Cooling thermal parameters and microstructure features of directionally solidi ﬁ ed ternary Sn – Bi – (Cu,Ag) solder alloys

Low temperature soldering technology encompasses Sn – Bi based alloys as reference materials for joints since such alloys may be molten at temperatures less than 180 °C. Despite the relatively high strength of these alloys, segregation problems and low ductility are recognized as potential disadvantages. Thus, for low-temperature applications, Bi – Sn eutectic or near-eutectic compositions with or without additions of alloying elements are considered interesting possibilities. In this context, additions of third elements such as Cu and Ag may be an alternative in order to reach sounder solder joints. The length scale of the phases and their proportions are knowntobethe mostimportant factorsaffectingthe ﬁ nal wear,mechanicalandcorrosionspropertiesofternary Sn – Bi – (Cu,Ag)alloys.Inspiteofthispromisingoutlook,studiesemphasizinginterrelationsofmicrostructurefea-tures and solidi ﬁ cation thermal parameters regarding these multicomponent alloys are rare in the literature. In the present investigation Sn – Bi – (Cu,Ag) alloys were directionally solidi ﬁ ed (DS) under transient heat ﬂ ow conditions.Acompletecharacterizationisperformedincludingexperimentalcooling thermalparameters,segregation (XRF), optical and scanning electron microscopies, X-ray diffraction (XRD) and length scale of the micro-structural phases. Experimental growth laws relating dendritic spacings to solidi ﬁ cation thermal parameters have been proposed with emphasis on the effects of Ag and Cu. The theoretical predictions of the Rappaz-Boettinger model are shown to be slightly above the experimental scatter of secondary dendritic arm spacings for both ternary Sn – Bi – Cu and Sn – Bi – Ag alloys examined. © 2016 Elsevier Inc. All rights reserved.


Introduction
Sensitive components for electronics; step soldering and soldering LEDs may require the usage of low temperature soldering metallic alloys. In this case, Sn-Bi based alloys may be applied with a view to covering such applications. However, inadequate microstructures and/or high level of segregation can be faced if non-optimized solidification parameters are chosen [1][2][3][4]. Insufficient mechanical response of soldered joints of Sn-Bi alloys can be perceived, which in some extreme cases may not allow the strength requirements to be achieved.
It is known that the dendritic microstructure of a solder alloy is affected by the cooling rate experienced during liquid-to-solid transformation of a solder fillet [5][6][7][8][9]. The alloy strength is a consequence of the microstructure characteristics. Recently, experimental growth laws were proposed for the directionally solidified (DS) Sn-52 wt.%Bi solder relating the primary/tertiary, and secondary dendrite arm spacings, to the experimental thermal parameters V L (growth rate) and Ṫ L (cooling rate), by power functions having −1/2 and −1/4 exponents, respectively. Experimental eutectic laws, λ coarse = 1.1 × (V L ) −1/2 and λ fine = 0.67 × (V L ) −1/2 , were shown to represent the evolutions of the eutectic spacings due to the presence of different sizes (coarser and finer) eutectic arrangements.
Until nowadays, lead-based alloys have been extensively used as solders for the electronic boards in the electronic industry. However, the new legislations worldwide have restricted the use of lead due to its negative impact on the human health and environment. Sn-Bi-Ag solders are one of the most familiar materials used for various microelectronic connections in the computer industry [10]. The intensive interest in these solder alloys is attributed to their positive material properties including high superplastic properties, low melting temperature, wettability [10][11][12].
According to Takao and co-authors [13] an optimized composition of Sn-Bi-Cu may improve the ductility as compared with that of Sn-Bi alloys, since Cu 6 Sn 5 fine particles are randomly distributed throughout the microstructure. In this case, such particles have been recognized as part of a ternary eutectic formed by Sn + Bi + Cu 6 Sn 5 . The ductility of the Sn-40 wt.%Bi-0.1 wt.%Cu alloy, for instance, is reported to be more than 2.5 times higher than that characterizing the conventional Sn-37 wt.%Pb solder [13]. McCormack et. al. [14] found that additions of small amount of Ag, about 0.25-0.5 wt.%, in near-eutectic Bi-Sn solder alloys improve the ductility by a factor of more than three and reduce the strain-rate dependent deformation behavior. This increase in ductility caused by Ag additions is attributed to the substantial refinement of the solidification microstructure. However, in the aforementioned studies the length scale and proportion of the phases forming the microstructure have not been determined. The effects of cooling rate on the formation and evolution of the microstructure of these ternary alloys have not been completely investigated so far. In the case of Sn-based solder alloys, it is common to note dendritic microstructures within the solder fillets. Hence, it is essential to characterize the scale of dendritic spacings. In contrast, other research studies with other ternary solder alloys systems focused on change of microstructure due to applied cooling rate. For instance, a single ratio between the cooling rate (Ṫ L ) and the alloy silver content (C 0 -Ag) of 0.45 in Sn-0.7Cu-XAg alloys was shown to be the parametric factor associated with the beginning of the growth of tertiary dendritic branches [15]. Moon and coauthors [16] stated that small Pb additions as 1.0 wt.% to Sn-Bi alloys can drastically expand the freezing range due to the formation of a ternary eutectic at 95.3°C. Moreover, Sakuyama and collaborators [17] stressed that as the degree of supercooling under the cooling conditions increases, the eutectic structure gets finer. Since the ternary eutectic is thermodynamically stable in the liquid state, the ternary eutectic is easier to form on further supercooling than a binary eutectic.
All Sn-Bi-Ag solders studied by Sebo et. al. [18] consist of a Sn matrix + Bi particles (big particles surrounded by small ones) + Ag 3 Sn compounds in the form of needles. Increase in Ag content causes the increase in the size of the needles. According to Shen et. al. [19] a small amount of Cu 6 Sn 5 particles with odd-shape was shown to be entrapped into the Bi-rich phase during solidification of a Sn-40 wt.%Bi-0.1 wt.%Cu solder alloy, which promoted the refinement of Bi dendrites. Such Cu 6 Sn 5 particles are commonly called H-shaped [20] or M-shaped [21] Sn-Cu intermetallics.
Vianco and Rejent [22] produced a series of Sn-Ag-Bi alloys with the Bi content ranging from 1 wt.% to 10 wt.%. The alloy composition containing 4.8 wt.%Bi was shown to have a short range Bi segregation, which was enough to modify the thermal behavior of this alloy as compared with the others. Both solid solution and precipitation strengthening were observed to occur due to the presence of Bi, which resulted in improved strength properties exhibited by the ternary alloys. The precipitation of Bi is a feature which deserves attention as explained by He and Acoff [23], who state that the strengthening effect increases with the Bi addition. However, when the Bi addition is more than 5 wt.%, there is no further increase in hardness for Sn-Ag-xBi solders.
Even though studies on the phase equilibria and solidification behavior of alloys of the Sn-Bi-Ag system were previously performed [24][25][26][27], metallographic analyses of the equilibrium alloys have been barely shown. Other than, lack of investigations on microstructure formation and evolution of Sn-Bi-Ag and Sn-Bi-Cu solder alloys is also noted for non-equilibrium cooling conditions. According to Kattner [26], a ternary eutectic for Sn-Bi-Ag occurs at 138.4°C where the liquid decomposes into two terminal solid solutions, (Bi) and (Sn), and the Ag 3 Sn phase. However, under non-equilibrium solidification the ternary eutectic is reported to occur even considering compositions for which the last liquid to solidify is expected for higher temperatures [27]. It can be inferred that both cooling rate and segregation could play a fundamental role on the formation of this eutectic.
The present study aims to investigate the microstructural evolution of ternary Sn-Bi-(Cu,Ag) solder alloys under a wide range of cooling rates, using a water-cooled directional solidification system. Microstructure and thermal parameters (growth and cooling rates) are aimed to be determined with a view to establishing experimental growth laws. The influences of alloying additions and macrosegregation on the length scale of the microstructure will be examined.

Experimental procedure
The solidification setup used in the experiments allows a unidirectional extraction of heat through a water-cooled bottom made of low carbon steel (SAE 1020), promoting vertical upward directional solidification. Details on the used casting assembly can be found in previous investigations [28,29]. A stainless steel split mold was used having an internal diameter of 60 mm, a height of 157 mm and a wall thickness of 5 mm. The lateral inner mold surface was covered with a layer of insulating mass silica-alumina ceramic to minimize radial heat losses. The bottom part of the mold was closed with a thin (3 mm thick) steel sheet. The upward solidification experiments were carried out with the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2.0 wt.%Ag alloys, thus imposing thermally and solutally stable solidification conditions. Continuous temperature measurements in the casting were monitored during solidification via the output of a bank of fine type J thermocouples sheathed in 1.5 mm outside diameter stainless steel tubes, and positioned at different positions from the heat-extracting surface at the bottom of the directionally solidified (DS) casting. All thermocouples were connected by coaxial cables to a data logger interfaced with a computer and the temperature data were acquired automatically. Experiments were carried out with superheats of 20% above the liquidus temperature for each alloy examined.
Selected transverse (perpendicular to the growth direction) and longitudinal samples of the DS Sn-Bi-Cu and Sn-Bi-Cu alloys castings were polished (solution of alumina 1 μm) and etched with a solution of 2 mL HCl, 10 mL FeCl 3 and 100 mL H 2 O applied during 10-30 s to reveal the microstructures. An optical image processing system Olympus, GX51 (Olympus Co., Japan) was used to acquire the images. The primary, tertiary (λ 1 , λ 3 ) and secondary dendrite arm spacings (λ 2 ) were measured on transverse and longitudinal sections of the casting, respectively. The triangle method was employed to measure λ 1 , whereas λ 2 and λ 3 were measured by the intercept method, as reported by Gündüz and Çadirli [30]. At least 40 measurements were performed for each selected position along the length of the DS castings. The microstructures of the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2.0 wt.%Ag alloys were analyzed from bottom to top with a view to determining the point where tertiary branches develop from the secondary dendritic branches. Furthermore, microstructural characterization in more details was performed using both a Field Emission Gun (FEG)-Scanning Electron Microscope (SEM-EDS) Philips (XL30 FEG) and a SEM-EDS FEI (Inspect S50L). Cross section samples of both ternary alloys were examined by SEM and elemental mapping was performed to determine the relative distribution of the elements.
Transverse samples were extracted from different positions along the length of the castings, and investigated by a fluorescence spectrometer (XRF), model Shimadzu EDX-720, to estimate local average Bi, Cu and Ag concentrations. The thicknesses of these samples were preserved for at least 4 mm. The X-ray diffraction (XRD) patterns for phases formed in ternary Sn-Bi-Cu and Sn-Bi-Ag alloys examined have been acquired by a Siemens D5000 diffractometer with a 2-theta range from 20°to 90°, CuKα radiation and a wavelength, λ, of 0.15406 nm. XRD patterns were recorded at a scan speed of 2°/min.
The thermophysical properties of the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2.0 wt.%Ag solder alloys, used to perform calculations with a predictive dendritic growth model for multicomponent alloys, are summarized in Table 1 [31][32][33][34]. Fig. 1 shows the macrostructures of the DS castings of the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2.0 wt.%Ag solder alloys. Columnar grains prevailed along the entire length of the solidified castings, which means that vertically aligned grains have grown from the bottom of the casting. This orientation permits a reliable determination of microstructural parameters as the dendritic arm spacings (λ 1 , λ 2 , λ 3 ).

Results and discussion
The thermocouples readings, collected during solidification, were used to generate plots of position (P) from the cooled surface of the DS casting as a function of time (t) corresponding to the liquidus front of every alloy passing by each thermocouple. A numerical technique, based on the minimum square method, was used to fit mathematical power functions of the form P(t) = a · t b (where "a" and "b" are constants) on these experimental plots. The derivative of these functions with respect to time gave values for the growth rate (V L ). Furthermore, the data acquisition system employed permitted accurate determination of the slope of the experimental cooling curves. Hence, the cooling rate (Ṫ L ) was determined along the castings lengths, by considering the thermal data recorded immediately before and after the passage of the liquidus front by each thermocouple.
Similarly to the Sn-Bi-Cu system, Fig. 4 shows the ternary phase diagram (Fig. 4a) and the Sn-rich part of the liquidus projection (Fig. 4b) of the Sn-Bi-Ag system. In this case, it can be verified the presence of two ternary eutectics. The first occurs near 136.5°C with the phases: [(Sn) + (Bi) + Ag 3 Sn] (indicated by gray dashed circle), which is the ternary eutectic of interest for the present investigation. In this case, Kattner [21] stated a temperature of 138.4°C regarding this reaction. The second ternary eutectic occurs at higher Bi content with prevalence of the following phases: [(ζAg) + (Bi) + Ag 3 Sn] (indicated by black dashed square). Thus, Ag additions should be carefully controlled, since the formation of primary Ag 3 Sn intermetallic particles could prevail. In this case, mechanical properties of the Sn-Bi-Ag alloy may be negatively affected due to the presence of this brittle phase [6,14,18,35,36,37]. According to Fig. 4 [38], the sequence of phases formed during solidification of the Sn-33 wt.%Bi-2 wt.%Ag is: (i) → L, (ii) → L + (primary Ag 3 Sn), (iii) → L + (primary Ag 3 Sn) + (primary Sn-rich), (iv) → L + (primary Ag 3 Sn) + (primary Sn-rich + Bi precipitates) and (v) → (primary Ag 3 Sn) + (primary Snrich + Bi precipitates) + ternary eutectic: [(Sn) + (Bi) + Ag 3 Sn], where L is liquid phase.
Sebo et. al. [18] investigated the effect of Ag on melting kinetics, wetting properties and shear strength of Sn-Bi-Ag alloys and Cu/solder/Cu joints. These authors stated that the microstructure is formed by dispersed fine particles of Ag 3 Sn within Sn and Bi phases. Sn and its alloys are well known by the reactions with copper forming Cu-Sn intermetallic compounds (IMC) during soldering operations. The presence of those intermetallic particles may decrease the solder joint reliability due to their brittleness and weakness. The control of the kinetics of the interfacial reactions is considered a key aspect in order to determine the final soundness of the joint.
The representative longitudinal and transversal microstructures and the corresponding growth (V L ) and cooling rates (Ṫ L ) of the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag alloys can be seen in Figs. 5 and 6, respectively, at some selected positions (P) from the cooled bottom of the DS castings. A fully dendritic arrangement can be observed for both Sn-Bi-Cu and Sn-Bi-Ag alloys. Three positions in casting having quite distinct microstructural length scales have been chosen for each alloy examined. This is mainly connected with the experimental profiles of growth rate and cooling rate, which exhibit higher values close to the bottom and lower values toward the top of the casting. The longitudinal microstructures show that primary dendritic branches may grow aligned with the direction of heat extraction.
The microstructures of the three Sn-Bi-Cu and Sn-Bi-Ag solders were monitored from bottom to top with a view to determining the point where tertiary branches begin to grow from the secondary branches. The tertiary dendrite arms have been found at specific positions of the DS Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag alloys castings. Typical tertiary dendrite branches are indicated by gray arrows in Fig. 7. The onsets of tertiary dendritic arms have been experimentally determined at positions P = 30 mm (λ 3 = 17.5 μm) and P = 15 mm (λ 3 = 14 μm) for Sn-Bi-Cu and Sn-Bi-Ag alloys, respectively. The growth and cooling rates values associated with these positions are 0.35 mm·s −1 /0.45°C.s -1 and 0.5 mm·s −1 /1.3°C.s -1 for Sn-Bi-Cu and Sn-Bi-Ag alloys, respectively.
A simple parametric factor relating Ṫ and the alloy Ag or the alloy Cu content (C 0-Ag or C 0-Cu ) seems to be able to denote the start of tertiary branches (since the Bi content is roughly kept constant for both alloys examined). Such a parametric factor of 0.65 seems to determine the conditions for the growth of tertiary branches for the two alloys examined, i.e. for ð _ T=C 0−Ag;Cu Þ≤0:65 the dendritic array will be formed by λ 1 , λ 2 and λ 3 . A factor of 0.45 following the same approach is accomplished in the case of ternary Sn-0.7 wt.%Cu-1.0, 2.0 and 3.0 wt.%Ag alloys as reported in the literature [15]. Fig. 8 shows the experimental macrosegregation profiles of Bi, Cu and Ag along the length of the DS Sn-Bi-Cu and Sn-Bi-Ag alloys castings. It can be seen that for the Sn-34 wt.%Bi-0.7 wt.%Cu solder alloy, a slight Bi positive macrosegregation profile is observed (Fig. 8a), i.e., regions closer to the top of the casting have Bi contents that are lower than that of the nominal alloy composition and positions closer to the cooled surface of the alloy casting refer to concentrations that are higher than the nominal one. In the case of the Sn-Bi-Ag alloy (Fig. 8c) it can be seen that the Bi concentration is essentially constant, with an average content close to 33 wt.%Bi despite slight fluctuations along the length of the casting. Meanwhile, the Cu concentration (Fig. 8b) remains practically constant along the alloy casting with an average content close to 0.7 wt.%Cu. A very similar profile has been reported for a DS binary Sn-0.7 wt.%Cu alloy casting [7] with the mean composition considered essentially constant.
The experimental macrosegregation profile of Ag (Fig. 8d) shows that this element is segregated toward the bottom part of the Sn-33 wt.%Bi-2 wt.%Ag alloy casting, reaching a content two times higher than the nominal value, which is typical of a positive macrosegregation profile. The higher proportion of Ag 3 Sn particles (see Fig. 6) at positions close to the bottom of the casting (0-35 mm) may be associated with the corresponding Ag enrichment. This may be due to the higher liquid density of Ag compared with that of molten Sn [39], which makes the Ag-enriched melt ahead the solidification front to flow through the interdendritic channels toward the water-cooled surface of the casting. In contrast, Bi participates mainly from the eutectic reaction, which is basically dominated by the lateral diffusion-coupled growth between phases. Indeed, the area fraction of the Sn-Bi eutectic mixture is quite representative for the ternary Sn-Bi-Ag alloy, as can be seen in Fig. 7a. The dashed lines in the Fig. 8 are indicative of the average composition of Bi and Cu elements for different positions along the Sn-Bi-(Cu,Ag) alloys castings. In the case of Ag, the line represents the positive macrosegregation profile. The microstructures of transverse sections of the DS castings show the constituent phases of the ternary Sn-Bi-Cu and Sn-Bi-Ag alloys (Fig. 9). The as-cast microstructures for both Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag solder alloys are arranged by Sn-rich dendrites with Bi precipitates in their own core surrounded by a complex eutectic mixture (Bi-rich and Sn-rich phases) and their respective intermetallic compounds, Cu 6 Sn 5 and Ag 3 Sn, as shown in Fig. 9a and b, respectively. The decrease in solid solubility of Bi in Sn during cooling explains the Bi precipitation within the Sn-rich regions [40]. The primary Cu 6 Sn 5 and Ag 3 Sn intermetallic particles are non-homogeneously Fig. 9. SEM and optical images of transverse sections detailing the microstructures found for: (a) Sn-34 wt.%Bi-0.7 wt.%Cu alloy, (b) Sn-33 wt.%Bi-2 wt.%Ag alloy; (c) morphologies of Bi precipitates within the Sn-rich dendritic matrix. P is the position from the metal/mold interface. distributed along the microstructure. This result is also in agreement with the distribution observed by Takao et. al. [13] for the Cu 6 Sn 5 intermetallics.
As can be seen through the SEM images in Fig. 9a and b, the Cu 6 Sn 5 and Ag 3 Sn intermetallics have plate-like morphologies, a typical characteristic of faceted phases. In general, the Cu 6 Sn 5 particles are diamond shaped (shorter or elongated) and the Ag 3 Sn are rectangle shaped (shorter or elongated). Kim and coauthors [41] observed similar morphologies for the Cu 6 Sn 5 and Ag 3 Sn intermetallics in Sn-Ag-Cu solder alloys.
The precipitation of Bi, as aforementioned [22,23], is an important feature that requires attention in Sn-Bi based alloys. The distribution and fraction of Bi precipitates within the Sn-rich dendrites appear to be uneven along the entire length of the solidified casting. In Fig. 9a and b, white arrows indicate the regions of lower volume fraction and white squares those of higher density of precipitates. This same behavior has been reported for a hypoeutectic Sn-52 wt.%Bi alloy directionally solidified under transient heat flow conditions [42]. Moreover, these particles have spherical and ellipsoidal morphologies as shown in Fig. 9c.
The X-ray diffractograms for the Sn-Bi-Cu and Sn-Bi-Ag solder alloys examined are shown in Fig. 10. Three different positions were examined along the length of each alloy casting encompassing a representative range of cooling rates. Fig. 10a shows the presence of peaks associated with the Cu 6 Sn 5 and Cu 3 Sn intermetallic compounds (IMCs), Sn-rich and Bi-rich phases for the ternary Sn-Bi-Cu alloy. Although the Cu 3 Sn particles could not be identified by microscopy, characteristic X-ray peaks have been identified. In general, the X-ray spectra of Fig. 10a shows that the Sn-rich and the Bi-rich phases corresponding to peaks' intensities, increase and decrease slightly, respectively, as the cooling rate decreases. There are no clear tendencies regarding the intensities of peaks for the different positions from the casting cooled surface for Cu 6 Sn 5 and Cu 3 Sn IMCs. Fig. 10b depicts the presence of peaks of Ag 3 Sn IMCs, Sn-rich and Bi-rich phases. In the case of Ag 3 Sn IMCs the intensities of peaks decreased for regions closer to the top of the casting. This result is in agreement with Fig. 8d, which shows that the Ag content decreased for those regions. In a similar manner for the Sn-Bi-Cu alloy, the intensities of peaks associated with the Bi-rich phase decreases also with decreasing cooling rate, whereas for Sn-rich phases there is not a definitive tendency. Figs. 11 and 12 show the elemental SEM-EDS mapping for the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag alloys. For the ternary Sn-Bi-Cu alloy (Fig. 11), Sn contrast (in green) is higher in Sn-rich dendrites and in the intermetallic compound, Cu 6 Sn 5 , with a lower intensity in the eutectic mixture. Bi (in red) is either concentrated in the eutectic mixture, or as particles dispersed in the Sn-rich matrix. These results show clearly that Cu (in blue) was consumed only in the formation of Cu 6 Sn 5 IMCs. This is befitting with the stoichiometry of these IMCs. SEM images depict that Cu 6 Sn 5 particles are either encompassing the Sn-rich matrix and eutectic or within the eutectic mixture, giving rise to the ternary Sn-Bi-Cu 6 Sn 5 eutectic. This ternary eutectic appears as isolated "islands" between the binary eutectic (Sn-Bi) and Sn-rich dendrites. The binary Sn-Bi eutectic is predominant in the whole microstructure. Fig. 12 shows that Sn (in blue) is heavily concentrated in the Sn-rich phases, but with less intensity in the eutectic region. Bi (in red) is noted likewise described for the Sn-Bi-Cu alloys. In the case of Ag (in green), it can be noted that this element was consumed basically in the formation of Ag 3 Sn particles. It can be seen through of these mappings and by microstructural observations that there is a transition of either morphology or preferential growth region concerning the Ag 3 Sn intermetallic particles. These transitions start to happen around the position P = 50 mm from the metal/mold interface, which corresponds to experimental cooling rate and growth rate of 0.17°C/s and 0.32 mm/s, respectively. For higher values of these solidification thermal parameters and considering Ag content higher than 2 wt.%Ag, the presence of larger plate-like Ag 3 Sn particles dispersed through the Sn-rich matrix and the eutectic mixture may prevail. In contrast, finer Ag 3 Sn particles dispersed only in the eutectic regions as shown in Fig. 12 tend to grow for Ṫ L b 0.17°C/s associated with an alloy Ag concentration around 1.0 wt.%.
Thus, the comprehension of solidification thermal parameters affecting the eventual prevalence of one of the cited Ag 3 Sn morphologies is of prime importance considering the final control of the solder microstructure, as well as of the application properties (mechanical, chemical and physical) and reliability of the Sn-Bi-Ag alloys.
A comparison of the experimental profiles for cooling and growth rates during solidification of the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag alloys can be observed in Fig. 13. In general, the experimental results regarding Ṫ L permit to stress that a large range of cooling rates is possible to be obtained by directional solidification of the ternary alloys. The experimental readings for the two alloys resulted in solidification cooling rates varying from 9.2°C/s to 0.05°C/s, which is a typical range of cooling rates in soldering processes.
It can be observed that Ṫ L and V L experimental values for the ternary Sn-Bi-Cu and Sn-Bi-Ag directionally solidified solders are close. Experimental fittings have been inserted in Fig. 13 in order to give a realistic indication of possible values considering positions which were not monitored within the castings. Fig. 14 shows the evolutions of primary (λ 1 ), tertiary (λ 3 ) and secondary (λ 2 ) dendritic arm spacings as a function of cooling rate (Ṫ L ) and growth rate (V L ) for both the Sn-34 wt.%Bi-0.7 wt.%Cu and the Sn-33 wt.%Bi-2 wt.%Ag alloys. Each average microstructural spacing has been determined along with its standard variation. The coefficients of determination, R 2 , have been inserted in order to indicate how well the linear relationships fitted the experimental scatter. The lines represent empirical power laws fitted to the experimental points. Also, the experimental laws suggested by Osório et al. [6] for the binary Sn-40 wt.%Bi solder have been inserted into the graph of Fig. 14c for comparison purposes. Also, the effects of Ag and Cu could be inferred through such comparison.
It is worth noting that both ternary alloys have developed a substantial fraction of eutectic mixture during solidification. So, the classic Jackson and Hunt growth law for eutectics [43] seems to be applicable to these alloys. Considering that Ṫ L is given by a constant × V L 2 [44], and substituting this expression on that of Jackson and Hunt, λ 1 becomes directly proportional to − 1/4 (Fig. 14a). As λ 3 has the same nature of growth found in primary dendritic branches, the same exponent becomes valid. In the case of the ternary Sn-Bi-Ag alloy a second experimental growth law is required for λ 1 due to the highly Agsegregated region close to the cooled surface of the casting. For cooling rates higher than 0.3°C/s a −0.55 exponent must be adopted. In sum, the Sn-33 wt.%Bi-2 wt.%Ag alloy may be associated with two experimental fittings considering the complete range of experimental Ṫ L .
Higher values of Ṫ L and higher Ag content are conducive to a higher fraction of primary intermetallics, which consume the solute inducing the need for lateral segregation, thus increasing local instabilities leading to the formation of new primary branches. A single experimental power law, λ 3 = 20 (Ṫ L ) − 1/4 has been adopted to describe the tertiary dendrite arm spacing evolution with the cooling rate. For any Ṫ L value, λ 3 is roughly 2 and 3 times lower than λ 1 for Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag alloys, respectively. It seems that the magnitude of the tertiary arm spacing is not affected by the type and content of element added to the Sn-Bi alloy, i.e., Cu or Ag.
The secondary dendrite arm spacing (λ 2 ) is usually related to V L or t SL (local solidification time) as reported by Kurz and Fisher [45], Feurer [46], Mortensen [47] and Bouchard-Kirkaldy [48]. − 2/3 and − 1.1 power laws characterize the experimental variation of λ 2 with V L for the Sn-Bi-Cu and the Sn-Bi-Ag solder alloys, respectively. The − 2/3 for λ 2 × V L relationships is that suggested by the theoretical growth model of Bouchard and Kirkaldy [44], and by studies involving other lead-free solders as Sn-Ag-Cu [15], Sn-Bi and Sn-Ag alloys [6]. Even though the same exponent could be adopted for the binary Sn-40 wt.%Bi and the ternary Sn-34 wt.%Bi-0.7 wt.%Cu, another exponent (−1.1) is shown to characterize the ternary Sn-Bi-Ag alloy. In general,   by Rappaz and Boettinger [49]. These authors proposed the only predictive model found in the literature for secondary dendritic growth of multicomponent alloys, which is given by Eq. (1): where t SL is the local solidification time, and M is dependent on the Gibbs-Thomson coefficient (Γ), the liquidus slope (m L ), the final liquid composition (c f ) at the dendrite root (generally assumed to be a eutectic composition), the final solid composition (c o ), the diffusion coefficient in the liquid (D) and the partition coefficient (k). Detailed information concerning this model used in the present study can be found in the work by Rappaz and Boettinger." with " M is given by the Eq. (2): The aim is to verify if this model is able to encompass experimental results of unsteady-state solidification for the ternary Sn-Bi-Cu and Sn-Bi-Ag alloys, using the thermophysical properties listed in Table 1.
It can be seen that the predictions furnished by the model are generally located slightly above the experimental average λ 2 scatter for both Sn-Bi-Cu and Sn-Bi-Ag solder alloys. Indeed, this theoretical approach can reasonably represent the experimental λ 2 evolution.

Conclusions
From the results obtained in this study, the following conclusions can be drawn: • A fully dendritic arrangement can be observed for both ternary Sn-Bi-Cu and Sn-Bi-Ag solder alloys. The as-cast microstructures for these alloys are arranged by Sn-rich dendrites with Bi precipitates in their own core surrounded by a complex eutectic mixture (Bi-rich and Sn-rich phases) and their respective intermetallic compounds, Cu 6 Sn 5 and Ag 3 Sn. The primary Cu 6 Sn 5 and Ag 3 Sn IMCs are nonhomogeneously distributed along the microstructure. Also, distribution and fraction of Bi precipitates within of Sn-rich dendrites was shown to be uneven along the entire length of the solidified casting. Additions of 0.7 wt.%Cu and 2 wt.%Ag in the binary Sn-Bi alloy tends to decrease the secondary dendrite arm spacing; • The elemental mappings for the Sn-34 wt.%Bi-0.7 wt.%Cu and Sn-33 wt.%Bi-2 wt.%Ag solders showed that Bi is either concentrated in the eutectic mixture, or as particles dispersed in the Sn-rich matrix.
The third elements such as Cu and Ag were consumed in the  formation of Cu 6 Sn 5 and Ag 3 Sn intermetallic compounds. SEM images depict that both Cu 6 Sn 5 and Ag 3 Sn particles can give rise to ternary Sn-Bi-Cu 6 Sn 5 and Sn-Bi-Ag 3 Sn eutectics. These ternary eutectics appear as isolated "islands" between the binary eutectic (Sn-Bi) and Sn-rich dendrites, being the binary Sn-Bi eutectic observed predominantly in the whole microstructure. The alloy Ag content, Ṫ L and V L were found to be determinant on the growth of the Ag 3 Sn intermetallics, either as a coarse primary particle or being part of the eutectic mixture as a finer phase; • Experimental growth laws were proposed relating the primary/ tertiary, and secondary dendrite arm spacings, to the experimental thermal parameters, Ṫ L and V L , respectively. A single experimental interrelation has been proposed for the tertiary dendritic growth considering both alloys examined. In the case of the experimental evolution of λ 1 along the length of the Sn-Bi-Ag alloy casting, it has been found that two experimental fittings should be considered for the complete range of experimental Ṫ L . Higher values of Ṫ L and higher Ag content are conducive to a higher fraction of primary intermetallic particles, which consume the solute inducing the need for lateral segregation, thus increasing local instabilities leading to the formation of new primary branches. These laws are not the only approach permitting the design of the scale of the dendritic pattern of such alloys as a function of solidification thermal parameters, since the Rappaz-Boettinger model reasonably estimated the experimental scatter of secondary dendrite arm spacings.