A molecular theory of surface tension is developed for a liquid–gas interface of a one component system. The Helmholtz free energy, the quantity minimized in the van der Waals approach, is obtained here from a rigorous expansion in powers of derivatives of density ρ and is minimized by the calculus of variations. The coefficient A (ρ) of the term in the square of the density gradient is (kT/6) ℱdr r2C (r,ρ), C being the direct correlation function. In the case in which ρ varies in one direction x only, the solution of the Euler–Lagrange differential equation is analyzed in detail. This describes the cases of a single phase and of two coexisting phases and leads to the equal area Maxwell construction. The effect of an external field on the solution is discussed. The Euler–Lagrange differential equation provides a differential statement of Bernoulli’s theorem. In a three dimensional treatment the stress tensor formula is obtained from the corresponding Euler–Lagrange partial differential equation. A (differential) generalization of the Young–Laplace equation for a spherical interface is also derived. In addition, metastable regions are described and interpreted. The stress tensor for a system in the presence of any kind of conservative force field is also obtained.

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This paper reports on the successful fabrication of a multilayered hybrid printed circuit board (PCB) for applications in the consumer electronics products, medical technologies, and military equipment. The PCB was fabricated by screen-printing silver (Ag) flake ink, as metallization layer, and UV acrylic-based ink, as dielectric layer, on different substrates such as paper, polyethylene terephthalate, and glass. Traditional electronic components were attached onto the printed pads to create the multilayered hybrid PCB. The feasibility of the hybrid PCB was demonstrated by integrating an embedded microcontroller to drive an liquid-crystal display (160 × 100 pixels). In addition, the amount of the ink spreading after printing, the effect of bending on the printed lines, and the effect of the roughness of the substrates on the resistance of the printed lines was investigated. It was observed that the resistance of the lines increased by ≈1.8%, after 10 000 cycles of bending, and the lowest resistance of 1.06 was measured for the 600 μm printed lines on paper, which had a roughness of 0.175 μm. The advantage of fabricating PCBs on flexible substrates is the ability to fold and place the boards on nearly any platform or to conform to any irregular surface, whereas the additive properties of printing processes allow for a faster fabrication process, while simultaneously producing less material waste in comparison with the traditional subtractive processes. The results obtained show the promising potential of employing screen printing process for the fabrication of flexible and light-weight hybrid PCBs.

In recent years, traditional printing methods have been integrated to print flexible electronic devices and circuits. Since process requirements for electronics differ from those for graphic printing, the fundamentals require rediscovery mainly to optimize manufacturing techniques and to find cost reduction methods without compromising functional performance. In addition, alternative inks need to be formulated to increase the variety of functional inks and to pioneer new product developments. In this report, we investigate a thermoplastic-based nickel ink prototype to print electrodes using a screen-printing process. Process fundamentals are explored, and cost reduction methods are addressed by studying the effect of substrate roughness, print direction, and number of ink layers on the electrical performance of printed nickel. Multilayered electrodes are printed on paper and heat stabilized engineered film. A novel fundamental mechanism is found that explains the effect of substrate roughness on ink film roughness in screen printing, including the roughness measurement of the screen mesh wire that is reported for the first time. Results demonstrated that (i) surface roughness of substrates does not have significant effect on printed ink film roughness in screen printing; (ii) ink film thickness is higher on nonabsorbent materials, while line gain is higher on absorbent materials; (iii) the effect of electrode orientation on electrical performance is insignificant; and (iv) the effect of substrate roughness on the electrical performance for the first print layer can be eliminated by printing multiple layers. The results significantly affect substrate choice and number of ink layers, which are considered the major cost factors in the manufacturing of printed electronics.

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