Materials Science & Engineering A 592 (2014) 236–240
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Quasi-elastic strain ﬁelds in a lattice of SnSbCu alloy during creep Mohamed A. Abo-Elsoud n Department of Physics, College of University, Umm Al Qura University, Al Gunfuda, Saudi Arabia
art ic l e i nf o
a b s t r a c t
Article history: Received 17 May 2013 Received in revised form 31 August 2013 Accepted 15 October 2013 Available online 31 October 2013
Apparent steady state creep of Sn0:65 Sb0:25 Cu0:10 alloy is studied under different constant stresses ranging from 19.27 to 21.67 MPa near the transformation temperature ﬃ 443 K. Analysis creep data in the quasisteady state regime at 443 K reveals an anomalous creep behavior with a strain rate sensitivity parameter m changing from 0.15 70.01 to 0.3370.01 which points to cross-slipping dislocation mechanism. The activation energy for creep deformation, Qst , was determined to be 45.47 7 5.96 kJ/mole in the temperature region investigated. The experimental results obtained of Qst are compared with those predicted by Friedel–Jaffe–Dorn and Escaig cross-slip models. The results indicated that the rate controlling dominant mechanism for creep regime is cross-slip of screw dislocations. X-ray and microstructural analysis supports that the relaxation of the internal lattice strain ﬁelds takes place during apparent steady state creep. & 2013 Published by Elsevier B.V.
Keywords: Creep testing Structural relaxation Strain ﬁelds Phase transformations Cross-slip model analysis
1. Introduction Conventional soldering materials [1,2] such as tin–antimony (Sn–Sb), tin–zinc (Sn–Zn), lead–tin (Pb–Sn), and silver–tin (Ag–Sn) are metal–alloy mixtures. These materials have been widely used in electronics industry for the purpose of surface mount technology on a printed circuit board. Particularly, the use of solders in optoelectronic packaging offers advantages over the use of adhesives, speciﬁcally, the passive alignment of components using surface tension forces of the molten solder, high thermal conduction, and superior dimensional stability at elevated temperatures . It is well known that high reliability and dimensional stability [4,5] are both the special demands required for solder materials employed in optical and optoelectronic devices. Many researchers [5,6] have tried to develop composite solders by mixing ﬁne particles such as Cu-atoms. Cu-atoms are reacted with Sn in the eutectic solder paste to form an intermetallic compound that dispersed randomly through the microstructure . This effect is probably caused by the second-phase particles which cannot act as nuclei of new grains . The dispersoids alter the mechanical properties by (a) pinning grain boundaries and thus impeding grain boundary sliding, and (b) restricting the movement of dislocations. The dispersoids also alter the recrystallization kinetics such that the enhanced mechanical properties are maintained during thermal cycle. So, it is worthwhile to get an insight into the creep characteristics of one of these alloys.
n Correspondence address: Materials Science Laboratory, Physics Department, Faculty of Science, Beni-Suief, Cairo, Egypt. Tel.: þ 966 502 4858. E-mail address: [email protected]
0921-5093/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.msea.2013.10.047
Tin–antimony–copper alloys are very well hardened by continuous precipitation, whereas these alloys present a discontinuous precipitation with a weak hardening effect . Steady state creep characteristics are known to depend mainly on the working temperature, the applied stress, and the microstructure of the test sample . The strain rate of steady state dislocation creep ε:st is usually described in a constitutive equation as being proportional to some diffusion coefﬁcient and a power of the applied stress in metals and solid-solution alloys by the formula: DGb s ðm1 Þ Qst ð1Þ ε:st ¼ A exp kT G kT where A is a material parameter, D is the diffusion coefﬁcient, b is the Burgers vector, s/G is the normalized stress (G is the shear modulus, s is the applied stress), m is the strain rate sensitivity parameter (SRS), kT is the Boltzmann constant times the absolute temperature, and Qst is the activation energy of the steady state creep. According to empirical evidence, m varies between 0.1 and 1 depending on the material, the strain rate and the temperature . Using Eq. (1), m and Qst can be determined experimentally using the equations: ∂ log s m¼ ð2Þ : ∂ log εst T Qst ¼ k
∂ log ε:st ∂ð1=TÞ
Maximum elongations are frequently found under conditions of maximum m . It is not yet clear, however, which creep mechanism predominates for the present test Sn Sb Cu alloys
M.A. Abo-Elsoud / Materials Science & Engineering A 592 (2014) 236–240
when transformation deformation occurs. The most important proposed mechanisms are the non-diffusional controlled process as obstacle-glide or cross-slip models, dislocation creep and diffusional creep. In several reviews the above mechanisms have been discussed [12–14]. A single mechanism is not sufﬁcient to explain the large apparent deformation . The experimental activation energy for creep, Qst , in Sn0:65 Sb0:25 Cu0:10 alloy is reported here and is interpreted assuming a temperature dependent threshold stress. The results obtained by tensile creep tests are compared with those predicted by Friedel–Jaffe–Dorn [12,13] and Escaig  for cross-slip models. However, it was found that the values of Qst o Qℓ at intermediate temperatures between 0:46 and 0:72 T m where Qℓ is the activation energy for Sn-lattice and Tm is the absolute melting point of Sn0:65 Sb0:25 Cu0:10 alloy. The values of Qst o Qℓ have been interpreted from several different viewpoints depending on the diffusional  and non-diffusional  controlled process. Most diffusion controlled mechanisms assume that dislocation climb is aided by vacancy diffusion along dislocation core, so that it is expected that the values of Qst is dependent on the applied stress and the reciprocal of strain rate sensitivity parameter is equal to ðð1=mÞ þ2Þ for high temperatures . Alternatively the low values of Qst have been attributed to such non-diffusional mechanisms as cross-slip of screw dislocation and obstacle-controlled glide [12– 14], where these models predict a genuine stress-dependence activation energy for creep. The theoretical models for cross-slip  are often inadequate for comparison with experimental data due to the complex nature of the process . Theories involving the cross-slip of screw dislocation fall into two broad categories depending on whether the partial dislocations constrict along a length equal to that of a Frank–Read source  or to a length less than the Frank–Read source [13,14,21] prior to cross-slip onto the new slip plane. In each case, the unit dislocation dissociates into its partials in the cross-slip plane. The former process is energetically unfavorable by comparison with the latter so that it is generally not important. There are two variations of the second process which is based on the Friedel-model . Jaffe and Dorn  assumed that the partial dislocations constrict over a critical length of about 4dc prior to cross-slip, dc is the stress dependent stacking fault width given by  0:5 b πG ð4Þ dc ¼ 0:5 NP s where NP is the number of dislocations in a piled-up array. The activation energy for this process is then estimated to be [12–14] Qcs ¼ βU c
Sn0:65 Sb0:25 Cu0:10 solid-solution alloy. Also, the experimental results obtained, Qst , are compared with those predicted by the Friedel–Jaffe–Dorn for obstacle-controlled glide and Escaig crossslip models.
2. Experimental procedure High purity Sn, Sb, and Cu were alloyed by standard powder metallurgical techniques , with a nominal composition of Sn0:65 Sb0:25 Cu0:10 . The alloy was then drop cast into a 1 mm diameter copper mold. Right cylindrical wires from casting, ranging in length from approximately, 50 mm, were spark cut from the drop cast ingots for creep experiments. The creep experiments were done in a homemade tensile creep machine. Creep deformation tests were carried out at different temperatures in the range from 423 to 473 K under different constant applied stresses ranging from 19.27 to 21.67 MPa. Heating was accomplished in a vertical tube furnace in argon, and the temperature was monitored by a thermocouple with an accuracy of 71 K Elongation was measured using a strain gauge with an accuracy of 71 μm. Characterization of the samples was done by X-ray diffraction with CuKα radiation and scanning electron microscopy (SEM).
3. Results and discussion After creep deformation, of the sample Sn0:65 Sb0:25 Cu0:10 at working temperature 443 K under constant stress ﬃ19:95 MPa, the Cu3 Sn-phase constituency was determined by X-ray diffraction (see Fig. 1). Unmarked peaks index to the Sn0:65 Sb0:25 Cu0:10 phase. The dissolution of the impurity phases occurred as a result of the high temperatures reached, up to transformation point ﬃ 443 K; during the creep experiments. Isothermal creep curves of Sn0:65 Sb0:25 Cu0:10 alloy were studied using constant different applied stresses from 19.27 to 2167 MPa at different working temperatures from 413 to 473 K in steps of 10 K. The sequence of creep curves with respect to temperature was irregular at 443 K (see Fig. 2a and b). An anomalous behavior in the creep curves might be attributed to the fact that the alloy is near the transformation temperature ﬃ 443 K. This behavior can be explained by the dislocation motion across precipitates , which might be one of the important mechanisms under the considered conditions, and which could be affected in this temperature region by the formation and dragging of a solute cloud . On increasing the temperature, Cu, Sb solid solubility in Snmatrix are increased and dissolved in α-phase of Sn and disappear completely at ﬃ 443 K. The new nuclei grains of the product
where β is a constant which is approximately 1:0 and 2:0 for low and high stacking fault energy materials, respectively, and Uc is given by  0:5 2 ð6Þ U c ¼ 0:016Gb dc lnðdc =bÞ Using Eqs. (4)–(6) the calculated values of activation energy, Qcs , for this process is  Qcs ¼ ð0:014β Gb =N p ÞðG=sÞ0:5 ½ lnð0:89=N p ÞðG=s0:5 Þ0:5 3
It is evident from the dependency of dc on the inverse square root of s that Qcs decreases with increasing stress. Eq. (7) predicted a nonlinear dependence of Qcs on stress. The main scope of our work is to study the stellar role of internal quasi-elastic strain ﬁelds in the lattice of a Sn0:96 Sb0:3 Cu0:1 alloy during steady state creep regime. In this paper a speciﬁc recovery model for subgrain-bearing materials is brieﬂy described and applied to a class of material characterized as
Fig. 1. X-ray diffraction scan of a Sn0.65Sb0.25Cu0.10 sample after creep deformation at 443 K under constant stress of 19.95 MPa. The peak marked with an arrow is a peak greatly affected by phason strain for Cu3Sn-product phase.
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Fig. 2. Creep curves for Sn0.65Sb0.25Cu0.10 alloy at various creep temperatures for (a) 19.27 and (b) 20.56 MPa.
phase are considerably smaller in size and elongation quasi-lattice strain ﬁelds than in ordinary crystals . The aperiodicity of quasi-lattice complicates the dislocations motion. Because dislocations in the new nuclei grains have both phonon and phason components, the difﬁculty of moving dislocations in quasi-lattice is increased . In the sequence of creep tests, at 443 K the minimum creep rate ε:st is varied with the applied stresses (Fig. 3a) where the phase transformation occurred at sb ﬃ 19:95 MPa (sb is called the recovery back stress). Tests of the type done to show that sb increases with formation of the subgrain structure during steady state creep regime . The steady state creep was commonly considered as due to some sort of balance between strain hardening and recovery. Fig. 3a shows that the strain rate sensitivity parameter, m, derived from the slopes of straight lines relating log s and log ε:st (see Eq. (2)), decreases from 0.33 to 0.15 as the stress increased beyond a transition back stress ðsb Þ. This suggests a transition from slip-dislocation along shear planes to crossslipping viscous glide mechanism [12–14]. The temperature dependence of the strain rate sensitivity parameter m can be seen as shown in Fig. 3b. The parameter m increased extensively with increasing working temperature and exhibited peak value at ﬃ443 K. It was found to have obtained values ranging from 0.15 70.01 to 0.32 7 0.01 (Fig. 3b). This behavior can be explained on the basis of processes associated with the release of stored deformation energy. These processes cause a redistribution of dislocations in the network at transformation and formation of Frank–Read sources . The relative decrease in creep characteristics might be attributed to the dissolution of the solute cloud phase, thus setting free Sb and Cu atoms which consequently move towards dislocation lines pinning them and thus causing retardation of their movements. The m values in Fig. 3 are considerably smaller than the values between 0.2 and 0.5 which were suggested for steady state creep theories based on a dislocation climb mechanism which were investigated by [11,29]. Those values may be in a good agreement with the cross-slipping dislocation mechanism . The activation energy of the apparent steady state creep, Qst , was calculated from the slopes of the straight lines relating ðlog ε:st Þ versus ð1000=TÞ given in Fig. 4, (see Eq. (3)). The activation energy of the steady state creep in the transformation
Fig. 3. Log–log plot of the quasi-creep steady state strain rate versus stress at 443 K. (a) Two mechanisms are active, with the transition occurring at E19.95 MPa, (b) the temperature dependence of strain rate sensitivity parameter m for Sn0.65Sb0.25Cu0.10 alloy.
temperature region was found to be 45.47 kJ/mole. Besides, the present results yield activation energies of 39.51 and 51.43 kJ/mole for steady state creep in the low, and high temperature regions, respectively, shown in Fig. 4. Our results showed that in the low temperature range, the steady state creep is governed by a thermally activated, stress assisted process for which 443 K is a critical temperature (Fig. 4a) coincident with the eutectic temperature of the Sn–Sb–Cu system, and with the recovery process of back stress (19.95 MPa) that increases with formation of the subgrain structure during creep regime. It occurs by overcoming of obstacles by cross-slip which is affected by the recovery process of back stress through the change of the elongation quasi-elastic internal strain ﬁelds between the dissolved precipitates by a thermally activated process. The higher value of steady state creep activation energy (51.43 kJ/mole) indicates that the supposed mechanism may be viscous ﬂow creep and/or overcoming of obstacles such as small precipitates. Clearly, this concept might be changed when the applied stress is changed at T exceeding the transition temperature region ﬃ 453 K as in Fig. 4c. From the same Fig. 4, we except that the activation energy for the steady state creep in the temperature range to the eutectic temperature of Sn0:65 Sb0:25 Cu0:10 alloy may be due to dislocation climb along the shear glide planes [12,13] and/or cross-slipping dislocation mechanism . These calculated values of the activation energies characterize the mechanism of dislocation climb along the grains boundaries with cross-slipping thus causing grain boundary
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Fig. 4. Plot of the natural log of the strain rate versus 1/T. The slope of the lines is –Qst /k, (a) in the low temperature range, (b) in the intermediate temperature range, and (c) in the high temperature range at constant different applied stresses sa ¼19.27, 19.95, and 20.56 MPa, respectively.
Table 1 Comparison of experimental values of activation energy with those estimated for the Friedel–Jaffe–Dorn (β¼ 1) and Escaig cross-slipping models as a function of normalized stress and the stress-dependent stacking fault width. r=G 4
3 10 5 10 4 10 10 4 20 10 4 a b c
Fig. 5. Comparison of the experimental activation energies for Sn0.65Sb0.25Cu0.10 alloy with those predicted by the Friedel–Jaffe–Dorn [12,13] and the Escaig  cross-slip models. The values of β ¼1 and 2 are for low and high stacking fault energies, respectively.
sliding and migration of the dissolved grains into new grains of product phases ðCu3 SnÞ precipitates at 443 K. The experimental results obtained (Qst ) and calculated values (Qcs ) with those predicted by Friedel–Jaffe–Dorn and Escaig crossslip models for creep activation energies at transformation temperatures region of Sn0:65 Sb0:25 Cu0:10 alloy are plotted as a function of s/G (Fig. 5), where the horizontal broken line represents Qℓ ¼ 210 kJ=mole . Table 1 presents the predicted values of dc and Qcs for different normalized stresses assuming G ¼ 3:6 104 MPa, b ¼ 2:56 10 10 m and Np ¼5 , the experimental values of Qst and the ratio of Qst /Qcs . Although Qcs decreases with increasing s/G (Fig. 5 and Table 1), there is a large consistency in the trend between the experimental and predicted values of the activation energies. Therefore, it is concluded that Friedel–Jaffe–Dorn, (β ¼1), and Escaig cross-slip models are important in a solder Sn0:65 Sb0:25 Cu0:10 alloy at intermediate temperatures region as the dominant deformation mechanism [12–14,30]. However, cross-slip by some other mechanisms may account for
50 47 42 32
2.6 2.0 1.6 1.5
54 50 40 29
1.08 1.06 0.95 0.91
Experimental values were reported. Estimated from Eq. (7), using b ¼ 2.65 1010 and Np ¼ 5 . Calculated from Eq. (4), using G ¼ 3.6 104.
the occurrence of wavy-slip in a limited range of stress, strain and temperature [18,33]. When comparing the high and low temperature regions (i.e., below and above 443 K) the activation energies are different by a factor of approximately 6.0, and the strain rate sensitivity parameter differs by a factor of almost 0.15. The much higher activation energy for the high temperature process reﬂects a much barrier to creep. This may be due to the climb component of the deformation, the activation of additional slip systems, or a combination of the two. The microstructure, also show that the strain rate sensitivity parameter m determined here appear to be reasonably representative of the various deformation mechanisms models. For example, models typical associate dislocation motion with a strain rate sensitivity parameter of approximately 0.33. Since strain rate sensitivity parameter cannot singly be used to deduce deformation mechanisms, it is necessary to examine the microstructural features which result from the deformation to fully elucidate deformation mechanisms. SEM was used for microstructural analysis to indentify the microscopic nature of the deformation mechanisms active, that reﬂect the generation and motion of dislocations during creep regime at the various temperatures . The microstructure of Sn0:65 Sb0:25 Cu0:10 sample deformed after creep experiments at 443 K under 19.95 MPa was found to contain very contract typical of phason strain in the new product phases ðCu3 SnÞ which appear star-shaped ; the darker background is the plastic matrix with the light hard cube-shaped crystals being the compounds SnSb, (Fig. 6a). This microstructure is reminiscent of microstructures where the dominant deformation mechanism is cross-slipping– dislocation glide along the shear planes [12–14,30]. SEM investigation revealed that deformation mechanisms occurred in the new nuclei grains of product phases ðCu3 SnÞ [26,34]. The micrograph in Fig. 6b displays a dislocation substructure consisting of low grain
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Fig. 6. Micrograph of a Sn0.65Sb0.25Cu0.10 sample after creep deformation at 443 K and 19.95 MPa showing, (a) the product phase (Cu3Sn) which appears star-shaped ; the darker background is the plastic matrix with the light hard cube-shaped crystals being the compounds SnSb, and (b) very few dislocation generations for cross-slipping and low grain boundaries for dislocation climb.
boundaries of the sample deformed at 443 K under constant stress, 19.95 MPa. The microstructure indicated that dislocation generation and motion has occurred and is the mode of deformation. The low grain boundaries also indicate that dislocation climb, which is a thermally activated (i.e. diffusion controlled) process, has occurred. The different dislocation structures in the micrograph in Fig. 6 are due to the inhomogeneous deformation that occurs in polycrystalline material . With the generation of dislocations in the deformed samples, a corresponding increase in phason strain occurs since dislocation strain ﬁelds are comprised of both phonon and phason components . Since the difﬁcultly of moving dislocations in Sn0:65 Sb0:25 Cu0:10 alloy is increased with decrease in quasi-elastic lattice strain and a recovery back stress. 4. Conclusions 1. A detailed analysis of the creep data on Sn0:65 Sb0:25 Cu0:10 alloy, obtained at intermediate temperatures regime between 0:46 and 0:72T m of the absolute melting point, showed that anomalous behavior for steady state creep regime at ﬃ 443 K where a relaxation of the quasi-elastic lattice strain ﬁelds with a recovery of back stress, sb took place during the transformation temperature region. 2. A recovery model for quasi-steady state creep behavior, obtained at 443 K under 19.95 MPa, is based on the premise of quasi-elastic lattice strain ﬁelds that cell boundaries in the subgrain interior act as sources and obstacles for dislocations. 3. The results indicated that the rate-controlling mechanism for quasi-steady state creep regime is cross-slipping-dislocation climb. 4. The transition from slip dislocation climb to cross-slipping models occurs at an essentially constant value of a recovery
back normalized stress, sb ﬃ 5:5 10 4 MPa, for new grains material irrespective of the stacking fault energy. The large values of dislocation width dc (listed in Table 1) are the possible reason for increasing the difﬁcultly of moving dislocations in Sn0:65 Sb0:25 Cu0:10 alloy and decreasing the elongation quasi-elastic lattice strain ﬁelds at transformation temperatures region. The strain rate sensitivity parameter m changes from 0.15 7 0.01 to 0.33 70.01 which points to cross-slipping dislocation climb rate-controlling mechanism. The activation energies of steady state creep rate, Qst , in the temperature ranges 413– 443 K and 453–473 K were found to be 39.51 and 51.43 kJ/mole as characteristic of the diffusion of Cu precipitates in the Snrich phase and dislocation climb, respectively. Besides, the activation energy for deformation was found to be 45.47 kJ/ mole in the transformation temperature region investigated. Microstructure analysis revealed both a substantial amount of dislocation and dislocation substructures, including low grain boundaries. This microstructure is typical of deformation controlled by dislocation climb and cross-slipping. The experimental results are compared with those predicted by the Friedel–Jaffe–Dorn for obstacle-controlled glide and Escaig cross-slip models during the creep process at transformation temperatures region, where cross-slipping models are important as the dominant deformation process.
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