Prediction of Two-Phase Creep Behavior from Constituent Phase Behavior in the Bi-Sn System
Chris H. Raeder, David Mitlin, and R.W. Messler, Jr.
Center for Integrated Electronics and Electronics Manufacturing and Department of Materials Engineering
Rensselaer Polytechnic Institute, Troy, NY 12180-3590
Constitutive equations derived from the creep behavior of 99.999wt.%Bi and Sn-xwt.%Bi single phase solid solutions are applied in the continuum mechanics creep model of Tanaka et al. to successfully predict the previously anomalous creep behavior of eutectic Bi-42wt.%Sn alloy. Steady-state rate, normal transient, and positive and negative recovery behaviors are predicted by the model. Experimental results differ from model predictions in that age-coarsened Bi-42wt.%Sn creeps more rapidly than unaged alloy. This is explained in terms of the increasing amount of Sn Sn and Bi-Sn boundary sliding absent in the unaged material.
The eutectic Bi-42wt.%Sn alloy (Tm = 138deg.C) is attractive for soldering the growing number of temperature-sensitive electronic assemblies, for use in step soldering, and from an environmental standpoint to replace Pb-containing alloys. An accurate phenomenological description of the time-dependent deformation behavior of this key Bi-Sn solder alloy is the first step to understanding and predicting the thermal fatigue process assumed to be life limiting in most electronics packages. The Bi-42wt.%Sn eutectic microstructure is composed of nearly equal volume fractions of Bi, which has minimal solubility of Sn, and an Sn-xBi saturated solid solution, where x varies between 2 wt.% at room temperature and 19 wt.% at the eutectic temperature of 138deg.C. Previous studies [2,3] have demonstrated the existence of three regions of different stress dependence in the steady state creep rate data at stresses and temperatures relevant to the expected service environment. To the authors' knowledge, a satisfactory explanation of this behavior has not yet been reported. The present study focuses on determining the sources of the three regions of stress dependence by examining the creep behaviors of the individual Sn and Bi phases.
The deformation of two-phase composites and metal alloys is described by numerous authors [1, 4-10]. Approaches vary widely from the use of the mechanical law of mixtures [4-7], to finite element methods , to continuum mechanics models [1,9,10]. All methods have enjoyed some success. From those approaches listed above, the model of Tanaka, Sakaki and IIzuka  was chosen to describe the behavior of the Bi-Sn eutectic because it phenomenologically predicts transient, steady-state, and recovery creep behaviors of ductile two-phase alloys from simple, steady-state constitutive creep equations derived from the bulk creep properties of the individual phases.
The model  assumes both phases are elastically and plastically isotropic, i.e., , and that creep strains are uniform in the individual phases. Internal stresses exist due to the strain difference, , between the phases arising during the early stages of creep. The components of internal stress averaged over the matrix and second-phase are calculated by Eshelby's equivalent inclusion method [11,12l and Mori-Tanaka's average internal stress concept . Following simplification, the triaxial stresses acting on each phase are converted with the von Mises relation to equivalent stresses, given by:
(1) (2) where is the applied stress, f is the volume fraction, K is a shape factor, E is the elastic modulus of the matrix phase, and x is the strain difference, . The equivalent stresses are applied in the creep laws for the individual phases. The calculated creep rates are then multiplied by the time increment (in the numerical simulation) to get the creep strains of the individual phases. The creep strain of the alloy is, therefore, given by:
The phase strain difference is summed with the previous difference, the stress distribution is modified, creep rates are again calculated, and so on in an iterative fashion to construct the complete creep response.
The present study details the experimentally measured steady-state creep behavior of 99.999wt.%Bi and Sn-l5wt.%Bi, and the transient and steady-state creep properties of the Bi-Sn eutectic. Constitutive equation creep constants derived from the Bi and Sn-15Bi steady state data are used in the continuum mechanics composite creep model (CMCC) to describe the Bi-Sn eutectic steady-state creep rates over a wide range of stress and temperature as observed in an earlier study . The role of the shape factor and microstructural variability on the predicted and observed creep behaviors are discussed as well as agreement, disagreement, and novel deformation behaviors predicted by the model.
Essentially pure 99.99wt.%Sn and 99.999wt.%Bi were used to produce all alloys. Sample preparation and test procedures are described elsewhere . Creep tests on Sn-lSwt.%Bi were performed at 1 20deg.C, a temperature at which there is complete solid solubility of Bi in Sn, ensuring a single solid solution phase. Creep tests were conducted at constant applied load at low strain rates (<10-4 /sec), and at a constant displacement rate at higher strain rates. A single sample was used to collect data at various stresses at a single temperature. These tests were run in order of increasing load, so as to avoid the effects of a stable high stress microstructure on low stress tests. For constant displacement rate tests, a single sample was used for each test. The strain rate reported corresponds to the strain rate at which the maximum stress was achieved.
To determine the effects of Bi-Sn eutectic morphology on the transient and steady-state creep behavior, the creep behavior of as-quenched samples was compared to that of samples quenched and then aged at 130deg.C for 22 days. In these tests, a separate sample was used for each test as not only the steady-state behavior was recorded but also the transient behavior was analyzed for comparison with model predictions. Also, on some samples, a facet was polished into the gage section prior to testing to later observe in the SEM the effects of creep on the surface structure of the sample. These surface observations are then linked to the deformation characteristics of the bulk material.
Figure 1 shows the steady-state creep data of the 99.999wt.%Bi. These data show power law behavior and were fit with an equation of the form:
where Q is the activation energy = 58 kJ/mol, is the applied stress, n is the stress exponent = 8.2, E is Young's modulus = 32.6xl03 - 46.8T MPa (where T is temperature in deg.C; the temperature dependence was determined in the course of the present study), and C is the creep constant = 103deg. J/mol-MPa-sec.
The Sn-l5wt.%Bi alloy was tested at 120deg.C only. Figure 2 compares the steady-state data obtained for the Sn-l5wt.%Bi alloy to that of the pure Bi and Bi-42wt.%Sn eutectic [2,3], all at 120deg.C. At stresses greater than 3 MPa, Sn-l5wt.%Bi has a lower strain rate than pure Bi. Below 3 MPa, pure Bi has the lower strain rate. To a first approximation, it is assumed that Sn-l5wt.%Bi would be the load bearing phase at stresses above 3 MPa and nearly pure Bi would be load bearing below 3 MPa in a composite of these two phases. Figure 2 shows that the steady-state creep behavior of the Bi-Sn eutectic in fact closely parallels the behavior of the individual pure Bi and Sn-xBi phases. The Bi-Sn eutectic creeps faster than the constituent phases because in the eutectic the equivalent stress in the load bearing phase is much higher than the applied stress.
To apply the continuum mechanics composite creep (CMCC) model over the four test temperatures of [2,3], the creep activation energy of the Sn-xBi phase must be known. Determination of the creep activation energy of the Sn-rich phase in the Bi-Sn eutectic is complicated because, in the Bi-Sn eutectic, the composition of the Sn-rich phase is strongly temperature dependent. For the present purposes, it was assumed that the creep rate of the Sn-rich phase is limited by the rate of diffusion of the Bi solute in the Sn matrix, as is expected in concentrated solid solutions. Coarsening studies of the Bi-Sn eutectic  give a tracer diffusion activation energy of 73 kJ/mol for Bi in Sn. Figure 2 shows an exponential stress dependence in the Sn-15wt.%Bi alloy so the Sn-xBi data were fit using a sinh stress dependence as follows:
where the Young's modulus of the Sn-xBi phase as a function of temperature,
deg.C, and equilibrium Bi concentration is 55,580 - 8.31T - 1.14T2 MPa .
The constants C, a and n were determined from the Sn-l5wt.%Bi creep curve to be
80 J/mol MPa-sec, 1900 and 2.1, respectively.
Figure 3a shows the Bi-Sn eutectic steady-state creep data and the
steady-state rates obtained from the CMCC model. What was previously referred to as three regions of stress dependence is now plainly seen as the exponential stress dependence
of the Sn-xBi phase at stresses greater than 6 MPa, and the transition to the
pure Bi phase as the load bearing phase at low stresses. This is shown
quantitatively in Figure 3b where the normalized effective stress is plotted
vs. applied stress for the individual phases of the Bi-Sn eutectic at 120deg.C,
as predicted by the CMCC model. The transition stress, given by the point of
intersection of the effective stress curves, was determined to be only a weak
function of temperature in the case of the Bi-Sn eutectic.
The Bi-Sn eutectic steady-state data of  were recorded for samples quenched from the molten state and then aged 10 days at 120deg.C. To determine the possible effects of
microstructure on the creep behavior, two extreme microstructures were tested. Figure 4a shows the as-quenched Bi 42wt.%Sn microstructure and Figure 4b shows the quenched
microstructure after 22 days aging at 130deg.C. There are at least two
significant differences produced by aging which should be noted. First, aging
produces significant coarsening of the structure, and second, in the
as-quenched state neither phase stands out as the matrix phase, whereas, in the
aged condition, the Sn-rich phase (darker phase) assumes the role of matrix and
Bi appears as an increasingly included second phase. This has been noted in a prior study [ 16] . Figure 5 shows the room
temperature steady-state creep rates from these two different sets of samples.
Aged samples showed consistently higher steady state creep rates than the
For each test, the creep transient was deconvoluted from the total strain response assuming linear addition of the elastic, transient, and steady-state strain components. The transient was then best-fit by:
where is the total accumulated transient strain for a given test, t is the time from the start of the test, and is the transient time constant. The insert of Figure 5 shows the time constants from these series of tests. The aged samples show consistently shorter time constants, indicating a more rapid transient response.
The CMCC model was successful in describing the steady-state creep rates of the Bi 42wt.%Sn eutectic from the creep rates of pure Bi and Sn-15wt.%Bi saturated solid solution. The activation energy for Bi interdiffusion  appears to apply to the creep deformation as well. Figure 6 details the transient and also the recovery behavior (when the load is removed upon reaching steady-state) predicted by the CMCC model at 90deg.C, 8 MPa. Figure 6a shows the effective stresses in the Sn-xBi solid solution and pure Bi phases. At 8 MPa, the Bi phase is the weaker phase and carries a disproportionately small load at steady-state. Upon removal of the load at 25 seconds, the low, but tensile, effective stress in the Bi phase becomes compressive; the Sn-xBi phase remains under a reduced, tensile effective stress. Figure 6b shows the corresponding predicted strain rates during the transient and recovery portions of the creep curve. Upon application of the load, the Bi phase (and, therefore, the composite) first creeps rapidly but then slows as load redistributes to the Sn-xBi phase, where the inverse is true. Upon removal of the load at t=25 sec, the Bi phase, under compressive effective stress, recovers, while the Sn-xBi phase, under tensile stress, continues to strain in the tensile direction. During recovery, the tensile strain rate of the Sn-xBi phase is greater in magnitude than the compressive strain rate of the Bi phase. Figure 6c shows the complete creep curve predicted by the CMCC model at 90deg.C, 8 MPa. There is a normal transient, followed by a short region of steady state creep. Once the load is removed, the model predicts a region of positive recovery.
Different recovery behaviors are predicted depending upon the initial applied stress. Where Bi is the load bearing phase, normal recovery is predicted. At the transition stress, no recovery is predicted, as there is no internal stress at steady state. Due to the very different stress exponents of the Sn-xBi versus pure Bi phases, at stresses just higher than the transition stress, positive recovery is predicted. At still higher stresses, an initially normal appearing recovery (negative and decreasing strain rate) is predicted, but at long times the recovery strain rate goes positive due to more rapid relaxation of compressive stress in the Bi phase than tensile stress in the Sn-xBi phase.
It should be noted that the shape factor of equations (1) and (2) does not effect the steady state creep rate predicted from the individual component creep equations, but that the phase strain difference for an applied stress and temperature is explicit and inversely proportional to the shape factor. The phase strain difference is manifested in the creep transient of the composite alloy. Explicit expressions for the shape factors of a variety of second-phase particle morphologies are given by Tanaka and Mori . The shape factor is a function of the elastic moduli and Poisson's ratios of the component phases and is given in Table l for the Bi-Sn system for common second phase morphologies
Second Phase Shape Factor K=A + B
Morphology Sn-Matrix Bi-Matrix Spherical 0.58 1.04 II Discs 0.58 1.76 Discs 0.60 1.33 II Needles 0.59 1.76 Needles 0.58 0.89
In the Bi-Sn system, the Sn phase is the high modulus phase, and, with Sn as the matrix phase, Bi-Sn composites will not be strengthened by a Bi second-phase of any morphology indicated by the consistent, 0.6, shape factor. On the other hand, a Bi matrix is strongly affected by the Sn phase morphology. Parallel discs or needles of Sn-xBi phase will result in a shape factor 3x higher or about one-third the phase strain difference of the Sn-matrix material. Other second phase geometries strengthen (i.e., decrease the transient), but to a lesser degree.
Figures 4a and 4b show the as-quenched and aged Bi-Sn eutectic microstructures tested. Assignment of a shape factor is somewhat subjective and simpler for the aged condition than for the as-quenched condition. In the aged condition, Sn is clearly becoming the matrix phase and, according to Table 1, the shape factor will vary little once this is achieved. A shape factor of 0.6 is probably appropriate. In the as-quenched state, the pure Bi phase appears slightly more continuous than the Sn-rich solid solution, though both are completely intertwined. If the matrix is Bi, and the disc and spherical shape factor are averaged, the quenched shape factor is approximately 1.3. According to the model, then, the quenched material should have half the phase strain difference of the aged material, and, in turn, show about half as much transient strain in half the time, . This is not what is shown in Figure 5. In fact, just the opposite was observed.
As noted above, in the CMCC model, microstructural morphology does not effect steady state creep rate, whereas the results reported in Figure 5 show that steady-state creep rate is, in fact, significantly affected by shape and coarseness of the microstructure. The coarsened, spheroidized microstructure creeps more rapidly. These two discrepancies between model and experiment may result from the same phenomena.
The shortcoming of the model may be that no boundary sliding is taken into account and, yet, may occur in the Bi-Sn system. To test for this possibility, deformation on the surface of a polished sample was recorded. In a single photomicrograph, deformation in coarsened and in fine, interwoven regions could be seen. These are presented in Figure 7, taken from a sample aged at 130deg.C for 22 days. Figures 7a and 7b show the same region after 5% strain at 40 MPa, 20deg.C. The coarsened regions show far more displacement at phase boundaries and at Sn-Sn boundaries than do the finer regions. Slippage at Bi-Bi boundaries is absent. This distinction between Sn-Sn and Bi-Bi boundaries may further explain why the age-coarsened Sn-matrix structure shows higher strain rates than the as quenched Bi-matrix structure. The faster transient in the aged microstructure is probably primarily due to the same phenomena which causes the faster steady-state creep behavior.
The study has explained the sigmoidal shape of the steady-state creep rate vs. stress curve of the Bi-Sn eutectic alloy. The CMCC model successfully predicts the eutectic creep rate from the component phases, as follows:
1. At low stress (< 3 MPa) in the Bi-42wt.%Sn eutectic alloy, the pure Bi phase is load carrying, has a high stress exponent, and dominates the behavior exhibited by the alloy. At higher stresses, the exponential stress dependence of the Sn-rich phase dominates the behavior, yielding an initially low stress exponent at intermediate stresses and increasing at higher stresses.
2. The model predicts normal transient strain behavior at all stresses, but both positive and negative recovery strains depending upon applied stress.
3. Not predicted by the CMCC model is the effect of microstructure on the steady-state creep rate. Age-coarsened Bi-42wt.%Sn alloy creeps at a higher rate than as-quenched alloy. Coarsening results in a Sn-rich matrix phase and an increased amount of Bi-Sn and Sn-Sn boundary sliding, whereas in the as-quenched state, the Bi is the more continuous phase and no Bi-Bi sliding is observed.
The authors wish to thank the members of the Electronics Manufacturing Program of the C.I.E.E.M. at R.P.I.; AT&T, The Boeing Co., U.S. Navy Mantech, Northern Telecom, Thomson Electronics, and the U.S. Army, without whose support this work would not have been possible. We also would like to thank George Schmeelk for performing the creep testing and D. Lee and D. Van Steele for many helpful discussions.
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