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Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel \begin{document}$ K(j, k) $\end{document} . Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel \begin{document}$ (i) $\end{document} \begin{document}$ K(j, k) = b_{j}a_{k} $\end{document} and \begin{document}$ (ii) $\end{document} \begin{document}$ K(j, k) = ja_{k}+b_{j}+\varepsilon\beta_{j}\alpha_{k} $\end{document} . For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass \begin{document}$ \rho_{s} $\end{document} above which there is no equilibrium. We prove that if the system has an initial mass below \begin{document}$ \rho_{s} $\end{document} then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above \begin{document}$ \rho_{s} $\end{document} then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.

Purpose This paper aims to demonstrate the improved functionality of additive manufacturing technology provided by combining multiple processes for the fabrication of packaged electronics. Design/methodology/approach This research is focused on the improvement in resolution of conductor deposition methods through experimentation with build parameters. Material dispensing with two different low temperature curing isotropic conductive adhesive materials was characterised for their application in printing each of three different conductor designs, traces, z-axis connections and fine pitch flip chip interconnects. Once optimised, demonstrator size can be minimised within the limitations of the chosen processes and materials. Findings The proposed method of printing z-axis through layer connections was successful with pillars 2 mm in height and 550 µm in width produced. Dispensing characterisation also resulted in tracks 134 µm in width and 38 µm in height allowing surface mount assembly of 0603 components and thin-shrink small outline packaged integrated circuits. Small 149-µm flip chip interconnects deposited at a 457-µm pitch have also been used for packaging silicon bare die. Originality/value This paper presents an improved multifunctional additive manufacturing method to produce fully packaged multilayer electronic systems. It discusses the development of new 3D printed, through layer z-axis connections and the use of a single electrically conductive adhesive material to produce all conductors. This facilitates the surface mount assembly of components directly onto these conductors before stereolithography is used to fully package multiple layers of circuitry in a photopolymer.

The exchange-driven growth model describes the mean-field kinetics of a population of composite particles (clusters) subject to pairwise exchange interactions. Exchange in this context means that upon interaction of two clusters, one loses a constituent unit (monomer) and the other gains this unit. Two variants of the exchange-driven growth model appear in applications. They differ in whether clusters of zero size are considered active or passive. In the active case, clusters of size zero can acquire a monomer from clusters of positive size. In the passive case they cannot, meaning that clusters reaching size zero are effectively removed from the system. We show that the large-time behavior is very different for the two variants of the model. We first consider an isolated system. In the passive case, the cluster size distribution tends towards a self-similar evolution and the typical cluster size grows as a power of time. In the active case, we identify a broad class of kernels for which the the cluster size distribution tends to a nontrivial time-independent equilibrium in which the typical cluster size is finite. We next consider a nonisolated system in which monomers are input at a constant rate. In the passive case, the cluster size distribution again attains a self-similar profile in which the typical cluster size grows as a power of time. In the active case, a surprising new behavior is found: the cluster size distribution asymptotes to the same equilibrium profile found in the isolated case but with an amplitude that increases linearly with time.

We study the linear stability of the Vlasov–Poisson system in one dimension on a finite interval and half line with the in-flow and specular reflection boundary conditions. Conditions for the existence and non-existence of stationary solutions are obtained. The behavior of the perturbed solutions is investigated in different contexts. We show that, if the equilibrium distribution μ is a strictly decreasing function of local energy density, then the Vlasov–Poisson system on a finite interval is stable in the (weighted) L 2 sense for the in-flow and specular reflection boundary conditions. For the case μ is strictly increasing, we prove a similar result on a finite interval under some extra assumptions. Finally, in the last part of the article, the spectral stability of the Vlasov–Poisson system on an unbounded interval is studied with similar monotonicity assumptions on μ .

This work reports the laser surface modification of 304S15 stainless steel to develop superhydrophobic properties and the subsequent application for homogeneous spot deposition. Superhydrophobic surfaces, with steady contact angle of ∼154° and contact angle hysteresis of ∼4°, are fabricated by direct laser texturing. In comparison with common pico-/femto-second lasers employed for this patterning, the nanosecond fiber laser used in this work is more cost-effective, compact and allows higher processing rates. The effect of laser power and scan line separation on surface wettability of textured surfaces are investigated and optimized fabrication parameters are given. Fluid flows and transportations of polystyrene (PS) nanoparticles suspension droplets on the processed surfaces and unprocessed wetting substrates are investigated. After evaporation is complete, the coffee-stain effect is observed on the untextured substrates but not on the superhydrophobic surfaces. Uniform deposition of PS particles on the laser textured surfaces is achieved and the deposited material is confined to smaller area.

The stationary solutions of the Vlasov–Poisson system for a plasma are studied with general diffusive boundary conditions. The distribution function \(f(x,v)\), which depends on the local energy and angular momentum, is determined uniquely under certain integrability and decay assumptions on the diffusive kernels and the particle injection intensities. The resulting nonlinear Poisson equation is then solved for the electric potential \(\phi (x)\). We study the existence and uniqueness of its solutions in one and higher dimensions under a variety of settings.

Dynamic control of the distribution of polystyrene suspended nanoparticles in evaporating droplets is\ud investigated using a 2.9 mm high power laser. Under laser radiation a droplet is locally heated and fluid\ud flows are induced that overcome the capillary flow, and thus a reversal of the coffee-stain effect is\ud observed. Suspension particles are accumulated in a localised area, one order of magnitude smaller than\ud the original droplet size. By scanning the laser beam over the droplet, particles can be deposited in an\ud arbitrary pattern. This finding raises the possibility for direct laser writing of suspended particles through\ud a liquid layer. Furthermore, a highly uniform coating is possible by manipulating the laser beam diameter\ud and exposure time. The effect is expected to be universally applicable to aqueous solutions independent\ud of solutes (either particles or molecules) and deposited substrates.

This work demonstrates a novel technique for fabricating surfaces with roughness and wettability gradients\ud and their subsequent applications for chemical sensors. Surface roughness gradients on brass sheets\ud are obtained directly by nanosecond laser texturing. When these structured surfaces are exposed to air,\ud their wettability decreases with time (up to 20 days) achieving both spatial and temporal wettability\ud gradients. The surfaces are responsive to organic solvents. Contact angles of a series of dilute isopropanol\ud solutions decay exponentially with concentration. In particular, a fall of 132◦ in contact angle is observed\ud on a surface gradient, one order of magnitude higher than the 14◦ observed for the unprocessed surface,\ud when the isopropanol concentration increased from 0 to 15.6 wt%. As the wettability changes gradually\ud over the surface, contact angle also changes correspondingly. This effect offers multi-sensitivity at\ud different zones on the surface and is useful for accurate measurement of chemical concentration

Additive Manufacturing processes can facilitate the rapid iterative product development of electronic devices by optimising their design and functionality. In addition, these methods present a number of potential advantages for improving the production speed and complexity of mass customised and bespoke electronics. In this paper, we present a new digitally driven hybrid fabrication process chain, capable of producing functional, multilayer electronics embedded within geometrically complex 3D printed structures. This has been achieved by interleaving stereolithography, micro-dispensing and surface mount assembly. The resultant combination of different template-less manufacturing techniques enables both the formation of multi-material circuits (conductors and dielectrics) and where the package housing encapsulates the electronics and forms part of the final 3D device. This paper also details the developments around depositing novel freestanding z-axis interconnects. A 555 timer circuit with flashing LED manufactured within a 3D pyramid was used as a demonstrator. The demonstrator contained circuits with feature sizes down to 170µm, and packaged components of 0603 size, a Small Outline Integrated Circuit (SOIC) and a SMD LED. In addition, flip chip packaging on 3D printed substrates has been demonstrated.

An alternative formulation of the phase field method is utilized from an integral equation perspective. The technique allows one to derive macroscopic conditions at the interface from the microscopic potentials. Differential geometry and asymptotic analysis yield interface conditions, in arbitrary spatial dimension, for interactions that may include anisotropy as well as non-local potentials. The interface conditions can be expressed in various formulations, for example, in terms of the principal curvature directions of the interface, or the second order directional derivatives of the (signed) distance function and the Hessian of the surface tension.

We derive a set of higher order phase field equations using a microscopic interaction Hamiltonian with detailed anisotropy in the interactions of the form $a_{0}+\delta\sum_{n=1}^{N}{a_{n}\cos( 2n\theta) + b_{n}\sin( 2n\theta) }$ where $\theta$ is the angle with respect to a fixed axis, and $\delta$ is a parameter. The Hamiltonian is expanded using complex Fourier series, and leads to a free energy and phase field equation with arbitrarily high order derivatives in the spatial variable. Formal asymptotic analysis is performed on these phase field equation in terms of the interface thickness in order to obtain the interfacial conditions. One can capture $2N$-fold anisotropy by retaining at least $2N^{th}$ degree phase field equation. We derive, in the limit of small $\delta,$ the classical result $( T-T_{E} ) [s]_{E}=-\kappa {\sigma( \theta ) + \sigma^{''}( \theta) }$ where $T-T_{E}$ is the difference between the temperature at the interface and the equilibrium temperature between phases, $[s]_{E}$ is the entropy difference between phases, $\sigma$ is the surface tension and $\kappa$ is the curvature. If there is only one mode in the anisotropy [i.e., the sum contains only one term: $A_{n}\cos( 2n\theta) $] then this identity is exact (valid for any magnitude of $\delta$) if the surface tension is interpreted as the sharp interface limit of excess free energy obtained by the solution of the $2N^{th}$ degree differential equation. The techniques rely on rewriting the sums of derivatives using complex variables and combinatorial identities, and performing formal asymptotic analyses for differential equations of arbitrary order.

This work demonstrates superhydrophobic surfaces on copper and brass fabricated by a cost-effective and compact nanosecond fiber laser. The processing rate is particularly high, about two orders of magnitude faster than that using femtosecond lasers, which is favourable for large area processing. The laser processed copper and brass surfaces exhibit superhydrophilic behavior directly after fabrication but this changes to superhydrophobic (contact angle ∼152°) over time under ambient conditions. Applications of the superhydrophobic brass surfaces for chemical sensing, which is based on the decrease of contact angle with liquid concentration, are presented. Furthermore, a novel approach for making roughness gradients on brass surface is shown, thus enabling surfaces with wettability gradients. This approach could have potential applications for multi-sensitivity chemical sensors, liquid transportation and microfluidics.

A novel hybrid additive manufacturing (AM) technology combining digital light projection (DLP) stereolithography (SL) with 3D micro-dispensing alongside conventional surface mount packaging is presented in this work. This technology overcomes the inherent limitations of individual AM processes and integrates seamlessly with conventional packaging processes to enable the deposition of multiple materials. This facilitates the creation of bespoke end-use products with complex 3D geometry and multi-layer embedded electronic systems. Through a combination of four-point probe measurement and non-contact focus variation microscopy, it was identified that there was no obvious adverse effect of DLP SL embedding process on the electrical conductivity of printed conductors. The resistivity maintained to be less than 4 × 10−4 Ω centerdot cm before and after DLP SL embedding when cured at 100 °C for 1 h. The mechanical strength of SL specimens with thick polymerized layers was also identified through tensile testing. It was found that the polymerization thickness should be minimised (less than 2 mm) to maximise the bonding strength. As a demonstrator a polymer pyramid with embedded triple-layer 555 LED blinking circuitry was successfully fabricated to prove the technical viability.

– Additive manufacturing, an umbrella term for a number of different manufacturing techniques, has attracted\ud increasing interest recently for a number of reasons, such as the facile customisation of parts, reduced time to\ud manufacture from initial design, and possibilities in distributed manufacturing and structural electronics. Inkjet printing\ud is an additive manufacturing technique that is readily integrated with other manufacturing processes, eminently\ud scalable and used extensively in printed electronics. It therefore presents itself as a good candidate for integration with\ud other additive manufacturing techniques to enable the creation of parts with embedded electronics in a timely and cost\ud effective manner. This review introduces some of the fundamental principles of inkjet printing; such as droplet generation,\ud deposition, phase change and post-deposition processing. Particular focus is given to materials most relevant\ud to incorporating structural electronics and how post-processing of these materials has been able to maintain compatibility\ud with temperature sensitive substrates. Specific obstacles likely to be encountered in such an integration and\ud potential strategies to address them will also be discussed.

We adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second-order equation. In particular, we study equations involving the derivative of a double-well potential such as u − u3 or − u + 2u3. Using momentum (Fourier) space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation, once length is properly renormalized, yielding an approximation to solutions of the higher order equation from the second order. The results are confirmed using numerical computations. Additional numerics confirm that the main role of the higher order derivatives is in rescaling the length.

We present an alternative formulation of the phase field method for modeling interfaces. In this approach one works directly with integral equations instead of differential equations. The technique allows one to derive, from microscopic potentials, the surface tension and macroscopic conditions at the interface. In the last section a numerical example is given to illustrate the use of the technique. For a given form of interaction potential, the surface tension and its anisotropy are calculated from which we construct the equilibrium shapes.

Exchange-driven growth is a process in which pairs of clusters interact and exchange a single unit of mass. The rate of exchange is given by an interaction kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field kinetic equations of this process for the first time. We find two different classes of behaviour depending on whether $K(j,k)$ is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in the sense that we show for a large class of initial conditions with kernels satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$ $\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$ ($\beta>2).$ In the intermediate regime $3, Comment: Mistakes in the uniqueness proofs are fixed. Some typos are corrected. Some references are added