Chaotic dynamic analysis of viscoelastic plates

The dynamic buckling of viscoelastic plates with large deflection is investigated in this paper by using chaotic and fractal theory. The material behavior is given in terms of the Boltzmann superposition principle. in order to obtain accurate computation results, the nonlinear integro-differential dynamic equation is changed into an autonomic four-dimensional dynamical system. The numerical time integrations of equations are performed by using the fourth-order Runge-Kutta method. And the Lyapunov exponent spectrum, the fractal dimension of strange attractors and the time evolution of deflection are obtained. The influence of geometry nonlinearity and viscoelastic parameter on the dynamic buckling of viscoelastic plates is discussed.

Statistical-theory of multiphoton excitation of polyatomic-molecules based on the wigner function

This paper is aimed at establishing a statistical theory of rotational and vibrational excitation of polyatomic molecules by an intense IR laser. Starting from the Wigner function of quantum statistical mechanics, we treat the rotational motion in the classical approximation; the vibrational modes are classified into active ones which are coupled directly with the laser and the background modes which are not coupled with the laser. The reduced Wigner function, i.e., the Wigner function integrated over all background coordinates should satisfy an integro-differential equation. We introduce the idea of ``viscous damping'' to handle the interaction between the active modes and the background. The damping coefficient can be calculated with the aid of the well-known Schwartz–Slawsky–Herzfeld theory. The resulting equation is solved by the method of moment equations. There is only one adjustable parameter in our scheme; it is introduced due to the lack of precise knowledge about the molecular potential. The theory developed in this paper explains satisfactorily the recent absorption experiments of SF6 irradiated by a short pulse CO2 laser, which are in sharp contradiction with the prevailing quasi-continuum theory. We also refined the density of energy levels which is responsible for the muliphoton excitation of polyatomic molecules.

Effects of Thermocapillary Convection on the Growth Process in Industrial 8-inch Silicon Czochralski Growth

Czochralski (CZ) crystal growth process is a widely used technique in manufacturing of silicon crystals and other semiconductor materials. The ultimate goal of the IC industry is to have the highest quality substrates, which are free of point defect, impurities and micro defect clusters. The scale up of silicon wafer size from 200 mm to 300 mm requires large crucible size and more heat power. Transport phenomena in crystal growth processes are quite complex due to melt and gas flows that may be oscillatory and/or turbulent, coupled convection and radiation, impurities and dopant distributions, unsteady kinetics of the growth process, melt crystal interface dynamics, free surface and meniscus, stoichiometry in the case of compound materials. A global model has been developed to simulate the temperature distribution and melt flow in an 8-inch system. The present program features the fluid convection, magnetohydrodynamics, and radiation models. A multi-zone method is used to divide the Cz system into different zones, e.g., the melt, the crystal and the hot zone. For calculation of temperature distribution, the whole system inside the stainless chamber is considered. For the convective flow, only the melt is considered. The widely used zonal method divides the surface of the radiation enclosure into a number of zones, which has a uniform distribution of temperature, radiative properties and composition. The integro-differential equations for the radiative heat transfer are solved using the matrix inversion technique. The zonal method for radiative heat transfer is used in the growth chamber, which is confined by crystal surface, melt surface, heat shield, and pull chamber. Free surface and crystal/melt interface are tracked using adaptive grid generation. The competition between the thermocapillary convection induced by non-uniform temperature distributions on the free surface and the forced convection by the rotation of the crystal determines the interface shape, dopant distribution, and striation pattern. The temperature gradients on the free surface are influenced by the effects of the thermocapillary force on the free surface and the rotation of the crystal and the crucible.