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This research investigates the free vibration of a rotating annular microplate under the flexoelectric effect. Initially, the Kirchhoff plate theory assumptions are used to express the displacement fields. After considering the displacement field, strains and their gradients are derived and substituted into the electric enthalpy and kinetic energy expressions. Subsequently, by applying Hamilton’s principle to the aforementioned equations, the electric and mechanical equations are computed. To derive the equations of motion, initially, the polarization vectors and their gradients are derived from the electric equations and associated boundary conditions. Subsequently, these are incorporated into the mechanical equations, which also encompass electric components. It is notable that by removing the time-dependent terms from the in-plane equations of motion, the static displacement due to rotation at each speed is obtained. After deriving the equations of motion and boundary conditions, these equations are non-dimensionalized using non-dimensionalizing relations. In the next step, Hamilton’s principle is used to discretize the equations and boundary conditions. Consequently, by applying the generalized differential quadrature method and extracting the stiffness and mass matrices resulting from the transverse equation of motion and boundary conditions, the natural transverse frequency of the rotating annular microplate under the flexoelectric effect is calculated. The results of this research are useful for promoting the use of rotating annular microplants under flexoelectric effect for microelectromechanical systems designers with high efficiency.

The Monte Carlo method is used to study the effect of boundary scattering on the temperature dependent part of the resistivity of thin metal films. A computational scheme is used that realistically simulates electron scattering mechanisms in the semi-classical context. As a test of the accuracy of the method, comparison is made between the present method at absolute zero (impurity and boundary scattering only) and the analytical results of Fuchs, which rely on the relaxation time approximation (RTA). The inclusion of phonon scattering provides a measure of the size induced deviations from Matthiessen’s Rule (SIDMR). At low temperatures phonon scattering cannot be adequately described using the RTA and the numerical technique presented shows good promise of overcoming this problem. Calculated SIDMR results are compared with some recent data on Al and Ga films.

Inspired by the work by Kämmerer, Dünweg, Binder, and d'Onorio de Meo we investigated the spin- Ising antiferromagnet on a fcc lattice in a homogeneous external magnetic field H with uniform nearest-neighbor interaction J by Monte Carlo simulations. In order to answer the open question whether this system exhibits a triple point or a multicritical point for positive H, we mainly studied lattices with 4 · 32³ and 4 · 72³ spins. We found a multicritical point at (H_c, k_BT_c) = (3.49 ± 0.02, 1.02 ± 0.01) J in the thermodynamic limit.

A model describing size-dependent melting temperature and thermal conductivity of nanosemiconductors is proposed based on Lindermann's melting criterion and Debye model. By the atomic thermal vibration consideration and by introducing intrinsic size effect of phonon velocity and mean free path combined with surface scattering effect, the model predicts that the melting temperature and thermal conductivity of nanosemiconductors decrease as the size reduces. The size effect depends on such material parameters as the vibration entropy, mean free path, the characteristic crystal size and surface roughness. The predictions are in agreement with experimental results of Si nanoparticles, nanowires and thin films.

With the miniaturization of parts, size effect occurs. The isothermal forming processes are performed to obtain homogenizing deformation. In the paper, a serial of isothermal upsetting tests are carried out with billets of different dimensions. Difference of flow stress is accepted as the parameter to evaluate the size dependence of flow stress on billets dimensions. The experimental results show that size effect occurs clearly. With the increasing of temperature, the difference of flow stress becomes smaller, which means that the degree of size dependence is reduced. Scatter of flow stress is observed in the tests at room temperature. When the deformation temperature is raised, the fluctuation of flow stress tends towards decreasing. In order to investigate the effect of grain size, different grain size is obtained with the heat treatment process. At the same temperature, the difference of flow stress increases with the increasing of grain size. These phenomena can be explained from the viewpoint of polycrystalline structure of material. The anisotropy of individual grain is appeared obviously, which leads the fluctuation of flow stress. In the isothermal deformation, the effect of single grain is smaller than that at room temperature.

In the present study, nanoindentation studies of the 95.8Sn-3.5Ag-0.7Cu lead-free solder were conducted over a range of maximum loads from 20 mN to 100 mN, under a constant ramp rate of 0.05 s^-1. The indentation scale dependence of creep behavior was investigated. The results revealed that the creep rate, creep strain rate and indentation stress are all dependent on the indentation depth. As the maximum load increased, an increasing trend in the creep rate was observed, while a decreasing trend in creep strain rate and indentation stress were observed. On the contrary, for the case of stress exponent value, no trend was observed and the values were found to range from 6.16 to 7.38. Furthermore, the experimental results also showed that the creep mechanism of the lead-free solder is dominated by dislocation climb.

Free-standing sub-micron Ti-5Al single crystal square pillars were fabricated along double slip and [0001] twinning orientations using FIB fabrication processes. Samples in range of 0.4 to 2.0µm were compressed. The yield stress increases much higher than their bulk counterpart as the specimen width decreases. The tendency of "smaller is stronger" is displayed in Ti-5Al single crystals loaded along and [0001] orientations. The number of slip systems is restricted by specimen physical size as it declines from 2µm to 0.5µm, when the specimens were subjected to double slip loading. Meanwhile, when sample size is less than 1.0µm, micro-pillars along twinning orientation have to compensate the incomplete twinning deformation via shearing due to geometrical restriction and dislocation starvation effects. This variation of deformation mode could be attributed to the starvation effect of dislocations.

With the help of the Euler–MacLaurin formula, the statistic characteristics of quantum gases confined in a rectangular box are investigated by using a unified way. Three components of the pressure tensor and the internal energy of the system are analytically derived. The important relation between the pressure tensor and the internal energy is strictly proved. It is pointed out that these results are different from those obtained in literature and textbooks under the thermodynamic limit condition. Moreover, the pressure of the system is rationally introduced with the help of the property of the diagonal tensor, so that some thermodynamic functions of the system may be conveniently obtained. The statistic characteristics of quantum and classical gases confined in some different containers are discussed, the influence of the size effect of the containers on the properties of gases is analyzed and some new significant results are obtained.

Molecular dynamics (MDs) simulations were used to explore the thermal stability of Au nanoparticles (NPs) with decahedral, cuboctahedral, icosahedral and Marks NPs. According to the calculated cohesive energy and melting temperature, the Marks NPs have a higher cohesive energy and melting temperature compared to these other shapes. The Lindemann index, radial distribution function, deformation parameters, mean square displacement and self-diffusivity have been used to characterize the structure variation during heating. This work may inspire researchers to prepare Marks NPs and apply them in different fields.

Herein, a corrected theoretical model is proposed for modeling the static and dynamic behavior of electrostatically actuated narrow-width nanotweezers considering the correction due to finite dimensions, size dependency and surface energy. The Gurtin–Murdoch surface elasticity in conjunction with the modified couple stress theory is employed to consider the coupling effect of surface stresses and size phenomenon. In addition, the model accounts for the external force corrections by incorporating the impact of narrow width on the distribution of Casimir attraction, van der Waals (vdW) force and the fringing field effect. The proposed model is beneficial for the precise modeling of the narrow nanotweezers in nano-scale.

We used the first principle of density functional theory to perform detailed calculations regarding the structure, and the electronic and magnetic properties of MX (M=Ga, In; X=S, Se, Te) nanoribbons. The armchair nanoribbons (ARNs) are nonmagnetic semiconductors, which have even or odd oscillations of bandgaps. All small-sized zigzag nanoribbons (ZRNs) were found to break the six-membered ring structure and move to the center, thereby exhibiting nonmagnetic semiconductor behavior owing to the quantum confinement effect. However, among the large ZRNs, which are all metals, MTe ZRNs are nonmagnetic; this differs from the case of graphene, MoS₂ and Ti₂CO₂ nanoribbons. MX (M=Ga, In; X=S, Se) ZRNs exhibited ferromagnetism owing to the presence of the unpaired electrons on the metal-edge side and the magnetic moment of each pair of molecules, which was controlled by the size of the nanoribbons. The results provided a theoretical reference that can be used in the future to produce MX materials for application in low-dimensional semiconductor devices, spin electron transport devices and new magnetoresistance devices.

The crystallization and sintering process of RuO${}_2\cdot$xH₂O nanoparticles have been investigated in this paper. The effect of RuO₂ crystallinity and particle size on the sheet resistance of RuO₂-based resistor paste after being co-fired with CaO–B₂O₃–SiO₂ green tapes has been reported. The results show that the nanoparticles are not fully crystallized below 600${}^{\circ }$C, and the effect of high-density defects, such as grain boundaries and dislocations on the residual resistance ($R_r)$ of RuO₂ particles, is significant. So, the sheet resistance decreases with the increase of crystallinity and the weakening of electron wave scattering. Above 600${}^{\circ }$C, the effect of crystal imperfections on $R_r$ is greatly weakened. However, the number of conductive chains formed in co-fired resistor decreases with the increase of particle size, thus the sheet resistance gradually raises. When the dwelling time is increased to 3 h, the RuO₂ is mostly crystallized and the effect of crystal imperfections on $R_r$ is negligible, and the sheet resistance mainly depends on particle size of RuO₂. Using RuO₂ particles with low crystallinity, small size and narrow particle-size distribution as conductive phase is expected to solve the problem of poor sheet resistance uniformity in high resistance paste for LTCC application.

The size effect of nano-grain BaTiO₃ ceramics has been studied in this paper. We assume that the surface charge does not fully compensate long-range Coulomb interaction, so that 180 degree nano-domains still exists in small BaTiO₃ particles. We have calculated the long-range interaction in BaTiO₃ nanocrystals with 180 degree nano-domains, and obtained the relationship between grain size and domain size. The relation between transition temperature and grain size is also obtained by incorporating the domain-wall energy into the Landau–Ginzburg free-energy density. The results show that with decreasing grain size the transition temperature of cubic-tetragonal phase decreases, while those of tetragonal-orthorhombic and orthorhombic-rhombohedral phases increase. Our model predicts that with further reduction of grain size, the structures of ferroelectric phase becomes unstable and gradually disappears, leaving only one stable structure of cubic phase. In addition, an interesting conclusion is that a quadruple phase point exists in the phase diagram of the transition temperature versus grain size. The results are compared with experimental data.

In confined space, the thermodynamic potential is shape-dependent. Therefore, the pressure of ideal gases in confined space is anisotropic. We study this anisotropy in a thermodynamic manner and find that the thermodynamic pressures usually depend on the form of deformations, and hence are not equal to each other which is a natural representation of the anisotropic mechanical properties of a confined ideal gas. We also find that the boundary effects are much more significant than the statistical fluctuations under low-temperature and high-density conditions. Finally, we show that there is little difference between the boundary effects in 2D space and those in 3D space.

In this paper, a semi-empirical molecular dynamics model is developed. The central collisions of C₆₀ + C₆₀ and X@C₆₀ + X@C₆₀ (X = He, Ne, Ar) at various incident energy are investigated within this model. The fullerene dimers like a "dumbbell" can be formed by a self-assembly of C₆₀ fullerene and X@C₆₀ (X = He, Ne) endohedral fullerenes, and the new fullerene structure like "peanut" can be formed by a self-assembly of Ar@C₆₀. It is found that Ar atom plays a great role in the collision of Ar@C₆₀ + Ar@C₆₀ because of its size effect. The energy effect is found that various incident energies cannot change the final structure at low energies if they are below a certain energy.

A simple theoretical model is developed to study the size and shape dependence of Debye temperature and Raman frequency of nanomaterial. We have studied the effect of size and shape on Debye temperature of nanocrystalline Fe, Co, Al and Ag. The model is extended to study the effect of size and shape on the Raman frequency of nanocrystalline SnO₂, CeO₂ and CdSe. The results obtained are compared with the available experimental data. A good agreement between the theory and experimental data supports the validity of the model developed. We also report the results for nanowire and nanofilm in the absence of experimental data, which may help the researchers engaged in the experimental studies.

With molecular dynamics simulations, the growth of face-centered-cubic nanocrystalline materials Ni and Ni₃Al has been studied. It is found that grain-rotation induced grain coalescence and curvature-driven grain-boundary migration are dominant mechanisms in the nanograin growth. A detailed comparison of the nanograin growth between the two systems is discussed in terms of grain rotation and grain sliding. We also study the temperature effect and the size effect in the nanograin growth. The tendency of twinning in the nanograin growth is discussed. It is found that in Ni₃Al, it seems more possible for nanograins to grow into twin-like structures than single crystal unless at very high temperatures.

The mechanism of plastic deformation under tensile and compressive loading of hexagonal close-packed (HCP)/face-centered cubic (FCC) biphasic titanium (Ti) nanopillars at different temperatures (70 K, 150 K, 300 K and 400 K) and different FCC phase sizes (2 nm, 4 nm, 6 nm and 8 nm) was investigated by molecular dynamics (MD). The plastic deformation is mainly concentrated in the FCC phase during compression loading. The HCP/FCC interface is the main source of $\frac{1}{6}〈\bar{1}\bar{2}1〉$ Shockley partial dislocations. As the temperature increases, the dislocation nucleation rate increases and the surface dislocation source is activated. During tensile loading, it is more likely that the Shockley partial dislocations react with each other in the FCC phase to form Lomer–Cottrell sessile dislocations and stacking fault (SF) nets. When the temperature is reduced to 70 K, tensile twins are formed at the phase interface. The plastic deformation is dominated by twins and $〈\mathrm{\text{c}}+\mathrm{\text{a}}〉$ dislocation slip occurs in the HCP phase. The effect of the FCC phase size on the plastic deformation mechanism of the nanopillar is strong. The FCC phase is transformed into the HCP phase when the FCC phase size in the nanopillar is reduced to 4 nm under compressive loading. However, twin deformation occurs at the HCP/FCC interface when the FCC phase size is reduced to 2 nm under tensile loading.

The effect of model size on fluid flow through fractal rough fractures under shearing is investigated using a numerical simulation method. The shear behavior of rough fractures with self-affine properties was described using the analytical model, and the aperture fields with sizes varying from 25 to 200mm were extracted under shear displacements up to 20mm. Fluid flow through fractures in the directions both parallel and perpendicular to the shear directions was simulated by solving the Reynolds equation using a finite element code. The results show that fluid flow tends to converge into a few main flow channels as shear displacement increases, while the shapes of flow channels change significantly as the fracture size increases. As the model size increases, the permeability in the directions both parallel and perpendicular to the shear direction changes significantly first and then tends to move to a stable state. The size effects on the permeability in the direction parallel to the shear direction are more obvious than that in the direction perpendicular to the shear direction, due to the formation of contact ridges and connected channels perpendicular to the shear direction. The variances of the ratio between permeability in both directions become smaller as the model size increases and then this ratio tends to maintain constant after a certain size, with the value mainly ranging from 1.0 to 3.0.

Hall–Petch strengthening has been widely used in materials science, but its mechanism is not very clear yet, some inverse phenomena were observed. This paper gives a fractal approach to explanation of the Hall–Petch effect, revealing the value of the fractal dimensions is the key factor: when it is larger than one, the Hall–Petch effect is predicted; while when it is smaller than one, an inverse Hall–Petch relationship is obtained. The fractal theory can also explain the nano-effort (size effort) in nanotechnology and spider silk’s strength.