Solder joints are utilized in microelectronic and solar photovoltaic (PV) panel manufacturing industries to provide electrical, thermal and mechanical continuities in electronic devices and assemblies. Overall integral performance of devices and circuits is dependent on the strength and reliability of solder joints. The growing popularity of Pb-free solder alloys is attributed to the benefit they provide to the environment and human-health. A large number of lead- free solders have been developed to replace Sn-Pb solders and among them Sn-xAg-yCu (SAC) solders are the most widely studied one. As compared to the traditional Sn-Pb solder alloys, Sn-xAg-yCu solder contains a significantly higher proportion of Sn (usually larger than 90 wt%) [1]. When SAC/Cu system is used for forming solder joints, the larger proportion of Sn in the SAC solder (compared to Sn-Pb system) causes a more pronounced growth of Cu₆Sn₅ intermetallic compound (IMC) at the interface. As interfacial IMCs are brittle [2], and their occurrence is more prominent in context of SAC solders; any research work studying about interfacial IMC contributes to the understanding of the reliability of solder joints. IMC occupy an increasing proportion of the SAC solder volume in miniaturized Pb-free SAC joints, and thus impact the reliability of the electronic devices [3].

The mechanical behavior of a solder joint is affected by the microstructure and morphology of the interfacial intermetallic compounds (IMCs). The shear strength of a solder joint is dependent on the microstructure not only of the bulk solder matrix but also of the interfacial IMCs. In the shear test performed in the solder joints at a shear height of 10 μm, Yang et al [4] have found out that the shear force for solders with prismatic IMCs is 17.22 N, whereas the shear force for those with scalloped IMCs is 16.03 N. Thus, the solder joints with prismatic IMCs have higher or better shear strength than those with scalloped IMCs.

Laser soldering is a joining technology of solder/substrate characterized by the application of heat only on the desired spot precisely. The moving heat source during laser processing is controlled by a computer and this highly advanced selective soldering can enhance the production rate in joining technology [5]. Especially in context of joint formation at printed circuit boards (PCBs) which consists of many thermally sensitive components, laser soldering has proved to be advantageous over conventional reflow soldering. The attributes such as precisely focused laser beam, localized and non-contact heating, rapid rise and fall in temperature, large cooling rate, short duration etc. make laser soldering a better choice over reflow soldering [[6], [7], [8]]. Nishikawa and Iwata [9] have found out that the solder joints processed by laser has superior impact reliability than those prepared by conventional reflow soldering. Moreover, the growing popularity of laser in flexible electronics [10], additive manufacturing (3D printing) [[11], [12], [13], [14]] and biomaterials fabrication [15] sectors, has attracted a great attention from researchers, and study of laser heat source on any materials system can aid in understanding the mechanism of laser-materials interaction.

The microstructure evolution at the interface of the solder is dependent on the laser-material interaction. The laser processing parameters namely power (P), scan speed (V_scan) etc. and materials composition mainly define the resulting interfacial microstructure of the SAC/Cu system. The transient and spatially non-uniform temperature profiles during laser soldering lead to interfacial microstructural evolution totally different from that of conventional isothermal reflow soldering [16]. The formation of interfacial IMCs of two different morphologies of IMC-prismatic and scalloped, at the interface of Sn/Cu and Sn-3.5Ag-0.5Cu/Cu systems, due to the alteration in the value of laser power and scan speed has been reported in [17]. However, the work uses only two solder compositions and so the solder alloy composition dependence of IMC could not be outlined quantitatively from the study. Provided that solder compositions of more types are introduced for performing laser experiments, a data-driven method can be utilized to infer the exact relationship between the laser parameters, solder composition and the resultant IMC morphology. Lee et al. [13] have utilized data analytics approach for melt-pool geometries in additive manufacturing. The targets for the melt pool geometries (width, depth, area, height etc.) and input features such as powder chemistry, thermal properties, powder size distribution and laser parameters (powder, scan speed, energy density etc.) were obtained from the experiments. In the research of Wang et al. [18] finite element method -generated (FEM-generated) data was utilized for performing the machine learning (ML) task of thermal transport analysis in Aluminum alloys with precipitate morphology.

In this present study, experimental works are performed to observe the morphology of interfacial Cu₆Sn₅ IMC in Sn-xAg-yCu solders (with 6 different alloy composition features) which are processed at different laser power and scan speed to form solder joints with Cu substrate. In line with the experimental laser power, scan speed and solder composition, finite element analysis is performed to create 51 observations based datasets on laser parameters, solder composition and temperature dependent Jackson parameter (αJ). The fem-obtained datasets are used in a neural network (NN) to predict the IMC morphology based upon the numerical values of αJ. The machine learning based prediction of IMC morphology is validated with experimental results, and critical scan speed required for morphology transition is determined. As this research work integrates together the results from experiments, computational work and data analysis; it can be helpful in the sector of advanced soldering materials design in accordance to the objective of materials genome initiative (MGI) [19].

In SAC solder alloys consisting of Ag, another intermetallic compound, namely Ag₃Sn is formed along with Cu₆Sn₅ compound in the aftermath of laser soldering [17]. Ag₃Sn intermetallic nanoparticles are found to be adsorbed on the top of micrometer sized Cu₆Sn₅ IMC. Although Ag₃Sn intermetallic compound influences the growth kinetics of the overall Cu₆Sn₅ IMC layer, its role on morphology transition of Cu₆Sn₅ IMC grains is not described adequately in the existing research works. Thus, in this work, the role of Ag₃Sn formation with regard to the morphological evolution of Cu₆Sn₅ IMC grains of SAC solders is assumed negligible. The assessment of Ag₃Sn nanoparticles in relation to the morphology of individual Cu₆Sn₅ IMC grains is a topic beyond the scope of this present work. Similarly, Cu₃Sn intermetallic compound is formed between Cu layer and Cu₆Sn₅ IMC layer only after prolonged aging for several hours at temperature below melting point of solder, and thus Cu₃Sn intermetallic formation can be neglected for experimental samples characterized immediately at the aftermath of laser soldering (with total duration of about a few seconds). Thus, only Cu₆Sn₅ compound is studied as the intermetallic compound at the interface of solder and Cu substrate in the present study. With this consideration, the acronym IMC used in this manuscript, solely represents Cu₆Sn₅ intermetallic compound and not the other types.

The schematic sketch for the experimental setup for laser soldering is shown in Fig. 1. Pb-free solder sheet with dimensions of 4 mm × 2 mm × 0.2 mm was placed over polycrystalline Cu substrate (5 mm × 5 mm × 0.1 mm). In our previous work [17], only pure Sn and Sn-3.5Ag-0.5Cu solders were considered for study, whereas this works aiming to infer the effect of solder composition of a generic Sn-xAg-yCu solder, additionally considers Sn-0.7Cu, Sn-3.5Ag, Sn-3.0Ag-0.5Cu and Sn-3.5Ag-0.7Cu solders. Thus, the types of solders considered in this study are outlined in Table 1, with their respective Ag and Cu weight fraction based (m_i) composition. Since the sum of the weight fraction of n components in a multicomponent system is equal to 1, n-1 components are sufficient to describe the composition of the system. In a ternary Ag-Cu-Sn system, m_Sn = 1-(m_Ag+m_Cu); the weight fractions of Ag and Cu are sufficient to describe the composition of Sn-xAg-yCu solder types.

An SP-50C Fiber laser (wavelength = 1528-1565 nm, pulse frequency = 5000 Hz, focal beam diameter = 25 μm, and pulse width = 2.0 × 10^-4 s) was utilized for heating the solder surface. As shown in Fig. 1, the laser beam was aligned perpendicular to the top surface of the Sn-xAg-yCu solder material. Laser output power (P) and scan speed (V_scan) are the two major laser processing parameters, which when varied can influence the materials properties and interfacial microstructure of the SAC/Cu interface. In this study, the values of power were varied as 30 W, 40 W and 50 W. Similarly, depending upon the input power, the magnitudes of scan speed were varied from 10-240 mm/min with a step of 10 mm/min. In order to prevent ablation, scan speed greater than 70 mm/min were used with 50 W power. In the same way, for assurance of joint formation, V_scan ≤ 150 mm/min were used with 30 W power (verification needed). For an example, combination of 50 W and 10 mm/min would lead to the ablation of solder surface whereas 30 W and 240 mm/min could lead to the absence of Cu₆Sn₅ IMC formation at the interface (no joint formation).

For laser processing of SAC solders with Cu substrate at suitable P and V_scan, the interface melts thereby accelerating the interfacial reaction between the solder and substrate. In such condition, Cu₆Sn₅ intermetallic compound (IMC) is the predominant one, and for the rest of this work, IMC implies the Cu₆Sn₅ compound. After the laser soldering of different Sn-xAg-yCu solder types processed with different combinations of P, V_scan, top view morphology of IMC was characterized for observation with scanning electron microscope (SEM). For this purpose, the laser processed samples (joints) were etched in 10% HNO₃ solution, and then mounted in epoxy. These samples were polished with SiC sandpapers (different grit sizes), followed by synthetic diamond polishing paste.

Fig. 2 shows the classification of top view morphologies of Cu₆Sn₅ IMCs in accordance to two stereotypes, namely scalloped and prismatic morphologies for various experimental solder types (Sn, Sn-0.7Cu, Sn-3.5Ag and Sn-3.5Ag-0.7Cu). For the given magnitudes of power (P) and scan speed (V_scan) mentioned in Fig. 2(a), all of the 4 types of solders possess IMCs with scalloped morphology. However, when P and V_scan of values mentioned in Fig. 2(b) are utilized during the laser processing, IMCs of all these solder alloys are found to be of prismatic morphology. Since, larger magnitude of V_scan is responsible for reducing temperature (T) at interface, whereas increase in P is favorable for increasing T (the reason is provided in section: Numerical); these parameters have opposite effects when it comes to IMC morphology. In context of pure Sn solder, Cu₆Sn₅ IMC has scalloped morphology when laser power is 30 W and scan speed is 40 mm/min. Now, when V_scan = 50 mm/min, the increase of power to 40 W has caused the IMC to bear prismatic morphology. In case of Sn-0.7Cu solder, at P = 30 W; the IMCs possess scalloped and prismatic/faceted morphologies at scan speeds of 70 mm/min and 30 mm/min, respectively. In other words, for 30 W of input laser power, the increase in V_scan from 30 to 70 mm/min caused the transition in IMC morphology from prismatic to scalloped. For Sn-3.5Ag solder alloys, at 30 W and very low scan speed of 20 mm/min, the morphology is prismatic. However, even at a very high magnitude of power (P = 50 W), the IMCs are scalloped at extremely larger scan speed of 220 mm/min. Finally, for Sn-3.5Ag-0.7Cu solder laser processed at 50 W, the IMCs transition from prismatic to scalloped when V_scan is reduced from 130 to 90 mm/min.

The intertwined relationship between solder composition and laser processing parameters (P, V_scan) is revealed also in Fig. 3. In case of Fig. 3(a), an experiment performed in Sn-0.7Cu/Cu system at 30 W and 70 mm/min reveals an IMC with scalloped morphology. Now, keeping power constant, if V_scan is reduced to 50 mm/min, the morphology is found to be prismatic. On the other hand, at original scan speed of 70 mm/min and increasing the power to 40 W also makes the morphology appear prismatic. As shown in Fig. 3(b), for laser processing at constant power of 40 W; Sn-3.0Ag-0.5Cu/Cu system produces IMCs with prismatic morphology at V_scan = 100 mm/min, whereas those of Sn/Cu system have a scalloped morphology even at a significantly lower scan speed of 70 mm/min. For experiments performed at constant power of 50 W, IMC corresponding to V_scan = 130 mm/min in case of Sn-3.5Ag-0.7Cu solder alloy possesses scalloped morphology whereas solders such as Sn-3.5Ag and Sn-3.5Ag-0.5Cu have prismatic IMCs even at relatively larger scan speeds of 160 mm/min and 140 mm/min respectively. Thus, Fig. 3(b) suggests that in addition to laser processing parameters, the composition of Sn-xAg-yCu solder has a significant role in deciding the morphology of the resultant IMC during laser soldering of Sn-xAg-yCu/Cu system.

In conventional reflow soldering of a given solder/substrate system, the morphology transition regime for IMCs from scalloped to prismatic or vice-versa can be described by a single magnitude of isothermal reflow temperature. For N experiments of a solder alloy reflowed at m isothermal reflow temperatures, ith temperature is defined as a critical temperature for IMC morphology transition if the IMC bears scalloped morphology below that temperature and prismatic or faceted morphology above that temperature [4]. In case of laser soldering, the temperature in the solder domain is neither uniform nor steady. The spatial and temporal variation in alloy temperature during laser processing, makes it difficult to define a transition temperature. The temperature of the melt pool (near the laser beam spot) is far larger than the distant interface; and even the temperature within the interface is changing with time. So, in this study, instead of reaction temperature, the magnitudes of laser processing parameters (P and V_scan), which influence the temperature of the solder medium, are used as markers for morphology transition. For a given solder composition, and at a given power, the increase in scan speed above a threshold value (V_scan,cr) will cause the morphology transition from prismatic to scalloped. Similarly, for given solder composition and scan speed, an increase in power above the critical value P_cr will cause morphology transition from scalloped to prismatic.

The attainment of a given IMC morphology at a Sn-xAg-yCu/Cu interface can be best defined by Jackson parameter (αJ) [20]. In accordance to the work of Choi et al [21], αJ is an indication of the degree of attachment of atoms to the growing interface, and this parameter predicts whether the interface becomes scalloped (αJ<2) or faceted (αJ>2) In other words, the Cu₆Sn₅ crystals with αJ<2 are known to have atomically rough interfaces and grow with a non-faceted front, whereas the IMC crystals with αJ>2 possess atomically smooth interfaces with ledges and grow with a faceted front [3]. For the purpose of defining Cu₆Sn₅ compound, Jackson parameter can defined mathematically by the following expression [4,21]:

where ΔH_ir is the enthalpy increment or change during the interfacial reaction of liquid solder and the Cu substrate to form IMC, T_if is the temperature at the interface of liquid solder and Cu during laser soldering, and it is not a constant, R is the universal gas constant, $\xi$ is a fraction of the total number of nearest neighbors in a plane parallel to the considered interface; and this parameter depends on the crystal structure of IMCs and random oriental relationship between adjacent IMC grains. Although there is a common agreement in several research works [21,22] that the value of $\xi$ is always greater than 0 and less than 1, the exact value of this parameter has never been reported. In [4,21,22], mentioning random oriental relationship between evolving IMCs as the major hindrance to calculate the exact value of $\xi$, an average value of 0.5 has been assigned to this parameter.

The enthalpy increment (ΔH_ir) corresponding to interfacial reaction at solder substrate interface, can be expressed as a polynomial function of T_if [23,24] :

Combining Eqs. (1) and (2), Jackson parameter can be rewritten as:

Using the experimental values of ΔH_ir for Cu₆Sn₅ IMC formation from the works of Choi et al. [21], Flandorfer et al. [25] and Yang et al. [4] have computed the polynomial coefficients A, B, C of Eq. (2) and reported A, B and C to have the values of 2.074 × 10⁴ J/mol, -83.25 J/(mol K) and 0.121 J/(mol K²) respectively. Putting these values in Eq. (3), it can be outlined that for any T_if ≤ 414 K, αJ monotonically increases with temperature (T_if).

With the predetermined values of R, A, B, C and $\xi$, Eq. (3), can be utilized to numerically predict the resulting morphology of IMC during laser soldering if T_if is known. Unlike isothermal reflow soldering, the presence of dynamic heat source from laser beam during processing renders the T_if non-uniform and transient. The measurement of such variable is very difficult and one of the possible ways to determine the transient and non-uniform temperature is using finite element analysis (FEA). The numerical results of temperature at a localized region of the interface during the entire timesteps can be averaged to produce the effective temperature denoted as $\overline{{{T}_{\text{if}}}}$.

As shown in Fig. 1, the laser beam is focused on the top surface of the solder alloy, and since the main purpose of numerical analysis is to determine the temperature at the bottom part of the solder (interface between the solder and Cu); a 2D rectangle of length 4 mm and depth (height) 0.2 mm (200 μm) is sufficient to represent the computational domain of the numerical simulation. Thus, the green colored region in Fig. 1, representing the solder alloy is the computational domain of the finite element model. It is assumed that the temperature profiles are symmetrical along the width of solder (that is in the direction perpendicular to laser motion). Triangular mesh elements are used for constructing the finite element mesh.

The partial differential equation (PDE) for phase change based heat transfer during laser soldering in a Sn-xAg-yCu solder [17]:

Where in T is temperature and t is time. In the equation, ρ is the density of the solder alloy. The effective heat capacity (C_eff) of the Sn-xAg-yCu alloy is a function of enthalpy (H) and is described as follows [17]:

The enthalpy in Eq. (5) is defined as $H(T)=\underset{0}{\overset{T}{\mathop \int }}\,\left( \rho {{C}_{\text{p}}}+\rho {{L}_{\text{f}}}\frac{\partial f}{\partial \tau } \right)\text{d}\tau$, where C_p is the specific heat capacity of solder alloy at constant pressure, L_f is the latent heat of fusion of the alloy and f(T) is the temperature dependent liquid-fraction. Enthalpy values for Sn, Sn-0.7Cu, Sn-3.5Ag, Sn-3.0Ag-0.5Cu, Sn-3.5Ag-0.5Cu and Sn-3.5Ag-0.7Cu alloys have been computed as function of T using OpenCalphad software [26,27], and the results are shown in Fig. 4 below. The computed values are in agreement with the experimental results of Morando et al [28]. As depicted in the H-T plot, the vertical lines of the H curves which represent the phase change regime correspond to different T values, implying that the solder alloys have different melting temperatures (T_m). Pure Sn has the highest melting temperature of 505.15 K, whereas Sn-Ag-Cu solders have the lowest melting point of 491 K. Sn-0.7Cu and Sn-3.5Ag alloys are shown in the figure to have vertical lines intermediate between pure Sn and Sn-Ag-Cu solders, and thus have intermediate melting temperatures. This difference in melting temperature is very significant for the temperature evolution during laser heating. It is important to note that the difference in enthalpies measured by the length of the vertical lines of the H-T plot in the two phase regimes, is defined as the latent heat of fusion for a given solder alloy. Within the left hand side of Eq. (4), the second term inside the brackets represents the convection heat transfer (which is more dominant in the region around the melt pool). The fluid flow regime described by velocity (($\vec{u}$) and pressure (p) variables, in the laser melted solder regime is mathematically expressed by the incompressible Navier-Stokes equations assuming Boussinesq approximation, mentioned as following.

The viscosity of the fluid is denoted by ν in Eq. (7). During laser processing, at a given time (t), the regimes melted by laser heat are in liquid phase, whereas some portion of the solder quite distant may still be in BCT phase, and thermophysical properties for the two phases are quite different. The enthalpy method [29], described by Eq. (4) uses the enthalpy difference between liquid and Body-Centered Tetragonal (BCT) phases, to distinguish the phases in the numerical model. The temperature dependent materials properties can then be implemented separately for the solid BCT and liquid phases. The first term in the right-hand side (RHS) of Eq. (4) is the thermal convection term, and k_th is the thermal conductivity of the solder alloy. The second term in the RHS of the heat equation is the heat source term due to laser irradiation, and thus Q_laser is the laser heat generation [30]. Assuming the intensity profile of the laser beam exhibits the Gaussian function, and the laser material interaction is characterized by Beer’s law, Q_laser for the context of 2D simulation, can be expressed by the following equation [31]:

where Ω in Eq. (8) is the absorption coefficient of the solder material (μm^-1) [32], P is the input power and r₀ is the radius of the laser beam. For a laser initially positioned at x₀ and moving with a given scan speed (V_scan), the radial distance (r) from the axis of the beam is defined as r = (x₀ + tV_scan). Now, it is obvious from Eq. (8) that the laser power (P) increases the magnitude of Q_laser, whereas scan speed appearing in the negative exponent term, produces the opposite effect. Subsequently, in accordance to Eq. (4), increasing P helps to raise the medium temperature whereas increasing V_scan will lower the solder temperature. Hence, a combination of the effects of P and V_scan, affect the interfacial reaction temperature (T_if), which will subsequently guide the numerical value of Jackson parameter.

The materials properties used in the energy equation (Eq. 4), continuity and momentum equations (Eqs. (6) and (7)), namely, density, thermal conductivity and viscosity are treated as temperature dependent functions. As defined in [29], ρ and k_th are assumed to be have linear dependence on T. The density is assumed to be a continuous function of temperature in both solid and liquid phases. Thermal conductivity function on the other hand, having continuity in each of solid-state temperatures and liquid state temperatures, is treated to have discontinuity in the two phase regime. In accordance to the experimental results of thermal conductivity measurements by Meydaneri et al. [33], the liquid phase in equilibrium with the BCT phase, in the two phase regime, is assigned to have its thermal conductivity 1.10 times higher than that of solid phase even at the same temperature. Arrhenius equation is utilized to describe the temperature dependence of ν in liquid phase, asymptotic viscosity is utilized in mushy region (two-phase region) and BCT phase is described as a region with a very high viscosity [29]. In enthalpy method, which is a typical Eulerian method, the numerical magnitudes of liquid phase viscosity (≈ 1 mPa S) and solid state BCT phase (> 1 Pa S) renders the effective implementation of Navier-Stokes equation in liquid phase only. The values of material properties for the Sn-xAg-yCu solders provided in Table 2, are obtained from [29,[32], [33], [34], [35], [36], [37], [38], [39]]. The thermal expansivity (β) of solders [17], appearing in body force term (F_boussinesq) of Eq. (7), is utilized as a constant in the model. The absorption coefficient (Ω) is taken to be same for all solder alloys, and is chosen for a solder paste containing flux [32].

The initial temperature of the computational domain is taken as 298.15 K. The velocity of the medium is taken as zero initially, and pressure is assumed to be equal to that of atmosphere.

Robin boundary conditions (BCs) are imposed for T variable at the boundaries of the numerical domain, in accordance to the following equation:

where h_n is the convection heat transfer coefficient for a given boundary material n, and T_ext is the external temperature of the air film. In the top, left and right sides of the domain exposed to air, the, convection heat transfer coefficient (h_air) is taken to be equal to 15 W/(m² K). In the bottom side of the domain touching the Cu substrate, the following time-dependent interfacial heat transfer coefficient (h_int) is employed in Eq. (9).

Assuming Cu as a moderately smooth surface, amplitude $\theta$ = 10,000 and the time-exponent m = 0.03 [40].

In context of velocity variable, no flux boundary condition (($\frac{\partial \vec{u}}{\partial n}=0$) is chosen for the top side of the solder exposed to air. No slip BC is applied at the bottom side of the solder. With the kinematic condition of no fluid flow across the rigid boundary at the bottom, no slip BC can be mathematically written in the following simplified form:

With Cu substrate at rest (u→substrate = 0), Eq. (11) can be rewritten as:

The set of Eqs. (4),(6) and (7) along with the provided material properties, initial conditions and boundary conditions, are solved using finite element method (FEM) in Elmer Multiphysics software [[41], [42], [43], [44]]. Since the thickness of the solder is 200 μm and the length is 4 mm; the simulations have been performed at the length scale of 1.0 × 10^-4 m; and corresponding to this length scale, the maximum timestep size for Elmer Solver based simulations was kept as 5.0 × 10^-2 s. 51 finite element simulations were performed with variation of solder types, P and V_scan. The total or gross simulation time set for the simulations ranged from 1 s (corresponding to largest V_scan = 240 mm/min), and 12.0 s (for the lowest scan speed of 20 mm/min). Using the corresponding solder alloy’s m_Ag and m_Cu and applied heat source’s P and V_scan in Eq. (4), a total number of 51 finite element method-based simulations were performed to determine the temperature profile in the solder models.

Fig. 5 shows the temperature distribution in context of pure Sn solder at t = 1.0 s for several combinations of P and V_scan. As shown by the H-T curves for pure Sn in Fig. 4, the melting point of Sn is around 505.15 K, and thus the red colored regions in Fig. 5 located at the vicinity of laser beam spot are all in the molten state. Even the yellow and yellow-green colored portion (corresponding to temperature beyond 505.15 K) is in liquid state. The blue-green and blue colored portion, which denotes T sufficiently less than 500 K, is clearly illustrates that the material is in solid state Body-Centered Tetragonal (BCT) phase. The figure shows that largest mass of liquid solder attains higher temperature (in accordance to the size of red colored regime) in context of laser parameters of 40 W power and 60 mm/min scan speed (case of highest P magnitude). It is followed by the model with P = 30 W and V_scan = 50 mm/min (case of lowest V_scan). The suppressing effect on T raise, due to lowering of power, while keeping the scan speed constant at 60 mm/min can be realized by observing the sizes of red colored regime in the two images- the second image from top (30 W, 60 mm/min) and the bottom-most image (40 W, 60 mm/min). Finally, when the power is kept the lowest at 30 W and scan speed is kept far larger (170 mm/min), where there are no red and yellow colored regimes in the region around the laser beam. A small part of the liquid around the beam spot reaches just above the melting temperature (yellow-green part), and the initial part of the solder is already at solid state.

The interfacial Cu₆Sn₅ grows at the bottom part of the solder, and the temperature profile (T_if) at the point $\chi$ of Fig. 1 is the property of significant important in this study. Fig. 6 shows the temporal variation of T_if at the bottom point $\chi$ for pure Sn processed with the previously discussed four combinations of P and V_scan. Although the temperature on the top surface at the vicinity of the laser beam is the highest for the case 40 W, 60 mm/min, the time averaged bottom point T_if is the largest for the one with smaller scan velocity (case 30 W, 50 mm/min represented by black colored curved in the figure). Although the T_if values for case 40 W, 60 mm/min (blue colored curve) are initially larger, the larger scan velocity effectively puts the laser beam radially farther from point $\chi$, thus causing a decrease in heating rate in accordance to Eq. (8). The bottom point interface temperature at the same point, is significantly lower in case of laser processing with 30 W, 60 mm/min (red colored curve), but still the solder material reaches the temperature beyond the melting point. With 30 W, 170 mm/min, the average value of T_if is lesser than 400 K, far below the melting temperature of the solder. Although the region around the vicinity of laser beam spot melts, it is clear from the figure that the bottom part of the solder does not reach its melting point. The time-averaged values of bottom corner temperature ((${{\bar{T}}_{\text{if}}}$) for the solder models are 562.73 K (30 W, 50 mm/min), 510.46 K (30 W, 60 mm/min), 389.64 K (30 W, 170 mm/min) and 556.74 K (40 W, 60 mm/min). Fig. 7 presents the temperature distribution at t = 2.0 s in different solder types for P = 30 W, V_scan = 60 mm/min. As the laser beam is already halfway from the initial position at 2 s, the left end extremities of the solders have already cooled down significantly. Thus, to observe the minute differences between the temperature values at the left extremities, the temperature color bar is scaled within the range of 500-600 K for the domain within the first 1 mm from the left and is shown in Fig. 7(a). The temperature profile for the complete length of the solders is exhibited in Fig. 7(b). From Fig. 7(a), it can be understood that extremities temperature is the largest for Sn-0.7Cu solder and the smallest for pure Sn. The Sn-3.5Ag and Sn-3.5Ag0.7Cu solders have the temperature values at the left end (x = 0) intermediate between Sn-0.7Cu and pure Sn alloys. The interfacial region’s temperature i.e. T_if for point χ for these solders is graphically plotted along with time in Fig. 8. It can be noticed in this figure, that for 0.5 s < t < 1.5 s, the temperature at the left bottom corner corresponding to pure Sn solder is in the range 504-506 K (melting point of pure Sn = 505.15 K); whereas in the same duration the temperature values of other solder types are quite higher and they have lower melting point than that of pure Sn. The highest temperature value corresponds to that of Sn-0.7Cu solder, which reaches 550 K at 1.5 s. The time-averaged values of bottom corner temperature ((${{\bar{T}}_{\text{if}}}$) for the different Sn-xAg-yCu solder types at P = 30 W and V_scan = 60 mm/min are 510.46 K (pure Sn), 533.59 K (Sn-0.7Cu), 519.20 K (Sn-3.5Ag) and 519.04 K (Sn-3.5Ag-0.7Cu).

As outlined in Section 4.3, the different magnitudes of ${\bar{T}}_{{if}}$ corresponding to laser soldering models of different solder types at same laser processing parameters (P, V_scan) or same solder types subjected to different P and V_scan, would surely impact the rate and behavior of Cu₆Sn₅ IMC grain growth at the interface. This situation will also naturally imply to different solder types irradiated with varying conditions of P and V_scan. In order to understand how the value of interfacial temperature affects the grain growth behavior of IMCs, a multiphase field simulation [45,46] is performed at two constant interface temperatures T_if of 453.15 K and 523.15 K as shown in Fig. 9, Fig. 10. For the simplification of the model, the content of Ag in the phases is considered to be zero, so that the system can be assumed as a binary Cu-Sn system. The phase field simulation is done using finite element method in Multiphysics Object Oriented Simulation Environment (MOOSE) Framework [47,48]. A rectangle of size 0.17 μm × 0.4 μm is taken as the size of the interface consisting of Sn-rich BCT/Liquid phase (yellowish region), IMC grains 1 and 2 (yellow-green region) and Cu-rich FCC (blue region) phases in Fig. 9. The phases are mathematically represented by non-conserved order parameters ((${{\eta }_{i}}$). The initial size of the identical IMC grains (i.e. at t = 0) are 40 nm × 40 nm (0.040 μm × 40 μm). The FCC phase is considered to have an initial height of 0.15 μm. Parabolic free energy expressions are utilized to describe the bulk free energy of the phases. The values of diffusivities for the phases at 453.15 and 523.15 K are used as provided in [49]. The diffusivity of liquid phase at 523.15 K is 2.0 × 10^-12 m²/s whereas that of BCT phase at 453.15 K is 2.0 × 10^-16 m²/s. The large difference in the magnitudes of phase diffusivities, corresponding to solder being whether in liquid phase or BCT phase, influences the growth rate of IMC grains. In the figure, the vertical lengths of a grown IMC grain at t = 1.75 h and t = 10 h (T_if =453.15 K) are 0.067 μm and 0.105 μm respectively. On the other hand, the vertical length of grain at t = 0.51 s (T_if = 523.15 K) is about 0.097 μm, and this is just slightly lower than the solid-state grain’s length attained at 10 h. The corresponding length of a grain at t = 1.01 s (T_if = 523.15 K) is 0.153 μm. In the case of grain growth in liquid solder media, since the solid IMC grains provide relative barrier to each other on lateral expansion, the vertical length of two identical grains grows at a faster rate than the lateral growth. In contrary to this, the grain growth rate in solid state solder is nearly equal in both directions. The differential rate of grain growth behavior in lateral and vertical directions for simulation performed at two different temperatures, is even more vividly revealed by the graphical plot of sum of squares of all the order parameters (($\underset{i}{\mathop \sum }\,{{({{\eta }_{i}})}^{2}}$) (shown in Fig. 10). The actual laser soldering process lasts for only several seconds, and for such small-time durations, the IMC grains have practically negligible growth rate at 453.15 K. However, it is noteworthy that number of nearest neighbors around the IMC grain corresponding to t = 0-10 s is far lower for interface temperature of 453.15 K than of interface at T = 523.15 K. This definitely lowers the value of $\xi$ in case of interface with smaller temperature. At t = 4 s (end of laser soldering time), the two grains corresponding to interface at 523.15 K, will almost touch each other laterally and undergo ripening; while growing unperturbed in the vertical direction. This produces compressive stress in one hand, and induces screw dislocation mechanism. Now, at higher interface temperatures, the screw dislocation mechanisms are pronounced to produce faceted or prismatic IMCs upon cooling.

With the heat source term defined by P and V_scan, laser soldering produces transient temperatures at the interface. Lower scan speed or higher power are associated with overall larger interfacial temperatures, whereas higher scan speed or lower power result into smaller interfacial temperatures. Thus, the results of the phase field simulations performed at isothermal interfacial temperature values can only be utilized for qualitative understanding of IMC grain growth behavior at such interfaces.

So far now, it has been outlined that the solder alloy composition (m_Ag, m_Cu) and the laser processing parameters (P, V_scan) can influence the values of interface temperature in the solder/substrate. FEM-generated datasets have been prepared to correlate m_Ag, m_Cu, P and V_scan with T_if. The change in values of T_if directly affects the change in enthalpy (ΔH_ir) during IMC formation. The use of different solder types may require the use of different $\xi $ in Eq. (1). It can therefore, be understood that αJ is a continuous function of m_Ag, m_Cu, P, and V_scan. Since the absolute magnitude of Jackson parameter is an indicator of the morphology of IMC (scalloped or faceted), the mapping of solder composition and laser processing parameters to αJ through machine learning, can subsequently help in quantitative association of alloy composition plus corresponding laser processing parameters to the resulting morphology type.

In Eq. (1), αJ is a function of T_if, provided that ξ is determined for a given alloy. The structural parameter can be assumed to be a constant for Cu₆Sn₅ IMC formation in a given solder alloy. In this study, each solder was assigned constant values ranging from 0.5 to 0.75, with an interval of 0.01. Then the FEM calculated time-averaged ${{\bar{T}}_{\text{if}}}$ for point $~\chi $ at different laser processing parameters were utilized with each magnitude of $~\chi $ for a fixed solder composition, to compute αJ. The optimum $~\chi $ for the given calculations, is the value which yields Jackson parameter perfectly mapped to the experimental morphology. This optimization procedure is repeated to all the solder types. Thus, the value of $~\chi $ are 0.665, 0.734, 0.70, 0.675, 0.670 and 0.524 for pure Sn, Sn-0.7Cu, Sn-3.5Ag, Sn-3.0Ag-0.5Cu, Sn-3.5Ag-0.5Cu, Sn-3.5Ag-0.7Cu solder alloys interfaced with polycrystalline Cu substrate. With the use of fem-generated data and experimental data (scanning electron microscopy images of IMC morphology), the optimum values of $~\chi $ is determined.

The FEM-generated datasets with 51 observations, are employed to build a neural network (NN). With solder alloy composition variables (m_Ag, m_Cu) and laser processing variables (P, V_scan) as input features, a NN shown in Fig. 11 is constructed to estimate αJ as output feature. The hyperparameters of machine learning such as number of layers, number of neurons in each layer, activation function, number of epochs, type of cost function, etc. are then supplied to the model. The NN consists of an input layer, hidden layer(s) and an output layer. The input and output layers have neurons sizes of 4 and 1, respectively. In order to choose the NN with optimized predictability, the number of hidden layers and neurons in them is varied as shown in Table 3 below. As Jackson parameter can be considered to be a continuous function of the input variables, the data analysis task can be considered to be clearly a regression problem. Thus, the NN network is defined to work as a regression-based feed forward neural network. The major benefit of using regression NN is that it can provide fair accuracy even in context of tiny and small datasets [50,51]. Rectified linear unit (ReLU) activation function has been utilized to activate the neurons of a hidden layer or both hidden layers, whereas linear activation function is used at the output layer.

For an input z, ReLU activation function is defined as f(z) = max{0, z}. In this study, the weight fractions of Ag and Cu considered in the data for solder composition, are well within the range of 0-1. However, laser power is in the range of 30-50. The scan speed ranges from 20 mm/min to 240 mm/min in the datasets. In such contexts, the use of datasets in unprocessed or raw form, may lead V_scan data (having the largest range) to have dominance over the results of neural network analysis. In order to avoid such hegemony of a given input parameter over the result [52,53], and avoid erroneous prediction from the network; a process called normalization can be utilized. Considering that X_i is an element of a column array consisting an input or output parameter of the NN, then normalization of the data is expressed by the following formula:

where $\bar{X}$ is the normalized value for a parameter’s element of actual value X_i. X_max and X_min are respectively the maximum and minimum values of the parameter. With the normalization procedure via minmax scaler, every elements of m_Ag, m_Cu, P, V_scan, predicted αJ (NN-estimated) and expected αJ (FEM-generated), will be within the range of 0-1. In order to convert the predicted αJ in its absolute magnitude, Eq. (13), is utilized again to compute X_i from $\bar{X}$.It is essential to assess the performance of an artificial neural network (ANN) outside the original sample (training datasets) to new datasets (test or validation data). The method of examining the prediction performance of an ANN for a validation data beyond the training data is defined as cross-validation. The original data consisting of 51 observations was split into training and validation datasets in the ratio of 4:1. The mean square error (MSE) between the predicted output (y_p) of the network and FEM-computed Jackson parameter (expected αJ) of each dataset [54], is defined as following:

Now, this MSE (cost function of the neural network) can be utilized to evaluate the performance (cross-validation) of the ANN first for training and then for validation data. Adam optimizer has been used for the optimization of the NN. With these set of hyperparameters, the neural network model was built, compiled and run in TensorFlow [55] with keras front end [56]. The parameters such as weight matrix ({w_i}) and bias matrix ({b_i}) are learnt during the model run, and these parameters associate input variables with the output variables of the neural network. The MSE values for training and validation data at the end of 1000 epochs for 10 different NN configuration cases are presented in Table 3. From the table, it is obvious that Case VIII having 9 neurons in the first hidden layer and 5 neurons in the second hidden layer has the least MSE magnitudes of MSE for both the test and validation data. The value of MSE for training data for this neural network (Case VIII) is 2.0 × 10^-4, whereas the corresponding validation data has the MSE value of 3.81 × 10^-3. Hence, this neural network is selected for further operations on the datasets. For this chosen neural network, the decreasing trend of MSE values of the training and validation data for a total of 1000 epochs is graphically presented in Fig. 12. As the error reduces for both training and validation data, the neural network can be considered to have acceptable performance for further prediction work.

With successful cross-validation of the neural network, the same NN model is further utilized to estimate the αJ for new datasets consisting of input variables of larger data size. A total of 558 datasets consisting of input variables were employed to perform prediction task. The predicted values of Jackson parameters from the neural network are presented in Fig. 13. As the experimental works have more data of scan speed (ranging from 10 to 240 mm/min) than the laser input power (30-50 W), the scan speed is chosen as a quantity in x-axis of the images of the figure. The numerical values of predicted Jackson parameter are validated with the experimental morphologies corresponding to 51 datasets. The predicted αJ are found to conform with the experimental data (SEM images).

From Fig. 13(a)-(c) it can be inferred that at a fixed laser power magnitude and for a given solder composition, αJ decreases with the increase in scan speed. Moreover, as illustrated in the Fig. 13(a), Sn-0.7Cu solder alloy (red colored curve) has the largest αJ at the vicinity of the dotted horizontal line indicating that it has the highest likelihood to bear prismatic morphology at P = 30 W. However in contexts of P = 40 W (Fig. 13(b)) and P = 50 W (Fig. 13(c)), Sn-3.5Ag solder alloy (green colored curve) has the largest likelihood of bearing prismatic morphology. It can be noticed from Fig. 13(a)-(c), that Jackson parameter for every solder increases with the increment in laser power for a given magnitude of scan speed. For example, in context of Sn-3.5Ag sample processed with V_scan = 20 mm/min, αJ = 2.19 at P = 30 W (Fig. 13(a)), whereas this Jackson parameter has larger magnitudes at higher power values: αJ = 2.668 at P = 40 W (Fig. 13(b)) and αJ = 3.16 at P = 50 W (Fig. 13(c)). The increase in αJ with P, for a given V_scan has a significant implication on the morphological feature of the IMC during laser soldering. IMC transforms into faceted or prismatic grains at αJ > 2, and as scalloped grains at αJ < 2, and this distinction is highlighted as regions separated by a horizontal dotted line corresponding to αJ = 2 in Fig. 13. Thus, at the points of the curves lying above the dotted line IMCs are observed to be bear prismatic morphology whereas they have scalloped morphology at the regions below the dotted line. It is interesting to note that the increment in laser power values also causes an increase in the scan speed values in the x-axes corresponding to the intersection of the dotted lines (αJ = 2) for each Sn-xAg-yCu solder alloy curve. This would help us to understand that at P = 30 W, the interfacial IMCs corresponding to Sn, Sn-3.0Ag-0.5Cu, Sn-3.5Ag-0.5Cu, and Sn-3.5Ag-0.7Cu alloys in Fig. 13(a) are still scalloped for all scan speeds greater than 60 mm/min, whereas at P = 50 W, the IMCs of these 4 solder types have prismatic morphology for V_scan as high as 120 mm/min. The scan speed for a fixed power and solder composition, which acts as a transition boundary between scalloped and prismatic IMC morphologies, has been defined earlier as critical scan speed (V_scan,cr; and thus the increase in solder power will raise the magnitude of the V_scan,cr. The numerical determination of Jackson parameter with the help of neural network analysis, has enabled in numerical quantification of the critical scan speed for a laser processing with given laser parameters and solder/substrate power. In summary, increase in P is characterized with the raise in magnitude of Jackson parameter, whereas increase in V_scan lowers the Jackson parameter.

With the success in estimation of numerical value of Jackson parameter through machine learning, it is possible predict the morphology of a resultant interfacial IMC at a solder/substrate interface for given laser power and scan speed. The prediction of IMC morphology is done by classifying the αJ - V_scan curves of Fig. 13 into two regions. As mentioned in the previous section, the classification is done with the help of horizontal line (Jackson parameter = 2). In another way, the classification can be done by a line AB perpendicular to the scan speed axis of Fig. 14, that intersects a Sn-xAg-yCu curve at Jackson parameter = 2. The scan speed corresponding to this line AB that intersects the curves at αJ = 2 is called the critical scan speed. Any points on the curve located in a region at right side of the line has scalloped morphology, whereas that to the left has a prismatic morphology. Also, any point on the curve at a region upper to the horizontal line (αJ = 2) has prismatic morphology, whereas that below the line has scalloped morphology. With these observations, it can be inferred that a point on the curve lying within yellow region of Fig. 14 is characterized with prismatic Cu₆Sn₅ IMC and a point on the curve within light pink region is featured with scalloped IMC.

Since scan speed is a laser processing parameter, it is intuitive to utilize critical scan speed (V_scan,cr) as a design parameter for interfacial IMC morphology. The scan speed (V_scan) values, depicted in Fig. 13, have positive values and increase along the positive x-axis. Thus, any scalloped interfacial IMC in the solders lying to the right of line AB. of Fig. 14 has V_scan larger than V_scan,cr whereas any prismatic IMC located to the left of the line has V_scan smaller than V_scan,cr. If V_scan,cr can be predicted for a given solder material processed at a given power, it can be used as a design tool to predict the IMC morphology in advance. In order to estimate the value of V_scan,cr for a given solder material at a given P, the prediction NN network is run by using datasets localized at the vicinity of αJ = 2. For a given solder type at a given power, two input datasets of the previous prediction NN-one yielding αJ just greater than 2 and having least difference with 2.0; and another producing αJ just smaller than 2 and having least difference with 2.0 are selected. These two datasets representing a fixed solder composition and fixed power possess same value of m_Ag, m_Cu and P, and they differ by V_scan. Let V_max,local and V_min,local be lower and higher limits of scan speed of these chosen two datasets. A total of 100 datasets are constructed and fed into the predictor NN. In each of these 100 datasets m_Ag, m_Cu and P do not vary; but V_scan lying between V_max,local and V_min,local are changed with a stepping of 0.1 mm/min. The scan speed of the datasets, which predicts the output having the least difference with 2.0 i.e. which corresponds to the smallest value of $\left| {{\alpha }_{\text{J},\text{predict}}}-2 \right|$, is then considered as the optimized value of critical scan speed. The similar process is repeated for different P magnitudes and different solder types.The predicted values of V in different solder types at different laser input power P, are summarized in Fig.15 and Table 4. For all solder types, V_scan,cr is raised with the increase in input laser power, and this increases the probability of occurrence of prismatic IMC morphology at larger laser input power. In case of pure Sn, the critical scan speed at 30 W power is 21.0 mm/min, which means the interfacial IMCs for this condition can retain prismatic morphology only at scan speed lower than 21.0 mm/min. For Sn-3.5Ag/Cu system processed with laser at 50 W, the critical scan speed is 184.3 mm/min, thereby indicating that its interfacial IMCs bear prismatic morphology for all scan speeds lower than 184.3 mm/min; and transform to scallop morphology beyond this critical scan speed. The critical scan speed can be utilized as a reference design parameter to set the power and scan speed parameters for a given alloy, and obtain a desired IMC morphology. From Fig. 15 and Table 4, it can be inferred that Sn-0.7Cu solder has the largest magnitude of critical scan speed (V_scan,cr = 54.4 mm/s) at 30 W input power whereas Sn-3.5Ag alloy has the largest values of critical scan speeds at 40 W and 50 W input power (114.3 mm/min @40 W and 184.3 mm/min @50 W). It implies that these solders retain prismatic IMC morphology even at larger values of scan speed for corresponding power; and thus can be selected as the most preferable solder types in context of applications where joints are exposed to shear loads. The other solders having lower magnitude of critical scan speed are less preferable for applications involving larger shear stress. Since the optimized values of critical scan speed presented in the chart can be utilized for promoting the reliability based design of solder joints in context of shear loads; Fig. 15 can be referred to as a design chart for solders with laser as source of heat.

Similar, to critical scan speed; another design parameter called critical laser power (P_cr) can be also correlated to Jackson parameter = 2.0. The future works on the determination of critical laser input power, relies on the availability of larger experimental data of input power. Moreover, in future, the effect of other processing parameters of the laser, e.g. wavelength, frequency, beam spot diameter, laser types etc. can be varied additionally in the input datasets to assess their influence on the IMC morphology.

This study integrates experiments with finite element analysis and machine learning to design the interfacial Cu₆Sn₅ IMC at the Sn-xAg-yCu/Cu system. The major conclusions from this research work are presented as following:

(1) Interfacial Cu₆Sn₅ morphology was investigated for numerous laser processing experiments at Sn-xAg-yCu/Cu interface. The magnitudes of laser processing parameters-power (P) and scan speed (V_scan) and the composition of the solders-weight percentages of Ag (x = m_Ag) and Cu (y = m_Cu) are found to influence the morphology of the IMC. The morphology of experimental IMCs affected combinatorically by these variables are classified into two types-prismatic and scalloped.

(2) The Jackson parameter (αJ) is chosen to marker to predict the occurrence of two different morphology. The fact that αJ is the mathematical function of interfacial temperature (T_if) of the solder/substrate interface, it is necessary to associate experimental variables with the interfacial temperature. The solder composition influences the enthalpy and other material properties of the heated system, whereas P and V_scan influence the source term of the heat transfer equation.

(3) A 2D finite element method is developed to simulate the spatial and temporal evolution of temperature for 6 solder types at different power and scan speeds. The temperature field at a point in the bottom interface is time-averaged to obtain average temperature of the solder/substrate interface.

(4) The fem-generated datasets is employed to construct a neural network (NN) model and this regression based NN is used to predict Jackson parameter of an IMC profile from the 4 input variables - namely m_Ag, m_Cu, P and V_scan. The condition that prismatic IMCs have αJ > 2 and scalloped ones have αJ < 2 is used for the validation of the prediction results of NN with experimental IMC morphologies. At 30 W, Sn-0.7Cu solder alloy are predicted to have the larger αJ for a given scan speed at the vicinity of morphology transition regime (1.9 < αJ < 2.1). Similarly, at 40 W and 50 W, the Sn-3.5Ag solder alloys is characterized with the largest values of αJ for all values of scan speeds within the regime 1.8 < αJ < 2.2.

(5) The NN predicted αJ values are mapped to the input variables to determine the critical scan speed for a given solder composition exposed to a given laser power. When the laser power is increased, the corresponding critical scan speed for a solder is higher thereby implying the likelihood of prismatic IMC formation. At 30 W input power, Sn-0.7Cu solder has the largest value of critical scan speed (V_scan,cr = 54.4 mm/s). In context of 40 W and 50 W input power, Sn-3.5Ag alloy has critical scan speeds of 114.3 mm/min and 184.3 mm/min respectively, thereby making it the solder type of having the largest V_scan,cr in these two power values. In accordance to this prediction, Sn-0.7Cu/Cu and Sn-3.5Ag/Cu solder joints processed by laser heating in the power range of 30-50 W, are considered the most preferable for applications requiring larger shear strength.

(6) Future work must be creating more experimental data of input laser power, so that the critical power for morphology transition, can also be designed for a broader range of scan speed and solder materials.