The application of an analytical model to solve an inverse heat conduction problem: Transient solidi ﬁ cation of a Sn-Sb peritectic solder alloy on distinct substrates

Three distinct alloy/substrate couples were considered. In order to treat the reaction interface problem e ﬀ ectively, sheets of commercially pure copper (Cu), electrolytic nickel (Ni) and low carbon steel were chosen so that solidi ﬁ cation of a Sn-Sb peritectic alloy could be evaluated comprehending very di ﬀ erent conditions. A straightforward view of the mechanisms a ﬀ ecting the heat transfer e ﬃ ciencies was consistent with a number of techniques applied in the present investigation, which includes directional solidi ﬁ cation experiments, analytical modelling, wettability analyses and characterization of the reactions between the alloy and the substrates. The proposed analytical model was perceptive to these reactions. For the Cu substrate, the motion of Cu towards the alloy was more e ﬀ ective as compared to the motion of Ni from the Ni substrate. As a consequence, the alloy/Cu interface presented a higher level of Kirkendall voids. The higher fraction of voids at the interface resulted in lower interfacial thermal conductance for the Sn-Sb/Cu couple. Hence, the present experimental-theoretical approach is useful to indicate the solder joint integrity in terms of the presence of empty spots. Despite the higher thermal conductivity of Cu and lower contact angle between the alloy and the Cu in comparison to the Ni substrate, the high porosity at the Cu interface during alloy soldering was shown to reduce the heat transfer capability.


Introduction
Pb-based alloys are applied to soldering operations of electronic circuits components.The most applied compositions are those containing 85-97 wt.% of lead (Pb).Such alloys may be applied in high temperature soldering in the manufacture of power semiconductor packaging as well as for soldering consecutive joints (step soldering), interconnection of semiconductor devices and soldered connections [1].Such powers are employed in the automotive, space and aerospace industries [2].
Considering the recent restrictions imposed on the use of Pb due its toxicity, an extensive search has been made for alternative materials to replace Pb solders [3][4][5][6].Within this scope, proposals have been attained considering alternative Sn-based peritectic alloys [2,[7][8][9][10][11].In spite of the potential application and the importance of alloys undergoing peritectic transformation, there is few understanding of the behavior of peritectic reactions.They occur in applications related to steels, magnetic materials and non-ferrous systems [12,13].Indeed, although there have been significant research efforts in the search for Pb-Free Solder Alloys (LFSA) in the last years, only a small portion of these studies has been related to high temperature solder alloys.It is worth noting that such alloys are crucial in many industrial assemblies, such as in the electronic industry for the attachment of semiconductor devices in metallic substrates [1].
In the range of high temperature solder alloys, reliable LFSA have been proposed in order to replace Pb-Sn alloys (from 85 to 97 wt.% of Pb).Consequently, materials formed by stable microstructures without the presence of brittle intermetallic compounds and with melting temperatures between 268 °C and 314 °C have been displayed.High temperature Pb-free solder alloys must be able to prevent soldered joints from remelting during the reflow process.They may also be used in packaging components and optical modules, including LEDs and laser devices [14].
Among several substitute solder alloys (e.g., Au-Sn, Sn-Cu, Au-Ge, Sn-Ag, Zn-Al, Sn-Mg, Zn-Sn, Bi-Ag and Sn-Sb alloy systems), Sn-Sb alloys are recommended candidates to replace high-Pb solders.This is explained by either microstructural stability and/or suitable mechanical properties characterizing such alloys.Sn-Sb alloys for high temperature soldering remain scarcely comprehended especially regarding their non-equilibrium solidification microstructures [1].The Sn-5.5 wt.% Sb solder alloy, considered in the literature as being close to the peritectic composition, has showed higher values of mechanical properties (mechanical strength and ductility) at room temperature and suitable creep resistance [15,16] in relation to Sn-Pb alloys [9].
The properties of soldered joints depend on the final microstructure after the solidification process.The heat transfer across the interface flowing from the solder alloy in direction to the substrate, directly influences the formation of the soldering microstructures.Cheung and collaborators [17] investigating the importance of the transient interfacial heat transfer coefficient at mold walls with different nature influencing the solidification of aluminum alloys evolvement, reported that two distinct surfaces in contact (alloy and substrate) creates a temperature drop.In fact this also depends on other factors such as the thermophysical properties of the contacting materials, the finishing grade of substrate contacting surface, the melt superheat, the solidification range and the wettability.It was demonstrated that the nature of the substrate can significantly affect the kinetics of solidification.
The solder alloy/substrate in soldered joints is the region in which the solder alloy reacts with the substrate leading (in most of the cases) to the formation of intermetallic compounds with different sizes, distribution, chemical stoichiometry and morphology [18,19].Therefore, the metallic substrate fundamentally defines the reliability of a solder joint with a electronic component since it directly affects the formation of the soldered joint.This, in turn, requires understanding the involved metallurgical phenomena, where heat and mass transfer are included.However, the effects of the type of substrate on solder/substrate heat transfer efficiency are scarcely reported in the literature, and absent for Sn-Sb solder alloys to the best knowledge of the present authors.
The ability of a liquid to spread over a solid material is known as wettability [20], which is measured by the contact angle (θ) formed at the solder/substrate [15].Few works in the literature reported contact angle values for Sn-Sb alloys.Kolenak and Kostolny [2] studied the wettability of the Sn-5 wt.%Sb alloy in silver (Ag), copper (Cu) and nickel (Ni) substrates, using a protective atmosphere (90% N 2 + 10% H 2 ).The lowest contact angle values (θ) were obtained for the Cu substrate, whereas for the Ag and Ni substrates the θ values were higher than 90°.Šebo et al. [21] described the wetting behavior of the Sn-5at.%Sb alloy in a Cu substrate, considering two atmospheres: ambient environment condition with flux and deoxidizing gas (N 2 +10H 2 ).The θ values remained quite unaltered if both atmospheres are compared, i.e., θ values between 26.0°and 27.5°.Mahidhara and coauthors [15] reported a wetting angle of 43°in a Cu substrate for the Sn-5 wt.%Sb alloy.El-Bahay et al. [16] measured contact angles of the Sn-5 wt.%Sb solder alloy at 573 K for 60 s in Cu and CuZn substrates.The θ values reached 40°and 41°, respectively.
Previous researches demonstrated that quantitative wetting data (wetting angles values-θ) may be related to the heat flux within the alloy/substrate pair, with heat flux being represented by a global heat transfer coefficient-h g .θ values/ h g /thermal parameters interrelations for LFSA have been determined and reported in the literature for: Zn-Sn [22], Sn-Cu [18,23] and Bi-Ag solder alloys [19].
The present work aims to contrast experimental findings for the wettability, the interface reaction (constituted by substrate -reaction layer -solder alloy) and results of simulations obtained by an analytical heat flow model for the Sn-5.5 wt%Sb alloy solidified over three different substrates: commercially pure copper (Cu), electrolytic nickel (Ni) and low carbon steel.The method of heat transfer solved by using an inverse heat transfer problem solution will be discussed regarding each tested couple.Moreover, an analysis of microstructural features related to the reaction layer such as size, morphology and nature of intermetallic compounds (IMCs) will be outlined.

Solidification experiments
The compositions of tin (Sn) and antimony (Sb) used to make the Sn-5.5 wt% Sb alloy are presented in Table 1.An upward vertical solidification device was used to promote directional solidification of the alloy in the unsteady state regime (Fig. 1).The device consists of the following main parts: heat resisting material to avoid heat losses; controlled electrical resistances used to achieve a desired superheat (10 °C over liquidus temperature) of the molten alloy; a cylindrical mold of stainless steel having 55 mm (internal diameter), 110 mm (height), and 5 mm (wall thickness); interchangeable sheets of different materials (carbon steel, copper, nickel) with a thickness of 3 mm coupled to the bottom part of the mold; a water impingement system providing cooling at the sheet when the molten alloy achieves the superheat and the electrical resistances are switched off.
The temperatures of the casting during solidification were monitored by a group of J type thermocouples, positioned along the casting, having the reference from the heat-extracting surface at the bottom of  the casting.The temperature data were acquired automatically, at the frequency of 5 Hz, through a data logger system connected to the thermocouples.Samples obtained along the length of the directionally solidified (DS) casting were polished and subjected to metallographic etching (with a solution of 2% HCl, 3%NHO 3 and 95% alcohol).The corresponding partial Sn-Sb phase diagram, adapted from Okamoto [24], is shown in Fig. 2. It includes the selected alloy composition (Sn-5.5 wt.%Sb) studied in this work.It is worth to mention the tiny solidification interval of only 3 °C characterizing such alloy (i.e., liquidus and solidus temperatures of 242 °C and 239 °C, respectively).

Wettability tests
Wettability measurements by using a Krüss GmbH goniometer give a guide to the relative Sn-Sb alloy wetting tendencies in Cu, Ni or steel substrates.This can be crucial in the selection of relative alloy /substrate heat transfer coefficients in order to generate data relevant to the soldering practices with the examined alloy.The measurement of the contact angles (θ) using the goniometer (Fig. 3) applied to each situation of solder alloy/ substrate is based on the following procedures: i. Purge of the furnace chamber from oxygen avoiding oxidation on the tested couples, by introducing Argon into the furnace chamber to saturate it with an inert (passive) atmosphere; ii.Set of constant heating rate of 10 K/min in order to attain a steady temperature stage at 290 °C, which was maintained for 20 min, being higher than the liquidus temperature of the alloy; iii.Keeping the solder/substrate couple inside the chamber after covering the solder alloy with flux (rosin mildly activated, RMA); iv.Natural cooling down inside the chamber until room temperature is attained.
Fig. 3 shows the illustration of the method given by the application of a light source in the sample in order to allow the optical system coupled to a camera to process the shape of the molten material.This process permits that the equipment continuously follows the evolvement of the droplet, being accompanied by the measurements of the contact angles.The mentioned analysis was carried out by means of the tangent method 2 of the Kruss drop-shape Analysis program.
The contact angles recorded in the first stages of melting will be considered for the comparative analysis and referred as θ I .The ending stage of the wetting curves may be associated with an equilibrium regime.
Sample preparation, as well as test conditions, are very important since wettability is sensitive to oxidation of the surfaces and to temperature.As such, the surfaces finishing of the substrates used in the goniometer had their finishing standardized with that employed during directional solidification, that is, that provided by a #1200 grit sandpaper.This is important to assuredness when correlating experimental contact angles and the heat transfer coefficient.Three substrate's materials were chosen, that is, commercially pure copper (Cu), electrolytic nickel (Ni) and low 1020 carbon steel.These substrates were produced from the same materials used to machine the sheets assembling the base of the directional solidification setup.Thus, a comparative analysis might be conceivable between the interfaces at the alloy and the substrates in both directional solidification and wettability experiments.Wettability tests performed in triplicate for each alloy/substrate couple summed nine (9) tests.So, nine cylindrical samples were extracted from the DS Sn-5.5 wt%Sb alloy casting and were machined to the dimensions of a 4 mm in diameter and 4 mm in height.

Inverse heat conduction solution: application of an analytical model
The mathematical procedure applied in this work was the inverse method of heat conduction, where the thermal profiles obtained experimentally along the solidification of a Sn5.5 wt% Sb alloy casting are compared to the simulated thermal profiles in order to ascertain the metal-substrate (mold) heat transfer coefficient (h i ) and the cooling rate at the solid-liquid interface (T ˙).
For this proposal, an heat-transfer analytical model developed by [25], which is able to describe the temperature distribution in both liquid and solid states during unidirectional plane-front solidification (e.g.pure metals and eutectics) in cooled molds, was implemented in C ++ programming language and solved in an interactive way.An interactive procedure is used for pursuing a solution for an inverse heat transfer problem based on the minimization of the difference between simulated and experimental thermal profiles.It has succeed on determining the transient heat transfer coefficients.The use of this analytical model was possible considering that the alloy concentration of interest is associated with a very small solidification interval (ΔT≅ 3 K).So, a unique boundary between solid and liquid could be assumed.In addition, the liquid/solid phase transformation occurs much outside thermodynamic equilibrium once a water-cooled mold is employed, thus favoring the liquidus and solidus isotherms to become closer [26].
The analytical model is based on the solution of unsteady-state regime of heat conduction, which is given by the Fourier ' s equation of heat conduction [27]: where T = temperature, t = time, a = thermal diffusivity and x = the heat flow direction.
The development of the exact analytical model by [25] considers a finite virtual thermal resistance at the metal/substrate (mold) interface (R i ) composed by the addition of a virtual layer of previously solidified metal (S o ) to encompass the heat transfer at this boundary.With the use of this pre-existing layer of metal -equivalent to the thermal resistance R i -the differential heat conduction equation can be exactly applicable including the influence of the metal/mold interfacial resistance, since the thermal profile becomes continuous and can be analytically solved.In the original model, the liquid was considered to be at the melting temperature.However, it can be extended to encompass a physical situation in which a temperature profile exists in the liquid from a superheating (ΔT) above the alloy melting temperature T f , leading to the following equations: where T S is temperature in the solid layer, T L is temperature in the molten liquid, T V is the beginning melt temperature, T i is the temperature at the metal/mold interface, T P is T f + ΔT, x indicates the position taking the metal/mold interface as reference, n = (a s /a L ) 1/2 , a s is the solid diffusivity, a L is the liquid diffusivity and φ the argument of solid/liquid displacement obtained by solving the equation: , k s is the solid thermal conductivity, k L is the liquid thermal conductivity and c s is the specific heat of the solid.
The pre-existing layer of metal (S o ) is given by an equation related to the metal-substrate (mold) heat transfer coefficient (h i ): The overall interfacial coefficient (h ov ) can be described in terms of thermal resistances as indicated in Fig. 4.These resistances can be represented by an overall resistance, given by: R ov = R w +R m +R i (6) or, where e = thickness of the bottom sheet (m), k M = sheet thermal conductivity (Wm −1 K −1 );h W = heat transfer coefficient concerning mold/ cooling fluid (Wm The air gap thermal resistance is about 84% of 1/h ov [28].Hence, it is reported that h ov follows the h i behavior which is defined as [17]: where a and b are constants, being b < 0.5.Since h w is high due to the water turbulent regime imposed by the experimental cooling system, the thermal resistance R W can be neglected.Therefore, isolating h i from Eq. 7: The interfacial thermal resistance, R i =

is formed by two parallel resistances represented by contact spots
1 and by air gaps where A is area; subscripts c and g correspond to contact sites and gap, respectively.
The cooling rate determination at the solid-liquid interface can be obtained by the multiplying the thermal gradient (K/s) and the growth rate (m/s), i. e. = T G V ˙. .The derivative of the T s equation (Eq.2) with respect to x, permits G to be determined and, the cooling rate can be expressed as [25]: The thermo-physical properties of the Sn-5.5 wt.%Sb alloy used in the simulations are given in Table 2. Specific heat and latent heat were calculated by ThermoCalc, a computational thermodynamics software, basing on the PChat database.The properties of pure Sn and Sb, taken from the literature [29], were used to compose thermal conductivity and density of the alloy.The thermal conductivity was calculated by weighted average, having as weight basis the nominal alloy composition.The alloy density was calculated using the equation provided by [30]: where C Sn and C Sb are respectively the Sn and Sb contents of the alloy.

Thermal analysis of the solder/substrate interface and of the solidification progress
As shown in the previous section, a set of thermal resistances governs the heat transfer during the solidification process.The efficiency, at which heat is extracted, is directly dictated by the interfacial coefficient at the interface (h i ).The analytical mathematical model used for solving the inverse heat conduction problem, i.e., for determining the overall heat transfer (h ov ), was based on the reduction of the difference between simulated and experimental thermal profiles, considering this difference not more than 1.5%, for several thermocouples positions taking as reference the heat extraction surface at the base of the casting, as shown in Figs.5(a-c).
The overall heat transfer coefficients profiles (h ov ), given by h ov =3100 t −0.1 , h ov =7300 t −0.1 and h ov =9200 t -0.1 for steel, Cu and Ni substrates, respectively, where h ov [W/m 2 K] and t [s], are shown in Fig. 5.It can be realized from the cooling profiles of the alloy solidified over the three substrates, that Cu and Ni substrates have a closer heat extraction efficiency over time, while the steel one, has the worst (Fig. 6).The solidification conditions for the three castings were standardized just having as difference, the nature of the substrate.Curiously the Ni substrate is related to a higher heat extraction capability if compared with other substrates, however, it was envisaged that the Cu substrate should be the best because its thermal conductivity is 6x greater than that of Ni [31].Fig. 7 shows simulated cooling rate profiles against experimental data for copper, nickel and steel substrates.Experimental cooling rates are taken as the time derivative of the thermal profiles provided by the thermocouples (slope of the cooling curve) at the liquidus temperature.It can be realized that the theoretical profile fits well the experimental data, which gives consistency to the higher simulated cooling rate values for positions near to the metal/bottom-sheet region.This particular region remains object of interest for soldering, since in industrial environments any thermal instrumentation based on contact would be  highly intrusive in small volumes, as those of soldering balls.The mathematical model emerges as a useful tool to provide an idea of scale at which the cooling rates develops.In this sense, it is possible to affirm that cooling rates of the magnitude of 70 °C/s and 30 °C/s are reached for the cases of solidification against both copper/nickel and steel substrates, respectively.Figs.7(a-c) include also some Sn-5.5 wt%Sb alloy transverse microstructures of some positions in the casting.It can be seen that the microstructures are mostly alike for a same cooling rate value.A more careful analysis at the metal/bottom sheet region will be presented in Sub-section 4.3.

Wetting behavior of the tested couples
Wetting of solder alloys against surfaces is an intricate and important phenomenon that changes the interfacial microstructure and hence the reliability of a solder joint.The solder alloy may react with a small amount of the base metal.Consequently, it wets the metal inducing intermetallic compounds (IMC) to be formed.The importance of wetting in soldering practice goes further than that.If the solid is properly wet, it means that the solder alloy may reach an appropriate fluid state and may collate satisfactorily to pads and component leads.If some degree of non-wetting is obtained, only partial attach of the solder to a surface will be ensued, leaving exposed substrate behind.
The wetting curves for spreading of the Sn-5.5 wt.%Sb alloy on Cu, Ni and steel substrates are presented in Fig. 8.Each experiment was repeated three times for approximately 600 s.After this period the change in the contact angle can be considered negligible.For all the examined substrate surface materials, decrease in contact angle of the Sn-5.5 wt.%Sb alloy was relatively sharp for the first 50 s and then the spreading of the solder practically ceased.The initial contact angle of the Sn-Sb alloy exhibited variation with the substrate surface materials.
The initial contact angles, θi, are inserted in the plots as a comparison between the three analyzed conditions.At the first wetting stages, changes on the alloy/substrate interaction might be very representative of the degree of wet as well as the attachment reliability between the two exposed surfaces.
It is well known that a reliable joint depends on good wetting between the solder alloy and work pieces.The initial contact angles determined in this investigation for the Sn-5.5 wt%Sb alloy in copper, nickel and steel are 25.7°, 39.1°and 45.1°, respectively.Equilibrium contact angle values of about 15°, 27°and 21°were obtained on Cu, Ni and steel substrates.Hence, the degree of wetting of the couples in the last stages of contact can be considered roughly following the same relative Sn-Sb alloy wetting tendencies in Cu, Ni or steel substrates in the first stages.It appears that the spreading of molten solder on Cu and Ni substrates occurred in a more uniform way as compared to that associated with the Sn-Sb/steel couple.This is because the θ scatter plot in Fig. 8(c) showed some non-uniformity during the equilibrium stage as well as larger standard deviation error bars.Some non-uniformity really happened for the Sn-Sb/steel couple.Similar experimental fluctuations were reported by Zhang et al [32].This study demonstrated that some non-stability may be due to convection currents within the Fig. 6.Sn-5.5 wt.%Sb overall heat transfer coefficients behavior along the time, against nickel, copper and steel substrates.alloy drop.The Sn-Sb alloy/steel interface is the only with no formation of a reaction layer among the three conditions evaluated in the present investigation.This could favor convection currents to happen more intensely.As a result, a final stage of higher unsteadiness could be achieved during measurements.
The lowest wettability behavior has been realized for the Sn-Sb alloy/steel couple since it showed the highest contact angle of the three examined couples at the first stages.According to this criterion, it can be said that the highest wettability refers to the alloy/copper couple.
Even though previous works demonstrated that the initial wetting stages are used to be correlated with heat transfer capability between eutectic forming alloys and substrate [22,23], it does not appear to be possible to use such approach here.The correlation between solder/ substrate heat transfer coefficient, h OV (as determined in the last section) and the wetting of the couple is very weak.Hence, lower θ I related to copper may not sustain higher heat transfer efficiency, which in the present tests, is associated with the solder/nickel couple.
The inverse heat transfer solution as showed in the previous sections of this study permitted time dependent functions of the form h ov =a (t) −b to be generated in the examined conditions.The multipliers 'a' in steel, nickel and copper are 3100; 9200 and 7300 W.m -2 .K -1 whereas the contact angles are 45.1°, 39.1°and 25.7°, respectively.
The use of h-related multipliers to express the wettability of liquid layers in contact with substrates is not possible considering the present results with a peritectic alloy.This opens for discussions on the behavior of h with respect to the kind of material employed for substrate.The next section of this article will bring suitable explanations of the interfacial mechanisms that resulted in h SnSb/Ni > h SnSb/Cu > h SnSb/Steel .
It is important to mention that the literature related to the behavior of solder alloys comprising a peritectic reaction is scarce on correlations showing the effects of the wettability in the interfacial heat flux as a function of the material nature.It can be denoted a gap to be explored concerning the microstructure evolution of the solder joint as a function of the heat flux through the solder alloy/substrate interface.

Formation of reaction layers, their IMCs and defects
Fig. 9(a) is the SEM image of the Sn-5.5 wt%Sb/Ni couple at the joint of the directionally solidified sample.As shown in Fig. 9(a), two reaction phases are observed.According to EDS analyses, their compositions are Sn-62 at.%Ni (point #1), and Sn-5.1 at.%Sb-42 at.%Ni comprising 58 at.% (Sn + Sb) and 42 at.%Ni (point #2).By comparing to the results demonstrated by Chen and Chen [33], the darkest and thinnest reaction phase adjacent to the Ni substrate (point #1) is found to be the Ni 3 Sn 2 IMC.In this case, the EDS data appears to suggest a transitional zone due to diffusion from Ni towards the IMC layer.The following reaction phase is found to be the Ni 3 Sn 4 phase with a thickness of about 3.1 μm.Due to the content of Sb, it could be described as the Ni 3 (Sn, Sb) 4 phase.
For regions farther from the interface (i.e., point # 3 and point #4 in Fig. 9(a)), the bulk alloy compositions (at.%) are found to be Sb19Sn76.5Ni4.5 and Sb17Sn81.3Ni1.7.Ni is not found for upper positions.This means that Ni dissolution into the alloy penetrated of about 10μm thickness from the solder/sheet interface.
Fig. 9(b) shows the Sn-Sb/Cu couple SEM microstructure at the joint of the directionally solidified sample.It can be observed a formation of one reaction layer of more irregular morphology.A composition of Sn-2.0 at.%Sb-35.7 at.%Cu at point #1 suggests that the lighter phase adjacent to the Cu substrate is the Cu 6 Sn 5 IMC with 2.0 at% Sb, although the analysis shows a higher Sn-content than supposed to [34].The average thickness of the Cu 6 Sn 5 layer is of about 7.0 μm.The same analysis conducted at the region of the metal/substrate was not able to be performed for solidified alloy against the steel sheet because they do not generated a joint, i.e., the solder alloy did not bond to the steel, which was already expected, since it is not a substrate material for soldering.
The Cu content for a region farther from the interface (that is, 18μm thickness from the Cu interface in Fig. 9(b)) is found to be 8.1 at.%Cu.Cu atoms seem to diffuse faster into the solder when compared to those of Ni.This appears to be related with the higher thickness of the intermetallic compounds generated at the interface in the Sb-Sn/Cu system.
Beyond the nature of the formed phases, the integrity of the interfaces must be examined permitting a more complete evaluation in terms of heat transfer efficiency.The defects at the interface can affect the local thermal resistance and, as a consequence, the heat transfer efficiency.Based on this evidence, the measurement of the areas occupied by the cavities formed at interface (Fig. 9) was performed.The average areas occupied by the cavities were determined by quantifying the area fractions of the alloy/nickel and alloy/copper interfaces using the ImageJ processing software, which is an image processing program and based on open source Java inspired by NIH Image [35].The software has a threshold tool, which is able to minimize the image noise and measurements of the delimited areas have been performed in 10 different regions at the interface.The area fraction is calculated by dividing the total measured areas for each interface.It was found that the fraction of cavities is 45% larger in the alloy/copper interface.
According to the reference [31] copper, nickel and steel has respectively thermal conductivities of 385, 60.7 and 25.3 W.m −1 .K −1 , the time dependent h i profiles (Eq.9), i.e., excluding the thermal resistance of the mold, are shown in Fig. 9(c).The increase in cavities (term A g of Eq. 10) impairs the connection between the alloy and the substrate, i.e., reduction in the A c term of Eq. 10, provoking the decrease of the heat transfer capacity through the interface.
On the basis of the evidence currently available, it seems fair to suggest that Cu atoms diffuse faster across a layer of Cu-Sn IMC than the Sn atoms.This phenomenon is denominated as the Kirkendall effect [36].The missing Sn-sites on the IMC-layer become vacancies, which, in order to reduce the total Gibbs energy, tend to accrete, forming the interfacial voids or cavities.Fig. 10 presents a simplified sketch illustrating the Kirkendall phenomenon in the case of the substrate of copper.The motion of copper appears to be much more effective than that of nickel.This is also proved when one contrasts the Ni dissolution in the liquid of 10μm thickness from the interface with the dissolution of 18μm thickness of Cu.This occurs because of the difference in diffusion rates of the involved metal atoms.
Even counting with a higher wettability and higher thermal conductivity, the higher density of cavities across the alloy/copper interface induced lower heat transfer efficiency (i.e., h i ) to be attained [37,38] as compared to h i related to the alloy/nickel interface.In this sense, it can be affirmed that the application of a heat transfer mathematical model is sensible to reactions occurring at the metal/substrate region becoming an important tool to indicate the integrity of the soldering joints in terms of the presence of voids.
All phases identified constituting the IMC interfaces of the Sn-Sb/Ni  and Sn-Sb/Cu couples are ratified by XRD analyses in Fig. 11.Fig. 11 c shows the XRD diffractogram of the sample solidified against steel and as expected, only phases the Sn-Sb system have been detected, indicating that dissolution of the steel has not occurred as well no IMC having iron in its composition has been detected.

Conclusions
• Sn-5.5 wt.% Sb solder alloy / substrate heat transfer coefficients were determined for three different substrate materials.For that purpose, an analytical model was successfully developed permitting the inverse heat conduction problem to be solved.The overall coefficients were given by: Steel substrate: h ov = 3100 t −0.1 ; Copper substrate h ov = 7300 t −0.1 ; Nickel substrate h ov = 9200 t −0.1 where h ov [W/m 2 K] and t [s].
• The multipliers of the proposed equations for h ov may not enable an obvious correlation with the contact angles between the Sn-5.5 wt.% Sb solder alloy and the tested substrates, which were 25.7°, 39.1°a nd 45.1°in copper, nickel and steel substrates, respectively.It can be inferred that the alloy/copper couple characterizes the highest wettability.In the case of this set of soldered couples, the h ov multipliers may not be admitted as indicative parameters of the wettability.
• The prevalent intermetallic layer of the Sn-5.5 wt.%Sb alloy/Ni couple was shown to be formed by the Ni 3 Sn 4 phase while a thicker Cu 6 Sn 5 film developed at the Cu interface.For both tested couples during solidification, occurred dissolution of Ni or Cu to the molten Sn-Sb alloy.However, the Cu 6 Sn 5 IMC layer permitted a higher intensity of dissolution with respect to the molten alloy.
• After careful examination of the solder/substrates interfaces, Kirkendall forces were observed to be the dominant factor generating vacancies, so that the quality of the interface was reduced in the alloy/Cu connection.In contrast, a reaction layer was not observed at the Sn-Sb alloy/steel interface.The higher area fraction of cavities in the interface region of the Sn-Sb/Cu connection resulted in lower heat transfer efficiency as compared to that characterizing the Sn-Sb/Ni interface.

Fig. 1 .
Fig. 1.Schematics of the cylindrical split mold used to generate upward directional solidification and the interchangeable bottom sheets made of copper, nickel and 1020 AISI carbon steel.

Fig. 5 .
Fig. 5. Experimental cooling profiles and numerical simulations utilized for ascertain the transient h ov functions of the Sn-5.5 wt.%Sb alloy considering different substrates (a) Nickel; (b) Copper; (c) AISI 1020 steel.

Fig. 7 .
Fig. 7. Comparison between simulated and experimental cooling rates for the Sn-5.5 wt.%Sb alloy solidified against (a) Nickel; (b) Copper; (c) AISI 1020 steel substrates.Transverse microstructures of some positions in the casting.

Fig. 10 .
Fig. 10.Illustrative demonstration of atomic fluxes for a single droplet on a substrate considering the case of a Sn-rich alloy/copper couple.

Table 1
Chemical compositions (wt.%) of metals used to make the Sn-5.5 wt.%Sb alloy.