Narrow Results

Existence of random dynamical systems for a class of coalescing stochastic flows on $ℝ$ is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

Molecular dynamic (MD) method is used to study the coalescence and fusing process of Au and Cu nanoclusters. The results show that shear deformation, surface and interface diffusion play important role in different stages of all simulation procedure. In most cases, shear deformation produces the twin boundary or/and stacking fault in particles by particle rotation and slide. The angle between the {111} of Au and Cu particles decrease with increasing temperature, which promotes the formation of the stable interface. Furthermore, the coalescence point and melting temperature increase as cluster diameter increases. For the other cases, there are no particle rotation and slide phenomenon in the elevating temperature process because the stable interface can be formed by forming twin boundaries once two particles contact.

The coalescence, the initial stage of sintering, of two contacted Cu nanoparticles is investigated under different heating rates of 700, 350 and 233 K/ns. The nanoparticles coalesced rapidly at the initial stage when the temperature of the system is low. Then, the nanoparticles collided softly in an equilibrium period. After the system was increased to a high temperature, the shrinkage ratio, gyration radius and atoms’ diffusion started to change dramatically. The lower heating rate can result in smaller shrinkage ratio, larger gyration radius and diffusion of atoms. However, the growth of sintering neck is hardly influenced by the heating rate. The results provide a theoretical guidance for the fundamental understanding and potential application regarding nanoparticle sintering.

In this paper, a new notion of super coalescence hidden-variable fractal interpolation function (SCHFIF) is introduced. The construction of SCHFIF involves choosing an IFS from a pool of several non-diagonal IFS at each level of iteration. Further, the integral of a SCHFIF is studied and shown to be a SCHFIF passing through a different set of interpolation data.

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty$ and Fuchsian singularity at $z=0$, holomorphically depending on parameter $t$ within a polydisk in ${ℂ}^n$ centered at $t=0$. The eigenvalues of the leading matrix at $\infty$, which is diagonal, coalesce along a coalescence locus $\mathrm{\Delta }$ contained in the polydisk. Under minimal vanishing conditions on the residue matrix at $z=0$, we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.] that isomonodromic deformations can be extended to the whole polydisk, including $\mathrm{\Delta }$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of $\mathrm{\Delta }$, where it simplifies. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting $\mathrm{\Delta }$, and if suitable entries of the Stokes matrices vanish, then $\mathrm{\Delta }$ is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.

Molecular dynamics simulations technique is used to study the consolidation of two nanoparticles of Cu element. We have studied sintering processes of two nanoparticles at different temperatures. Two model systems with 4 and 10 nm diameter of particles are selected to study the sintering process of the two nanoparticles. Orientation effects on the physical properties of consolidation of two nanoparticles with respect to each other are investigated. Temperature effects on the consolidation of two nanoparticles are also studied. The order of the values obtained in the simulation for the constant volume heat capacity and latent heat of fusion is good agreement with the bulk results. Moreover, we have investigated the size effects on the consolidation of two different sizes of nanoparticles, that is, one particle of diameter with 10 nm is fixed while the other one is changing from 1 to 10 nm. Melting temperatures of the copper nanoparticles are found to be decreased as the size of the particle decreases. It is found that simulation results are compatible with the other theoretical calculations.

The coalescence of particles extensively exists in the industrial production and nature, which is of great research significance. This paper examined the alloying process of Cu/Au nanoparticles with different sizes by molecular dynamics (MDs) simulations. The coalescence process presents three stages which can be divided by the contact and fusion. The alloying processes of Cu/Au nanoparticles with different sizes had contacted with each other before the heating at 300 K. The Au atoms diffused through the outer area of the sintering neck before the nanoparticles were fused into one particle. The coalescence had become severe after the systems reached the melting temperature. The different systems showed different sintering rate.

This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order $\nu$ of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF.

The droplet coalescence phenomenon extensively exists in the industrial production and application, as well as in nature, which is of great research significance. This paper adopted the molecular dynamics (MDs) simulation method to investigate the behavioral characteristics of water/water, ethanol/ethanol and water/ethanol nanodroplets coalescence. The results suggested that, in water and ethanol nanodroplet coalescence process within the water/ethanol system, ethanol was always wrapped on the outer layer of water droplets. The droplet shrinkage in the water/water system was greater than those in the other two systems; meanwhile, that in the water/ethanol system rapidly increased after the contact of droplets, and subsequently surpassed that in the ethanol/ethanol system.

With the popularization of 3D printing technology, micro/nanoparticles sintering technology has drawn lots of attentions all over the world. Here, molecular dynamic simulation is employed to discuss the effects of different interfacial lattice structures, different diameter of nanoparticles, and different heating rates on the coalescence of metallic Cu nanoparticles. The results showed that the diameter of nanoparticles determine the melting point of the system. Besides, the interfacial lattice structure, diameter of nanoparticles, and heating rate have an influence on the initial sintering temperature. This is because the melting point is the inherent property of material which relies on the mass of substance. However, the initial sintering temperature is sensitive to many factors, including the temperature, interfacial, and intermolecular interactions.