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Marangoni and Variable Viscosity Phenomena in Picoliter Size Solder Droplet Deposition

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Marangoni and Variable Viscosity Phenomena in Picoliter Size Solder Droplet Deposition Proceedings of IMECE 2002 ASME International Mechanical Engineering Congress & Exposition November 17-22, 2002, New Orleans, Louisiana Paper No IMECE 2002-HT-32111 MARANGONI AND VARIABLE VISCOSITY PHENOMENA IN PICOLITER SIZE SOLDER DROPLET DEPOSITION M. Dietzel S. Haferl Y. Ventikos D. Poulikakos Laboratory of Thermodynamics in Emerging Technologies Institute of Energy Technology, Department of Mechanical and Process Engineering Swiss Federal Institute of Technology ABSTRACT was examined, thereby laying the focus on the effects of a This numerical investigation studied the effects which the temperature dependence of surface tension (temperature-induced temperature dependence of surface tension (Marangoni phenomenon) Marangoni effect) and viscosity. and viscosity has on the spreading, the transient behavior and final Earlier numerical and experimental investigations have shown post-solidification shape of a molten Sn63Pb solder droplet deposited that the impact velocity and surface tension are important parameters on a flat substrate. A Lagrangian finite element formulation of the determining the fluid mechanical behavior of the droplet upon impact complete axisymmetric Navier-Stokes equations was utilized for the within the parametric domain of solder jetting processes, [2,3]. The description of the droplet behavior. Linear temperature dependence low impact velocities of O (1m/s) ensure that no splashing occurs. for the surface tension and an exponential dependence for the Considering the importance of surface tension for the problem, a viscosity were assumed. The initial droplet temperature was varied variable surface tension may change the spreading behavior of the in 50K steps from 200ºC to 500ºC, whereas the substrate temperature droplet markedly. was kept constant at 25ºC. This varied the initial Reynolds number Marangoni convection was shown to be of crucial importance Re0 from 360 to 716 and the Marangoni number Ma from –9 to –49. in solidification processes leading to phase separation or to local The initial Weber number We 0 and initial Prandtl number Pr0 were concentration changes in alloys, Cao and Wang et al. [4]. Song and for all cases O(1) and O(10-2), respectively. The impact velocity and Dailey et al. [5] reported that internal droplet flows are often caused the droplet diameter remained unchanged in all cases examined at 1.5 by surface tension forces rather than by buoyancy forces. Ehrhard m/s and 80 microns. and Davis [6] discussed the spreading of droplets on a horizontal A major finding of the work was that, contrary to intuition, the plate under the presence of thermocapillary forces based on the Marangoni effect decreased droplet spreading monotonically. Due to lubrication theory. This work was extended by Braun and Murray et the Marangoni effect, surface tension forces instead of freezing al. [7] to account for solutal Marangoni effects in Pb-Sn alloy arrested spreading. Droplet receding during recoiling was aided by droplets with Ma of O(10-1) – O(10-2). Both investigations reported a the Marangoni effect. On the other hand, the change of viscosity reduced spreading if the droplet was heated from the substrate with temperature showed no significant influence on the outcome of underneath. The accurate prediction in spreading is of importance in the droplet impact. many applications in microchip manufacturing. To the best of our knowledge no work on inertia dominated droplet impact flow combined with Marangoni convection has yet been presented in the INTRODUCTION open literature, a problem which typically occurs in solder jetting. Free surface flows with thermal transport play an important The numerical approach herein is based on the Lagrangian role in a wide range of modern technical applications, such as spray formulation of the axisymmetric, unsteady Navier-Stokes equations, deposition, injection, casting, welding, soldering or extrusion and the energy equation, which are discretized utilizing the Galerkin processes, Maronnier and Picasso et al. [1]. In recent years, there has finite element method and a deforming, non-adaptive triangulation been an increasing interest in micro-scale liquid transport with many mesh. The implementation employed extends the methodology of [2] applications in MEMS and microelectronics manufacturing. to account for the temperature dependence of the surface tension and In this context, the deposition of molten picoliter size solder viscosity. droplets with diameter O(100µm) upon a flat conductive substrate 1 Copyright © 2002 by ASME NOMENCLATURE c Speed of sound [m/s] α Thermal diffusivity [m2/s] d Droplet diameter [m] δ Kronecker symbol [-] 2 g Gravity [m/s ] ε Navier-slip coefficient [-] γ H, H Curvature [1/m], [-] Surface tension [N/m] k Thermal conductivity [W/mK] µ Dynamic viscosity [kg/ms] L Latent heat [J/kg] ρ Density [kg/m3] M Mach number [-] Ma Marangoni number [-] Subscripts p, P Pressure [Pa], [-] 0 Initial r, R Radial coordinate [m], [-] 1 Droplet s, S Arc length [m], [-] 2 Substrate T, Θ Temperature [K, °C], [-] θ Azimuthal direction t , τ Time [s], [-] u, U Radial velocity [m/s], [-] Mathematical Operators v, V Axial velocity [m/s], [-] Dτ Lagrangian derivative towards dimensionless time xSn Molar fraction of tin [-] dT Ordinary derivative towards temperature z, Z Axial coordinate [m], [-] ∂ R,Z Partial derivative towards dimensionless coordinates R, Z SURFACE TENSION AND VISCOSITY CORRELATIONS −5 1/ 3 0.08849 ⋅ ρSn µ = 2.75 ⋅10 ρ exp Variations in surface tension cause a force acting tangentially to Sn Sn T + 273 the surface considered. Gradients in surface tension originate from (den Boer [8]) −5 1/ 3 0.0863⋅ ρPb • a temperature gradient, causing a thermocapillary effect µPb = 2.54 ⋅10 ρPb exp T + 273 (2b) • a concentration gradient, causing a destillocapillary effect • an electrical potential ρ = 7.142 ⋅103 − 0.6127 ⋅T Since Sn63Pb is an eutectic alloy, we concentrate in the following on Sn the thermocapillary effect, assuming that neither a marked 4 concentration gradient through a non-eutectic solidification nor an ρPb = 1.109 ⋅10 −1.3174 ⋅T electrical potential is involved in the solder droplet deposition process. This set of equations provides the viscosity of solder in [Pa·s] if the The most widely accepted correlation between surface tension temperature T is inserted in [°C]. It has the advantage towards γ and temperature is the following linear approach: conventional exponential approaches to provide the most realistic approximation for viscosity especially close to the solidification temperature. γ = γ ref + (d T γ )ref (T − Tref ) (1) One can show by the virtue of statistical mechanics that the approaches for surface tension and viscosity, which both rely on The use of a first order Taylor expansion for the surface tension macroscopic measurable properties, are well supported by theories on matches experimental data quite well, in particular if the reference the microscopic level. A detailed discussion can be found in Fowler point for determining the surface tension gradient (d γ ) is chosen T ref [11] and Born and Green [12]. Based on their work, Egry [13] appropriately within the parametric domain. pointed out that there is a fixed relation between viscosity µ and Experimental data show a simple exponential dependence of surface tension γ, since they have similar microscopic origins. This the viscosity µ on temperature. Koke [9] discussed a more justifies the simultaneous consideration of the temperature sophisticated approach for viscosity especially for alloys, in dependence of surface tension and viscosity in this study. particular solder. It is based on the work of Thresh and Crawley [10] The thermophysical properties of solder used and shown in and can be cited as follows: Table 1 are carefully chosen and attuned to several literature sources, [14-16]. µ = µ + x (µ − µ ) Solder Pb Sn Sn Pb (2) where 2 Copyright © 2002 by ASME Table 1 Thermophysical Properties of Sn63Pb @ 260°C r z u v R = ,Z = ,U = ,V = d d v v Liquid Solid 0 0 0 0 Density ρ [kg/m3] 8218 8240 t p − pamb τ = ,P = 2 Viscosity µref [Pas] 0.002237 - d0 / v0 ρv0 (7) Surf. Tens. γref [mN/m] 498.53 - 2 -3 σ ij = (σ ij + δij p0 )/ ρv0 Surf. Tens. Slope (dTγ)ref [N/mK] -0.214⋅10 - Heat Capacity c [J/kgK] 238 176 p Ti − min(T1,0 ,T2,0 ) Θi = Thermal Conductivity k [W/mK] 25 48 T1,0 − T2,0 Latent Heat of Fusion L [kJ/kg] 4.2 4.2 A sketch of the problem, including the definitions of the coordinates, Melting Point Tm [°C] 183 183 is shown in Fig. 1. The dimensionless stress components σ ij in the momentum equations are defined as follows: z,Z S/Smax = 1 2 σ = −P + ∂ U , RR Re R 2 σ RR σ RZ 0 σ ZZ = −P + ∂ ZV Re T = σ RZ σ ZZ 0 (8) Splat 2 U 0 0 σ σ = −P + , θθ θθ Re R Interface layer 1 S Substrate σ = ()∂ U + ∂ V RZ Re Z R S/Smax = 0 r,R where T is the dimensionless stress tensor. The characteristic Fig. 1 Schematic of the impingement process dimensionless numbers of the problem are the initial Reynolds, Froude, Mach, Weber and Peclet number defined as: 2 GOVERNING EQUATIONS AND SOLUTION PROCEDURE ρv d v v0 Re = 0 0 , Fr = 0 , M = 0 µ d g c 0 0 (9) 2 d v Set of Equations ρd0v0 0 0 We = , Pei = Re0i ⋅ Pr0i = The axisymmetric mathematical description of the problem in a 0 α γ 0 i dimensionless Lagrangian form of the Navier Stokes equations is: where α represents the thermal diffusivities of the different regions Continuity: i (i = 1 droplet, i =2 substrate). The dimensionless initial and 1 1 (3) boundary conditions have the following form: D P + ∂ RU + ∂ V = 0 τ 2 R ()Z M R 4 R-Momentum: U = 0, V = −1, P = , We 0 1 σ (4) τ = 0 : D U − ∂ ()Rσ − ∂ σ + θθ = 0 τ R RR Z RZ R R Θ1 = 0, Θ2 =1 (10) Z-Momentum: 1 1 (5) D V − ∂ ()Rσ −∂ σ − = 0 R = 0 : U = 0, ∂ V = 0, ∂ Θ = 0 τ R R RZ Z ZZ Fr R R Energy, i = 1 – Droplet, i = 2 – Substrate: Z = 0 : U = V = 0 2 1 2 (6) The initial substrate temperature is equal to the surrounding Dτ Θ i − ∂ R ()R∂ R Θ i + ∂ ZZ Θ i = 0 Pei R temperature and both the free surface of the droplet and the substrate are considered to be adiabatic.
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