Nicholas J. Nigro

Effect of Chip and Pad Geometry on Solder Joint Formation in SMT

An analytical model of solder joint formation during a surface mount reflow process is developed for two-dimensional fillets whose flow may be restricted due to “finite” metallizations on a leadless component and the printed circuit board. Although these height and length constraints on the fillet geometry may result in obtuse contact angles, the solution is obtained in the form of an explicit integral, similar to that previously derived by the authors for the case of acute contact angles. This solution may also be recast into the form of elliptic integrals of the first and second kinds, thereby permitting one to evaluate the fillet geometry using mathematical tables or special function software, if desired, rather than resorting to a computer-based numerical quadrature. In addition an approximate zero-gravity solution is given by means of simple closed-form expressions relating the height, length, contact angles, and cross-sectional area of the fillet. Numerical results generated by implementing the “exact” integral solution for the joint profile are given in the form of dimensionless plots, relating fillet geometry to the solder properties (surface tension and density), amount of solder, chip height, and pad length. Also presented in dimensionless form are the approximate results from the zero-gravity model, which are independent of solder properties, yet are of sufficient accuracy for “small” joints. Because of their dimensionless nature, the results of the present paper may be of maximum utility to process engineers aiming to achieve desired joint geometries (e.g., to maximize fatigue life or to eliminate bridging problems), or to board designers responsible for selecting efficient footprint patterns to maximize board density. Models of solder joint formation, such as the one presented here, may be of most value when used in conjunction with stress analysis packages (e.g., finite element programs) and appropriate fatigue models. In this way an integrated approach to the design of solder joints and circuit boards may be taken, resulting in improved product reliability and performance.

Numerical modelling and experimental verification of the formation of 2D and 3D brazed joints

This paper addresses the issue of determining the joint shape formed after molten metal re-solidification at the peak brazing temperature. A theoretical approach for modelling two- and three-dimensional joint shapes in situations where the mating surfaces are neither plane nor orthogonal is presented. The approach is based on a variational principle involving the minimization of the potential energy of the molten metal liquid just prior to the onset of solidification. The numerical solution of the variational problem is obtained by employing a parametric finite element method in conjunction with a direct optimization algorithm. The results from the theory are verified by comparison with experimental data obtained from a set of controlled atmosphere brazing tests of aluminium alloys. The results obtained from the theory are in good agreement with the experimental data and empirical evidence. An analysis of the influence of geometry, configuration and orientation of mating surfaces on both two- and three-dimensional brazed joint shapes is presented in an accompanying paper.

Resonant characteristics of rectangular microcantilevers vibrating torsionally in viscous liquid media

The resonant characteristics of rectangular microcantilevers vibrating in the torsional mode in viscous liquid media are investigated. The hydrodynamic load (torque per unit length) on the vibrating beam due to the liquid was first determined using a finite element model. An analytical expression of the hydrodynamic function in terms of the Reynolds number and aspect ratio, h/b (with thickness, h, and width, b) was then obtained by fitting the numerical results. This allowed for the resonance frequency and quality factor to be investigated as functions of both beam geometry and medium properties. Moreover, the effects of the aspect ratio on the cross-sections torsional constant, K, which affects the microcantilevers torsional stiffness, and on its polar moment of inertia, Jp, which is associated with the beams rotational inertia, are also considered when obtaining the resonance frequency and quality factor. Compared with microcantilevers under out-of-plane (transverse) flexural vibration, the results show that microcantilevers that vibrate in their 1st torsional or 1st in-plane (lateral) flexural resonant modes have higher resonance frequency and quality factor. The increase in resonance frequency and quality factor results in higher mass sensitivity and reduced frequency noise, respectively. The improvement in the sensitivity and quality factor are expected to yield much lower limits of detection in liquid-phase chemical sensing applications.

Study of the axially symmetric motion of an incompressible viscous fluid between two concentric rotating spheres

The research reported herein involves the study of the steady state and transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research is to study the use of an approximate analytical method for analyzing the transient motion of the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems include cases where: (a) one (or both) spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive torque.In this research, the coupled solid and fluid equations of motion are linearized by employing the perturbation technique. The meridional dependence in these equations is removed by expanding the dependent variables in a series of Gegenbauer functions with variable coefficients and employing the orthogonality property of these functions. The equations for the variable coefficients are solved by separation of variables and Laplace transform methods. Results for the stream function, circumferential function, angular velocity of the spheres and torque coefficient are presented as a function of time for various values of the dimensionless system parameters.

Lateral-Mode Vibration of Microcantilever-Based Sensors in Viscous Fluids Using Timoshenko Beam Theory

To more accurately model microcantilever resonant behavior in liquids and to improve lateral-mode sensor performance, a new model is developed to incorporate viscous fluid effects and Timoshenko beam effects (shear deformation, rotatory inertia). The model is motivated by studies showing that the most promising geometries for lateral-mode sensing are those for which Timoshenko effects are most pronounced. Analytical solutions for beam response due to harmonic tip force and electrothermal loadings are expressed in terms of total and bending displacements, which correspond to laser and piezoresistive readouts, respectively. The influence of shear deformation, rotatory inertia, fluid properties, and actuation/detection schemes on resonant frequencies (fres) and quality factors (Q) are examined, showing that Timoshenko beam effects may reduce fres and Q by up to 40% and 23%, respectively, but are negligible for width-tolength ratios of 1/10 and lower. Comparisons with measurements (in water) indicate that the model predicts the qualitative data trends, but underestimates the softening that occurs in stiffer specimens, indicating that support deformation becomes a factor. For thinner specimens, the model estimates Q quite well, but exceeds the observed values for thicker specimens, showing that the Stokes resistance model employed should be extended to include pressure effects for these geometries.

Influence of joint topology on the formation of brazed joints

This paper discusses the influence of joint topology on the formation of brazed joints. For the purposes of this study, the joint topology is defined by dimensionless parameters that characterize: (i) the geometry of the mating surfaces (shape and configuration), (ii) the gap between bonded parts (clearance and tolerances), (iii) the joint orientation in the gravity field and (iv) the volume of the joint. The influence of these parameters is presented for several joints with a topology that is typically found in the manufacture of compact heat exchangers. The results were obtained by employing a method that is based on a variational principle and minimization of the potential energy of the molten aluminium liquid metal just prior to the onset of solidification. The method, which was verified by comparison with experimental data obtained from controlled atmosphere brazing of aluminium alloys, is discussed in an accompanying paper.

Timoshenko Beam Model for Lateral Vibration of Liquid-Phase Microcantilever-Based Sensors

Dynamic-mode microcantilever-based devices are potentially well suited to biological and chemical sensing applications. However, when these applications involve liquid-phase detection, fluid-induced dissipative forces can significantly impair device performance. Recent experimental and analytical research has shown that higher in-fluid quality factors (Q) are achieved by exciting microcantilevers in the lateral flexural mode. However, experimental results show that, for microcantilevers having larger width-to-length ratios, the behaviors predicted by current analytical models differ from measurements. To more accurately model microcantilever resonant behavior in viscous fluids and to improve understanding of lateral-mode sensor performance, a new analytical model is developed, incorporating both viscous fluid effects and “Timoshenko beam” effects (shear deformation and rotatory inertia). Beam response is examined for two harmonic load types that simulate current actuation methods: tip force and support rotation. Results are expressed in terms of total beam displacement and beam displacement due solely to bending deformation, which correspond to current detection methods used with microcantilever-based devices (optical and piezoresistive detection, respectively). The influences of the shear, rotatory inertia, and fluid parameters, as well as the load/detection scheme, are investigated. Results indicate that load/detection type can impact the measured resonant characteristics and, thus, sensor performance, especially at larger values of fluid resistance.

Effect of design parameters on the rotational response of a novel disk resonator for liquid-phase sensing: Analytical results

The advantages of a novel “all-shear interaction device” (ASID), consisting of a disk driven and supported by two tangentially oriented microcantilevers, are explored through the derivation of a new analytical model for the in-plane rotational response of the disk due to harmonic axial eigenstrains (e.g., electrothermal strains) imposed on the supporting legs. The resulting expressions for frequency response, resonant frequency, and quality factor (Q) explicitly show their relationships to the devices geometric and material parameters and the density and viscosity of the liquid. The validity of the model is confirmed by FEA and recent experimental data from the literature, thus demonstrating the potential for both the device and the model for use in liquid-phase sensing applications.

A modified finite element method for determining equilibrium capillary surfaces of liquids with specified volumes

This paper describes a modified finite element method (MFEM) for determining the static equilibrium shape of the capillary surface of a liquid with a prescribed volume constrained by rigid boundaries with arbitrary shapes. It is assumed that the liquid is in static equilibrium under the influence of surface tension, adhesion, and gravity forces. This problem can be solved by employing the conventional FEM; however, a major difficulty arises due to the presence of the volume (integral) constraint and usually requires the use of the Lagrange multiplier method, the sequential unconstrained minimization technique, or the augmented Lagrange multiplier method. With the MFEM, the space variables defining the equilibrium surfaces (or curves) are expanded in terms of parametric interpolation functions, which are designed such that the boundary conditions and the integral constraint equation are automatically satisfied during each iteration of a direct numerical search process. Hence, there is no need to include Lagrange multipliers and/or penalty factors and the problem can be treated more simply as one involving unconstrained optimization. This investigation indicates that the MFEM is more efficient and reliable than the other methods. Results are presented for several case study problems involving liquid solder drops. Copyright

Analytical Modeling of a Novel High- Q Disk Resonator for Liquid-Phase Applications

To overcome the detrimental effects of liquid environments on microelectromechanical systems resonator performance, the in-fluid vibration of a novel disk resonator supported by two electrothermally driven legs is investigated through analytical modeling and the effects of the systems geometric/material parameters on the dynamic response are explored. The all-shear interaction device (ASID) is based on engaging the surrounding fluid primarily through shearing action. The theory comprises a continuous-system, multimodal model, and a single-degree-of-freedom model, the latter yielding simple formulas for the fundamental-mode resonant characteristics that often furnish excellent estimates to the results based on the more general model. Comparisons between theoretical predictions and previously published liquid-phase quality factor (Q) data (silicon devices in heptane) show that the theoretical results capture the observed trends and also give very good quantitative estimates, particularly for the highest Q devices. Moreover, the highest Q value measured in the earlier study (304) corresponded to a specimen whose disk radius-to-thickness ratio was 2.5, a value that compares well with the optimal value of 2.3 predicted by the present model. The insight furnished by the proposed theory is expected to lead to further improvements in ASID design to achieve unprecedented levels of performance for a wide variety of liquid-phase resonator applications.