An analytical and numerical study of droplet formation and break-off for jetting of dense suspensions

Transcription

1 DEGREE PROJECT IN MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2016 An analytical and numerical study of droplet formation and break-off for jetting of dense suspensions KURIAN JOMY VACHAPARAMBIL KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

2 An analytical and numerical study of droplet formation and break-off for jetting of dense suspensions Thesis submitted to Royal Institute of Technology for the fulfillment of Master s degree in Fluid Mechanics AUTHOR: Kurian Jomy Vachaparambil SUPERVISOR: Dr.Gustaf Mårtensson, Mycronic AB,Täby, Sweden EXAMINER: Prof.Luca Brandt, Department of Mechanics, Royal Institute of Technology DATE OF SUBMISSION: 25 August 2016 Royal Institute of Technology Stockholm, Sweden

3 Abstract The jet printing of solder paste from a fluid dynamics perspective involves viscosity change due to varying shear rate and eventual break off of the ejected solder paste droplet from the fluid in the printer head. The ability to model the jetting process in a simulation package is important as it can be used as a tool for future development of the jetting device. The jetting process is modelled as a two phase (air - solder paste) flow with interface tracking performed using phase field method and temporal stepping based on a second-order Backward Difference Formula with relaxed tolerences. This thesis investigates the droplet morphology, volume and speed predictions for three different piston actuation modes and solder paste viscosity definitions given by the Carreau- Yasuda model. A Darcy condition with the porosity parameter Φ is calibrated equal to unity such that the droplet speed is within the realistic range of 20 m/s - 30 m/s. The simulations are compared against previous simulation results from IBOFlow, performed within a collaboration between Mycronic AB and Fraunhofer-Chalmers Centre. As the Carreau models cannot capture the dependence of the fluid viscosity of flow history, an indirect structure based viscosity model is used to compare the thixotopic behaviour. The expressions for the parameters of the structure based viscosity model are derived based on an analytical model which assumes that shear rate is constant. Experimental data for constant shear rate is curve fitted on a Carreau model and an initial estimate of the parameters are obtained. The parameters are then adjusted to match experimental thixotopic behaviour. This method can be used to obtain parameter values for structure based viscosity models for fluids with no previous data. Once the solder paste is ejected through the nozzle and the piston retracts, the fluid undergoes stretching. Studying filament stretching during jetting is difficult as it can be driven by both droplet and piston motion. The data from an extensional rheometer is analyzed to study the filament stretching phenomenon for solder pastes. An analytical model for the critical aspect ratio is derived for a Newtonian fluid filament undergoing a pure extension and modelled as a cylinder whose radius is decreases with time. The exponential decrease of the filament radius predicted by the analytical model is found to reproduce the experimental observations very well. The filament radius calculated based on the filament height from the experiments and analytical model shows that the model 1

4 captures the stretching process, but the formation of beads usually seen in suspensions is not accounted for. Keywords: Thixotropy, Jetting, COMSOL, Solder paste filaments 2

5 Acknowledgement I would like to thank Mycronic AB for providing the opportunity to work on my master thesis with them. I would like to thank my supervisor at Mycronic AB, Gustaf Mårtensson for his continuous support and mentoring throughout the duration of the thesis. My gratitude goes to Malin Wahlberg and Lars Essén for the help to setup and run the rotational rheometer. I would like to thank Kezhao Xing for providing rheological data for the solder pastes and Andrzej Karawajczyk for helping me out in understanding the results from IBOFlow. I would like to express my gratitude to Luca Brandt, KTH Mechanics who suggested this thesis opportunity and the rest of the R&D team who made my time at Mycronic memorable. I would also like to acknowledge the support given by Magnus Björkman and Niklas Rom, COMSOL AB to improve the setup of the simulations. And finally, I would like to thank my family who supported me during my masters education. 3

6 Contents List of Figures 6 List of Tables 8 1 Introduction Objectives & scope Outline of the thesis Theory Rheology of solder paste Carreau-Yasuda model Structure based viscosity models Thinning and breakdown of thixotropic fluids Interface tracking method Geometry and mesh generation Geometry Meshing Adaptive meshing Computational setup Material Numerical framework Two-phase laminar flow Interface thickness Boundary conditions Implementing the motion of the piston Scaling of variables Viscosity models Carreau-Yasuda model Indirect structure based viscosity model Fine tuning the parameters

7 5.2.2 Implementation in COMSOL Elongated filaments Critical aspect ratio Processing data from extensional rheometry Quality and independence studies Domain independence Mesh independence Effect of including the boundary layer Mass conservation Tolerances Temporal discretization Spatial discretization Comparison of methods to actuate the piston Results & discussion Comparison with IBOFlow simulations Effect of piston actuation Effect of Viscosity models Comparison of viscosity models implemented in COMSOL Indirect structure based viscosity model in COMSOL Filament dynamics Conclusions 63 5

8 List of Figures 1 Drawings of the jet printer head(obtained from[6]) Geometries of the printer head used in the simulations. All dimensions are in millimeters Sectioning of the domain used in meshing (not to scale) Adaptive refinement of mesh A image of solder sphere obtained from scanning electron microscope(sem) 21 6 Image of droplet taken by camera[10]. The image captures droplet at two different times using a stroboscope to extract the droplet speed Boundary conditions used in the simulations Piston actuation profiles Carreau model fitted to experimental steady state shear rate data Extensional behaviour of solder paste considered for the modelling Domain independence studies Mesh convergence studies Change in the droplet mass with time Effect of changing absolute tolerance on the solution Comparison of the discretization/element order Comparison of methods of piston actuation Comparison of IBOFlow and COMSOL Comparison of droplet volumes (V) for realistic piston actuation Formation of secondary droplet during piston actuation (Wave 1) Comparison of droplet velocities (U) for realistic piston actuation Comparison of droplet morphology for different piston actuations (not to scale) Comparison of solder paste jetting for different piston actuations Comparison of viscosity models Comparison of structure based viscosity models Comparison of viscosity models based on droplet speed Comparison of viscosity models based on droplet volume Comparison of droplet morphology for different viscosity models

9 28 Viscosity in the droplet Jetting modelled using structure based viscosity model using new parameter (Table.9) and ζ t=0 = Comparison of droplet volume for tuned and new parameter lists Necking when probe speed is 200mm/s Variation of critical aspect ratio with probe speed Filament thinning for a probe speed of 0.2m/s Comparison of necking Estimated break off time for the filament

10 List of Tables 1 Mesh parameters used to create the initial mesh Parameter list used in the simulations Scaling factors used in COMSOL Parameter list for Carreau-Yasuda model obtained from [10] Parameter list for Carreau-Yasuda model obtained from [11] Parameter list for Carreau-Yasuda model obtained through curve fitting 30 7 Initial estimate of parameters for Indirect structure based viscosity model 30 8 Fine tuned list of parameters for Indirect structure based viscosity model 31 9 New list of parameters for Indirect structure based viscosity model Variation of the radius of the filament based on the analytical relation Comparison between the experimental and analytical filament radius Comparison between the experimental and analytical relaxation time

11 1 Introduction The concept of Scaling of circuits was introduced by Douglas Engelbart in 1960[1] and later Gordon Moore predicted that by 1975 a silicon wafer one fourth of an inch would house components per integrated circuit as a result of miniaturization of components[2]. Dr.Moore revised the rate of circuit density doubling in Progress in Digital Integrated Electronics in 1975, which later came to be known as the Moore s Law [3]. These days with increasing circuit complexity, depositing the right amount of solder paste for components both large and small on the printed circuit board (PCB) has become extremely important. One of the most common methods to dispense solder paste is screen printing, due to its relatively lower costs and ease of controlling the ejected droplet volume. But statistical analysis has shown that defects in this method like misalignment accounts for 64% of process related errors[4]. An alternative to the screen printing is the jet printing, which involves ejecting solder paste out of the injector head by applying force. Thus, jet printing offers better control over the ejected volume of the solder paste and better repeatability which is critical for the mass production of PCB. Though there is continuous research and development in the field of jet printing, most of the techniques are patent protected due to the competitive nature of the electronics industry. Mycronic AB is a Swedish company that produces jet printers and constantly develops new techniques to enhance the precision and speed of jet printing [5][6][7][8]. The company introduced their first jet printer almost a decade ago and the latest addition to their product portfolio is the MY600 jet printer. MY600 has a printing frequency of more than one million dots per hour and the dot size can vary from a minimum of 330µm to maximum of 520µm. The jet printer s success is based on its ability of modulate pressure in the printing head and eject the predefined amount of solder paste while the printer head is in motion. The jetting process is driven by a piezo element that undergoes deformation through the application of an electrical signal, which driving the piston. The motion of the piston alters the pressure in the printer head resulting in ejection of the solder paste through the nozzle. To improve the performance of MY600, a deep understanding of the non-newtonian behaviour of the solder paste during the operating conditions must be gained. One of 9

12 the earliest work in this regard was based on the Rayleigh-Plateau instability to predict the relation between orifice sizes and the characteristic time for necking to simulate the delivery of molten solder from a piston driven system, which were compared to experimental data[9]. Another relevant numerical study is a collaboration between Mycronic AB and Fraunhofer-Chalmers Centre (FCC) that produced a simulation tool, IBOFlow, capable of modeling the jetting behaviour[10][11]. These simulations were able to capture the dynamics of droplet formation and reproduce droplet velocities similar to the experimental data. Other than these works, there is a serious dearth of publicly available research on numerical simulation of the jetting process in solder paste. 1.1 Objectives & scope The objective of this thesis is to provide a deeper understanding of the jetting process of the solder paste which is necessary for further advancement of the jet printer technology. Both numerical and experimental studies have been performed to understand the flow conditions in the printer head, droplet formation and the behaviour of the solder paste under operating conditions. The results from the numerical simulations are validated by results from IBOFlow and the analytical models are compared to experimental data. The research questions that this thesis hopes to answer are listed below. 1. Can COMSOL Multiphysics produce results comparable to IBOFlow? The first question is important because if proven to be capable, COMSOL can be used to model the jetting process. Since IBOFlow has already been validated against experiments, comparison against it can be seen as the test for COMSOL s modeling capability. 2. What are the effects of viscosity models on droplet morphology and speed? The second question is important as it sheds light into how the viscosity model affects the droplet and how well the thixotropic behaviour of the solder paste is captured. 3. How does the piston motion effect the droplet? The piston motion effects the droplet characteristics like volume, shape and speed, so the third question which compares multiple piston actuations is important. 10

13 4. How does the solder paste filament behave during extensional stresses? Although, a lot of studies have been performed on the behaviour of complex fluids undergoing extensional stresses, there has been no study that particularly worked with solder paste. The final question aims to provide a preliminary insight into the behaviour of solder paste undergoing extension. The thesis does not deal with the interaction between the solder paste droplet ejected from the printer head and the PCB, but focuses only on the ejected droplet. The main limitation of the thesis is that the solder paste though it consists of two separate phases (solder spheres and flux), is modelled as a continuous medium. This approach has been shown to work in replicating the shear dependent behaviour of the solder paste in [10]. While the simulations performed using IBOFlow have utilized Fluid- Structure Interaction to model the deformation of the printer head wall (especially the lower chamber wall), this thesis will not delve into that. The thesis only deals with the geometry where the lower chamber wall is 300µm, but the simulations in IBOFlow have been done for a wall thickness of both 300µm and 600µm. Due to the restrictions in the computational resources, the geometry used for the simulations has been simplified to a 2D axisymmetric model. 1.2 Outline of the thesis The thesis starts with a brief introduction into the rheological behaviour of solder paste and the interface tracking method used in the CFD simulations, after which the geometry of the printer head used for the analysis is explained. This is followed by the meshing strategy used in the computations. The material properties of the solder paste and the CFD methodology is mentioned in Computational setup. Thereafter parameters used in the viscosity models, such as the Carreau-Yasuda model and the Indirect structure based viscosity model, are explained. The analytical derivation to describe the critical aspect ratio is elaborated in Elongated filaments. The next section consists of the quality studies undertaken to understand the errors in the simulations. This is followed by a results and discussion of the data from the COMSOL simulations and the analytical model for critical aspect ratio. 11

14 2 Theory The jetting of solder paste occurs as the piston motion changes the viscosity of the fluid based on the shear rate in the chamber. The downward motion of the piston reduces the viscosity of the solder paste and ejects it out through the nozzle. Once the piston starts to retract, the fluid that has been ejected starts to stretch. In this flow regime, the extensional viscosity plays an important role. The first part of this section deals with a brief introduction to rheological behaviour of solder paste and concludes with some insight into the scheme used to track interface between air and solder paste. 2.1 Rheology of solder paste Solder paste normally consists of solder (alloys of Sn-Pb(Tin-Lead) and SAC (Tin- Silver-Copper)) added to a carrier liquid called flux. The solder paste is thixotropic in nature[12]. The most accepted definition of thixotropy is the gradual decrease of viscosity under shear stress, followed by a gradual recovery of structure when the stress is removed [13]. This behaviour of solder paste has been investigated through hysteresis-loop and steady shear rate tests in the work done by Durairaj et al.[14]. Thixotropic materials exhibit a dependence of microstructure on flow history. Microstructure refers to the spatial distribution of the particles (solder spheres in the case of solder paste) at a given time. This arrangement is based on equilibrium between Brownian motion, interparticle forces and hydrodynamic forces. At rest, the particles either assume a state of equilibrium which is dictated by the dominant Brownian motion or form structures based on the overall inter-particle forces. This results in extensive associations leading to higher viscosity. When forces are applied on the fluid, the associations formed previously are broken and particles are rearranged resulting in lower viscosity. For example, if at rest, the particles forms flocs due to interparticle attraction, the application of stresses would enable the breakdown of the flocs and reduce the viscosity[15]. Barnes et al. mentions the microstructure evolution from a Brownian motion dominant microstructure at rest to particle layers that results in reducing viscosity[13]. One of the important characteristics of thixotropic materials is that the time scales associated with breaking down a microstructure are much faster than rebuilding the microstruture. Further insight into 12

15 the phenomena of thixotropy, its manifestations and mechanism are available in several existing texts[15][16]. Modelling the thixotropic behaviour has been a very challenging problem due to its dependence on microstructure. Some of the common models used in predicting the viscosity variation with shear rate are explained below Carreau-Yasuda model The Carreau-Yasuda model is widely used to simulate the behaviour of non-newtonian fluids especially shear-thinning and is represented as Eq.1, where λ is a time constant, n is the exponent, η 0 is zero shear viscosity and η is the infinite shear viscosity. The infinite shear viscosity and zero shear viscosity are defined as shown in Eq.2 and Eq.3 respectively. This model does not consider the effects of microstructural changes. η = η + (η 0 η )(1 + (λ γ) 2 ) (n 1)/2 (1) σ lim γ γ = η (2) σ lim γ 0 γ = η 0 (3) The values of zero shear viscosity and infinite shear viscosity are determined based on these plateau regions, i.e. η is calculated from the high shear plateau and η 0 from low shear plateau Structure based viscosity models As explained before the inclusion of microstructure and its evolution is important to understand the time varying behaviour of the viscosity of the fluid. There has been continuous research in this area of thixotropy[15][17]. Two types of structure models are mentioned extensively in the literature, namely direct and indirect structure models[17]. Direct structural models use the number of links between the particles to compute the structural change, whereas indirect structural models use a scalar structural parameter ζ. The scalar structural parameter takes values between zero and one, where zero 13

16 corresponds to a total lack of a structural network and the value one to a continuous network[17]. An indirect structure model mentioned in the work of Mujumdar et al.[17] is shown in Eq.4 as the time derivative of ζ for indirect structure model. dζ dt = k f γ a (1 ζ) k b γ b ζ, (4) where k f, k b, a and b represent a formation rate constant, a breakup rate constant, the exponent for the shear-induced buildup and an exponent for the shear-induced breakup of the microstructure, respectively. Another example of microstructural evolution found in the literature considers Brownian buildup and shear breakdown[18]. 2.2 Thinning and breakdown of thixotropic fluids In Newtonian fluids, the break up of a fluid column is governed by capillary forces that attempts to reduce the surface area of the column which results in a column of liquid that collapses into droplets[19]. Whereas thixotropic fluids thins and breaks down, this process is driven by the relative magnitude of capillary forces, inertia, viscosity and the stresses in the filament[20]. Understanding the dynamics of the filament is important because it affects the quality of the droplet formed and determines the type of break off. A general insight into the different regimes in break off of the fluid filament, like dripping and jetting, based on Capillary velocity (V Ca = σ/η, for a viscous Newtonian fluid), can be found in [20]. Breakdown of filaments can be divided into the following regimes: filament elongation, stress relaxation, and filament breakup[21]. The first regime consists of an exponential decrease of the radius during the axial elongation. As the radius of the filament decreases, it reaches a point in time beyond which the extensional stresses and extensional viscosity of the fluid reaches a constant value. The second regime Stress relaxation follows, during which the radius of the filament remains almost constant. In this regime, the capillary pressure becomes dominant and results in the start of the Pinchoff regime, where the filament breaks off into a droplet, described by the last regime. During the Pinch-off regime, the radius of the filament decreases in a self-similar manner[21]. Scalable solutions for the pinching of an axisymmetric Newtonian fluid column were established by Jens Eggers[22]. Of late, there have been interesting papers that 14

17 investigate the behaviour of viscoelastic fluid filaments described by Herschel-Bulkley and Carreau-Yasuda rheologies near break-off[23][24]. In this thesis, the focus has been on the thinning or the extensional regime of the filament. One of the ways to study the dynamics of this regime is to study the evolution of liquid bridges which are formed purely by viscous, inertial and capillary forces like the ones formed in extensional rheometer[20]. 2.3 Interface tracking method Tracking interfaces can be done using the following two methods: moving-mesh and fixed grid. Moving-mesh methods involves the tracking of interface with Arbitrary Lagrangian-Eulerian (ALE) method, which are extremely computationally intensive and cannot handle topological changes like droplet formation[25]. The fixed grid method on the other hand uses a transport equations to track the position of the interface on a non-deforming mesh. Some of the commonly used fixed grid methods includes level-set (uses an additional one transport equation) and phase field (uses two additional transport equations)[25]. In this section, Phase field method proposed by Yue et al.[26] and its implementation in COMSOL (refer to COMSOL Help and [27]) is explained in detail. The Phase field model describes the interface motion by the free energy. A phase field variable φ is used to vary the concentration of each phase in the domain, and it adopts values from 1 to 1 each corresponding to a different phase. In COMSOL, this definition of φ is used to describe the volume fraction of each fluid and is given as V f1 = 1 φ, V f2 = 1 + φ 2 2 (5) The free energy is determined as a sum of the mixing energy, the bulk distortion energy and the anchoring energy. The bulk distortion energy and anchoring energy are taken into account when the fluid under consideration has particles like crystals in it. The mixing energy is written as a function of φ and its gradient. Since this method deals with mixing energies and not with interfacial surface tension like Level set methods, the relationship between the mixing energy density parameter λ and interfacial tension σ 15

18 can be described as σ = 23/2 3 λ ɛ, (6) where ɛ is the interface thickness parameter. Though not included in Eq.6, anchoring energy can contribute to surface energy, giving rise to an anisotropy in σ calculated[26]. The governing equation used to track the phase field interface is given by Cahn-Hilliard equation φ t + v. φ =.γλ ψ (7) ɛ2 ψ =.ɛ 2 φ + (φ 2 1)φ, (8) where ψ is called the phase field help variable, v is the velocity field and γ is the mobility parameter (γ = χɛ 2, χ is the Mobility tuning parameter). And surface tension is computed by the equation given below, where G is the chemical potential and f φ is the φ-derivative of the external free energy ( F st = G f φ ) φ (9) ( G = λ 2 φ(φ 1) ) φ + ɛ 2 + f φ (10) This method of computing the surface tension is the main advantage of using the phase field method as it eliminates the cumbersome process of calculating the surface normal and curvature by distributing the surface tension using ψ and the gradient of φ[26]. 16

19 3 Geometry and mesh generation 3.1 Geometry Before the geometry used in the simulations is mentioned, the jet printer head is explained to get some insight into the working principle of the device, see Fig.1. The solder paste stored in 1 is pressurized using a pressurization pump 2 and fed into the inlet of the feed screw. The rotational feed screw 3, powered by the electric motor 4 via the motor shaft 5, transports the solder paste from the inlet of the feed screw to the outlet 6. From the outlet, the solder paste is transported to the chamber 7 via channel 8. The piston consists of the portion that slides through the bore 9. The actuation of the piston is caused by the extension of the piezoelectric stack 10 which is controlled by 11 and connected to a support 12. The solder paste in chamber 7 is ejected through the nozzle 13 by the actuation of the piston 14. The details of the construction of the printer head has been obtained from a patent assigned to Mydata Automation AB [6]. (a) Sectional view of the jet printer (b) Enlarged view near the nozzle region Figure 1: Drawings of the jet printer head(obtained from[6]) For the simulations performed in the thesis, the chambers 7, 13 and the channel 8 are modelled to be 2D axisymmetric. This approximation of the model is valid if the entire length of the channel and the connection to outlet 6 are not considered. There are two types of geometries used in this thesis: first, the model consisting of the chambers 7 and 13 (see Fig.2a) called Model 1. This model does not account for the flow of solder paste that flows in/out of through the channel when the piston is actuated. Second, the channel 8 is modeled along with the chambers to account for the deficiency in the 17

20 previous geometry referred to as Model 2, see Fig.2b. The Model 1 is used for the independence studies and preliminarily studies, whereas Model 2 is used in the final simulations whose results are discussed later on. (a) Model 1 (b) Model 2 Figure 2: Geometries of the printer head used in the simulations. All dimensions are in millimeters For the region below the jet printer head which initially contains air, the height is determined from the IBOFlow simulations as 4.5mm from the nozzle of the printer head. And the width of this region is determined from the domain independence studies (mentioned in Section7.1) to be 0.6mm. 3.2 Meshing The flow phenomenon to be modelled is the droplet formation during the jetting of solder paste. There are two approaches while meshing the geometry for this sort of problems, where the first approach is to use uniform mesh that captures all the length scales associated with the problem, which can be extremely computationally intensive, and the second approach is to use an adaptive mesh, which is implemented in this thesis. The geometry is subdivided into several regions as shown in Fig.3 to minimize the number of elements created. As COMSOL supports adaptive mesh refinement only in triangular elements, the entire geometry consists of only triangular elements. The parameters used to create the initial mesh is tabulated in Table.1. As the flow of solder paste is a low Reynolds number case ( 1), the boundary layer is not resolved. The effect of inclusion of boundary layer in the simulations has been mentioned in Section

21 Figure 3: Sectioning of the domain used in meshing (not to scale) Table 1: Mesh parameters used to create the initial mesh Maximum element size (µm) Minimum element size (µm) Growth rate Region Region Region Region Adaptive meshing The refinement of the mesh used in thesis is performed along the longest edge of the element, which implies that the longest edge in the element is divided based on the maximum number of refinements mentioned. For two phase flows, the error based on which the mesh refinement takes place is indicated by a gradient of phase field variable in both r and z components. The maximum number of mesh refinement is set to 20 in order to keep the ratio between the maximum element size in the domain and number of mesh refinements close to one. Further information regarding adaptive mesh refinement can be found in [28]. The progressive refinement of mesh as the interface between the solder paste and air changes with time is shown in Fig.4. 19

22 (a) Original mesh (b) Mesh 1 (c) Mesh 2 (d) Mesh 3 Figure 4: Adaptive refinement of mesh 4 Computational setup 4.1 Material The solder paste is a dense suspension consisting of a resin carrier fluid and metal alloy spheres. The volume fraction of solder balls and flux is 89/11 by weight. The metal alloy composition can vary but a standard lead-free alloy consists of 3% of Silver, 0.5% of Copper and remaining of Tin (by mass) and a typical solder of this composition can be seen in Fig.5. The Type 5 solder paste produced by Senju Metal Industry Co. Ltd., used in this thesis, has solder spheres ranging between 15µm- 25µm and its relative density with respect to the flux is 8. The density of the solder paste used for the simulations is ρ =4000 kg/m 3. 20

23 Figure 5: A image of solder sphere obtained from scanning electron microscope(sem) 4.2 Numerical framework The phenomena under consideration involves an incompressible fluid motion modelled through continuity equation (Eq.11) and the Navier-Stokes equation (Eq.12). In the equations, ρ denotes the density, η is the dynamic viscosity, g denotes the acceleration due to gravity, p is the pressure term and F st is the surface tension acting at the interface modelled through Eq.9 mentioned in the Interface tracking method. The interface motion is captured by the Cahn-Hilliard equation (see Eq.7 and Eq.8). The dynamic viscosity η is a function of shear rate γ.. v = 0 (11) ρ v t + ρ( v. ) v =.( pi + η[ v + ( v)t ]) + F st + ρg (12) For two phase time dependent flow scenarios, COMSOL uses a Phase initialization to initially determine the distance to the interface and a Transient study to capture the time dependent dynamics. A fully coupled solver is used to handle solution for both phase initialization and time dependent studies. A constant Newton non-linear method 21

24 (damping factor α =1) is used in the fully coupled solver. In the Direct linear solver, the parallel sparse direct solver (PARDISO) is used as it is tends to the fastest amongst the available options. Tolerances are used to determine the termination criteria of Newton iterations, so both absolute tolerance and relative tolerance are set to a value of The tolerance level used is quite high to keep the simulation time realistic (in the order of days) and investigations into the influence of tolerance on the solution is discussed in Section7.5. The discretization used is P1+P1, which implies that both velocity and pressure fields use linear elements[29]. The effect of using higher order elements like P2+P1 on the solution is discussed in Section.7.7. The time stepping method used is Backward Difference Formula (BDF) with a second order interpolation. The initial time step is set to s and subsequent time steps are chosen by the BDF method. The non-linear controller is also selected to enable efficient time stepping Two-phase laminar flow The flow scenario to be modelled is a low Reynolds number flow ( 1) with two phases i.e. Air and Solder paste. The phase field method (explained in Section2.3) is used to track the interface between the phases. When using 2 phase laminar flow module in COMSOL, few parameters used in the simulations for viscosity definition and phase field method must be defined by the user. These user defined parameters have been tabulated in Table.2. The f φ term takes into account the energy contribution due to bulk distortion energy and anchoring energy, applicable in rod-like particles[26]. Table 2: Parameter list used in the simulations Value Remarks Mobility parameter χ 1m.s/kg default value in COMSOL f φ 0 [26] Surface tension 0.1 N/m based on IBOFlow simulations Interface thickness 6µm Section Viscosity models - Section Interface thickness The interface thickness, required as an input in the phase field method, was determined through analyzing Fig.6, which is an image of droplet obtained using a stroboscope. 22

25 Based on the gradient of index values perpendicular to droplet motion, the interface thickness is calculated to be one pixel dimension which is 6µm. Figure 6: Image of droplet taken by camera[10]. The image captures droplet at two different times using a stroboscope to extract the droplet speed Boundary conditions The boundary conditions used in the simulation have been illustrated in Fig.7. The pressure outlet is assigned at the boundaries exposed to atmosphere and the gauge pressure is set equal to zero. The initial position of the interface between air and solder paste is set at the end of the nozzle. All the walls in the chamber/printer head, except for the piston head, are defined as no-slip walls. The boundary condition at the channel is defined using Darcy s law, v = Φ v i, where v is the Darcy s flux that passes through or the fluid velocity at the boundary, Φ the porosity and v i the fluid velocity obtained through computations. The value of porosity Φ used in the simulations is set equal to one such that the droplet speed falls within the realistic speed of 20 m/s-30 m/s. This is implemented as a velocity boundary condition at the outlet with v = (u, 0, w), where u and w are the r component, z component of the velocity obtained through the computations Implementing the motion of the piston The piston has been identified in Fig.7 and it can be modelled as a Moving wall or Velocity inlet. Using the moving wall technique requires the implementation of moving mesh that enables the deformation of the mesh as the piston moves. The alternate method approximates the piston motion by the introduction of a velocity 23

26 (a) Sectional view of the jet printer (b) Enlarged view near the nozzle region Figure 7: Boundary conditions used in the simulations inlet, i.e. the fluid enters the domain through the piston wall with velocity equal to the piston motion. The influence of both the methods on the droplet volume and speed has been included in Section.7.8 and from the study it was observed that the Velocity inlet predicts droplet volume and speed that is comparable to Moving mesh technique in much shorter time. The simulations discussed in the results have been run with a piston motion approximated by a velocity boundary condition. The Fig.8a represents the actuation mode used in all the quality and independence studies and Fig.8b corresponds to the realistic piston actuation which are results from a previous structural analysis study done at the Mycronic AB by Gunnar Gustafsson Scaling of variables COMSOL uses scaling of variables to form a well-scaled linearized system. Although the scaling is done automatically, the manual input of scaling factors has been found to reduce the solving time. The scaling factors are determined such that the ratio between 24

27 (a) Simplified piston motion (b) Realistic piston motion Figure 8: Piston actuation profiles the variable and the scale would be in the order of one. The scaling factors for variables presented in Table.3 have been determined from the results from IBOFlow[10]. Table 3: Scaling factors used in COMSOL Variable name Scale Pressure 10 6 Velocity 10 Phase field help variable 1 Phase field variable 1 5 Viscosity models This section describes the viscosity models used in the simulations and the methodology of obtaining the parameters of indirect structure-based model from experimental rheological data. 5.1 Carreau-Yasuda model The parameters of the Carreau-Yasuda model, presented in Table.5 and Table.4, are obtained from [10] and [11] where they were determined by curve fitting the model on shear sweep test data performed at 25 C. The parameters predicted by Table.5 differ from Table.4 because the former is fitted to include additional data in the higher shear rate regime. The Table.4 has been used for all the mesh and domain independence studies. From this point on, the Carreau-Yasuda models based on Table.4 and Table.5 25

28 will be referred to as Carreau 1 and Carreau 2 respectively. Table 4: Parameter list for Carreau-Yasuda model obtained from [10] Constant name Symbol Value Unit Zero shear rate viscosity η Pa.s Infinite shear rate viscosity η 4 Pa.s Time constant λ s Power law exponent n Table 5: Parameter list for Carreau-Yasuda model obtained from [11] Constant name Symbol Value Unit Zero shear rate viscosity η Pa.s Infinite shear rate viscosity η Pa.s Time constant λ s Power law exponent n Indirect structure based viscosity model In the first part of this subsection, the time derivative of ζ mentioned in Eq.13, which is the same as Eq.4 mentioned in the theory for structure based viscosity models, is solved analytically with the approximation that the shear rate is constant to obtain equations from which an initial estimate of the parameters can be made. Once the initial estimate is obtained, the parameters can be adjusted manually to match the experimental rheological data. This method can be seen as the first step to estimate the starting values of parameters for thixotropic fluids which has no available data, which can be further tuned using nonlinear parameter fitting methods. The first and second terms in the equation given below represents the shear rate related structural buildup breakdown[17]. The equation does not consider Brownian motion as the solder spheres are too massive (15µm-25µm) to be influenced by it. The Brownian motion is dominant in particle with sizes below 1µm[13]. dζ dt = k f γ a (1 ζ) k b γ b ζ (13) Rearranging the terms for the ease of integration results in dζ ( dt = (k f γ a + k b γ b k f γ a ) ) k f γ a + k b γ b ζ (14) 26

29 Applying limits for integration, time varies from 0 to t and correspondingly ζ changes from ζ t=0 to ζ. ζ t=0 is the value of the structure parameter at the beginning of the test or t = 0. ζ ζ t=0 ( dζ k f γ a k f γ a +k b γ b ) = ζ t 0 (k f γ a + k b γ b )dt (15) ( ln k f γ a ) k f γ a + k b γ b ζ ( + ln k f γ a k f γ a + k b γ b ζ t=0 The Eq.17 is the solution to Eq.13 when γ is a constant, ( ζ = k f γ a ) k f γ a + k b γ b ( + ζ t=0 ) = (k f γ a + k b γ b )t (16) k f γ a ) k f γ a + k b γ b e (k f γ a +k b γ b )t. (17) The structure parameter can be defined in terms of viscosity as given by Eq.18 [18] ζ p = η η η 0 η. (18) Now if Eq.17 and Eq.18 are combined, it results in a solution for viscosity varying with time as given below (( η = η + (η 0 η ) k f γ a ) k f γ a + k b γ b ( + ζ t=0 k f γ a ) ) k f γ a + k b γ b e (k f γ a +k b γ b p )t. (19) For any material, k f and k b relate to the interaction between the particles to form/break links between each other (based on inter-particle forces which can either be attractive or repulsive) and so it can be considered to be independent of shear. The exponent p shows the tendency of the material to move between zero shear viscosity and infinite shear viscosity plateau regions for varying shear rates, which again is directly related to the material. In the limit t, the exponent term disappears from Eq.19 and produces an equation for viscosity at steady state as given below ( k f γ a ) p. η = η + (η 0 η ) (20) k f γ a + k b γ b This equation describing the variation of viscosity at steady state is very similar to Carreau-Yasuda model described by Eq.1. Comparing the Carreau model (Eq.1) and 27

30 Eq.20, one obtains λ = ( kb ) 1 b a k f (21) b a = 2 (22) p = n 1 2 (23) Considering the variation of structure parameter with time under constant shear rate in Eq.17, the equation can be written as below when t ( k f γ a ) ζ t = ζ = k f γ a + k b γ b (24) ζ = ζ + (ζ t=0 + ζ )e (k f γ a +k b γ b )t. (25) If Eq.18 is used to to rewrite Eq.25 in terms of viscosity ( η η ) 1/p = e (k f γ a +k b γ b )t. (26) η 0 η We can rewrite Eq.26 as ln η η η 0 η = γ a (k f + k b γ b a )pt. (27) Substituting k b in terms of k f using Eq.21 ln η η η 0 η = γ a (k f + λ b a k f γ b a )pt (28) Viscosity measurements at different shear rate (at steady state) provides values of η 1 and η 2 for shear rates of γ 1 and γ 2, respectively. If they are compared at the same time t, one obtains (note that p is a constant according to Eq.23) ln η1 η η 0 η ln η2 η η 0 η = γ 1 a (1 + λ b a γ b a 1 ) γ a 2 (1 + λ b a γ b a 2 ). (29) 28

31 Since the factors b a and λ are known from curve fitting the Carreau model, a can be calculated from Eq.30. Once a is known, other parameters eg. b, k f and k b can be determined from Eq.22, Eq.28 and Eq.21, respectively. a = ( ln (1+λ b a γ b a 2 ) (1+λ b a γ b a 1 ) ( ln γ1 ln η 1 η η 0 η ln η 2 η η 0 η ) γ2 ) (30) This part of the subsection deals with the rheological measurements. During a shear sweep, the viscosity never actually reaches a steady value for any particular shear rate, so the viscosity-shear rate behaviour predicted by shear sweep is different in comparison to the viscosity predictions from steady shear rate experiments. To calculate the parameters, experiments performed at constant shear rates are used to curve fit a Carreau model. The rheological experiments are performed in a plate-plate type rheometer at 25 C with the distance between the plates set at d = 0.2 mm. The plate-plate type rheometer consists of two parallel discs, where the lower plate remains still and upper plate rotates at the angular speed Ω to produce the required shear rate, which is equal to the average shear rate along the radial direction of γ = Ωr/d, where r is the radius from the axis of rotation. The distance between the plates d is based on a rule of the thumb, that parallel plates must be separated by a gap of around ten times the diameter of the solder spheres. The solder paste is given a total rest period of 15 minutes in order to revive the structure after it is injected onto the plate of the rheometer and stabilize forces from the plate surfaces. It was observed that for shear rates above 300s 1 the viscosity observed is substantially lower, this is due to the loss of solder paste at higher angular speeds of the plate in the rheometer. One way to overcome this problem is to reduce the gap between the plates, but for the scope of this thesis the gap is kept constant. Curve fitting the data from the experiments is then used to determine the parameters of Carreau model which is given are the Table6 and compared to the experimental data in the Fig.9. For the calculations, consider the parameters of the Carreau model (Table6) and the two points from the rheological data: γ 1 =0.1 s 1 and γ 2 =5 s 1 which produced η 1 =4135 Pa.s and η 2 = 161 Pa.s respectively. Following the steps mentioned before, 29

32 Figure 9: Carreau model fitted to experimental steady state shear rate data Table 6: Parameter list for Carreau-Yasuda model obtained through curve fitting Variable name Symbol Value Unit Zero shear rate viscosity η Pa.s Infinite shear rate viscosity η 9 Pa.s Time constant λ s Power law exponent n the parameters a, b, k f, k b are calculated and tabulated in Table Fine tuning the parameters The initial estimates of the constants used in the indirect-structure based viscosity model are further adjusted based on experimental data. The experimental data used to adjust the parameter is obtained from a 3IT test done by Lars Essén at Mycronic. The test consists of an initial period of 16s when the shear rate is 1s 1, followed by shear rate of 100s 1 for next 98s and finally a shear rate of Table 7: Initial estimate of parameters for Indirect structure based viscosity model Variable name Value a b k f k b p

33 1s 1 for the last 49s. As measurements of viscosity at zero shear rate cannot be done on a plate-plate rheometer, the starting shear rate is chosen as 1s 1. The solder paste is allowed to reach a steady viscosity. This is used as the stating point of the calibration of the initial structure parameters. During the initial shear period of 16s, when the shear rate is the maintained at 1s 1, the structure parameter ζ is explicitly calculated using Eq.31 (from the measured viscosity) to be Once the shear rate changes, the equation describing the rate of structural breakdown (Eq.13) is discretized using an explicit simple backward difference scheme for time and can be rearranged to calculate ζ as shown below, ( ) 1/p ηn η η ζ n = 0 η ) ζ n 1 + t (k f γ n 1(1 a ζ n 1 ) k b γ n 1ζ b n 1 t n 16s t n > 16s, (31) where n denotes the steps such that (t 1 corresponds to 0s) and t = t n t n 1. The tuned parameters are obtained by manually trying out values to resemble the experimental data and is tabulated in Table.8. Table 8: Fine tuned list of parameters for Indirect structure based viscosity model Variable name Value a 1.8 b 0.5 k f k b 0.02 p Implementation in COMSOL The indirect structure model (Eq.13) is implemented through a module Coefficient form PDE. This equation for structure based viscosity model is solved only in the region with solder paste. The partial differential equation (PDE) is discretized using linear elements to reduce the computational time. To solve the PDE, the initial values of ζ and dζ/dt are defined as 0.1 and respectively. The partial differential equation implemented in Coefficient form PDE is coupled to the fluid properties using the Eq.18, which can be written as η = η + (η 0 η )ζ p. (32) 31

34 6 Elongated filaments The axial elongation of the solder paste during jetting is a result of two driving factors: droplet and piston motion. Understanding the elongation process becomes less complicated if it is driven by just one factor like in the case of extensional rheometer with one probe. The data from the extensional rheometer is used to compare with the analytical model to get a preliminary understanding of how the solder paste behaves during extensional stresses. For the ease of the calculations, the behaviour of the filament when undergoing extension in an extensional rheometer is approximated to behave as shown in Fig.10. The arrow head and U indicates the direction of motion and the speed of the probe respectively, which produces the elongation of the fluid column. The horizontal line at the bottom of the figure is the stationary wall where solder paste is applied. This model in reality Figure 10: Extensional behaviour of solder paste considered for the modelling can be used to describe the dynamics of the filament formed between the fluid blobs formed on the probe and stationary wall during the experiments. 6.1 Critical aspect ratio When a fluid filament undergoes extension in an extensional rheometer, after a certain aspect ratio the the capillary forces dominate, the critical aspect ratio can be used as a measure to identify this point. The velocity field in the fluid column undergoing 32

35 extension in z-direction can be written as (u, 0, w). Assume a constant rate of stretching E, such that the velocity along z axis can be written as w = Ez. By using continuity equation, the r component of velocity can be determined to be u = Er/2. The free surface of the filament,r = h(z, t), must satisfy the kinematic condition dh dt + w dh = u. (33) dz As the extension of the filament is modelled as a cylinder, the free surface h is independent of z and depends only on time. This can be used to solve Eq.33 and solution looks like h(t) = h 0 exp( Et/2), where h 0 is the radius of the filament at t = 0. In order to understand the parameter E in a different perspective, strain rate ɛ for a cylindrical fluid element (described by Eq.34) is introduced[20]. ɛ = 2 d ln h h 0 dt (34) On substituting the solution of Eq.33 into the above equation, we get ɛ = E, which can be used to rewrite the transient behaviour of h(t) as h(t) = h 0 e ( ɛt/2). (35) For a general non-newtonian fluid column, the force balance on the thinning filament (given by Eq.36) can be used to get general overview of break off[20]. The force balance equation given below does not include the effect of gravity and η s corresponds to viscosity at any slice S in the column along the elongation. The equation uses the radius R = R(z, t) as a function of axial direct and time, R = dr/dz, and the terms on the right hand side of the equation represents the viscous extensional stress, the non-newtonian tensile stress difference and the capillary pressure arising from radial and axial curvature. F z (t) πr 2 { 2 = 3η s R ( dr } + (τ P,zz τ P,rr ) + dt )Z0 σ { R 1 (1 + R 2 ) RR } (1 + R 2 ) 1.5 (36) 33

36 The viscosity across any slice S is based on the shear rate given in Eq.37, that can be simplified using the formulations of u and w derived earlier to get γ = ( ( du ) 2 ( u ) 2 ( du dr r dz + dw ) 2 ( dw ) 2 ) = 3 1/2 ɛ. (37) dr dz The viscosity definition given by Carreau-Yasuda model can be determined to be a constant ( ɛ has already been proved to be equal to the constant rate of stretching E) as shown below, η = η + (η 0 η )(1 + 3λ ɛ 2 ) (n 1)/2. (38) On applying the assumption that the fluid column is axially uniform, the above equation can be reduced to Eq.39 for constant viscosity across any slice (shown by Eq.38) also called as zero-dimensional form[20]. F (t) πh 2 = 3η { 2 h ( dh )} + (τ P,zz τ P,rr ) + σ dt h (39) The non-newtonian tensile stress difference can be written as τ P,zz τ P,rr = Trη ɛ, Tr = 3 for Newtonian fluids[30]. Substituting these into Eq.39 produces F (t) πh 2 = 3η { 2 h ( dh )} + ɛ(3η) + σ dt h. (40) Now substitute the solution h(t) = h 0 exp( ɛt/2) into the above equation and after some algebra, the equation can be written as F (t) = 6πη ɛh 2 + πσh. (41) From the expression it can be observed that surface tension forces becomes dominant when h is very small. When F (t) = 2πσh, applicable at the surface between the filament and the fluid blob, surface tension forces balance the resultant of viscous stresses and non-newtonian contribution to tensile stresses[20]. When the fluid column is thin and is moments before break off, the surface tension can be scaled approximately equal to the resultant of viscous stresses and non-newtonian contribution to tensile stresses which can be rearranged to get Eq.42. At this point, the height and radius of the column are 34

37 given by z c and h c respectively. h c = σ/(6η ɛ), (42) where η is defined by the Carreau-Yasuda model for extensional flows (Eq.38). In order to make calculations simpler, the viscosity under high extension is considered to be η, as at high shear rate the Carreau-Yasuda model gives viscosity at the high shear rate plateau. So for further calculations The strain rate can be assumed to be ɛ W/z c, since E = ɛ, where W is the speed of the probe in the extensional rheometer. After substituting the assumption of ɛ in Eq.42, we get the critical aspect ratio Γ c 6Ca, (43) where Γ c = z c /h c and Ca is the capillary number (Ca = η W/σ). 6.2 Processing data from extensional rheometry The data for different probe speeds is obtained from a work done by Ylva Langett at Mycronic AB. The camera used for imaging the dynamics of extension has a pixel resolution of 23µm and a imaging frequency of 300Hz. the diameter of the probe used for the measurements is 2mm and all the data from the camera is analysed using Matlab. The images from the extensional rheometer are converted into binary images from grayscale using im2bw based on a global threshold. From the binary image, the region withing the diameter of the probe is isolated as necking of the solder paste happens in this region. This isolated region is used to calculate the diameter at each slice and the minimum is taken as the necking diameter. For the critical aspect ratio calculations, the time step right before the break off is analysed. In the thesis, comparison between the experimental data and the analytical model is based on the radius of the filament. The aspect ratio Γ c is calculated based on the following values η =11.855Pa.s, W velocity of the probe and σ =0.1N/m. On analyzing the data from different experiments, it is observed that when the diameter of the filament undergoing extension is approximately 94µm the filament is roughly cylindrical (if the beads formed are neglected) and the analytical model can be applied 35

38 in this region. So height of this cylindrical region is measured from the experimental data as it is much larger than the pixel resolution of the camera used. And the critical height calculated is used in the analytical relation for critical aspect ratio to determine the critical radius. 36

39 7 Quality and independence studies As the simulations with the final configuration take extremely long to compute (in the order of weeks), the quality studies are performed on a coarser mesh and wider interface thickness ( 60µm)with absolute tolerance set at 10 1 to speed up the simulations. Also the simulations in this section (unless mentioned) are all based on moving mesh technique. 7.1 Domain independence The domain independence studies is important to ensure that the solution obtained is independent of the domain dimensions under consideration. As discussed in the geometry, the height of the region below the nozzle which initially contains just air is fixed by the geometry used in IBOFlow to 4.5mm. But the width of this region is determined through the domain independence study. The different values of width considered for the analysis are multiples of the radius of the nozzle i.e. 75µm. The Fig.11 shows the net mass flow rate of the solder paste through the outlet for various width of the domain. The difference between the mass flow rate predicted between the domain width of 300µm and 375µm is almost 19%, between 375µm and 525µm is around 5%. From the asymptotic behaviour of the curve, the domain width was set to 600µm or eight times the radius of the nozzle. Figure 11: Domain independence studies 37

40 7.2 Mesh independence The domain was split into four regions as shown in Fig.3(in Meshing). This subsection deals with the mesh convergence studies done for each of those sections. The Section 1 is the region where the solder paste flows through after exiting the nozzle. The mesh convergence for this region alone is preformed using adaptive meshing to save computational time. The maximum number of mesh refinements is set to 20 and the maximum element size in the domain is varied between 30µm, 50µm, 70µm. The Fig.12a shows the variation of net mass flow of solder paste through a section in the region and the difference between the data points is always less than 0.85%. From the graph it can be determined that the maximum element size in this region must be around 20µm so that the solution is not affected by the element size. The Fig.12b shows the comparison of net mass outflow of solder paste with varying element size for the Chamber or Section 2. The difference between the data points is always less than 0.5% and the maximum element size of 20µm can be assumed to be the size when the solution is independent of the mesh. The region 3 does not have any flow of solder paste directly in it as it falls right beside Section 1, but when the diffused interface (the interface thickness for this study alone is defined in the order of the maximum element size in the domain) is considered for analysis the mass flow rate expected through this domain is extremely less which is observed in Fig.12c. Since the influence of the mesh in this region on the solution is very minimal, the maximum element size in this region is set at 50µm. The mesh in the channel (4 ) must have a maximum element size of approximately 20µm to make sure that the solution is not affected by the mesh, based on the Fig.12d. But the difference between the mass flow of solder paste for different data points in the graph is always less than 0.1%. 7.3 Effect of including the boundary layer This subsection delves into the studies done to quantify the effect of resolving the boundary layer in the printer head. A mesh with the first layer thickness set at ( 0.4µm) and growth rate for 10 layers set at 1.1 was compared with mesh with local domain element size of 7.5µm. The difference observed for the difference in the solder paste that moved 38

41 (a) Section 1 (b) Section 2 /Chamber (c) Section 3 (d) Section 4 /Channel Figure 12: Mesh convergence studies through the domain was less than 0.15%. Another method to check the necessity of resolving the boundary layer is to analytically calculate the wall distance needed to obtain y+ = 1. The drawback associated with this approach is that the methodology is applicable for only flow over a flat plate. Assuming the flow at the nozzle (U =100m/s), viscosity based at the infinite shear rate region µ =11.855Pa.s, density ρ =4000kg/m 3 and reference length as the diameter of the nozzle (150µm) the wall spacing required for the y+ can be calculated to be m which is in larger than the diameter of the nozzle itself. The calculation of wall distance is done using an online Y+ calculator provided by Pointwise. These results supports the approach of not resolving the boundary layer in final mesh used for the simulations. 39

42 7.4 Mass conservation In this thesis, mass conservation is measured as the percentage change in mass of the droplet as it falls through the domain towards the outlet. The droplet once detached from the solder paste in the nozzle during the piston actuation mode Wave 2 is used in this analysis. The Fig.13 shows the variation of mass of the droplet over time and the total mass lost during this duration is less than 0.1%. Figure 13: Change in the droplet mass with time 7.5 Tolerances The governing equations are solved until solution reaches a set tolerance in COMSOL. So in order to obtain accurate solution the tolerances are tightened for a given mesh. In COMSOL both relative and absolute tolerances are used to control the steps taken by the solver. The steps are set such that the estimated error in each integration step satisfies the Eq.44 for all components of x i and x i e = max(ap, rl x i ) and e = max(av, rl x i ) (44) where ap is the absolute tolerance on the position components, av is the absolute tolerance on the velocity components and rl is the relative tolerance[29]. To check the effect of changing the tolerances on the solution, the absolute tolerance for velocity field is tightened. The effect of changing the absolute tolerance for velocity 40

43 field can be scaled similarly for parameters like pressure and phase field variable. Figure 14: Effect of changing absolute tolerance on the solution The Fig.14 shows the variation of total mass of solder that passed through the outlet during the simulation with absolute tolerance imposed on velocity field. The data shows a tendency to become asymptotic around 10 6, which should be used as the absolute tolerance for velocity field and parameters like pressure. As the error is based on maximum of absolute error and relative error (from Eq.44), if the relative error is much larger than absolute tolerance it would become the deciding criterion. So to tighten the convergence criteria the relative tolerance must also be set to a value comparable to absolute tolerance ( 10 6 ). 7.6 Temporal discretization The time stepping method used in the simulations is a implicit BDF (Backward Differentiation Formula) method. Although the scheme is very stable, it introduces dissipative errors in the first order. So the BDF method used for time stepping in this thesis is set an order of two. For the Steps takes by solver, Free is chosen to let the BDF method to choose the time step without any restriction. Beyond this, no further analysis regarding the effect of using a higher order BDF method has not been investigated in this thesis due to the lack of time. 41

44 7.7 Spatial discretization The discretization involves splitting the domain into elements and nodes on which the governing equations on. And the order of the node therefore determines how each edge in the element is split. For example, P1 uses linear elements which means that each edge would contain two nodes located at each end whereas P2 which uses second order elements has three nodes (two on the ends and one at the mid-point of the edge). More information regrading how finite element methods use the order of elements to compute the solution vector can be found in [31]. The Fig.15 compares the droplet velocity and volume predicted by P1+P1 and P2+P1 discretization schemes. P1+P1 as mentioned before uses linear elements for both pressure and velocity fields whereas P2+P1 uses second order elements for velocity field and linear elements for pressure field[29]. From the figure it can be seen that predictions from both the discretization schemes are substantially different (upto 6.6%) once the solder paste is ejected from the printer head, but the lower order scheme requires much lesser computational time. (a) Comparison of droplet velocity (b) Comparison of droplet volume Figure 15: Comparison of the discretization/element order 7.8 Comparison of methods to actuate the piston The Fig.16b and Fig.16a compares the droplet volume and speed predicted by Moving wall and Velocity inlet with P1+P1 discretization. The droplet volume predicted by Moving mesh reduces after reaching the maxima, but it remains constant for Velocity inlet and Moving mesh with P2+P1 discretization. This discrepancy could be attributed to the error associated with the lower element order. So Velocity inlet is 42

45 used to model the piston motion as it could capture the droplet parameters predicted by Moving mesh with P2+P1 discretization at lower degrees of freedom. (a) Comparison of droplet velocity (b) Comparison of droplet volume Figure 16: Comparison of methods of piston actuation 43

46 8 Results & discussion The simulations are compared based on two important parameters: droplet volume and droplet speed. A brief description on how both of them are computed is given below. COMSOL has an in-built integration operator intop1 that can be used for spatial integration over the domain Ω. The droplet volume (V) is computed using the Eq.45, where V f is the volume fraction of solder paste and integration is done for the part of the domain which is below 700µm (the nozzle end is located at z=-600µm). V = Ω 2πr(V f >= 0.5)(z < (700[µm])) dr dz (45) The droplet speed (U) is calculated based on Eq.46, where w is the z component of the velocity. 2πrw(V f >= 0.5)(z < (700[µm])) dr dz Ω U = V (46) 8.1 Comparison with IBOFlow simulations The IBOFlow simulations are done for different combinations of parameters like position of the meniscus, permeability, nozzle diameter and thickness of the lower plate and it is calibrated based on the droplet mass calculated from the experiments for lower chamber wall thickness of 600µm. The IBOFlow simulations has also been validated against experimental data for droplet shape and speed; so the simulations with other parameters values can be used to validate the COMSOL simulations. The setup for the IBOFlow is quite different from the setup used in this thesis i.e. IBOFlow uses a porous domain in the channel whose permeability is calibrated to match experimental droplet mass and the COMSOL setup uses Darcy s law which is applied to the boundary (mentioned in Section.4.2.3) such that the droplet speed is within the realistic 20m/s - 30m/s. The parameters used in IBOFlow computations for comparison are nozzle diameter is set at 150µm, thickness of the lower plate is 300µm, Carreau 1 fluid and Wave 1 piston actuation sequence. For the comparison of the results, the volume of the droplet estimated from IBOFlow for different permeability values is plotted in Fig.17a. Since the general trend on the data points show an exponential behaviour, to get a droplet 44

47 volume of 12.6nL on IBOFlow, the corresponding permeability can be calculated to be m 2, based on the function that describes the exponential decrease (through curve fitting). The permeability obtained from the droplet volume comparison can be used to determine the corresponding droplet velocity from IBOFlow. The permeability value is substituted in the exponential function for IBOFlow velocity data (parameters obtained by curve fitting) to get the corresponding droplet velocity (see Fig.17b). The droplet speed obtained from IBOFlow is 43m/s and COMOSL predicted a speed of 25.5m/s for a droplet volume of 12.6nL. When the droplet speed and volume is extracted from experiments (previously done at the company), the speed of the droplet of volume 13.4nL is 63m/s. The reduced velocity in COMSOL on comparsion with IBOFlow could be attributed to the approximations used in the setup like the linear spatial discretization, loose tolerances and approximation of the piston motion as velocity inlet. Based on the simulations done in the quality studies an reasonable estimate of the effect of approximations used in COMSOL on droplet speed can be calculated: if the spatial discretization is set to P2+P1 and piston is actuated using moving mesh the speed of droplet would increase by 6m/s and volume would increase by 0.6nL and if the tolerance is tightened especially for velocity field the mass flux of the solder paste increases by almost 20%. Although both COMSOL and IBOFlow has capabilities to predict the droplet volume observed in experiments, the droplet speed determined by both the models are still quite different from the actual measurement which could be because Carreau-Yasuda model cannot replicate the rheological behaviour of solder paste. (a) Droplet volume comparison (b) Droplet speed comparison Figure 17: Comparison of IBOFlow and COMSOL 45

48 8.2 Effect of piston actuation The motion of the piston changes the viscosity of the solder paste in the printer head enabling it to get ejected through the nozzle. So the motion of the piston plays an important role in the droplet morphology and speed. In order to compare the effect of the piston motion on the droplet, Carreau-Yasuda model with parameters described in Table.4, Carreau 1 fluid is used for comparing realistic piston actuation i.e. Wave 1, Wave 2 and Wave 3 (mentioned in Fig.8). Figure 18: Comparison of droplet volumes (V) for realistic piston actuation The Fig.18 compares the droplet volume predicted by the realistic piston actuations. The droplet produced by Wave 1 is 13nL, compared to Wave 2 which is produces 10nL than Wave 3 which produces the least at 8nL, this is expected as the the net displacement of the piston for each actuation profile follow a similar trend. The droplet volume for Wave 1 drops after s because the droplet goes out of the computational domain. The droplet volume reduces slightly over time due to numerical dissipation associated with the linear order elements used for spatial discretization. The slope of the graphs are different due to the difference in the piston actuation and the the slight increase of droplet volume around 90µs is due to due the second downward motion of the piston. This increase in droplet volume is not observed in Wave 1 because the secondary droplet formed by the motion of piston downward never reaches the main 46

49 (a) t =58µs (b) t =75µs (c) t =91µs (d) t =108µs Figure 19: Formation of secondary droplet during piston actuation (Wave 1) droplet as the subsequent upward motion of the piston pulls the secondary droplet back into the nozzle as seen in Fig.19. The velocity of the droplet produced by the piston actuation is compared in Fig.20. Since the speed of the droplet is computed based on the volume of the droplet, for Wave 1 speed of the droplet reduces after s as the droplet goes out of the computational domain. The droplet speed determined for Wave 1, Wave 2 and Wave 3 are 25.5 m/s, 9.2 m/s and 13 m/s respectively. On comparing the effect of the piston actuation on droplet speed, the latter is predominantly affected by the magnitude of the initial downward acceleration of the piston i.e. Wave 1, Wave 2 and Wave 3 reaches an acceleration of almost 8000g, 2000g and 5000g where g = 9.81m/s 2. Small increase in the droplet speeds observed for Wave 2 and Wave 3 is due to the merger of the solder paste produced by the secondary downward motion of the piston that joins the solder paste ejected in the initially. Another important parameter to that must be analyzed is the droplet morphology. Comparison between the different piston actuation is done qualitatively after the droplet is formed in Fig.21. The droplet formed with Wave 1 has an extremely long tail, this can be attributed to the shear thinning of the solder paste during the longer upward motion of the piston. The bulge at the end of the droplet s tail for Wave 2 and Wave 3 is the additional solder paste added to the droplet due to the second downward motion of the piston. As Wave 3 is similar to Wave 1 but with a smaller magnitude, the tail of the droplet formed also has a tail that is shear thinned. The droplet formed by Wave 2 has much smaller change in the radius of the droplet compared to the droplets from 47

50 Figure 20: Comparison of droplet velocities (U) for realistic piston actuation (a) Wave 1 (b) Wave 2 (c) Wave 3 Figure 21: Comparison of droplet morphology for different piston actuations (not to scale) Wave 1 and Wave 3, due to the relatively lower rate of change of speed of the piston. The droplet lengths produced by Wave 1, Wave 2 and Wave 3 are 4.23mm, 1.55mm and 1.26mm respectively. The most observable difference is that Wave 1 produces a droplet that is almost thrice as long as Wave 2 and Wave 3. Since the piston actuations are different from each other, it is interesting to dynamically compare the jetting process. The Fig.22 compares the jetting for Wave 1, Wave 2 and Wave 3 at the following time during jetting: 17µs, 33µs, 50µs, 66µs, 83µs, 100µs and 133µs based on the volume fraction of solder paste. An interesting observation is the formation of large pockets of air when piston moves upwards in Wave 1 and Wave 3. 48

51 Wave 1 and Wave 3 also show the formation of solder paste finger in the printer head towards the last moments of the piston actuation. (a) Wave 1 (b) Wave 2 (c) Wave 3 Figure 22: Comparison of solder paste jetting for different piston actuations 49

52 8.3 Effect of Viscosity models The viscosity models sheds light into the rheological properties, so it is critical to understand the limitations of each of the model. So before discussing the results from CFD, an insight into the behaviour of each viscosity model is compared against experimental data. The Fig.23 compares Carreau model (Carreau 1) and structure based viscosity model (based on tuned parameters) against experimental data. The plot shows the shear rate ( in red dash-dotted line) varying from 1s 1 to 100s 1 and back to 1s 1 (see Section for more details). The Carreau model does not include the time dependent nature of the viscosity and microstructural changes in its definition, so it fails to predict any dependence of viscosity on the flow history. On the other hand the structure based viscosity model captures the transient nature of viscosity quite reasonably. Figure 23: Comparison of viscosity models Although the structure based model with the tuned parameters resemble the thixotopic behaviour of the solder paste, it does not quite capture the drastic change in viscosity when the shear rate is altered. So a new list of parameters in Table.9 is used to capture the drastic change in viscosity. The main difference between the parameter lists considered for the viscosity model are the changes associated with the microstructure breakdown parameters like k b and b. The Fig.24 compares the structure based viscosity models i.e. Table.7, Table.8 and 50

53 Table 9: New list of parameters for Indirect structure based viscosity model Variable name Value a 1.8 b 0.5 k f k b 0.05 p Figure 24: Comparison of structure based viscosity models 51

54 Table.9 against experimental rheological data. The untuned parameters i.e. Table.7, poorly predicts the viscosity and it has a low breakdown rate which shows that it requires longer to reach a constant viscosity. But the tuned parameters i.e. Table.8, though it has a higher breakdown rate, it does not predict the drastic change in viscosity when shear rate is altered. But the steady state values reached by the model is remarkably close to the experimental results. The Table.9 has breakdown rates in concurrence with the experimental data, but it fails to reach the same steady state values after the shear rate is reduced back to 1s Comparison of viscosity models implemented in COMSOL The viscosity models used for comparison are Carreau 1 (Table.4), Carreau 2 (Table.5) and a Newtonian fluid of viscosity 4P a.s which undergoes piston actuation based on Wave 1. The Fig.25 and Fig.26 shows the comparison of viscosity models based on the droplet speed and droplet volume predicted by the models respectively. The droplet speed and volume predicted by Carreau 1 is 25.6m/s and 12.6 nl respectively. For the Newtonian fluid the droplet speed and volume is 25.4m/s and 12.4nL respectively. The droplet produced by Newtonian fluid of η =4Pa.s is almost the same as the droplet produced by Carreau 1 fluid. The decrease of droplet volume and speed after s is because the droplet goes out of the domain, as explained in the previous section. For both the viscosity models, the solder paste ejected by the second downward motion does not reach the ejected droplet as it is pulled back in due to the subsequent upward motion. For Carreau 2, the droplet speed is 12.7m/s and the droplet volume is 11.8nL. Interestingly, the solder paste ejected during the secondary downward motion of piston reaches the initially ejected droplet, which causes the small increase observed in droplet speed and volume plots unlike the droplets produced by Newtonian and Carreau 1 fluids. The Fig.27 compares the droplet shape predicted by Newtonian, Carreau 1 and Carreau 2 fluids with an image of the droplet taken for the same piston actuation experimentally. The Carreau 2 model resembles the experimental droplet shape, especially near the tail of the droplet, amongst the three models considered. The droplet length produced by Newtonian, Carreau 1 and Carreau 2 are 4mm, 4.23mm and 1.93mm respectively. As 52

55 Figure 25: Comparison of viscosity models based on droplet speed Figure 26: Comparison of viscosity models based on droplet volume 53

56 (a) Expt.[10] (b) Newtonian (c) Carreau 1 (d) Carreau 2 Figure 27: Comparison of droplet morphology for different viscosity models expected the droplet lengths from Newtonian and Carreau 1 are almost the same and is twice of the length of the droplet predicted by Carreau 2. The main difference between the droplet formed by the Newtonian definition and Carreau model is that former uses a constant viscosity in the fluid and latter sets the viscosity based on the local shear rate. As the solder paste is ejected out of the printer head, the shear rate experienced by the head and tail of the droplet becomes different, this results in local spatial change in the viscosity of the droplet. The Fig.28 compares the viscosity predicted by the Carreau models for the predicted droplet. The image uses a contour based on the volume fraction of the solder paste to outline the droplet shape (in gray scale) and surface plot which is non zero only inside the droplet (in cyclic scale). The droplet head is predicted to have a viscosity around 70Pa.s and 100Pa.s based on Carreau 1 and Carreau 2 respectively. Both the Carreau models show that the tail of the droplet has a viscosity close to η and the region right after the head has viscosity varying between 40-50Pa.s approximately. 54

57 (a) Carreau 1 (b) Carreau 2 Figure 28: Viscosity in the droplet Indirect structure based viscosity model in COMSOL When using this definition of viscosity, η = η + (η 0 η )ζ p, in N-S equation, the term undergoes a spatial derivative producing the a term whose coefficient looks like pζ p 1. As p <1, ζ must be much greater than zero to prevent the term from going to infinity and subsequently crash the solver. This puts a the lower limit to which η t=0 can be defined for the CFD model. From the rheological data used in comparing the viscosity models ζ t=0 can be explicitly calculated to estimate the working structural parameter to be for a η t=0 =488Pa.s, which is very close to zero. (a) t =25µs (b) t =33µs (c) t =50µs (d) t =108µs Figure 29: Jetting modelled using structure based viscosity model using new parameter (Table.9) and ζ t=0 =0.1 Before analyzing the results from the simulations, when ζ t=0 =0.1 is used as the initial condition for solving the equation of structural breakdown, the viscosity of the solder paste corresponds to 54782Pa.s. As the viscosity is very high initially, the solder paste 55