Step stiffness

The relative lifetimes of the two terraee types at any one saddle point location has been measured[31] to differ by a factor of 6 at 1060C. The change in terrace type occurs by the bridging of the short dimension by step fluctuations. Since the probability of a fluctuation of a particular amplitude depends linearly on the step stiffness[8] the observed lifetime ratio is consistent with measured step stiffnesses[37] and the geometrical picture given above[38]. 

To extract a value of the step-mobility h from the grating relaxation experiments [12], we must evaluate the strength of the step-step interaction y. Computational work suggests that ydue to elastic interactions between Si(OOl) steps is 0.2 eV run [29], while, we estimate that the entropic interaction is 10 times larger. (We use a step stiffness P calculated from the geometric mean of P for Sa and Sb steps given in Ref [30] P, 0.03 eV mn-. ) Therefore, entropic repulsion should dominate, and... 

We have assumed e l, corresponding to a large step stiffness ( > >1), so thatmnltiple kinks of a step in the x direction are negligible. Less stiff steps would cross the terraces more easily and lead to a larger decay rate of the grooved surface. 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

Here the expression in parentheses is called the step stiffness Both strrface and step stiffnesses may be changed by adsorption. 

Following the analysis proposed by Giesen et al. [30], at any point on its perimeter an island in equilibrium has a constant chemical potential ji, which is related to the stiffness coefficient y and the local curvature. In addition, the stiffness coefficient depends on the step orientation 0 (see Fig. 3.5) and the line tension y. The authors derive a simple relation between the shape coordinates of islands and the energy parameters. Using the coordinate system and the island orientation shown in the figure, a point of minimum curvature exists at y, V = 0, and for reasons of symmetry, also at -y, x = 0. Line tension and step stiffness are then related by... 

All in all, the simulations reviewed provide a consistent picture of the surface dynamics. With increasing field, the surface becomes more mobile, which entails larger step fluctuations and a decrease of the step stiffness. At the same time, the island shapes become more rounded and the coarsening faster. The same effects occur with increasing temperature. It has often been observed that in certain electrochemical experiments, the potential plays a similar role to that of the temperature in UHV. Thus, electrochemical desorption spectra obtained by a potential sweep bear a certain similarity to thermal desorption spectra in UHV. 

Laminate failure analysis was performed using a step-by-step stiffness reduction pro-... 

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

To be more precise, this error occurs in the limit /c — oo with Ef = 0(1) and step-size k such that k /ii = const. 3> 1. This error does not occur if Ef = 0 for the analytic problem, i.e., in case there is no vibrational energy in the stiff spring which implies V,. = U. 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

Step 1 To solve a Stokes flow problem by this program the inertia term in the elemental stiffness matrix should be eliminated. Multiplication of the density variable by zero enforces this conversion (this variable is identified in the program listing). 

Step 2 General structure of stiffness matrices derived for the model equations of Stokes flow in (x, 3O and (r, z) formulations (see Chapter 4) are compared. 

Step 3 Comparing systems (7.2) and (7.3) additional terms in the members of the stiffness matrix corresponding to the axisymmetric fon-nulation are identified. Note that the measure of integration in these tenns is (r drdz). 

Most commercial processes involve copolymerization of ethylene with the acid comonomer followed by partial neutralization, using appropriate metal compounds. The copolymerization step is best carried out in a weU-stirred autoclave with continuous feeds of all ingredients and the free-radical initiator, under substantially constant environment conditions (22—24). Owing to the relatively high reactivity of the acid comonomer, it is desirable to provide rapid end-over-end mixing, and the comonomer content of the feed is much lower than that of the copolymer product. Temperatures of 150—280°C and pressures well in excess of 100 MPa (1000 atm) are maintained. Modifications on the basic process described above have been described (25,26). When specific properties such as increased stiffness are required, nonrandom copolymers may be preferred. An additional comonomer, however, may be introduced to decrease crystallinity (10,27). 

The resins used in air and oil filters are moderate-to-low molecular weight, catalyzed by caustic in one step 10—20% alcohol is added soHds content is in the range of 50—60%. These resins are designed to penetrate the sheet thoroughly, yet not to affect the porosity of the paper. In the B-stage, the resin must have sufficient flexibiHty to permit pleating the C-stage should have stiffness and resistance to hot oil. 

The question of stiffness then depends on the solution at the current time. Consequently nonhuear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stabihty and accuracy of the numerical scheme and the step size used. 

At the other extreme of Distefano s sample problems, for the largest initial charge, the maximum-stiffness ratio is of the order of 1500, which is considered to be a relatively large value. In this case, more than 10,000 time steps are required to distih 90 percent of the initial change, and the problem is better handled by a stiff integrator. 

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