Diffusion creep is more dominant at low stress and high homologous temperature. Under the driving force of applied stress and high homologous temperature, atoms diffuse from the one side of the grain to the other side. The grains become elongated axially, and the process becomes faster at high temperature due to the presence of more vacancies. Rate of atomic diffusion in one direction is same as vacancy diffusion in other direction. The mechanism is called Nabarro–Herring creep [5]. The jump frequency of atoms and vacancies is higher along the disordered grain boundaries than within the lattice structure of the grain for Coble creep [11], whereas the jump frequency of atoms and vacancies is higher inside the grain for Nabarro–Herring creep. The rate-controlling mechanisms in both cases are vacancy diffusion or self-diffusion. Equivalent creep strain rate due to these two diffusional mechanisms is given by Display Formula

(3)$ε.NH=ANHDvd2σΩkT; ε.C=ACδDgbd3σΩkT$

where subscript NH refers to Nabarro-Herring creep and subscript C refers to Coble creep, ANH and AC are the material constants, Dv is the diffusion coefficient for volumetric self-diffusion through the bulk material, Dgb is the diffusion coefficient for self-diffusion along the grain boundary, Ω is the atomic volume (b3), δ is the grain-boundary thickness, σ is the applied von Mises' stress, d is the grain size, k is the Boltzmann constant, and T is the absolute temperature. In order to understand the relative dominance of these different mechanisms, one needs to look at the Ashby deformation mechanism map for the particular material of interest [12]. In this paper, authors will provide a comprehensive understanding of phenomenological creep constitutive models for solder joints in Sec. 2, and state-of-the-art mechanistic physical models will be discussed in detail in Sec. 3. Finally, the paper will conclude with a brief summary in Sec. 4.

## Phenomenological/Empirical Constitutive Creep Models for Solders

In this section, the authors will present a review of the empirical models reported over the years in the literature, to describe creep behavior of different materials (Sec. 2.1). Then, the models which have been extensively used for modeling the creep behavior of solder interconnects are discussed. In Sec. 2.2, authors review the effect of varying weight percent of silver and copper on creep constitutive properties of solder material. In Sec. 2.3, the effect of aging on creep constitutive properties of solder interconnect materials is reviewed. In Sec. 2.4, some of the sources of variability in the creep constitutive properties of electronic interconnect materials are reviewed and the limitations of phenomenological models in capturing the creep behavior of lead-free solder materials are discussed.

###### Phenomenological/Empirical Constitutive Models for Primary/Secondary Creep.

Several empirical constitutive models of varying degrees of sophistication and different ranges of applicability are available for describing the creep deformation behavior of materials. More than 32 creep models, ranging from the classical simple Kelvin–Voigt linear viscoelastic creep model [13] to more complex nonlinear state-variable approaches like the Ramaswamy–Stouffer model [14], have been identified by reviewing the literature. More importantly, each of the models identified uses different strategies to capture the dependence on one or more of the four following groups of creep drivers:

• Strain–time dependence (strain hardening or time hardening) in creep models.

• Stress dependence in creep models.

• Temperature dependence in creep models.

• Dependence on other state variables, such as back stress and dislocation drag stress.

The dependence of the creep deformation on each of these four types of creep drivers (time, stress, temperature, and state variables) has been modeled by different researchers with exponential, logarithmic, sine-hyperbolic, and power-law models, as shown in several important examples in Table 1. In all these models, unless otherwise specified, σ is the von Mises' stress, ε is the equivalent strain (work conjugate of von Mises' stress), and t is the time.

Since solder creep behavior is typically important at or above 0.4 × Tm [26] (where Tm is the absolute melting temperature), most of the above models can be used for describing creep deformation behavior of solder alloys. In case of solders, secondary creep (also called steady-state creep) is widely studied since it dominates the creep-rupture life of the solders. Most popular models used for describing the secondary creep rate of solders are classified and summarized in Table 2.

There have been numerous studies [3647] on the creep constitutive properties of solder materials because most life-cycle deformation histories in solder joints are dominated by creep mechanisms, in view of the high homologous temperature for most solders at typical environmental conditions. Tribula and Morris [48] presented steady-state creep rate data for several solder compositions, including the binary eutectic alloy and Pb–Sn solder alloyed with small amounts of Bi, Cd, In, and Sb, in a joint configuration. Constitutive behavior and low-cycle thermal fatigue behavior of 97Sn–3Cu solder joints was extensively studied by Jih and his research group in Ford Motor Company [49,50]. Darveaux and Banerji [51] first came up with the extensive database of creep properties of Sn36Pb2Ag, Sn40Pb, Sn3.5Ag, Pb2.5Sn, and Pb5Sn solder by conducting shear tests on bulk solder specimens. They used the Garofalo steady-state creep model to come up with the relevant secondary creep constants. A popular critical review of microstructure and creep process activation energy values for a number of lead-free solder alloys like Sn–Ag, Sn–Bi, and Sn–In was presented by Igoshev and Kleiman [52]. This review revealed significant variability in the experimental data for the same solder material sets reported by different authors.

In 2002, a huge database of solder properties with emphasis on new lead-free solders was documented by NIST and Colorado School of Mines [53]. Solders of interest were Sn4.0Ag0.5Cu, Sn2.0Cu0.8Sb0.2Ag, Sn36Pb2Ag, and Sn37Pb. They also documented stress exponents and activation energies for the modified Dorn equation for four lead-free solder alloys Sn2.6Sb, Sn5.0Sb, Sn7.8Sb, and Sn2.0Sb2.0In. Another extensive study was conducted by Shi et al. [54], where they carried out extensive creep tests on eutectic Sn37Pb solder at various stresses between 1.25 and 70 MPa over a wide temperature range of −40 °C to 150 °C. The stress exponent and the activation energy were both observed to be dependent on temperature and stress, and the Arrhenius model was employed to describe the creep flow of the solder. The next comprehensive study was done by Motalab et al. [55], where they studied the creep properties of Sn3.5Ag, Sn3.0Ag0.5Cu, and Sn0.7Cu solder interconnects. Secondary strain rate of Sn3.9Ag0.6Cu alloy was studied at varying stress levels (2–40) MPa at temperatures (−25 °C to 160 °C) by Vianco et al. [56].

An analytical framework to predict failure of solders under creep conditions was proposed, and a creep-rupture model for two-phase eutectic solders, based on both micromechanics and fracture mechanics, was developed by Wong et al. [57] at the Hughes Aircraft Research wing. The creep behavior of air-cooled and liquid nitrogen-quenched soldered joints of 60Sn40Pb at 65 °C has been studied by Mei et al. [58]. The stress exponent, n, in the power law changes from a value of about 6 to values of 2–3, as the strain rate drops below 1 × 10−4/s. This result, combined with the authors' previous stepped load creep test results, indicates a transition of the creep deformation mechanism from conventional dislocation climb to superplastic grain-boundary sliding. The superplastic creep of the soldered joints is ascribed to their nonlamellar microstructure due to the fast cooling rate. During creep deformation, recrystallization of the solder material is observed to cause softening. Another significant contribution was made by Syed [59,60], where he developed a model which can predict creep damage accumulation in solder joints during thermal cycling. Effect of both grain-boundary sliding and matrix creep is considered for eutectic Sn–Pb solder. The model was validated by correlating the predicted fatigue life of solder joints for leadless ceramic chip carriers with the published measured data for different test conditions. Recently, there have been two very comprehensive reviews of creep properties of lead-free solders conducted by Ma [61] and Clech [62]. Constitutive properties of selected lead-free solders were also reported by Zhang and Dasgupta [63]. Huge discrepancies were reported in both studies for secondary creep strain rate data for several SAC305 solders reported by different authors at a temperature of 125 °C (plotted in Fig. 2).

This variability can also be seen in the values of the viscoplastic creep model constants of these solder materials, as reported in Table 3. This discrepancy is mainly caused by differences in specimen geometry (tabulated in “specimen type” in Table 3), testing methodology, and test conditions (varying temperature and stress levels). Readers are encouraged to refer to Ref. [61] for more details on test conditions and test methodology for each solder alloy model constant. Therefore, empirical models used for capturing the constitutive behavior from the creep data available need careful examination. In Sec. 3.1, mechanistic models for creep will be reviewed, which attempt to connect the microstructure explicitly to the creep response, using fundamental mechanics principles. Such models not only provide physical insights into the creep deformation process but also provide understanding of the sources of variabilities in the creep response.

###### Effect of Composition on Creep Behavior of Sn-Based Solder.

Although the creep behavior of Sn–Ag–Cu based solders has been the subject of many recent studies, the impact of Ag and Cu content, and associated microstructural changes, on the creep response of near-eutectic Sn–Ag–Cu ternary alloys has received relatively less attention. Ag and Cu weight percent can have a huge effect on the creep behavior of Sn-based Pb-free solder alloys, due to the changing volume fraction of the eutectic phase and Sn dendrite with increasing alloy content in Sn matrix.

The creep behavior of Sn1.0Ag0.5Cu, Sn2.5Ag1.0Cu, and Sn4.0Ag0.5Cu ball grid array solder balls and 99.99% pure polycrystalline bulk Sn was studied using impression creep and related to the microstructure by Chen and Dutta [64]. As discussed earlier in Sec. 1, SAC solders generally consist of a pro-eutectic primary phase of Sn dendrites and a eutectic phase comprising nanoscale Ag3Sn IMC precipitates embedded in a pure β-Sn phase. With increasing concentrations of Ag in the alloy, the proportion of the eutectic phase in relation to the primary pro-eutectic β-Sn phase increases. In pure Sn and Sn1.0Ag0.5Cu, the β-Sn grains form the continuous matrix, whereas in Sn2.5Ag1.0Cu and Sn4.0Ag0.5Cu, the eutectic phase forms a continuous network around the β-Sn dendritic colonies. In general, the steady-state creep rate decreased with increasing Ag content, and in particular, with the increase in volume fraction of Ag3Sn and Cu6Sn5 precipitates. The rate-limiting creep mechanism in all these materials investigated here was dislocation climb because of dislocation motion impediment from precipitates.

###### Effect of Isothermal Aging on Creep Behavior of Sn-Based Solder.

The effect of isothermal aging on the creep constitutive response of SAC solder interconnects has been an important topic of interest in the electronic packaging community because solder interconnects experience significant amount of thermal exposure during the life-cycle. Several studies have demonstrated how the creep resistance of SAC solders decreases with the increase in duration and temperature of isothermal aging. A comprehensive review of the effect of aging on the creep behavior of lead-free solders (Sn3.0Ag0.5Cu and Sn4.0Ag0.5Cu) was presented by Ma et al. [65], where isothermal aging was conducted for varying durations (3 days, 6 days, 9 days, 21 days, 42 days, and 63 days) at room temperature (RT). For both the above alloys, RT aging affects both the strain rate in the secondary creep region and the elapsed time before tertiary creep and rupture. For the highest stress level considered, the creep strain rate increased by factors of 16× for Sn4.0Ag0.5Cu and 73× for Sn3.0Ag0.5Cu during the 63 days of RT aging. The aging effects were similar for the two lead-free alloys. However, the data clearly showed that Sn4.0Ag0.5Cu creeps less than Sn3.0Ag0.5Cu for the same stress levels and aging conditions. In another study by the previous group [66], they also studied the effect of elevated temperature aging (80, 100, 125, and 150 °C) on the creep resistance of Sn3.0Ag0.5Cu and Sn4.0Ag0.5Cu lead-free solders. The samples were aged for various durations (0–6 months). Elevated temperature aging for longer durations was observed to drastically reduce the creep resistance of the above alloys. The decrease in creep resistance was hypothesized to be due to significant coarsening of second phase particles at high temperature, thus reducing the dispersion strengthening from the second-phase particles. The same research group also extended their previous studies by including two more low silver content alloys (Sn2.0Ag0.5Cu and Sn1.0Ag0.5Cu) and increased the aging duration from 6 to 12 months [67]. As expected, the creep rates evolved more dramatically when the aging temperature was increased and have detrimental effect on thermomechanical fatigue life of solder joint [68]. It was also observed that lowering of the silver content in SAC alloy leads to increase in the creep rates for all the aging conditions. A revised Anand viscoplastic model was proposed by Motalab et al. [55], which includes material parameters which evolve with thermal aging of the solder material. The results show that two of the nine model constants remain essentially constant during aging, while the other seven model constants show large changes (30–70%) with up to 6 months of aging.

Similar observations have been also reported by Xiao et al. [69] for Sn3.9Ag0.6Cu and Sn37Pb alloys which were aged at 180 °C. Additionally research conducted by Chauhan [70] provides the relation between isothermal aging and the thermal cycling reliability of select Sn-based solders. The Sn-based solders with 3%, 1%, and 0% silver content that have replaced tin–lead are studied and compared against tin–lead solder. The activation energy and growth exponents of the Arrhenius model for the intermetallic growth in the solders are provided. An aging metric to quantify the aging of solder joints, in terms of phase size in the solder bulk and interfacial IMC thickness at the solder–pad interface, is established in the study. Microstructural coarsening occurs throughout the life, driven by plastic deformation, and does affect fatigue damage initiation. Microstructural coarsening increases with increasing strain rate in a wide variety of two-phase alloys [71]. Several theoretical models have been proposed to model the kinetics of the grain growth. The coarsening theory of Senkov and Myshlyaev [72] has been successfully applied by several researchers, e.g., Ref. [73]. In particular, the solder microstructure has been experimentally observed to coarsen in accordance with the cubic coarsening model by several researchers (e.g., Refs. [7375]). All the above studies have been purely experimental and do not provide much insight into the physics of microstructural evolution occurring during isothermal aging at either RT or high temperature.

###### Limitations of Phenomenological/Empirical Constitutive Models.

The significant scatter in creep constitutive properties of Pb-free solder joints discussed in Sec. 2.1 has a strong dependence on the size of the test specimens (polycrystalline bulk versus multicrystalline joint size), specimen fabrication process, reflow profile for the solder, cooling rate during solidification, and different testing methods. However, even when the above factors are accurately controlled by a single investigator, significant piece-to-piece scatter in constitutive properties is evident when testing small test specimens that are of the same length scale as functional solder joints [76]. This scale effect is attributed to the fact that there are only a few large highly anisotropic Sn grains in such joints [76], thus making every joint unique in terms of microstructure and the resulting mechanical response.

Both the elastic modulus and the coefficient of thermal expansion (CTE) of Sn single crystal vary considerably as a function of crystal orientation [77]. The CTE of Sn increases by a factor of 2–3 between [100/010] and [001] directions and Young's modulus varies by a factor of ∼3 on the (001) plane (between [100] and [110] directions). Sn has a body-centered tetragonal (BCT) structure with additional atoms located at (0.5, 0, 0.25) and (0, 0.5, 0.75) coordinates. These extra atoms form a tetrahedral bonding arrangement with the body-center atom. Sn behaves stiffest in the direction in which it expands the most, leading to substantial internal stresses in polycrystals that cause heterogeneous deformation near grain boundaries during thermal cycling. Thus, deformation behavior of lead-free solder joints in electronic circuits is very sensitive to microstructure and crystal orientations of Sn grains, in contrast to Sn–Pb solder joint, where the softer Pb islands mostly accommodate isotropic Sn deformations.

Lee et al. [78,79] also demonstrated that during a thermal cycle typical of electronic systems, stresses well above the yield stress could result in two adjacent, misoriented crystals. Matin et al. [80] recently published on anisotropic analysis of polycrystalline tin deformation during thermal cycling. Damage was identified in grain-boundary regions that correlated with large stress states resulting from CTE mismatch arising from the intrinsic elastic anisotropy of tin. Park et al. [81] cross sectioned and characterized eutectic SAC solder balls having different microstructures using cross-polarized light microscopy and used differential image correlation technique to measure localized von Mises strain resulting from thermal cycling. Dramatic strain gradients were observed at the intersection of grain boundaries of Sn grains. Such spatial variation of von Mises strain was not evident in a single grained SAC joint studied by them. Therefore, continuum phenomenological model with average model constants is not sufficient to explain the variability in the creep constitutive properties of SAC solder joints. A brief summary (see Table 4) is presented below for the above experimental/modeling efforts in Secs. 2.3 and 2.4 and their respective limitations.

It is recommended that physical models based on underlying microstructure and physical deformation mechanisms be used to aid in the development of new phenomenological models that better address these issues and to provide guidance in the optimum selection of material parameters for these models.

## Mechanistic/Physical Constitutive Creep Models

In the case of dispersion-hardened materials such as SAC solder, the intermetallic particles serve as obstacles to the path of dislocation glide. The interaction between the particle and dislocation can be either attractive or repulsive. The dislocations upon reaching the particle have two options: either change its slip plane by cross-slip or climb around the particle and continue movement along the same path. When the dislocation cuts through the particle, it is termed as “cutting mechanism.” In some cases, the dislocation line wraps around the particle and moves ahead leaving behind loops called Orowan loops. These alternate mechanisms largely depend on the line tension of the dislocations and the strength of the particle and the stress level and temperature. If temperature is high enough, then climb dominates even if the stresses are lower than the Orowan stresses. Assuming that the particles are strong enough to resist cutting by the dislocations, then dislocation loops are left behind by the passing dislocation. The effective creep rate is dictated by the difference between the applied stress and the back stress formed from the Orowan loops. In the limit, the two may reach comparable magnitudes, thus acting as a creep arresting mechanism. However, this type of creep hardening occurs only at low temperatures. At high temperatures, these Orowan loops attain sufficient energy to climb to the poles of the obstructing particles and self-annihilate, thereby allowing deformation again. For a steady-state of creep deformation, it is logical to expect that the formation rate of loops should equal the annihilation rate of the loops. Wiese and Wolter [82] reported from experimental testing that the creep deformation in eutectic SAC solder is dominated by dislocation climb. Chen and Dutta [64] found (for temperatures between 298 K and 398 K) that dominant creep mechanism was dislocation climb aided by lattice diffusion at lower stresses while at higher stresses combined dislocation glide and climb occurred. Similar observations were made by Dutta et al. [83], Kerr and Chawla [84], Mathew et al. [85], and Ochoa et. al. [86], for Sn3.5Ag. Furthermore, Ochoa et al. [86] found that grain-boundary sliding was found to have a significant contribution to total creep strain in Sn3.5Ag bulk solder specimen. Mathew et al. [85] found that the creep mechanism at temperatures in the range 0.58 < Th < 0.8 was dominated by dislocation climb.

Furthermore, transmission electron microscopy studies conducted by Arzt and Göhring [87] on other dispersion strengthened materials such as aluminum and titanium alloys suggest that dislocation climb occurs in multiple stages: approach, climb, and detachment. A strong attractive interaction between dislocation and the obstacle is observed in postclimb phase of the dislocation, and the detachment of the departing dislocation rather than climb is seen to be the rate-limiting mechanisms for deformation [88]. The above mechanism was also evident in other studies by Kerr and Chawla [84] and Chen and Dutta [64] that captured the contributions of the eutectic Sn–Ag phase and nanometer dimension IMC length scales. Hence, dislocation climb and detachment are found to be the primary creep mechanisms in SAC solder under certain stress and temperature conditions. The high secondary creep stress exponents of SAC solder reported in the literature [76] support the hypothesis that dislocation climb and detachment over obstacles are the dominant rate-governing creep mechanism in SAC solder. Therefore, to capture the above phenomena, state-of-the-art mechanistic models and its limitations will be discussed in Sec. 3.1. In Sec. 3.2, new techniques to model the effect of anisotropy of Sn on mechanistic creep response of SAC solder will be discussed and then finally in Sec. 3.3, new techniques to model the effect of grain morphology in SAC solder on its mechanistic creep response will be discussed. Inherent limitations of these mechanistic models along with the possible future work directions will be discussed in Secs. 3.2 and 3.3, respectively.

###### State-of-the-Art Mechanistic Constitutive Models for Creep.

The mechanistic or physical models used to capture the creep constitutive response of SAC solder available in literature are discussed here. The microstructural length scales in SAC solder joints vary from nanometer-sized Ag3Sn intermetallics in Sn–Ag eutectic region to the structural dimensions of Sn grains (millimeter-sized) that contribute to heterogeneous behavior of microscale SAC joints. In the case of microscale SAC solder joints, the grain configurations have an additional influence on the constitutive response. The microstructure of Sn dendrites and IMCs is dependent on various parameters like manufacturing profile, loading history, and weight fraction of Ag and Cu in SAC alloys [88]. Therefore, comprehensive understanding of the influence of each of these microstructural features through purely experimental parametric studies is both costly and time consuming. Moreover, such experimental understanding of the constitutive response based on empirical models cannot be extrapolated to other test conditions or other material systems. In order to obtain insights into the physics of deformation induced by each of these microstructural features, mechanistic multiscale modeling framework, that captures the dominant creep mechanisms in terms of key microstructural features, is needed.

Dutta et al. [83] suggested that the creep response of Sn–Ag solders is controlled by the behavior of the eutectic structure which consists of Ag3Sn IMC particles embedded in a Sn matrix. The total creep rate of Sn–3.5Ag solder was written as the sum of two simultaneous mechanisms: glide–climb mechanism (where either viscous glide or climb is the rate-controlling mechanism) and particle-limited climb mechanism. The authors proposed that glide–climb mechanism dominates the observed creep behavior at low stresses, whereas the particle-limited climb mechanism dominates at high stresses. Gong et al. [89] utilized above microscale dislocation climb models to capture the creep response of Sn–3.5Ag solder by considering the dispersion strengthening from nanoscale Ag3Sn IMCs. However, the model assumes that the behavior is dictated only by the dispersion strengthening of nanoscale Ag3Sn IMCs in the Sn–Ag eutectic. Only the eutectic Sn–Ag region is modeled, without any load sharing between Sn dendrites and Sn–Ag eutectic. In another study, Chawla and Sidhu [90] reviewed the literature on analytical and numerical techniques available to model the heterogeneous microstructure of multiphase solder materials. Models presented in their review in 2007 mostly simplify the heterogeneous microstructure of multiphase solder materials. They addressed the critical link between microstructure and deformation behavior, by using two-dimensional (2D) and three-dimensional (3D) virtual microstructures as the basis for a robust model to simulate damage caused by deformation. Their model was unable to capture all the dominant physical mechanisms during the deformation process because homogenized properties were considered for different phases in the solder during finite element analysis. These simplifications definitely make modeling and analysis more efficient and straightforward, but fail to accurately predict the effective properties and local damage behavior which are dependent on the detailed microstructure of SAC solder. Another interesting study by Pei and Qu [91] evaluated the effective creep properties of Pb-free SnAg solders using homogenization methods based on the measured properties of individual phases (eutectic Sn–Ag phase and Sn dendritic phase). However, their technique does not utilize mechanistic equations to capture individual IMC phase properties and thus lacks the predictive capability for extrapolating the creep response to other microstructural states of SAC solder. Studies involving mechanistic modeling to capture the physics of the underlying viscoplastic creep mechanisms of SAC solder are very limited in the literature. Furthermore, studies that provide the influence of the Sn dendrites, micron-scale IMCs, and the eutectic Sn–Ag region (lower length scales) on the creep behavior are required. Table 5 provides a brief summary of some of the mechanistic models reviewed in this paper.

To address the above issues, Cuddalorepatta and Dasgupta [88] proposed an isotropic mechanistic secondary creep model based on Rosler and Artz's dislocation detachment model [9598] for SAC solders that captured the two lowest length scales in SAC solders: the smallest length scale modeled was that of nanoscale IMCs via analytical dislocation creep models, and the next length scale was that of the micron-scale Sn dendritic colonies, via isotropic micromechanics homogenization theory. Since secondary creep measurements show much less sensitivity than primary creep, to the coarse-grained anisotropic Sn microstructure in the Sn3.0Ag0.5Cu specimens [76], geometric modeling of the Sn grains and grain boundaries was not necessary. Furthermore, isotropic modeling was considered to be adequate when developing analytical formulation of the secondary creep deformation at the two lowest length scales. The response of the eutectic Sn–Ag region is evaluated by assuming Sn as the matrix and IMCs of nanometer dimension Ag3Sn as inclusions causing dispersion hardening by obstructing dislocations. At the second level, the effective creep properties of SAC composite are evaluated by considering the load sharing between the eutectic Sn–Ag region embedded with micron-scale Cu6Sn5 IMCs as the matrix and the micron-scale softer pro-eutectic pure Sn dendrites as the inclusions. The model accounted for motion of a single dislocation front and inherently assumed that an attractive interaction exists between the dislocation and the nanometer dimension Ag3Sn IMC dispersoids. The eutectic Sn–Ag shear creep rate was taken to be due to a combination of the Sn matrix shear creep rate and the Ag3Sn IMC dispersion-hardening shear creep rate. Effect of Sn dendritic colonies was taken into account by using self-consistent homogenization schemes from micromechanics theories used in composite materials. The Sn dendrite lobes were assumed to be spherical-shaped inclusions (with effective isotropic properties) and were hence accounted for only through their volume fractions in Eshelby's tensor.

Although the model proposed by Cuddalorepatta and Dasgupta [88] provides theoretical insights into the effects of key microstructural features, e.g., effect of micro-alloying on creep response [92] and effect of aging on creep response [93], it is still an isotropic model and cannot capture the strong anisotropy seen in transient creep behavior of SAC single crystals. This becomes important in primary creep since experiments reveal that the single-crystal anisotropy and the grain structure have very strong influences on primary creep behavior [94]. Furthermore, anisotropy introduced by the nonspherical shape of dendritic lobes also becomes important. Hence, studies that address the impact of the experimentally observed nonspherical Sn dendritic configurations on the measured creep response are necessary. The sensitivity of the model to the nanoscale IMCs is extremely high. In its present form, the above model only provides a “dilute approximation” since it does not explicitly account for the interactions between neighboring dislocation fronts at high dislocation densities [99]. Furthermore, the dislocation density used in Cuddalorepatta's study is based on a hypothetical assumption that all the slip planes in BCT Sn are completely saturated with dislocations when the creep deformation advances to the secondary creep regime. The dislocation density in all the dominant slip systems of Sn is thus considered to be extremely high (∼1 × 1021 m−2). This uniform saturation of all the dominant slip systems is the mechanistic underpinning in Cuddalorepatta's study for justifying the use of the isotropic model and therefore cannot be used to model the experimentally observed anisotropic primary and secondary creep behavior of solder alloys. Research on the microstructure of SAC solder [100,101], its evolution [102,103], and mechanical properties [104110] has been extensively conducted, but most of the studies do not provide any insights into a detailed understanding of the heterogeneous behavior of this anisotropic material, since most of the studies consider the solder joint to be isotropic.

###### Modeling the Effect of Sn Anisotropy on Creep Response of Sn-Based Solder.

Erinç et al. [111] simulated fatigue damage in an SAC solder joint taking into account anisotropic elastic properties of Sn grains. It has been reported in the literature that the anisotropic character of the β-Sn crystal structure (BCT) can lead to stress concentrations at Sn grain boundaries during thermal cycling [78,80]. Furthermore, localized deformation has been shown to occur at the grain boundaries of Sn-rich solders upon thermal loading [112]. Thermal fatigue in Sn-rich alloys has also been reported to be influenced by the anisotropic crystallinity of the microstructure [113]. Recently, experimental evidence has been presented that these stress concentrations can actually lead to damage, purely due to the influence of thermal cycling without any external mechanical constraints [80]. Chen and Li [114] showed, using an anisotropic elastic finite-element model of a multicrystal system, that stress concentrations arise at certain grain boundaries and triple junctions. Matin et al. [80] showed in a comparison between experimentally obtained data and a simulation for a thermally cycled SAC specimen, using an elastic analysis including anisotropic elastic and thermal expansion properties, that the locations of predicted stress concentrations correlated reasonably with damaged areas found in the experimental observations.

As evident from the literature, most of the modeling efforts have taken into account only the anisotropic elasticity because the viscoplastic anisotropic properties of individual phases have not been modeled or characterized before. Therefore, modeling the effect of Sn anisotropy and grain morphology to explain the variability observed in creep response of SAC305 solder of varying grain structures is necessary.

In order to understand the effect of anisotropy of Sn and grain orientation on the variability of creep response of these SAC solder interconnects, a multiscale modeling approach has been proposed by Mukherjee et al. [94], where the microstructure can be classified into five distinct length scales (refer to Fig. 3). Here, anisotropic models of dislocation interactions with particles that depend on spacing/size are appropriate for describing the dispersion strengthening. Tier 0 refers to dislocation slip systems needed for modeling in tier 1. Predicted creep rates along dominant slip systems of single crystal eutectic Sn–Ag phase in tier 1 length scale has been used in conjunction with those for pure Sn dendrites to predict the anisotropic creep response of SAC single crystal (tier 3) using anisotropic micromechanics homogenization techniques (tier 2).

Line tension and mobility of dislocations (tier 0) in dominant slip systems of single crystal Sn are captured using elastic crystal anisotropy of BCT Sn and Stroh's matrix formalism [115]. The anisotropic creep rate of the eutectic Sn–Ag phase of tier I is then modeled using above inputs, and the evolving dislocation density is estimated for dominant glide systems. The evolving dislocation density history is estimated by modeling the equilibrium between five competing processes: (1) dislocation generation; (2) dislocation impediment due to forest dislocations; (3) dislocation recovery (by climb and diffusion) from forest dislocations; (4) dislocation impediment caused by back stress from pinning of dislocations at IMC particles; and (5) dislocation recovery due to climb/detachment from Ag3Sn IMC particles. Of these mechanisms, the third and fifth mechanisms are modeled to be the rate-governing mechanisms for anisotropic creep deformation processes for the Sn matrix and the eutectic Sn–Ag alloy, respectively. Orowan climb mechanism and dislocation detachment mechanism from nanoscale Ag3Sn particles are found to be the rate-controlling creep mechanisms in the eutectic Sn–Ag phase of SAC single crystal depending on applied stress level (refer to Fig. 4). Of these two mechanisms, the one that contributes to faster viscoplastic flow of dislocations in the material is the one that governs the effective creep rates.

The creep response of the eutectic phase (from tier 1) is combined with creep of ellipsoidal Sn lobes at tier 2 using the anisotropic Mori–Tanaka homogenization theory [116120], to obtain the creep response of SAC305 single crystal along global specimen direction and is calibrated to experimentally obtained creep response of the same SAC305 single crystal specimen. The Eshelby strain concentration tensors required for this homogenization process are calculated numerically for ellipsoidal Sn inclusions embedded in anisotropic eutectic Sn–Ag matrix. The orientations of SAC single crystal specimens with respect to loading direction are identified using orientation image mapping (OIM) using electron backscatter detector and then utilized in the model to estimate resolved shear stress along dominant slip directions.

The proposed model is then used for investigating the variability in transient and secondary creep response of Sn3.0Ag0.5Cu (SAC305) solder, based on grain orientation. Transient creep strain rates along [001] direction of SAC305 single crystal #1 is predicted to be 1–2 orders of magnitude higher than those along [100]/[010] direction. The proposed model is able to quantitatively capture the creep response of two SAC305 single crystals and two bicrystal specimens reasonably well. Parametric studies have also been conducted to predict the effect of changing orientation, aspect ratio, and volume fraction of Sn inclusions on the anisotropic creep response of SAC single crystal. The model is also able to capture the decrease in creep resistance of SAC305 single crystal, when the volume fraction of Sn inclusion in the SAC single crystal increases. Transient creep strain along the most facile slip system (110)[001] is predicted to increase by (1–2) orders of magnitude as the interparticle spacing between nanoscale IMCs increases by a factor of three. Similarly, increase in the volume fraction of nanoscale Ag3Sn IMCs is predicted to decrease the transient creep rates along (110)[001], which is in agreement with experimental observations.

Although the proposed mechanistic model is able to provide significant insights into the effect of Sn anisotropy on the creep response of SAC single crystal, the model still has certain simplifying assumptions and needs further refinement in future. Contribution of cross-slip phenomena in screw dislocations to transient creep response has been ignored in this study. Molecular dynamics simulations need to be carried out in future to estimate the activation energy for cross-slip so that it can be incorporated into the dislocation density evolution and dislocation recovery models. The analytical dislocation density evolution model proposed in this study [94] should be calibrated with experimental creep data of eutectic Sn–Ag phase in single crystal SAC solder for better resolution of calibration parameters. Since, fabricating single crystal solder joints is difficult, the analytical model can also be calibrated to creep response from discrete dislocation dynamics simulation of eutectic Sn–Ag phase in single crystal Sn. In dislocation climb and detachment model, the dislocations are assumed to be randomly distributed in the crystal, so that their stress fields upon the dislocation overcoming the dispersoids are neglected. However, in most practical applications, resulting spatial distribution of lattice dislocations is not random and the stress fields from these pile-ups cannot anymore be assumed to cancel at all the points within the material. These stress fields will affect the lead pile-up dislocation pinned by the dispersoid and either aid or hinder the bypassing processes (Orowan climb or detachment process) by which the controlling dislocation overcomes its obstacle and thus will increase or decrease the value of the threshold stress. The effect of pile-up stress on the effective stress for Orowan Climb and detachment stress for Rosler's model needs to be modeled in future. Interactions between neighboring Sn dendrites have not been explicitly modeled in this study [94]. As a result, this study is appropriate for dilute concentration of dendrites only and loses accuracy as the dendrite concentration increases.

###### Modeling the Effect of Grain Morphology on Creep Response of Sn-Based Solder.

Matin et al. [80] reported that anisotropy in thermal expansion and elastic properties of Sn induces significant stresses at Sn-grain boundaries during thermal cycling. Fatigue damage was shown to occur in a mechanically unconstrained Sn-rich solder under thermal cycling between 293 K and 353 K. The microcracks were found to be localized mainly along high grain boundaries. A combination of experiments using orientation imaging microscopy and numerical methods was used to find a strong correlation between large stresses caused at grain boundaries and thermomechanical anisotropy of Sn. Zhao et al. [121] also studied the effect of anisotropy of Sn elastic properties and thermal expansion on Sn whisker growth during temperature cycling (−55  °C to 85  °C), using computational modeling. The grain structure was explicitly captured in the model by using the Voronoi cell method. Sn elasticity of each grain was modeled using the elastic stiffness matrix of Sn single crystal. The crystal orientations were assigned to the grains according to the information collected by X-ray diffraction. The stress along with the strain energy density distributions in grains was calculated in Ref. [122] using finite-element methods, for different grain structures of Sn microstructure. The effect of Sn plasticity was explored using a simple bilinear isotropic hardening stress–strain relation because anisotropic plastic properties were not available during the study. Creep response was not addressed in this study. The strain energy density criterion was selected to predict Sn whisker growth driving force. Strain energy density was found to be highest in the Sn grains with high angle grain boundaries due to significant anisotropy in elastic and CTE properties of Sn grains.

In another study by Subramanian [123], the CTE mismatch at the interface of Sn grains was found to be significantly higher compared to bulk solder–IMC interface or IMC–Cu pad interface, due to inherent anisotropy of the Sn phase. Significant stresses that can develop from thermal excursions in Sn-based solders can cause extensive grain-boundary sliding and grain-boundary decohesion. The coalescence of such cracks in the highly constrained region of the solder near the solder/IMC interface can develop into catastrophic cracks that can degrade the mechanical integrity of the solder joint. Telang and Bieler [124] characterized the microstructure and crystal orientation of Sn phase in single shear lap Sn3.5Ag solder joint specimens. They found that solder joints are typically made up of at most a few dominant grain orientations, having low angle boundaries, implying a single or multicrystalline rather than a polycrystalline texture. The OIM maps from two opposite sides of a joint showed that the same crystal orientation is highly probable throughout the joint, implying that surface OIM scans are probably sufficiently representative of the microstructure in the interior of the joint. The dominant orientations obtained after solidification from several specimens indicated some preferred orientations that may be a consequence of rapid crystal growth in [110] directions that are aligned with the heat flow direction. They hypothesized that lack of orientations with the [001] c-axis in the plane of the joint might have resulted from the large difference in thermal expansion between tin in the [001] direction and the copper substrate. A 3D elastoviscoplastic damage model has been used to investigate the possible influence of the mechanical and thermal anisotropy of Sn on fatigue damage by Ubachs et al. [125]. They modeled the elastic behavior and the thermal expansion coefficients as anisotropic, whereas the viscoplastic and damage behavior were considered to be isotropic. The simulations showed stress and viscoplastic strain concentrations at grain boundaries, dependent on the misorientation between the grains and their orientation with respect to the grain boundary. A consistent agreement was found between the numerically obtained strain distribution field and the experimentally observed damaged areas. The authors concluded that anisotropy indeed plays an important role in determining the thermal fatigue life of SAC solders.

In another study by Park et al. [126], a multigrain solder interconnect was modeled to study the effect of anisotropy on thermal cycling reliability of the joint. They used a global-local approach where the global model addressed the entire assembly of the component, substrate, and interconnects; while the local model addressed an individual interconnect in much greater detail. Solder material properties used in the assembly level global model were considered to be homogenous and isotropic. Anisotropic elastic material properties of tin were used in the numerical local model of the single multigrained interconnect. Stress concentrations were observed at the grain boundaries, depending on the orientation of the connected grains, their interfaces, and the orientation mismatches between the adjoining grains. Comparison of the numerical results with the experimental observations indicated a reasonable agreement with the strain distribution. The results showed higher strain localization not only along the pad–solder interfaces but also along the grain boundaries. Those were the primary areas of energy consumption during fatigue loading.

In another recent study by Zamiri et al. [127], crystal plasticity finite-element (CPFE) analysis was used to evaluate stress and strain resulting from a 165 °C temperature change in a single-crystal joint using two simplified geometries appropriate for realistic solder joints. Phenomenological flow models for ten slip systems were estimated based upon semiquantitative information available in the literature, along with known anisotropic elastic property information. The results showed that the internal energy of the system is a strong function of the tin crystal orientation and geometry of the solder joint. The internal energy (and presumably the likelihood of damage) is highest when the crystal c-axis lies in the plane of the substrate, leading to significant plastic deformation. When the a-axis is in the plane of the interface, deformation due to a 165  °C temperature change is predominantly elastic.

As evident from the above studies, majority of the studies have focused on anisotropy present in elastic properties and CTE of Sn grains in their approach while modeling the thermal cycling reliability of SAC solder interconnects. None of the studies considered the anisotropic creep properties, which is one of the most important properties, since these solder materials demonstrate significant amounts of creep deformation even at RT. The main reason that previous authors have not been able to use anisotropic creep properties of SAC solder interconnects in their studies is because creep properties of single crystal SAC solders are not available in literature. Also, the microstructural length scales in the above studies are limited to Sn grains and contributions from smaller features such as Sn dendrites and bulk IMCs are not explicitly captured in their creep constitutive models of SAC solder.

The model proposed by Mukherjee et al. [94] can be used to include the effect of anisotropic creep behavior to model the interaction between the individual Sn grains. Their study predicted that transient shear creep strain along global specimen direction may vary by 1–3 orders of magnitude due to change in one of the Euler angles (φ1) in one of the SAC305 single crystal specimens, which is in agreement with the experimental observations of the authors. The proposed model is then further used to predict the steady-state creep rates of few selected (single crystal and bicrystal) SAC305 specimens. Very good agreement was observed between the predicted steady-state creep rate of two SAC305 single crystal specimens and measured creep rates of the same specimens. For the bicrystal specimens, experimentally measured steady-state creep rates are found to fall between the predicted creep rates for individual crystals in the joint; pointing toward possible grain-boundary sliding during the creep deformation process of multicrystal specimens. The predicted steady-state creep rate in one of the SAC305 single crystal specimen is found to vary by almost one order of magnitude due to systematic variation of the orientation of Sn dendrites with respect to the loading direction. Contribution of grain-boundary sliding to total creep strain has not been modeled in this study. The scope of this work [94] is limited to the anisotropic primary and secondary creep modeling of SAC single crystals. To extend this study in future to predict the anisotropic creep response of coarse-grained SAC solder joint, the model will need to include the mechanics of grain-boundary sliding. The outputs of this study [94] can be used as inputs for CPFE modeling in future to capture the thermomechanical cyclic fatigue response of SAC solder joints using anisotropic elastic, plastic, and creep properties of single crystal SAC solder.

## Summary and Conclusions

Phenomenological and mechanistic models used in the literature for predicting creep response of both bulk and grain scale solder joints are reviewed in this paper. Empirically calibrated phenomenological models are compact, easy to use, and necessary for such applications as finite-element modeling. The phenomenological approach is very convenient for electronic packaging analysts and designers to predict the expected deformation and durability of soldered assemblies. Phenomenological approaches characterize the mechanical behavior of lead-free solder joints using homogeneous constitutive models containing empirically determined coefficients and exponents. However, it is clearly difficult to make the phenomenological models sufficiently detailed to include all the important microstructural and loading parameters in a complex material system like solder. It is also a challenge to run a sufficient number of tests to find all the model constants with sufficient accuracy, across a sufficiently wide range of stress conditions and microstructural variations.

On the other hand, the mechanistic models are intrinsically more capable of addressing all the fundamental drivers of creep deformation. Mechanistic approaches model all of the interactions taking place in the solder joint that contribute to creep deformation, including individual grain structure, sliding grain boundaries, and the propagation of dislocations around, over, or through IMC particles of various compositions. Such modeling approaches provide a rich understanding of the creep behavior of complex material systems like solders, but are complex and usually do not provide a single closed-form equation at the end that can be included in such tools like finite-element material libraries. Industry is moving toward a hybrid approach where the mechanistic studies are used to gain fundamental understanding of the underlying deformation mechanisms and to gain quantitative insights into the effect of various driving forces and various microstructural features on the overall creep response of new solders. This insight is invaluable to the developers of phenomenological models because it provides the information necessary to create more realistic compact material models that are easy to use by designers and can be applied at the length-scale of the solder joint. Mechanistic modeling approaches can also serve as virtual testing tools to supplement experimental studies, thus reducing the total experimental time needed for characterizing the creep response of new solders and obtaining the model constants for improved phenomenological models.

## Acknowledgements

This work was sponsored by the members of the CALCE Electronic Products and Systems Consortium at the University of Maryland, College Park, MD.

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## Figures

Fig. 1

Typical creep curve for a viscoplastic material

Fig. 2

Discrepancies in creep data for models used by several authors [61]

Fig. 3

Multiple length scales in SAC solder

Fig. 4

Recovery of dislocations from nanoscale Ag3Sn dispersoids by (a) Orowan climb mechanism and (b) dislocation detachment mechanism

## Tables

Table 1 General classification of primary and secondary creep models with dependencies on strain–time, stress, temperature, and state variables
Table 2 Creep constitutive models commonly used for solder materials
Table 4 Summary of limitations of phenomenological studies
Table 5 Mechanistic models for SAC solder alloys
Table 3 Secondary creep constitutive constants for different solder alloys [61]

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