Effect of solder volume on joint shape with variable chip-to-board contact pad ratio

The objective of this paper is to investigate the effect of the pad size ratio between the chip and board end of a solder joint on the shape of that solder joint in combination with the solder volume available. The shape of the solder joint is correlated to its reliability and thus of importance. For low density chip bond pad applications Flip Chip (FC) manufacturing costs can be kept down by using larger size board pads suitable for solder application. By using “Surface Evolver” software package the solder joint shapes associated with different size/shape solder preforms and chip/board pad ratios are predicted. In this case a so called Flip-Chip Over Hole (FCOH) assembly format has been used. Assembly trials involved the deposition of lead-free 99.3Sn0.7Cu solder on the board side, followed by reflow, an underfill process and back die encapsulation. During the assembly work pad off-sets occurred that have been taken into account for the Surface Evolver solder joint shape prediction and accurately matched the real assembly. Overall, good correlation was found between the simulated solder joint shape and the actual fabricated solder joint shapes. Solder preforms were found to exhibit better control over the solder volume. Reflow simulation of commercially available solder preform volumes suggests that for a fixed stand-off height and chip-board pad ratio, the solder volume value and the surface tension determines the shape of the joint.

On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Role of the fibula in the stability of diaphyseal tibial fractures fixed by intramedullary nailing

Background: For tibial fractures, the decision to fix a concomitant fibular fracture is undertaken on a case-by-case basis. To aid in this clinical decision-making process, we investigated whether loss of integrity of the fibula significantly destabilises midshaft tibial fractures, whether fixation of the fibula restores stability to the tibia, and whether removal of the fibula and interosseous membrane for expediency in biomechanical testing significantly influences tibial interfragmentary mechanics. Methods: Tibia/fibula pairs were harvested from six cadaveric donors with the interosseous membrane intact. A tibial osteotomy fracture was fixed by reamed intramedullary (IM) nailing. Axial, torsion, bending, and shear tests were completed for four models of fibular involvement: intact fibula, osteotomy fracture, fibular plating, and resected fibula and interosseous membrane. Findings: Overall construct stiffness decreased slightly with fibular osteotomy compared to intact bone, but this change was not statistically significant. Under low loads, the influence of the fibula on construct stability was only statistically significant in torsion (large effect size). Fibular plating stiffened the construct slightly, but this change was not statistically significant compared to the fibular osteotomy case. Complete resection of the fibula and interosseous membrane significantly decreased construct torsional stiffness only (large effect size). Interpretation: These results suggest that fixation of the fibula may not contribute significantly to the stability of diaphyseal tibial fractures and should not be undertaken unless otherwise clinically indicated. For testing purposes, load-sharing through the interosseous membrane contributes significantly to overall construct mechanics, especially in torsion, and we recommend preservation of these structures when possible.