Identities for droplets with circular footprint on tilted surfaces

Exact mathematical identities are presented between the relevant parameters of droplets displaying circular contact boundary based on flat tilted surfaces. Two of the identities are derived from the force balance, and one from the torque balance. The tilt surfaces cover the full range of inclinations for sessile or pendant drops, including the intermediate case of droplets on a wall (vertical surface). The identities are put under test both by the available solutions of a linear response approximation at small Bond numbers as well as the ones obtained from numerical solutions, making use of the Surface Evolver software. The subtleties to obtain certain angle-averages appearing in identities by the numerical solutions are discussed in detail. It is argued how the identities are useful in two respects. First is to replace some unknown values in the Young–Laplace equation by their expressions obtained from the identities. Second is to use the identities to estimate the error for approximate analytical or numerical solutions without any reference to an exact solution.

The energy of fcc and hcp foams

We present Surface Evolver evaluations of the difference in energy between face-centred cubic and hexagonal close-packed foams for liquid fractions from the dry to the wet limit, together with explicit expressions derived from first principles.

Study on the Spreading Characteristics of the Solder Sn-3.0Ag-0.5Cu on an Inverted V-shaped Substrate

The impact of the contact angle of a droplet, the included angle of a substrate and the droplet volume on the morphology and profile of the droplet is discussed, and the spreading characteristics of lead-free solder on an inverted V-shaped substrate are studied, which provides theoretical guidance and data support for a comprehensive study of the interface reaction and wetting mechanism between solder and substrate, and helps improve the brazing process to adapt to complex welding operations. Based on the method of finite element simulation, different contact angles, included angles of inverted V-shaped substrate, droplet volumes and other variables are entered in the model; the relevant physical parameters are defined according to the surface tension and density of solder Sn-3.0Ag-0.5Cu at temperature of 490K; the theoretical spreading results of the droplet are simulated and calculated by Surface Evolver by using the principle of energy minimization and the method of gradient descent; and the spreading distance, contact area and energy equivalence of the droplet are read out by program, which helps to investigate the spreading behavior and wetting characteristics of the droplet.

Stacking Yield Prediction of Package-on-Package Assembly Using Uncertainty Propagation Analysis—Part II: Implementation of Stochastic Model

The stochastic model for yield loss prediction proposed in Part I is implemented for a package-on-package (PoP) assembly. The assembly consists of a stacked die thin flat ball grid array (TFBGA) as the top package and a flip chip ball grid array (fcBGA) as the bottom package. The top and bottom packages are connected through 216 solder joints of 0.5 mm pitch in two peripheral rows. The warpage values of the top and bottom package are calculated by finite element analysis (FEA), and the corresponding probability of density functions (PDFs) are obtained by the eigenvector dimension reduction (EDR) method. The solder ball heights of the top and bottom package and the corner pad joint heights are determined by surface evolver, and their PDFs are determined by the EDR method, too. Only 137 modeling runs are conducted to obtain all 549 PDFs in spite of the large number of input variables considered in the study (27 input variables). Finally, the noncontact open-induced staking yield loss of the PoP assembly is predicted from the PDFs.

Elastic Wave Propagation in Open-Cell Foams

This work examines elastic wave propagation phenomena in open-cell foams with the use of the Bloch wave method and finite element analysis. Random foam topologies are generated with the Surface Evolver and subsequently meshed with Timoshenko beam elements, creating open-cell foam models. Convergence studies on band diagrams of different domain sizes indicate that a representative volume element (RVE) consists of at least 83 cells. Wave directionality and energy flow features are investigated by extracting phase and group velocity plots. Explicit dynamic simulations are performed on finite size domains of the considered foam structure to validate the RVE results. The effect of topological disorder is studied in detail, and excellent agreement is found between the wave behavior of the random foam and that of both the regular and perturbed Kelvin foams in the low-frequency regime. In higher modes and frequencies, however, as the wavelengths become smaller, disorder has a significant effect and the deviation between regular and random foam increases significantly.

DLITE: Dynamic Local Intercellular Tension Estimation

The shape of cell-cell interfaces and the forces resulting from actomyosin interactions, mem-brane tension, or cell-cell adhesion are closely coupled. For example, the tight junction protein, ZO-1, forms a link between the force-bearing actin cortex and the rest of the tight junction protein (TJP) complex, regulating epithelial cell differentiation and the flux of solutes across epithelia. Here we introduce a method for Dynamic Local Intercellular Tension Estimation (DLITE) to computationally infer the evolution of cell-cell forces from a mechanical model of collective cell behaviour. This builds upon prior work in the field (CellFIT, Brodland et al., PloS one 9.6 (2014): e99116). We validate our estimated forces against those predicted by Surface Evolver simulations. Inferred tensions of a cell colony rearranging over time correlate better with the ground truth for our method (DLITE) than for prior methods intended for single time-points. DLITE is robust to both skeletonization errors and topological changes. Finally, we used DLITE in WTC-11 human induced pluripotent stem (hIPS) cells endogenously expressing ZO-1 GFP to find that major topo-logical changes in cell connectivity, e.g. mitosis, can result in an increase in tension. This suggests a correlation between the dynamics of cell-cell forces and colony rearrangement.

Equilibrium configurations of drops or bubbles in an eccentric annulus

The purpose of this paper is to find the zero-gravity equilibrium configurations of liquid drops or bubbles that have sufficient volume to form large-aspect-ratio bridging segments or occluding slugs in the eccentric annulus between two cylinders. In zero gravity, the static problem depends on the contact angle of the fluid segment on the solid support, and two geometric parameters: the radius ratio and the dimensionless distance between the cylinder centres. For both non-wetting and wetting liquids, we find regions of geometric parameter space where only occluding configurations occur, a bistable region where either configuration can occur, and a region where only the non-occluding bridging configuration can occur. For the non-occluding cases, we applied a large-aspect-ratio free-energy minimization approach to predict the cross-sectional shape of the liquid, and a finite element method was used to compute the interface shape of the occluding cases. A Surface Evolver model was used to simulate the three-dimensional shape of both occluding and non-occluding configurations. The simulation results support the theoretical predictions well. The fractional open area of the conduit was determined for both highly wetting and highly non-wetting minority phases. Optimization of the geometric parameters for a given wetting condition could facilitate the segregation and transport of two fluid phases in applications involving large aspect ratios and small pressure driving forces.

Design of Solder Connections for Self-Assembly of Optoelectronic Devices

In surface mount assembly, advantage is taken of the high surface tension of molten solder to self-align ball grid array packages and flip chip die. However, in these applications, the volume of solder applied as paste by stencil printing is not sufficiently well controlled to achieve the precise alignment required for optoelectronic devices. We believe that the requirement on solder volume control for assembly of optoelectronic devices can be relaxed by designing the bond pads so that the height or alignment of connections is controlled by the surface tension of the solder rather than its volume. Our design approach to accomplishing this is to connect auxiliary pads to the primary attachment pad, which act as solder reservoirs. Surface tension causes the solder to be redistributed among these pads to achieve a uniform pressure throughout the solder volume. This phenomena is governed by the Young-Laplace equation, ΔP = γκ, in which ΔP represents the difference in pressure within and outside the solder, γ the surface tension of the solder, and κ the local curvature of the solder surface. Thus, the design of the set of primary and auxiliary pads is critically important to realizing the desired control of joint height. In this article, we describe the use of the Surface Evolver software package in combination with analytical models, to analyze the behavior of various connection configurations with respect to variations in printed solder volume. Specifically, we calculate the equilibrium shape of the solder surface over the connected set of pads and examine how control of joint height is affected by the number, size, and geometry of auxiliary pad configurations.